Keshab Chandra Bakshi [ Partially supported by DST INSPIRE Faculty grant DST/INSPIRE/04/2019/002754]  , Satyajit Guin [ Partially supported by SERB grant MTR/2021/000818]
Relative position between a pair of spin model subfactors

Jones pioneered the theory of subfactors, which may be regarded as a quantized version of closed subspaces in Hilbert space. It deals with the relative position of a single factor inside an ambient factor. The level of intricacy increases considerably if there are more than two factors involved. Indeed, Jones proposed the study of two subfactors of a $II_1$ factor as a quantization of two closed subspaces in a Hilbert space. The Pimsner-Popa probabilistic constant, Sano-Watatani angle, interior and exterior angle, and Connes-Størmer relative entropy (along with a slight variant of it) are a few key invariants for pair of subfactors that analyze their relative position. In practice, however, the explicit computation of these invariants is often difficult.

In this article, we provide an in-depth analysis of  a special class of two subfactors, namely a pair of spin model subfactors of the hyperfinite type $II_1$ factor $R$. We first characterize when two distinct $n\times n$ complex Hadamard matrices give rise to distinct spin model subfactors. Then, a detailed investigation has been carried out for pairs of (Hadamard equivalent) complex Hadamard matrices of order $2\times 2$ as well as Hadamard inequivalent complex Hadamard matrices of order $4\times 4$. To the best of our knowledge, this article is the first instance in the literature where the exact value of the Pimsner-Popa probabilistic constant and the noncommutative relative entropy for pairs of (non-trivial) subfactors have been obtained. Furthermore, we prove the factoriality of the intersection of the corresponding pair of subfactors using the `commuting square technique'. En route, we construct an infinite family of potentially new subfactors of $R$. All these subfactors are irreducible with Jones index $4n,n\geq 2$. As a consequence, the rigidity of the interior angle between the spin model subfactors is established. Last but not least, we explicitly compute the Sano-Watatani angle between the spin model subfactors.

AMS Subject Classification No.: 46L37, 46L55, 46L10, 37A35.

Keywords. Jones index, Connes-Størmer entropy, Pimsner-Popa probabilistic index, angle operator, spin model subfactor, complex Hadamard matrix, commuting square, commuting cube.

§.§ Notation

* Throughout the article underlying field is $\bbc$ and we simply write $M_n$ instead of $M_n(\bbc)$ to denote type $I_n$ factors. $\Delta_n$ will denote the diagonal subalgebra (Masa) in $M_n$.

* $M_n^{(k)}$ will denote tensor product of $M_n$ with itself $k$-times. For $A\in M_n,\,A^{(k)}$ will denote the matrix $A^{\,\otimes\,k}$.

* $k\times k$ diagonal matrices will be denoted by $\mbox{diag}\{\mu_1,\ldots,\mu_k\}$, where $\mu_j\in\bbc$.

* Block diagonal matrix of the form $\left[{\begin{smallmatrix}
A_1 &  &  & \\
& \ddots &  & \\
&  &  & A_k\\
\end{smallmatrix}}\right]$ will be denoted by $\mbox{bl-diag}\{A_1,\ldots,A_k\}$.

* For $A\in M_k,\,I_n\otimes A$ denotes the matrix $\mbox{bl-diag}\{A,\ldots,A\}$ in $M_{nk}$.

* The following Pauli spin matrices are used throughout the article on several occasions, especially in <Ref>,

\[
\sigma_1=\left[{\begin{matrix}
0 & 1\\
1 & 0
\end{matrix}}\right]\quad,\quad\sigma_2=\left[{\begin{matrix}
0 & -i\\
i & 0
\end{matrix}}\right]\quad,\quad\sigma_3=\left[{\begin{matrix}
1 & 0\\
0 & -1
\end{matrix}}\right]\,.
\]

* The projection $\frac{1}{n}\sum_{i,j=1}^nE_{ij}$ in $M_n(\bbc)$ will be denoted by $J_n\,$.

* Given an inclusion of finite von Neumann algebras $\mathcal{N}\subset\mathcal{M}$ with a fixed trace $tr$ on $\mathcal{M}$, the $tr$-preserving conditional expectation is denoted by $E^{\mathcal{M}}_{\mathcal{N}}$.

* For unital inclusion of finite von Neumann algebras $\mathcal{N\subset M}$ with a fixed trace $tr$ on $\mathcal{M}$, $L^2(\mathcal{M})$ (resp. $L^2(\mathcal{N})$) denotes the GNS Hilbert space corresponding to $tr$ (resp., $tr|_{\mathcal{N}}$). The Jones projection, denoted by $e_{\mathcal{N}}$, is the projection onto the closed subspace $L^2(\mathcal{N})$ of $L^2(\mathcal{M})$.

* We often denote a quadruple of von Neumann algebras

\[
\begin{matrix}
\mathcal{P} & \subset & \mathcal{M}\\
\cup & & \cup\\
\mathcal{N} & \subset & \mathcal{Q}
\end{matrix}
\]

by $(\mathcal{N}\subset\mathcal{P,Q}\subset\mathcal{M})$ for brevity. The Sano-Watatani angle is denoted by $\text{Ang}_M(P,Q)$ and the interior angle (resp. exterior angle) is denoted by $\alpha^N_M(P,Q)$ (resp., $\beta^N_M(P,Q)$).

* All the logarithms appearing in this paper are with base $e$.

* Throughout the paper, the notations $\lambda$ and $H$ (resp., $h$) will be reserved for the Pimsner-Popa probabilistic constant and Connes-Størmer relative entropy (resp., its variant) respectively.

§.§ Motivation

A subfactor $N$ is a unital subalgebra of a type $II_1$-factor $M$, which is itself a  $II_1$ factor, and $1_N=1_M$. An important invariant of a subfactor is the Jones index $[M:N]$ [25] which generalizes the subgroup index. The Jones index measures how much bigger $M$ is compared to $N$. In view of the fact that the subfactor theory deals with the relative position of a subfactor $N$ inside an ambient factor $M$, it is a fundamental question to consider the relative positions of multiple subfactors. In the simplest case, one considers an intermediate subfactor $N\subset P\subset M$, and this is relatively well understood. As the next level of difficulty, if one takes a pair of subfactors $P,Q\subset M$, the theory becomes complicated. The study of multiple subfactors was initiated by Ocneanu. He proposed the concept of `maximal atlas' for a compatible family of `finite-index' bimodules arising from the subfactors (see [35]). Ocneanu's work leads Jones to systematically study two subfactors [28]. The first difficulty one encounters is that $P\cap Q$ need not be a factor, and therefore we can not talk about the Jones index of $P\cap Q$ in $M$. However, this can be remedied using the `probabilistic index' $\lambda(M,P\cap Q)^{-1}$ due to Pimsner and Popa [37], which is a substitute of the Jones index for a non-factorial inclusion and coincides with the Jones index for subfactor, that is, if $N\subset M$ is a subfactor of type $II_1$ factors, then $\lambda(M,N)={[M:N]}^{-1}$.  We discuss a few key invariants for the theory of `two subfactors' as described in <Ref>. Note that we are not claiming here that this list of invariants is exhaustive. Discovering the `complete invariant' for `good class' of two subfactors seems open. Indeed, the formulation of `planar algebra' for a pair of subfactors – even for commuting square– remains elusive (see [29]).

rounded rectangle,
minimum width=2.5cm,
minimum height=1cm] (block1)  Key invariants
for two subfactors ;

left=of block1,
minimum width=3.5cm,
minimum height=1cm] (block2)  Pimsner-Popa constant ;

below=of block2,
minimum width=3.5cm,
minimum height=1cm] (block3)  Sano-Watatani angle ;

right=of block1,
minimum width=3.5cm,
minimum height=1cm] (block4)  Interior and exterior angle ;

below=of block4,
minimum width=3.5cm,
minimum height=1cm] (block5)  Connes-Størmer entropy ;

[-latex] (block1) edge (block2)
(block1) edge (block3)
(block1) edge (block4)
(block1) edge (block5);

Few key invariants for two subfactors

We begin with the first invariant, which is a generalization of the Pimsner-Popa probabilistic index $\lambda(M,N)^{-1}$ for $N\subset M$. Given a pair of subfactors $P,Q\subset M$ one can associate a number $\lambda(P,Q)\in[0,1]$, called the Pimsner-Popa probabilistic constant, which is an invariant to examine the relative position between the subfactors. Note that the definition makes sense for subalgebras of a finite von Neumann algebra as well. Recently in [4], given a pair of intermediate subfactors $P$ and $Q$ of a subfactor $N\subset M$ with $[M:N]<\infty$ and $N^{\prime}\cap M=\mathbb{C}$, the first author has given a formula of $\lambda(P,Q)$ in terms of the so-called `biprojections'. In general, given two subfactors $P,Q\subset M$, the formula for $\lambda(P,Q)$ is not known. As a next level of difficulty in the two subfactor theory, one sees that for a pair of subfactors $P,Q\subset M$ with $[M:P], [M:Q]<\infty$ it may very well happen that  $\lambda(M,P\cap Q)^{-1}=\infty$. Indeed, even  if we assume that both $P$ and $Q$ are of index  $2$, it can happen that $P\cap Q$ is of infinite (Pimsner-Popa) index in $M$ (see [28], for instance). In [27], Jones and Xu ask the following question:

Question (Jones and Xu). Consider a pair of finite index subfactors $P,Q\subset M$. Under what condition is the intersection $P\cap Q$ of a finite index?

They have shown that the finiteness of $\lambda(M,P\cap Q)^{-1}$ is equivalent to the finiteness of the spectrum of the so-called `angle operator', which leads us to our  second invariant for two subfactors.

Sano and Watatani introduced a notion of angle [43], denoted by $\text{Ang}_\mathcal{M}(\mathcal{P,Q})$, between two subalgebras of a finite von Neumann algebra $\mathcal{M}$, motivated by the angle between two subspaces (projections) in a Hilbert space. The angle between the subfactors $P\subset M$ and $Q\subset M$ determines the degree of non-commutativity of the two subfactors. Finding the possible value of the angle is known to be an important question in the  theory of subfactors. This has been highlighted by Grossman and Jones [16]. The computation of the angle, even in simple cases, is not easy. See [43, 16, 17] for many interesting applications and explicit computations of the angle operator. In another direction, to understand the relative position between a pair of intermediate subfactors $P$ and $Q$ of a finite index subfactor $N\subset M$ (i.e., $N\subset P,Q\subset M$), the notion of interior angle $\alpha^N_M(P,Q)$ and exterior angle $\beta^N_M(P,Q)$ between $P$ and $Q$ has been introduced in [2], which is our third invariant. This angle has been used to answer an open question by Longo by improving  the existing upper bound for the cardinality of the lattice of intermediate subfactors. Furthermore, a surprising connection between the intermediate subfactor theory and kissing numbers/sphere packing in geometry has been discovered.

Now we discuss the fourth and final invariant for two subfactors, namely the relative entropy of Connes-Størmer. In recent years, non-commutative entropy has become an active topic of research. Motivated by Shannon's information theory, Kolmogorov introduced (classical) conditional entropy, and subsequently, Sinai improved the definition. Conditional entropy is an important notion in classical ergodic theory. In the noncommutative situation, a notion of conditional (relative) entropy between two finite-dimensional von Neumann subalgebras of a finite von Neumann algebra was introduced by Connes and Størmer in [12]. They proved a Kolmogorov-Sinai type theorem using relative entropy as the main technical tool, and as an application, they proved that for $n\neq m$ the $n$-shift of the hyperfinite $II_1$ factor is not conjugate to the $m$-shift using an appropriate definition of  the entropy of an automorphism in the noncommutative framework of operator algebras. Later, Pimsner and Popa observed that relative entropy can be defined for arbitrary von-Neumann subalgebras of type $II_1$ factors (more generally, of tracial von Neumann algebras). More precisely, in analogy with the classical case, one can define $H(P|Q)$ (resp. $H(Q|P)$) which we refer to as the Connes-Størmer relative entropy. We often use a slight modification of $H(P|Q)$, denoted by $h(P|Q)$, due to Choda (see [10]) which is more calculable and provides an alternative invariant for two subfactors. Pimsner and Popa have discovered a surprising connection between relative entropy and the Jones index [37]. Subsequently, many interesting connections between the Connes-Størmer entropy and Jones' subfactor theory were found (see [38, 40, 42, 6, 7, 23], just to name a few). The relationship between the minimal index of subfactors (not necessarily of type $II_1$ factors) and entropy was clarified in [20, 21]. Furthermore, entropy for `canonical shifts' had been discussed in [8, 11, 22]. However, note that explicit computation of $H(P|Q)$ in general situations is often difficult, and one can safely say that noncommutative conditional entropy is not yet well-understood. To quote a $1992$ survey article by Størmer, “While entropy has for a third of a century been a central concept in ergodic theory, its non-abelian counterpart is still in its adolescent stage with only a few signs of mature strength". The major obstruction here is that in the non-abelian world, it is no longer true that two finite subsystems generate a finite subsystem. Another obstacle in the computation of $H$ is that entropy is not well-behaved with respect to the tensor products (see [19] for some interesting results along these lines). In Section $6$ [37], given a unital inclusion of finite-dimensional $C^*$-algebras, an explicit formula for $H$ has been provided. However, if we consider a pair of finite-dimensional subalgebras of a finite-dimensional $C^*$-algebra the situation becomes complicated. Indeed, an explicit formula for the Connes-Størmer relative entropy between a pair of Masas of a matrix algebra, that is $H(\Delta_n|u\Delta_n u^*)$ for $u\in\mathcal{U}(M_n)$, seems to be open even when $n=2$ (see [36]).

Given a pair of subfactors $P,Q\subset M$, computation of these invariants is hard. A relatively easier situation is the existence of a subfactor $N\subset M$ such that we have a quadruple of $II_1$ factors

P & \subset & M\\
\cup & & \cup\\
N & \subset & Q\\
\end{matrix}$

If the above quadruple is a commuting square, we know that the angle $\mathrm{Ang}_M(P,Q)$ is the singleton set $\{\pi/2\}$ and vice versa. In this situation, as a generalization of the notion of the non-degenerate commuting square, we have introduced the notion of the non-degenerate commuting cube (see <Ref>), a natural and relevant concept to deal with two subfactors. The basic construction of the non-degenerate commuting cube has been discussed. We have investigated a few of its properties. One of the applications of this notion is that if a quadruple of $II_1$ factors is obtained as an iterated basic construction of a non-degenerate commuting cube (see Sections <ref> and <ref> for examples of such), it becomes easy to check whether the quadruple we are dealing with is far from being a commuting square. Using this notion, the angle between the subfactors satisfying certain hypotheses has been computed. In this paper, this notion has been crucially used on several occasions in Sections <ref> and <ref>.

In the absence of a general theory for two subfactors, and observing the level of difficulty for its invariants, it is natural that one should start investigating some particular important class of subfactors to hope for building a general theory for two subfactors. Motivated by this goal, in this article, our focus is on a pair of `spin model subfactors'. Spin model subfactors were introduced by Jones ([29], see also [26]) as an important class of subfactors of the hyperfinite type $II_1$ factor $R$. Given a complex Hadamard matrix $u$ in $M_n$, one obtains a `spin model commuting square' and iterating the basic construction of such a square, one gets a hyperfinite subfactor $R_u\subset R$ with $[R:R_u]=n$ and $R_u^{\,\prime}\cap R=\bbc$. The importance of spin model subfactor has been emphasized by Jones (see [29]).  Not much is known about this subfactor. Indeed, computation of the relative commutants/standard invariant is very hard, and to compute even the first few relative commutants, computer assistance is required. The study of two spin model subfactors is even more intricate. If we start with a pair of distinct complex Hadamard matrices $u\mbox{ and }v$ of order $n\times n$, we (possibly) obtain a `pair' of spin model subfactors $R_u, R_v\subset R$. However, a number of caveats exist. The first one is that it is not guaranteed whether $R_u\neq R_v$ even if $u\neq v$. The second one is the factoriality of $R_u\cap R_v$, and finally, even if $R_u\cap R_v$ is a factor, it can be of infinite index in $R$. The starting point of this paper is the quest for the following problems:

* Problem 1. Given a pair of (distinct) $n\times n$ complex Hadamard matrices $u$ and $v$, characterize when the spin model subfactors $R_u$ and $R_v$ of the hyperfinite type $II_1$ factor $R$ are distinct.

* Problem 2. Suppose that we have a pair of spin model subfactors $R_u\subset R\mbox{ and }R_v\subset R$. Is $R_u\cap R_v$ a factor? Under what condition is it a finite index?

* Problem 3. Find the value of the following invariants for the spin model subfactors $R_u\subset R\mbox{ and }R_v\subset R$.

* Pimsner-Popa probabilistic constant $\lambda(R_u,R_v).$

* Sano-Watatani angle $\text{Ang}_R(R_u,R_v)$.

* The interior (resp., exterior) angle ${\alpha}^{R_u\cap R_v}_R(R_u,R_v)$ (resp., ${\beta}^{R_u\cap R_v}_R(R_u,R_v)$).

* Connes-Størmer relative entropy $H(R_u|R_v)$ (and its slight modification $h(R_u|R_v)$).

We have a complete answer to Problem $1$. We classify pairs $(u,v)$ of complex Hadamard matrices of order $n\times n$ such that they give rise to distinct spin model subfactors $(R_u\neq R_v)$. The classification is obtained in terms of an equivalence relation that is finer than the Hadamard equivalence. The Pimsner-Popa probabilistic constant and the Hamming numbers of the rows of the unitary matrix $u^*v$ play the lead roles. As a consequence of this, it turns out that inequivalent complex Hadamard matrices always give rise to pairs of (distinct) spin model subfactors. We have considered the cases of $n=2\mbox{ and }n=4$ for detailed investigation to incorporate both the instances of Hadamard equivalent and inequivalent matrices. Note that up to Hadamard equivalence, there is a single family of complex Hadamard matrices of order $2\times 2$, whereas, there exists a continuous, one-parameter family of inequivalent complex Hadamard matrices of order $4\times 4$. We take the liberty to remind our readers that the complete classification of complex Hadamard matrices is extremely difficult and unsolved in the literature. For example, even the case of $6\times 6$ is conjectured only. Problem $2$ and Problem $3$ have been solved completely in the cases of $n=2\mbox{ and }n=4$. We observe that while the factoriality and finite index of $R_u\cap R_v$ both hold in the case of $n=2$, they fail for certain situations in the $n=4$ case, and this case behaves rather erratically, which convinces us that Problem $2$ and Problem $3$ (fully) for the general $n\times n$ case seem beyond reach at the moment.

We now discuss our findings in this paper. First, we briefly discuss the case of $n=2$. We have overcome the first hurdle by proving that $R_u\cap R_v$ is a (non-irreducible) factor. More precisely, we have proved that $R_u\cap R_v\subset R$ is a diagonal subfactor with Jones index $4$ (see the above question due to Jones and Xu). It is indeed a curious fact that the intersection $R_u\cap R_v$ of the spin model subfactors $R_u\mbox{ and }R_v$ in this case turns out to be a vertex model subfactor of index $4$. We have also computed the value of $\lambda(R_u,R_v)$, the Sano-Watatani angle $\mathrm{Ang}_R(R_u,R_v)$, the interior angle $\alpha^{R_u\cap R_v}_R(R_u,R_v)$ and the exterior angle $\beta^{R_u\cap R_v}_R(R_u,R_v)$, and the modified Connes-Størmer relative entropy $h(R_u|R_v)$, which provides a legitimate bound for the Connes-Størmer relative entropy $H(R_u|R_v)$. It turns out that $h(R_u|R_v)$ is not constant; rather, it takes value depending on $u\mbox{ and }v$. We have characterized the quadruple of $II_1$ factors $(R_u\cap R_v\subset R_u,R_v\subset R)$ in terms of a single bi-unitary matrix in $M_4$ and completely determined when it is a commuting square. Thus, the relative position between $R_u\mbox{ and }R_v$ has been mostly understood in the case of $n=2$.

Now, we briefly discuss the case of $n=4$. Here, the complex Hadamard matrices $u\mbox{ and }v$ are Hadamard inequivalent. This case is more delicate and very interesting. It turns out that factoriality and the finite index of $R_u\cap R_v$ depend on whether certain rotations naturally arising from the circle parameters associated with $u\mbox{ and }v$ are rational or irrational. In the case of rational rotation, we construct en route an infinite family of potentially new subfactors of $R$. All these subfactors are irreducible with the Jones index $4n,n\in\bbn$ and $n\geq 2$. This has enabled us to conclude factoriality for the intersection $R_u\cap R_v$ for the spin model subfactors $R_u\mbox{ and }R_v$ in the case of rational rotation, and we observe that the intersection is not of the fixed index in $R$, unlike the situation of $2\times 2$. Moreover, $R_u\cap R_v$ is irreducible, and $[R:R_u\cap R_v]\in\{4n:n\in\bbn,\,n\geq 2\}$. Now, if the rotation is irrational, we observe that $R_u\cap R_v$ is of infinite Pimsner-Popa index in $R$. Unfortunately, we could not determine whether this is a factor in this case. Therefore, we see that the case of $4\times 4$ (inequivalent Hadamard matrices) is in sharp contrast to the case of $2\times 2$ (equivalent Hadamard matrices). To understand how the two subfactors `interact' we have computed explicitly the value of $\lambda(R_u,R_v)$, and in the case of rational rotation, the Sano-Watatani angle $\mathrm{Ang}_R(R_u,R_v)$, the interior angle $\alpha^{R_u\cap R_v}_R(R_u,R_v)$ and the exterior angle $\beta^{R_u\cap R_v}_R(R_u,R_v)$, and the exact value of the Connes-Størmer relative entropy $H(R_u|R_v)$. The rigidity of the interior angle has also been established, namely, that it is always greater than $\pi/3$. In the case of irrational rotation, a legitimate bound for the Connes-Størmer relative entropy $H(R_u|R_v)$ has been obtained. Finally, as an application of the commuting cube, we have completely characterized when the quadruple of $II_1$ factors $(R_u\cap R_v\subset R_u,R_v\subset R)$ for given $u\mbox{ and }v$ forms a commuting square. The major highlight of the $n=4$ case (rational rotation case) is that starting with two (Hadamard inequivalent complex Hadamard) $4\times 4$ matrices, one produces an infinite family of integer indices (in particular, $4n,\,n\in\bbn$ and $n\geq 2$) and irreducible subfactors of the hyperfinite type $II_1$ factor $R$. However, in the irrational rotation case, many questions remain unanswered.

We draw a table in <Ref> to highlight the contrast between the $n=2\mbox{ and }n=4$ cases. We only mention the rational rotation case in the $n=4$ situation, as the irrational rotation case remains mysterious except for the fact that $R_u\cap R_v$ is of infinite index.

$R_u,R_v\subset R$     $\,\,u,v\mbox{ are }2\times 2\,\,$     $\,\,\,\,u,v\mbox{ are }4\times 4\,,$ and rational rotation
$R_u\neq R_v$     only when $u\nsim v$     $\mbox{always}$

$R_u\cap R_v$      always a factor     always a factor

$[R:R_u\cap R_v]$     always $4$     $\{4n:n\geq 2\,\,\mbox{integer}\}$

$R_u\cap R_v\subset R$ is vertex model     yes     no

relative commutant     non-irreducible     irreducible

characterization of $R_u\cap R_v$      diagonal subfactor     unknown!

$\texttt{\#}\mathrm{Ang}_R(R_u,R_v)$ (set of angles)     singleton set     $\lfloor\frac{n}{2}\rfloor$, if $[R:R_u\cap R_v]=4n$

interior angle $\alpha_R(R_u,R_v)$     no rigidity     rigidity present, $\alpha>\frac{\pi}{3}$ always

exterior angle $\beta_R(R_u,R_v)$     different values     fixed value, $\beta=\arccos(\frac{1}{3})$

relative entropy $h$     different values     fixed value $\log 2$

Contrast between the cases of $n=2\mbox{ and }n=4$

Now it is not difficult to persuade why the general $n\times n$ case seems beyond our reach at the moment, and extensive investigation is needed. A major difficulty is the lack of complete classification of complex Hadamard matrices of order beyond $5\times 5$.

§.§ Statement of the main results

To introduce our main results, we briefly recall some well-known facts about complex Hadamard matrices that appear in many areas like quantum teleportation, coding theory, mutually unbiased bases, unitary error bases, operator algebras, Harmonic analysis, etc. In this paper, we provide yet another application of complex Hadamard matrices in von Neumann algebras.

A matrix with entries $\pm 1$ and mutually orthogonal rows and columns is called a Hadamard matrix. If $H$ is a Hadamard matrix and $H^{\intercal}$ denote its transpose, then $H H^{\intercal}=n I_n$. A complex Hadamard matrix is a generalization of Hadamard matrix.

A complex Hadamard matrix is a $n\times n$ matrix with complex entries of same modulus and $HH^*=nI_n$.

Note that $\frac{1}{\sqrt{n}}H$ is a unitary matrix. In this paper, by a complex Hadamard matrix we shall always mean the associated unitary matrix $\frac{1}{\sqrt{n}}H$. Observe that for every $n\geq 1$, the Fourier matrix $(F_n)_{i,j}:=\frac{1}{\sqrt{n}}\exp(2\pi \mathbf{i} (i-1)(j-1)/n)$ is a complex Hadamard matrix. Thus,

\[
F_1=[1]\,\,,\quad F_2=\frac{1}{\sqrt{2}}\begin{bmatrix}
1 & 1\\
1 & -1
\end{bmatrix}\quad \mbox{and}\quad F_3=\frac{1}{\sqrt{3}}\begin{bmatrix}
1 & 1 &1\\
1 & \omega & {\omega}^2\\
1 & {\omega}^2 &\omega
\end{bmatrix}.
\]

Two complex Hadamard matrices are called Hadamard equivalent, denoted by  ${\displaystyle H_{1}\simeq H_{2}}$, if there exist diagonal unitary matrices ${\displaystyle D_{1},D_{2}}$ and permutation matrices ${\displaystyle P_{1},P_{2}}$ such that

\[
{\displaystyle H_{1}=D_{1}P_{1}H_{2}P_{2}D_{2}\,.}
\]

It is known that for $n=2,3,5$, all complex Hadamard matrices are equivalent to the Fourier matrix $F_{n}\,$. We refer the reader to [18] for proof. In particular, any $2\times 2$ complex Hadamard matrix is of the form  $\frac{1}{\sqrt{2}}\begin{bmatrix}
e^{i\alpha_1} & e^{i (\alpha_1+\alpha_3)}\\
e^{i\alpha_2} & -e^{i(\alpha_2+\alpha_3)}
\end{bmatrix}$ for $\alpha_j\in[0,2\pi)$. However, it is known that there exists a continuous, one-parameter family of Hadamard inequivalent $4\times 4$ complex Hadamard matrices, and any element of this family is of the following form
u(z)= 1/2 [
1     1     1     1

1     iz     -1     -iz

1     -1     1     -1

1     -iz     -1     iz

where $\,z=e^{i\alpha}\in\mathbb{S}^1$ with $\alpha\in[0,\pi)$. It is known that the classification of complex Hadamard matrix in higher dimensions is extremely hard.

Our first major theorem in this regard is the following. Recall that given a pair of complex Hadamard matrices $u$ and $v$, one obtains a pair of spin model subfactors $R_u,R_v\subset R$ (see <Ref> for details). We have introduced an equivalence relation `$\sim$' finer than the Hadamard equivalence `$\simeq$' which completely characterizes when spin model subfactors $R_u,R_v$ arising from two $n\times n$ distinct complex Hadamard matrices $u,v$ are distinct $(R_u\neq R_v)$.

* For distinct $n\times n$ complex Hadamard matrices $u\mbox{ and }v$, the pair of spin model subfactors $R_u\mbox{ and }R_v$ of the hyperfinite type $II_1$ factor $R$ are distinct $(R_u\neq R_v)$ if and only if $\,u\nsim v$.

* If two $n\times n$ complex Hadamard matrices $u\mbox{ and }v$ are Hadamard inequivalent, then the corresponding spin model subfactors $R_u\mbox{ and }R_v$ of $R$ are always distinct $(R_u\neq R_v)$.

§.§.§ The case of $2\times 2$ complex Hadamard matrices

If $u$ and $v$ are two distinct $2\times 2$ complex Hadamard matrices, by <Ref> we have $R_u\neq R_v$ if and only if $v\neq uD$ (that is, $u\nsim v$), where $D$ is a unitary matrix of the form $\left[{\begin{matrix}
z & 0\\
0 & w\\
\end{matrix}}\right]$ or $\left[{\begin{matrix}
0 & z\\
w & 0\\
\end{matrix}}\right]\mbox{ for }z,w\in\mathbb{S}^1$. It turns out that without loss of generality, we can assume $u,v$ are of the following form
1     1

e^iα     -e^iα

1     1

e^iβ     -e^iβ

where $\alpha,\beta\in[0,2\pi)$ and $\alpha\neq\beta$. This situation gives rise to pair of (distinct) spin model subfactors $R_u,R_v\subset R$, and we have a priori a quadruple of von Neumann algebras

\[
\begin{matrix}
R_v & \subset & R\\
\cup & & \cup\\
R_u\cap R_v &\subset & R_u\\
\end{matrix}
\]

If $u$ and $v$ are as above, then the Pimsner-Popa constant $\lambda(R_u,R_v)$ is $\frac{1}{2}$.

The von Neumann subalgebra $R_u\cap R_v$ of $R$ is a $II_1$ subfactor with $[R:R_u\cap R_v]=4$ and $[R_u:R_u\cap R_v]=[R_v:R_u\cap R_v]=2$.

The interior and the exterior angle for the quadruple of $II_1$ factors $(R_u\cap R_v\subset R_u,R_v\subset R)$ are given by the following,

\[
\cos\big(\alpha^{R_u\cap R_v}_R(R_u,R_v)\big)=\cos\big(\beta^{R_u\cap R_v}_R(R_u,R_v)\big)=\cos^2(\alpha-\beta)\,.
\]

The quadruple $(R_u\cap R_v\subset R_u,R_v\subset R)$ is a commuting square (and consequently co-commuting square) if and only if $\,\alpha-\beta=\pm\frac{\pi}{2}$. Here, $\alpha\mbox{ and }\beta$ are associated with the complex Hadamard matrices $u\mbox{ and }v$ as described in <Ref>.

In this regard, we obtain a concrete example of a quadruple of $II_1$ factors (namely, $(R_u\cap R_v\subset R_u,R_v\subset R)$) which is non-degenerate (that is, $\overline{R_uR_v}=\overline{R_vR_u}=R$) but not co-commuting square.

The relative commutant ${(R_u\cap R_v)}^\prime\cap R$ is $\bbc\oplus\bbc$.

The following results completely characterize the quadruple of $II_1$ factors $(R_u\cap R_v\subset R_u,R_v\subset R)$.

The subfactor $R_u\cap R_v\subset R$ is a vertex model subfactor of Jones index $4$. More explicitly, we have the following.

* The pair of subfactors $R_u$ and $R_v$ of the hyperfinite type $II_1$ factor $R$ are conjugate to each other via a unitary in ${(R_u\cap R_v)}^\prime\cap R$.

* The bi-unitary matrix $\,u_2W_2V_2$ in $M_4$ generates (in the sense of basic construction) the composition of subfactors $R_u\cap R_v\subset R_u\subset R$, and consequently the quadruple $(R_u\cap R_v\subset R_u,R_v\subset R)$.

* The subfactor $R_u\cap R_v\subset R$ is isomorphic to the diagonal subfactor

\[
\left\{\begin{bmatrix}
x & 0\\
0 & \theta(x)
\end{bmatrix}:x\in R~\mathrm{and}~\theta\in\mathrm{Out}(R)~\mathrm{such~ that}~{\theta}^2=\mathrm{id}\right\}\subset M_2(\mathbb{C})\otimes R\,.
\]

Under this isomorphism, $R_u$ becomes $\big(\mathrm{diag}\{1,e^{i(\alpha-\beta)}\}\big)R_v\big(\mathrm{diag}\{1,e^{i(\beta-\alpha)}\}\big)$, where $R_v$ is given by the following

\[
\Bigg\{\begin{bmatrix}
x & y\\
\theta(y) & \theta(x)
\end{bmatrix} : x,y \in R\Bigg\}\,.
\]

The relative position of $R_u\mbox{ and }R_v$ are described by the following results.

We have the following.

* $H(R|R_u\cap R_v)=2\log 2\mbox{ and }H(R_u|R_u\cap R_v)=H(R_v|R_u\cap R_v)=\log 2$.

* $h(R_u|R_v)=\eta\big(\cos^2\big(\frac{\alpha-\beta}{2}\big)\big)+\eta\big(\sin^2\big(\frac{\alpha-\beta}{2}\big)\big)\,.$

* $\eta\big(\cos^2\big(\frac{\alpha-\beta}{2}\big)\big)+\eta\big(\sin^2\big(\frac{\alpha-\beta}{2}\big)\big)\leq H(R_u|R_v)\leq\log 2\,,$
and if $\,\alpha-\beta=\pm\frac{\pi}{2}$, then $H(R_u|R_v)=h(R_u|R_v)=\log 2$. Here, $\alpha\mbox{ and }\beta$ are associated with the complex Hadamard matrices $u\mbox{ and }v$ as described in <Ref>.

The Sano-Watatani angle between the subfactors $R_u\mbox{ and }R_v$ of $R$ is the singleton set $\{\arccos|\cos(\alpha-\beta)|\}$.

§.§.§ The case of $4\times 4$ complex Hadamard matrices

The case of Hadamard inequivalent $4\times 4$ (distinct) complex Hadamard matrices is more delicate and very interesting. Recall that any such complex Hadamard matrix is of the form described in <Ref>. By <Ref>, any two distinct elements $u\mbox{ and }v$ of this family give rise to a pair of spin model subfactors. That is, we have a quadruple of von Neumann algebras

\[
\begin{matrix}
R_v & \subset & R\\
\cup & & \cup\\
R_u\cap R_v &\subset & R_u\\
\end{matrix}
\]

The Pimsner-Popa constant for the pair of subfactors $R_u\mbox{ and }R_v$ of the hyperfinite $II_1$ factor $R$ is given by $\lambda(R_u,R_v)=\frac{1}{2}$.

It turns out that in this case, the (Pimsner-Popa) index of the intersection $R_u\cap R_v$ in $R$ can be both finite and infinite, depending on which $u\mbox{ and }v$ we are starting with. Moreover, in the case of finite index, it can be any number in the set $\{4n:n\in\bbn,\,n\geq 2\}$. Conversely, given any $n\geq 2$, there exist $u\mbox{ and }v$ such that index of $R_u\cap R_v$ in $R$ is equal to $4n$. Below we describe the scenario briefly.

Consider the following subgroup of $\mathbb{S}^1$
Γ:={ω∈𝕊^1:ω^m=1m∈2+2}∪{1} .

That is, $\Gamma$ consists of all even roots of unity. Let $u:=u(a)\mbox{ and }v:=u(b)$ be two $4\times 4$ Hadamard inequivalent complex Hadamard matrices parametrized by the circle parameters $a\mbox{ and }b$ respectively, and we have a pair of subfactors $R_u\mbox{ and }R_v$ of the hyperfinite $II_1$ factor $R$. We define a relation $u(a)\sim u(b)$ if and only if $\theta(a,b):=\overline{a}b\in\Gamma$, that is, $\theta(a,b)^{\,m}=1\mbox{ for some }m\in 2\bbn+2$. Note that we are considering pair $(a,b)$ such that $a\neq b$, and hence $\theta(a,b)\neq 1$. It is easy to check that `$\sim$' is an equivalence relation.

Caution: This equivalence relation has no relation with that in <Ref>.

Our first major theorem in this direction is the following.

If $u\nsim v$, then $\lambda(R,R_u\cap R_v)=\lambda(R_u,R_u\cap R_v)=\lambda(R_v,R_u\cap R_v)=0$. That is, $R_u\cap R_v$ is a von Neumann subalgebra of the hyperfinite $II_1$ factor $R$ with infinite Pimsner-Popa index.

If $u\sim v$, the outcome is completely opposite, and we obtain a series of potentially new integer index subfactors of $R$.

For a pair of Hadamard inequivalent $4\times 4$ complex Hadamard matrices $u=u(a)$ and $v=u(b)$ such that ${(\bar{a}b)}^m=1$, $R_u\cap R_v\subset R$ is an irreducible subfactor of $R$ with $[R:R_u\cap R_v]=2m$. Furthermore, $H(R_u|R_u\cap R_v)=H(R_v|R_u\cap R_v)=\log m-\log 2$ and also, $H(R|R_u\cap R_v)=\log 2+\log m$.

The following results describe the relative position of $R_u\mbox{ and }R_v$. The first few of them describe the angle between $R_u\mbox{ and }R_v$, whereas the last one gives relative entropy between these subfactors.

For the case of $u\sim v$, the interior angle $\alpha^{R_u\cap R_v}_R\big(R_u,R_v\big)$  is strictly  greater than $\pi/3$ and the exterior angle $\beta^{R_u\cap R_v}_R\big(R_u,R_v\big)=\arccos 1/3$. Furthermore, $\alpha^{R_u\cap R_v}_R\big(R_u,R_v\big)$ converges to $\pi/3$ decreasingly as $m$ tends to infinity.

The quadruple of von Neumann algebras

\[
\begin{matrix}
R_{u(b)} & \subset & R\\
\cup & & \cup\\
R_{u(a)}\cap R_{u(b)} &\subset & R_{u(a)}\\
\end{matrix}
\]

is a commuting square if and only if $(\overline{a}b)^4=1$, that is $b=\pm ia$.

Let $u=u(a)$ and $v=u(b)$ be distinct $4\times 4$ inequivalent complex Hadamard matrices parametrized by the circle parameters $a\mbox{ and }b$ such that $u\sim v$ with $(\overline{a}b)^m=1,\,m\in 2\bbn+2$. Then, we have the following.

* If $\,m=4$, then $\mathrm{Ang}_R(R_u,R_v)=\{\pi/2\};$

* If $\,m\geq 6$, then $\mathrm{Ang}_{R}(R_u,R_v)=\left\{\frac{2k\pi}{m}\,:\,k=1,2,\ldots, \lfloor\frac{m-2}{4}\rfloor\right\}$.

For a pair of Hadamard inequivalent $4\times 4$ complex Hadamard matrices $u\mbox{ and }v$,

* if $u\sim v$, then $H(R_u|R_v)=h(R_u|R_v)=\log 2;$

* if $u\nsim v$, then $0<\log 2+\frac{1}{8}\left(\eta|1+a\overline{b}|^2+\eta|1-a\overline{b}|^2\right)\leq H(R_u|R_v)\leq\log 2\,.$

The crucial ingredient that we have used to prove the above theorem is the existence of an intermediate subfactor $R_u\cap R_v\subset\mathcal{I}_{u,v}\subset R$ containing both $R_u\mbox{ and }R_v$. For the case of $u\nsim v$, we are not able to obtain the exact value of the relative entropy $H(R_u|R_v)$. Major difficulties are that $R_u\cap R_v$ is of infinite Pimsner-Popa index in $R$, and its factoriality is not known.

§.§.§ Results in finite dimensions

It turns out that both the Pimsner-Popa probabilistic constant and the Connes-Størmer relative entropy for a pair of subfactors behave well under `controlled' limits. This suggests that to achieve results in infinite-dimension, we should look at finite-dimensional situations. The same theme has been highlighted in Section $6$ [37] for the case of a single subfactor. In this paper, prior to the above results on infinite factors, we have en route a couple of results in finite dimensions that are instrumental in proving our results. These can also be treated as results of independent interest.

If $\Delta_n$ and $u\Delta_n u^*$ are two Masas in $M_n(\mathbb{C})$, $u$ being a unitary matrix in $M_n(\mathbb{C})$, then the Pimsner-Popa constant is given by the following:

\[
\lambda(\Delta_n\,,\,u\Delta_n u^*)=\min_{1\leq i\leq n}\,\big(\mbox{$h\left(u^*\right)_i$}\big)^{-1}
\]

where $\left(u^*\right)_i$ is the $i$-th column of $u^*$ and $h$ denotes the Hamming number of a vector in $\bbc^n$ (see <Ref>).

This result is instrumental in determining when spin model subfactors arising from distinct $n\times n$ complex Hadamard matrices are distinct (<Ref>).

For the subalgebras $M_n(\bbc)$ and $uM_n(\bbc)u^*$ of $M_n(\bbc)\oplus M_n(\bbc)$, where $\,u\in\Delta_2\otimes M_n(\bbc)$ is a unitary matrix given by $\,u=\left[{\begin{smallmatrix}
u_1 & 0\\
0 & u_2\\
\end{smallmatrix}}\right]$ with unitary matrices $u_1,u_2\in M_n(\bbc)$, we have the following :
λ(M_n() , uM_n()u^*) =
1        u_1^*u_2 , 1/2        u_1^*u_2 .

This result is instrumental in computing the value of the Pimsner-Popa constant between the pairs of spin model subfactors (both $2\times 2$ and $4\times 4$ situations in Sections <ref> and <ref>).

§.§ Outline of the paper

The organization of the paper is as follows: In <Ref>, the one following this Introduction, we discuss the key invariants for “two subfactors" introduced earlier. In <Ref>, we introduce the notion of “commuting cube", which is instrumental for this paper, as well as a general study of  two subfactors. In <Ref>, we concentrate on pairs of spin model subfactors, a particular situation of two subfactors. <Ref> contains a key result concerning the Pimsner-Popa constant in a finite-dimensional situation. Then, in <Ref>, a pair of subfactors arising from $2\times 2$ complex Hadamard matrices have been investigated. This is followed by the case of $4\times 4$ inequivalent complex Hadamard matrices in <Ref>, which is one of the major highlights of this paper. Finally, in <Ref>, we discuss a few open problems and possible directions for future work. In the Appendix (<Ref>), we have provided detailed proofs for the construction of the tower of basic constructions for the interested readers.

§ A FEW KEY INVARIANTS FOR TWO SUBFACTORS

In this section, we discuss a few key invariants for “two subfactors", namely, the Sano-Watatani angle, the interior and exterior angle, the Pimsner-Popa probabilistic constant, and the Connes-Størmer relative entropy along with a modified version of it. We also recall the basic construction of a non-degenerate symmetric commuting square, and a few essential results to be used throughout the article.

§.§ Angle between two subfactors and commuting square

In [39], Popa introduced a notion of orthogonality for a pair of von Neumann subalgebras $\mathcal{B}_1$ and $\mathcal{B}_2$ of a finite von Neumann algebra $\mathcal{M}$. Suppose that $\mathcal{M}$ is commutative, and let $(X,\mathcal{B},\mu)$ be a probability space such that $\mathcal{M}$ is isomorphic to $L^{\infty}(X,\mu)$. If $\mathcal{B}_1$ and $\mathcal{B}_2$ are von Neumann subalgebras of $\mathcal{M}$, then $\mathcal{B}_1$ is orthogonal to $\mathcal{B}_2$ if and only if the corresponding $\sigma$-subalgebras of $\mathcal{B}$ are independent. Thus, orthogonality is the non-commutative version of classical independence. As a marginal generalization of the notion of orthogonality, Popa also introduced the so-called `commuting square', which proves to be an indispensable tool in subfactor theory. Consider an inclusion of finite von Neumann algebras $\mathcal{N}\subset \mathcal{M}$ with a fixed trace $tr$ on $\mathcal{M}$ and intermediate von Neumann subalgebras $\mathcal{P}$ and $\mathcal{Q}$. Thus, we obtain a quadruple of von Neumann algebras

\[
\begin{matrix}
\mathcal{Q} &\subset & \mathcal{M} \cr \cup &\ &\cup\cr \mathcal{N} &\subset & \mathcal{P}\,.
\end{matrix}
\]

If $\mathcal{P}\vee \mathcal{Q}=\mathcal{M}$ and $\mathcal{P}\wedge \mathcal{Q}=\mathcal{N}$, then a quadruple is called a quadrilateral.

A quadruple

\[
\begin{matrix}
\mathcal{Q} &\subset & \mathcal{M} \cr \cup &\ &\cup\cr \mathcal{N} &\subset & \mathcal{P}
\end{matrix}
\]

of finite von Neumann algebras is called a commuting square if $E^{\mathcal{M}}_{\mathcal{P}}E^{\mathcal{M}}_{\mathcal{Q}}=E^{\mathcal{M}}_{\mathcal{Q}}E^{\mathcal{M}}_{\mathcal{P}}=E^{\mathcal{M}}_{\mathcal{N}}$. The quadruple is said to be non-degenrate if ${\overline{\mathcal{PQ}}}^{\,\tiny \mbox{SOT}}={\overline{\mathcal{QP}}}^{\,\tiny \mbox{SOT}}=\mathcal{M}$. A quadruple is called a non-degenerate commuting square or symmetric commuting square if it is a commuting square and non-degenerate.

For brevity, we sometimes write $(\mathcal{N}\subset\mathcal{P,Q}\subset \mathcal{M})$ to mean the quadruple

\[
\begin{matrix}
\mathcal{Q} &\subset & \mathcal{M} \cr \cup &\ &\cup\cr \mathcal{N} &\subset & \mathcal{P}
\end{matrix}
\]

Note that for a quadruple $(\mathcal{N}\subset \mathcal{P,Q}\subset \mathcal{M})$, if $E^\mathcal{M}_\mathcal{Q}(\mathcal{P})\subseteq \mathcal{N}$ (or $E^\mathcal{M}_\mathcal{P}(\mathcal{Q})\subseteq \mathcal{N}$), then it is a commuting square. Indeed, observe that for any $x\in M$, using the hypothesis $E^M_Q(P)\subseteq N$, we have the following
E_Q^ME^M_P(x)=E^Q_NE_Q^ME^M_P(x)=E^M_NE^M_P(x)=E^P_NE^M_P(x)=E^M_N(x) .

Moreover, the converse is also true.

[[40], [43],[44]]
A quadruple

\[
\begin{matrix}
\mathcal{Q} &\subset & \mathcal{M} \cr \cup &\ &\cup\cr \mathcal{N} &\subset & \mathcal{P}
\end{matrix}
\]

of finite von Neumann algebras with a fixed (finite, faithful, normal) trace $\tau^\prime$ on $\mathcal{N}^{\,\prime}$ is called a co-commuting square if the quadruple

\[
\begin{matrix}
\mathcal{P}^\prime &\subset & \mathcal{N}^\prime \cr \cup &\ &\cup\cr \mathcal{M}^\prime &\subset & \mathcal{Q}^\prime
\end{matrix}
\]

is a commuting square.

Since $\mathcal{M}^\prime=\mathcal{P}^\prime\cap\mathcal{Q}^\prime$, it is necessary for $\mathcal{M}=\mathcal{P}\vee\mathcal{Q}$ to hold. We refer to [13, 26, 40, 2] for a comprehensive treatment on commuting and co-commuting squares.

In general, co-commuting and non-degenerate need not be same. If a quadruple $(N\subset P,Q\subset M)$ of $II_1$ factors is a commuting square with $[M:N]<\infty$, then it is a co-commuting square if and only if non-degenerate (Theorem $7.1$ and Corollary $7.1$ in [43]). If the commuting square condition is absent, but irreducibility and finite-index of $N\subset M$ is given, then also this holds (Theorem $3.21$ in [16]). In the same spirit, if $(N\subset P,Q\subset M)$ is a commuting square of finite-dimensional $C^*$-algebras with $||\Lambda^M_Q||^2=[M:Q]=[P:N]=||\Lambda^P_N||^2$ (Watatani index), then the quadruple is non-degenerate (Lemma $3.10$ in [5]).

A quadruple need not always be a commuting square as there might have nontrivial `angle' between the subalgebras. Indeed, as a generalization of commuting square, Sano and Watatani  introduced a notion of “angle" between a pair of subalgebras of a given finite von Neumann algebra as the spectrum of certain angle operator [43]. This was motivated by relative position between two different (closed) subspaces $\mathcal{K}$ and $\mathcal{L}$ in a Hilbert space $\mathcal{H}$. Recall the angle operator $\Theta(p,q)$, where $p$ (resp. $q$) is the orthogonal projection onto $\mathcal{K}$ (resp. $\mathcal{L}$). The set $\text{Ang}(p,q)$ of angles between $p$ and $q$ is the subset of $[0,\pi/2]$ defined by the following (see Definition $2.1$ in [43]),

\begin{equation}
\mathrm{Ang}(p,q)=\begin{cases}
\text{sp}~\Theta(p,q), & \text{if}~~ pq\neq qp.\\
\{\pi/2\},& \text{otherwise.}
\end{cases}
\end{equation}

Note that $\Theta$ is a positive operator and the spectrum of $\Theta$ is contained in $[0,\frac{\pi}{2}]$, but $0\mbox{ and }\frac{\pi}{2}$ are not eigenvalues.
Let $\mathcal{M}$ be a finite von Neumann algebra with a faithful normal tracial state $\mathrm{tr}$ and $\mathcal{P,Q}$ be von Neumann subalgebras of $\mathcal{M}$. The trace $\mathrm{tr}$ determines the normal faithful conditional expectations $E^{\mathcal{M}}_{\mathcal{P}}:\mathcal{M}\to\mathcal{P}$ and $E^{\mathcal{M}}_{\mathcal{Q}}:\mathcal{M}\to\mathcal{Q}$. They extend to the orthogonal projections $e_{\mathcal{P}}\mbox{ and }e_{\mathcal{Q}}$ on the GNS Hilbert space $L^2(\mathcal{M})$. The angle $\mathrm{Ang}_{\mathcal{M}}(\mathcal{P,Q})$ between $\mathcal{P}$ and $\mathcal{Q}$  is defined as follows :

\[{\mathrm{Ang}}_{\mathcal{M}}(\mathcal{P,Q}):=\mathrm{Ang}(e_{\mathcal{P}},e_{\mathcal{Q}}).\]

The following facts are well-known (see [43]).

* For $pq\neq qp$, $\text{Ang}(p,q)\setminus\{\frac{\pi}{2}\}=\sigma\big(\mbox{arccos}\sqrt{pqp-p\wedge q}\big)\setminus\{\frac{\pi}{2}\}$.

* ${\mathrm{Ang}}_{\mathcal{M}}(\mathcal{P,Q})={\mathrm{Ang}}_{\mathcal{M}}(\mathcal{Q,P})$ and ${\mathrm{Ang}}_{\widetilde{\mathcal{M}}}(\mathcal{P,Q})={\mathrm{Ang}}_{\mathcal{M}}(\mathcal{P,Q})$ for any finite von Neumann algebra $\widetilde{\mathcal{M}}\supset\mathcal{M}$ with trace $\widetilde{\mathrm{tr}}$ such that $\widetilde{\mathrm{tr}}|_{\mathcal{M}}=\mathrm{tr}$.

* For quadrilateral of $II_1$ factors $(N\subset P,Q\subset M),\,\mathrm{Ang}_M(P,Q)$ is a finite set which contains neither $0$ nor $\frac{\pi}{2}$. Moreover, $\mathrm{Ang}_M(P,Q)=\sigma\big(\mbox{arccos}\sqrt{e_Pe_Qe_P-e_N}\big)\setminus\{\frac{\pi}{2}\}$.

In [27], Jones and Xu proved that finiteness of the angle (as a substet of $[0,\pi/2]$) is equivalent to finiteness of the (Pimsner-Popa)index of $P\cap Q$ in $M$. In [16], Grossman and Jones have considered a slight variant of the angle between two subfactors $P,Q\subset M$, namely the spectrum of the positive self-adjoint operator $E_PE_QE_P$ (on $L^2(M)$).

In another direction, to understand the relative position between a pair of intermediate subfactors $P$ and $Q$ of a finite index subfactor $N\subset M$ (i.e., $N\subset P,Q\subset M$), the first author along with Das, Liu and Ren have introduced a new notion of angle $\alpha^N_M(P,Q)$ between $P$ and $Q$ [2]. This angle has been used to answer an open question by Longo. More explicitly, the existing upper bound for the cardinality of the lattice of intermediate subfactors have been improved. Furthermore, a surprising connection between the intermediate subfactor theory and kissing numbers/sphere packing in geometry have been discovered.

[[2]](Angle between intermediate subfactors)
Let $P$ and $Q$ be two intermediate subfactors of a subfactor $N\subset M$. Then, the interior angle $\alpha^N_M(P,Q)$ between $P$ and $Q$ is given by

\[
\alpha^N_M(P, Q)=\cos^{-1}{\langle v_P,v_Q\rangle}_2\,,
\]

where $v_P:=\frac{e_P-e_N}{{\lVert e_P-e_N\rVert}_2}$ (and similarly $v_Q$),$\,{\langle x,y\rangle}_2:=tr(y^*x)$ and ${\lVert x\rVert}_2:=(tr(x^*x))^{1/2}$. The exterior angle between $P$ and $Q$ is given by $\beta^N_M(P, Q)=\alpha^M_{M_1}(P_1, Q_1)$.

The interior and exterior angle can be depicted pictorially as in <Ref>. Here, $P_1,Q_1,M_1$ denote the basic constructions of $P\subset M,\,Q\subset M,\mbox{ and }N\subset M$ respectively.

9 cm!

Interior and Exterior angles

The following results are well-known.
[[43, 2]]
Consider a quadruple

\begin{equation}\label{quadruple}
\begin{matrix}
\mathcal{Q} &\subset & \mathcal{M} \cr \cup &\ &\cup\cr \mathcal{N} &\subset & \mathcal{P}
\end{matrix}
\end{equation}

of finite von Neumann algebras with a fixed trace $tr$ on $\mathcal{M}$ and suppose that $E^{\mathcal{M}}_{\mathcal{Q}}, E^{\mathcal{M}}_{\mathcal{P}}$ and $E^{\mathcal{M}}_{\mathcal{N}}$ are the corresponding $tr$-preserving conditional expectations. The quadruple $(\ref{quadruple})$ is a commuting square if and only if $\mathrm{Ang}_M(P,Q)=\pi/2.$ Furthermore, if the quadruple $(\ref{quadruple})$ is a quadruple of $II_1$ factors with $[M:N]<\infty$, then it is a commuting square if and only if $\alpha^N_M(P,Q)=\pi/2$ and co-commuting square if and only if $\beta^N_M(P,Q)=\pi/2.$

Consider intermediate subfactors $P\mbox{ and }Q$ of $II_1$ factors $N\subset M$. Let $\{\lambda_i\}$ (resp., $\{\mu_j\}$) be (right) basis for $P/N$ (resp., $Q/N$). Then,

\[
\cos(\alpha^N_M(P,Q))=\frac{\sum_{i,j}\mathrm{tr}\big(E^M_N(\lambda_i^*\mu_j)\mu_j^*\lambda_i\big)-1}{\sqrt{[P:N]-1}\,\sqrt{[Q:N]-1}}\,.
\]

Below we mention a few well-known examples of commuting squares.

$(1)$ If $R$ is the hyperfinite $II_1$ factor and $G$ is a finite group with subgroups $H$ and $K$, then the quadruple

R\rtimes K &\subset & R\rtimes G \cr \cup &\ &\cup\cr R &\subset & R\rtimes H
\end{matrix}$$
is a commuting square. Thus, $\mathrm{Ang}_R(R\rtimes H, R\rtimes K)=\{\pi/2\}$.

$(2)$ Suppose that $K\subset L$ is an inclusion of finite von Neumann algebras with a  fixed trace $tr_L$ on $L$. Then, $M_n(L)$ is again a finite von Neumann algebra with the trace $tr_{M_n(\mathbb{C})}\otimes tr_L$, and it follows that the following quadruple

\[\begin{matrix}
M_n(K) &\subset & M_n(L) \cr \cup &\ &\cup\cr K &\subset & L\end{matrix}\]

is a commuting square with respect to the $\,tr_{M_n(\mathbb{C})}\otimes tr_L$-preserving conditional expectations.

Complex Hadamard matrices provide examples of  commuting and co-commuting squares (see [26], for instance), which will play a central role in this paper.

Consider the subalgebra $\Delta_n$ of diagonal matrices in $M_n(\mathbb{C})$, and let $u$ be a unitary matrix in $M_n(\mathbb{C})$. Then, the following conditions are equivalent.

* The quadruple
u\Delta_n u^* &\subset & M_n(\mathbb{C}) \cr \cup &\ &\cup\cr \mathbb{C} &\subset & \Delta_n
\end{matrix}$$
is a commuting and co-commuting square.

* $u$ is a complex Hadamard matrix.

§.§ Pimsner-Popa probabilistic constant

Given a subfactor $N\subset M$ of type $II_1$ factors, Jones [25] discovered a notion of index $[M:N]$ as the Murray-von Neumann’s coupling constant $\text{dim}_{N}(L^2(M))$. For von Neumann subalgebras $\mathcal{Q \subset P\subset M}$ of a finite von Neumann algebra $\mathcal{M}$, Pimsner and Popa [37] defined the following probabilistic constant

\begin{equation}\label{probabilistic index}
\lambda(\mathcal{P,Q})=\text{sup}\{\lambda\geq 0\,:\,E_\mathcal{Q}^\mathcal{P}(x)\geq \lambda x\,\,\forall\,x\in {\mathcal{P}}_{+}\}.
\end{equation}

Moreover, if $M$ is a type $II_1$ factor and $N\subset M$ is a subfactor, then $E_N(x)\geq [M:N]^{-1}x$ for all $x\in M_+$ (with the convention $\frac{1}{\infty}=0$) and

\begin{equation}\label{pimsnerpopa2}
\lambda(M,N)=[M:N]^{-1}.
\end{equation}

Thus, the Pimsner-Popa probabilistic constant serves as a substitute of the Jones index.

For von Neumann algebras $\mathcal{Q \subset P}$, we call $\lambda(\mathcal{P},\mathcal{Q})^{-1}$ the Pimsner-Popa index.

We would like to mention the following important properties of $\lambda$.

Property 1:  $\lambda$ does not respect tensor product. The simplest example is that $\lambda(M_2,\bbc)=\frac{1}{2}$, whereas $\lambda(M_2\otimes M_2,\bbc\otimes M_2)=\lambda(M_4,M_2)=\frac{1}{4}$ (see Sec. $6$ in [37]).

Property 2:  For $\mathcal{N\subset Q\subset P\subset M}$ we have $\lambda(\mathcal{P,N})\geq\lambda(\mathcal{P,Q})\lambda(\mathcal{Q,N})$. To see this, observe that $E^\mathcal{Q}_\mathcal{N}(x)\geq\lambda(\mathcal{Q,N})x$ for all $x\in \mathcal{Q}_+$ implies that $E^\mathcal{Q}_\mathcal{N}E^\mathcal{P}_\mathcal{Q}(y)\geq\lambda(\mathcal{Q,N})E^\mathcal{P}_\mathcal{Q}(y)$ for all $y\in \mathcal{P}_+$. This says that $E^\mathcal{P}_\mathcal{N}(y)\geq\lambda(\mathcal{Q,N})E^\mathcal{P}_\mathcal{Q}(y)\geq\lambda(\mathcal{Q,N})\lambda(\mathcal{P,Q})y$ for all $y\in \mathcal{P}_+$, and hence the result follows. We call it the submultiplicativity property of $\lambda$.

Property 3:  In general, $\lambda$ is not multiplicative. For example, take $\bbc\subset M_2\subset M_4$ and apply Theorem $6.1$ in Sec. $6$, [37] to observe that the inequality in property $2$ is strict.

Property 4:  $\lambda$ is not well-behaved with respect to (symmetric) commuting square. For example, consider the quadruple $\big(\bbc\otimes\bbc\subset\bbc\otimes M_2,M_2\otimes\bbc\subset M_2\otimes M_2\big)$ which is a (symmetric) commuting square. Although $\lambda(\bbc\otimes M_2,\bbc\otimes\bbc)=\lambda(M_2,\bbc)=1/2$, we have $\lambda(M_2\otimes M_2,M_2\otimes\bbc)=\lambda(M_4,M_2)=1/4$ by Theorem $6.1$ in Sec. $6$, [37].

The definition of $\lambda(\mathcal{P,Q})$ as in <Ref> works for general von Neumann subalgebras $\mathcal{P}$ and $\mathcal{Q}$ (not necessarily $\mathcal{Q \subset P})$ of a finite von Neumann algebra $\mathcal{M}$ as well.

Consider a pair of von Neumann subalgebras $\mathcal{P}$ and $\mathcal{Q}$ of a finite von Neumann algebra $\mathcal{M}$. The Pimsner-Popa probabilistic constant for the ordered pair $(\mathcal{P,Q})$  is defined as follows :

\[
\lambda(\mathcal{P,Q})=\sup\{t\geq 0\,:\,E_\mathcal{Q}^\mathcal{M}(x)\geq t x\,\,\forall\,x\in {\mathcal{P}}_{+}\}.
\]

Note that in general, $\lambda(\mathcal{P,Q}))\neq\lambda(\mathcal{Q,P})$. For a quadruple $(N\subset P,Q\subset M)$ of type $II_1$ factors with $[M:N]<\infty$, if the quadruple is a commuting square, then $\lambda(P,Q)=[P:N]^{-1}$ and $\lambda(Q,P)=[Q:N]^{-1}$. Moreover, if the quadruple is a co-commuting square, then $\lambda(P,Q)=[M:Q]^{-1}$ and $\lambda(Q,P)=[M:P]^{-1}$ (see Theorem $3.12$ in [4]). We mention that we always work with unital algebras, and we have $0\leq \lambda(\mathcal{P,Q})\leq 1$. Indeed, $E_\mathcal{Q}^\mathcal{M}(x)\geq t x$ implies that $||x||\geq ||E_\mathcal{Q}^\mathcal{M}(x)||\geq t ||x||$, and hence $\lambda(\mathcal{P,Q})\leq 1$. This justifies the term `probabilistic constant'. Henceforth, we drop the word `probabilistic' for brevity. For a type $II_1$ factor $M$ and subfactors $N_1,N_2\subset M$, we have $E_{N_2}^M(x)\geq [M:N_2]^{-1}x\,\,\forall\,x\in M_+$, and hence $\lambda(M,N_2)\leq\lambda(N_1,N_2)$. Thus, we always have a lower bound for $\lambda(N_1,N_2)$, namely $\lambda(N_1,N_2)\geq[M:N_2]^{-1}$. Computation of $\lambda(\mathcal{P,Q})$ is in general hard. If $(N,Q,P,M)$ is a quadruple of type $II_1$ factors with $[M:N]< \infty$ and $N^{\prime}\cap M=\mathbb{C}$, then the following formula

$\lambda(P,Q)=\displaystyle \frac{\mbox{tr}(e_Pe_Q)}{\mbox{tr}(e_P)}$

is known due to Theorem $3.3$ in [4]. However, if $N\subset M$ is not irreducible, then the situation is unclear. It seems to be a proper place to mention that we have some results in this direction in Sections <ref> and <ref>.

Consider a pair of von Neumann subalgebras $\mathcal{P}$ and $\mathcal{Q}$ of a finite von Neumann algebra $\mathcal{M}$. Then, $\lambda(\mathcal{P,Q})=1$ if and only if $\mathcal{P}\subseteq\mathcal{Q}$.

Assume that $\lambda(\mathcal{P,Q})=1$. Then, $E_\mathcal{Q}^\mathcal{M}(x)\geq x$ for all $x\in\mathcal{P}_+$. Since the trace `$\mbox{tr}$' on $\mathcal{M}$ is faithful, we have $\mbox{tr}(E_\mathcal{Q}^\mathcal{M}(x)-x)\geq 0$. However, $\mbox{tr}\circ E_\mathcal{Q}^\mathcal{M}=\mbox{tr}$ implies that $E_\mathcal{Q}^\mathcal{M}(x)=x$, and hence $x\in\mathcal{Q}$. Thus $\mathcal{P}_+\subseteq\mathcal{Q}$, and hence $\mathcal{P}\subseteq\mathcal{Q}$. The converse direction is obvious from the definition.

Adapting the proof of Proposition $2.6$ in [37], we get the following result which is very helpful in finding the value of $\lambda$.

* Let $\{M_n\}, \{A_n\}$ and $\{B_n\}$ be increasing sequences of von Neumann subalgebras of a finite von Neumann algebra $M$ such that $\{A_n\}, \{B_n\}\subset M$ and $M=\big(\bigcup_{n=1}^{\infty}M_n\big)^{\dprime}.$ If $A=\big(\bigcup_{n=1}^{\infty} A_n\big)^{\dprime}$ and $B=\big(\bigcup_{n=1}^{\infty} B_n\big)^{\dprime}$, then $\lambda(B,A)\geq \displaystyle \limsup\,\lambda(B_n,A_n).$

* If in adition, $E_{A_{n+1}}E_{M_n}=E_{A_n}$ and $E_{B_{n+1}}E_{M_n}=E_{B_n}$ for $n\in \mathbb{N}$, then $\lambda(B,A)=\displaystyle \lim\,\lambda(B_n,A_n),$ decreasingly.

First observe that for $x\in M$, $E_A(x)=\displaystyle \lim_{n \to \infty} E_{A_n}(x)$ and $E_BE_A(x)=\displaystyle \lim_{n\to \infty} E_{B_n} E_{A_n}(x)$ in strong operator topology. Put $\lambda_n=\lambda(B_n,A_n)$ and let $\lambda=\limsup \lambda_n$. Thus, for arbitrarily fixed $\epsilon>0$, we must obtain a subsequence $\{\lambda_{k_n}\}$ such that $\lambda_{k_n}\geq \lambda-\epsilon$, for $n\geq 1.$ Now, for any $x\in M_{+}$, we have

\[
E_{A_{k_n}}\big(E_{B_{k_n}}(x)\big)\geq \lambda_{k_n} E_{B_{k_n}}(x)\geq (\lambda-\epsilon) E_{B_{k_n}}(x)
\]

and letting $n\to \infty$, we obtain $E_A\big(E_B(x)\big)\geq (\lambda-\epsilon) E_B(x)$ for all $x \in M_{+}$. Therefore, $\lambda(B,A)\geq \lambda-\epsilon$. As $\epsilon>0$ is arbitrary, we conclude the first part.

Now suppose that $E_{A_{n+1}}E_{M_n}=E_{A_n}$ and $E_{B_{n+1}}E_{M_n}=E_{B_n}$ for $n\in \mathbb{N}$. To see that the sequence $\{\lambda_n\}$ is decreasing, consider $x\in (B_n)_+$ and observe that $E_{A_n}(x)=E_{A_{n+1}}(x)\geq \lambda_{n+1} x$, and hence $\lambda_n\geq \lambda_{n+1}$. Furthermore, we see that for any $n\geq 1, E_AE_{M_n}=E_{A_n}$ and $E_BE_{M_n}=E_{B_n}$. So, for any $x\in(B_n)_+$ we have $E_{A_n}(x)= E_A(x)\geq \lambda(B,A) x$. Thus $\lambda_n\geq \lambda(B,A)$, and hence we get $\lambda(B,A)\leq \displaystyle \liminf\,\lambda_n\,$ which concludes the second part.

Note that if $A_n\subset B_n$ for all $n\geq 1$ in the above, then we recover Proposition $2.6$ in [37].

§.§ Connes-Størmer relative entropy

To obtain an appropriate definition of the entropy of an automorphism in the non-commutative framework of operator algebras, Connes and Størmer [12] defined relative entropy $H(\mathcal{P|Q})$ between a pair of finite-dimensional von Neumann-subalgebras $\mathcal{P}$ and $\mathcal{Q}$ of a finite von Neumann algebra $\mathcal{M}$ equipped with a fixed faithful normal trace. This generalizes the classical notion of conditional entropy from ergodic theory. Using the relative entropy as the main technical tool, they have proved a non-commutative version of the Kolmogorov-Sinai type theorem (see [33] also). Pimsner and Popa [37] observed that the definition of the Connes-Størmer relative entropy does not depend on $\mathcal{P,Q}$ being finite-dimensional, so that one may also consider the relative entropy $H(\mathcal{P|Q})$ for arbitrary von Neumann subalgebras $\mathcal{P,Q \subset M}.$
Let $(\mathcal{M},\tau)$ be a finite von Neumann algebra and $\mathcal{P,Q\subseteq M}$ are von Neumann subalgebras. Let

\begin{align*}
\gamma &= \{x_j\in \mathcal{M}_+:\sum x_j=1,\,j=1,\ldots,n\}\,\,\,\mbox{ be a finite partition of unity},\cr
\eta &: [0,\infty)\longrightarrow\mathbb{R}\,\mbox{ be the continuous function }\,\,t\longmapsto -t\log t,\cr
H_\gamma(\mathcal{P|Q}) &:= \sum_{j=1}^n\left(\tau\circ\eta\,E_{\mathcal{Q}}^{\mathcal{M}}(x_j)-\tau\circ\eta\,E_{\mathcal{P}}^{\mathcal{M}}(x_j)\right).
\end{align*}

Then, $H(\mathcal{P|Q}):=\sup_{\gamma}\,H_\gamma(\mathcal{P|Q})$ is the Connes-Størmer relative entropy between $\mathcal{P}\mbox{ and }\mathcal{Q}$.

The relative entropy has the following important properties (see [12],[37],[33]).

Property 1:  $H(\mathcal{P|Q})\geq 0$ and $H(\mathcal{P|Q})=0$ if and only if $\mathcal{P}\subseteq\mathcal{Q}$.

Property 2:  $H(\mathcal{P|Q})\leq H(\mathcal{P|N})+H(\mathcal{N|Q})$ for $\mathcal{P,Q,N\subseteq M}$.

Property 3:  $H(\mathcal{P|Q})$ is increasing in $\mathcal{P}$ and decreasing in $\mathcal{Q}$.

Property 4:  In general, $H$ is not additive. For example, take $\bbc\subset M_2\subset M_4$ and apply ($6.5$ in Sec. $6$, [37]) to observe that $H(M_4|\bbc)=\log 4<H(M_4|M_2)+H(M_2|\bbc)=\log 4+\log 2$.

By Property $2$, for a pair of von Neumann subalgebras $\mathcal{P,Q\subseteq M}$ we always have an upper bound for the relative entropy $H(\mathcal{P|Q})$, namely $H(\mathcal{P|Q})\leq H(\mathcal{P|M})+H(\mathcal{M|Q})=H(\mathcal{M|Q})$ (using Property $1$ we have $H(\mathcal{P|M})=0$). In fact, $\mathcal{M}$ can be replaced by $\mathcal{P}\vee\mathcal{Q}$, that is, $H(\mathcal{P|Q})\leq H(\mathcal{P}\vee\mathcal{Q}|\mathcal{Q})$. We record the following useful result which shows that the relative entropy behaves well with respect to `controlled' limit, the proof of which follows from a minor modification of Proposition $3.4$ in [37], and hence we omit it.

* Let $\{M_n\}, \{A_n\}$ and $\{B_n\}$ be increasing sequences of von Neumann subalgebras of a finite von Neumann algebra $M$ such that $\{A_n\}, \{B_n\}\subset M$ and $M=\big(\bigcup_{n=1}^{\infty}M_n\big)^{\dprime}.$ If $A=\big(\bigcup_{n=1}^{\infty} A_n\big)^{\dprime}$ and $B=\big(\bigcup_{n=1}^{\infty} B_n\big)^{\dprime}$, then $H(B|A)\leq \displaystyle \liminf\,H(B_n|A_n).$

* If in addition, $E_{A_{n+1}}E_{M_n}=E_{A_n}$ and $E_{B_{n+1}}E_{M_n}=E_{B_n}$ for $n\in \mathbb{N}$, then $H(B|A)=\displaystyle \lim\,H(B_n|A_n)$ increasingly.

In most of the situations, finding exact value of $H(\mathcal{P|Q})$ is difficult. For example, if we consider the following simple situation

\[
\begin{matrix}
\Delta_n &\subset & M_n \cr
& & \cup\cr
&  & U\Delta_n U^*
\end{matrix}
\]

where $U\in M_n$ is a unitary and $\Delta_n\subset M_n$ is the diagonal subalgebra (Masa), then the formula for $H(\Delta_n|U\Delta_n U^*)$, even when $n=2$, seems to be hard and is open in the literature as mentioned in [36].

In literature, there is a modified version of the Connes-Størmer relative entropy which is `more' computable. We will take help of it at a place towards the end of the article to compute the value of the Connes-Størmer relative entropy, and hence discuss it briefly.

Suppose that $(\mathcal{M},\tau)$ be a finite von Neumann algebra and $\mathcal{P,Q\subseteq M}$ are von Neumann subalgebras. Let

\begin{align*}
\gamma &= \{x_j\in \mathcal{P}_+:\sum x_j=1,\,j=1,\ldots,n\}\,\,\,\mbox{ be a finite partition of unity},\cr
\eta &: [0,\infty)\longrightarrow\mathbb{R}\,\mbox{ be the continuous function }\,\,t\longmapsto -t\log t,\cr
h_\gamma(\mathcal{P|Q}) &:= \sum_{j=1}^n\left(\tau\circ\eta\,E_{\mathcal{Q}}^{\mathcal{M}}(x_j)-\tau\circ\eta\,(x_j)\right).
\end{align*}

Then, $h(\mathcal{P|Q}):=\sup_{\gamma}\,h_\gamma(\mathcal{P|Q})$ is the modified Connes-Størmer relative entropy between $\mathcal{P}\mbox{ and }\mathcal{Q}$.

If $\mathcal{M}$ is abelian, then $H(\mathcal{P|Q})=h(\mathcal{P|Q})$. Thus, $h$ also generalizes the classical relative entropy. Moreover, if $\mathcal{P}\subset\mathcal{M}$ then $H(\mathcal{M|P})=h(\mathcal{M|P})$. Also, it is known that $0\leq h(\mathcal{P|Q})\leq H(\mathcal{P|Q})$. Moreover, the following result holds (see [10]) to which we give a proof for the sake of completeness.

Let $(\mathcal{M},\tau)$ be a finite von Neumann algebra and the quadruple

\[
\begin{matrix}
P &\subset & M\\
\cup & &\cup\\
N &\subset & Q\\
\end{matrix}
\]

be a commuting square. Then, $H(\mathcal{P|Q})=h(\mathcal{P|Q})=H(\mathcal{P|N})=h(\mathcal{P|N})$.

Since $H(\mathcal{P|N})=h(\mathcal{P|N})$, it is enough to show that $h(\mathcal{P|Q})=h(\mathcal{P|N})$. This is because by Theorem $6$ in [44], it is known that $H(\mathcal{P|Q})=H(\mathcal{P|N})$ in the case of commuting squares, and we also have $H(\mathcal{P|N})=h(\mathcal{P|N})$ because $\mathcal{N}\subset\mathcal{P}$. Let $\mathscr{P}$ be the set of all finite partitions $\{(x_i)_{1\leq i\leq n}:\sum_{i=1}^nx_i=1\mbox{ and each }x_i\in\mathcal{P}_+\}$. Similar to the proof of Theorem $6$ in [44], we have the following,
h(𝒫|𝒩)    =    sup_(x_i)∈𝒫∑_iτηE^ℳ_𝒩(x_i)-τη(x_i)

=    sup_(x_i)∈𝒫∑_iτηE^ℳ_𝒬E^ℳ_𝒫(x_i)-τη(x_i)

=    sup_(x_i)∈𝒫∑_iτηE^ℳ_𝒬(x_i)-τη(x_i)

=    h(𝒫|𝒬)

which completes the proof.

Therefore, we may safely conclude that the modifed Connes-Størmer relative entropy $h$ is closely related to the Connes-Størmer relative entropy $H$. For the situation $\Delta_n,U\Delta_n U^*\subset M_n$ discussed above, the value of $h(\Delta_n|U\Delta_n U^*)$ is known to be $\frac{1}{n}\sum_{i,j}\eta(|u_{ij}|^2)$ for $U=(u_{ij})$ [9]. This shows that the importance of $h$ lies in the fact that it is a more computable quantity than $H$, and it always provides a lower bound for $H$. Moreover, for certain `nice' situation as indicated in <Ref>, it determines the exact value of $H$. Note that we are not claiming here computation of $h$ is easy, rather, we only want to emphasize the fact that $h$ is indeed a `good' invariant for pair of von Neumann algebras, which is closely related to $H$ and theoretically more computable. The Pimsner-Popa constant $\lambda(\mathcal{P,Q})$ defined earlier always provides an upper-bound for $h(\mathcal{P|Q})$.

At this stage, we would like to add a comment here. In Theorem $4.3$, [4] it is claimed that if $N\subset M$ is an irreducible subfactor with finite Jones index and $P$ and $Q$ are two intermediate subfactors, then $H(P|Q)=-\log\lambda(P,Q)$. However, there is a mistake in obtaining the upper bound in the first part of the proof (namely, $H(P|Q)\leq-\log\lambda(P,Q)$). The proof works for $h(P|Q)$ and not $H(P|Q)$. The correct replacement of Theorem $4.3$ in [4] are the following two statements.

If $\mathcal{P}$ and $\mathcal{Q}$ are von Neumann subalgebras of a finite von Neumann algebra $\mathcal{M}$, then $\,h(\mathcal{P|Q})\leq -\log \lambda(\mathcal{P},\mathcal{Q})$.

This will follow by the first part of the proof of Theorem $4.3$ in [4].

Suppose $N\subset M$ is an irreducible subfactor with finite Jones index and $P$ and $Q$ are two intermediate subfactors. Then $h(P|Q)=-\log \lambda(P,Q)$, and consequently $H(P|Q)\geq-\log \lambda(P,Q)$.

This will follow by the second part of the proof of Theorem $4.3$ in [4] together with <Ref>.

Remarkably, Pimsner and Popa in [37] had discovered that for a subfactor $N\subset M$ of type $II_1$ factors, $H(M|N)(=h(M|N))$ depends on both the Jones index and the relative commutant. In particular, for a finite index subfactor $N\subset M$ with $N^{\prime}\cap M=\mathbb{C}$ (such a subfactor is called irreducible) they proved that $H(M|N)= \log\,[M:N]$. More generally, the subfactor $N\subset M$ is extremal if and only if $H(M|N)=\log\,[M:N]$. For a finite index irreducible subfactor $N\subset M$ with intermediate subfactors $P$ and $Q$, the Pimsner-Popa constant $\lambda(P,Q)$ is closely related to both $h(P|Q)$ and $H(P|Q)$ (<Ref> and <ref>).

As an application of <Ref>, we now show that possible values of $h(P|Q)$ has certain gap in the irreducible situation. This is reminiscent of the Jones index rigidity.

Consider a quadrilateral of finite index subfactors

\[
\begin{matrix}
{Q} &\subset &{M} \cr \cup &\ &\cup\cr {N} &\subset &{P}
\end{matrix}
\]

with $N\subset M$ irreducible and suppose that $[P:N]$ and $[Q:N]$ are less than $4$. Then, either

$h(P|Q)\in\big\{\log\big(4~{\cos}^2(\pi/n)\big)\,:\,n\geq 3\big\}$

$h(P|Q)=h(Q|P)\in\{\log\big({4~{\cos}^2 (\pi/2k)-1}\big)\,:\,k\geq 3\}.$

Consider the dual factors $\bar{P}$  and $\bar{Q}$  from the basic constructions $P\subset M$ and $Q\subset M$, respectively (with the corresponding Jones projections $e_{\bar{P}}$ and $e_{\bar{Q}}$). Then, we have

\[
tr\big(e_{\bar{P}}e_{\bar{Q}}\big)=\frac{[M:P]}{[Q:N]} tr\big(e_Pe_Q\big).
\]

Furthermore, if $N_{-1}\subset N\subset M, P_{-1}\subset N\subset P$ and $Q_{-1}\subset N\subset Q$ are instances of downward basic constructions with the corresponding Jones projection $e_{N_{-1}}, e_{P_{-1}}$ and $e_{Q_{-1}}$ respectively, then we also have

\[
tr\big(e_{P_{-1}}e_{Q_{-1}}\big)=\frac{[M:Q]}{[P:N]} tr(e_Pe_Q).
\]

By Lemma 3.16 in [16], we get that $\,tr(e_Pe_Q)=\big(\mbox{dim}_M L^2(\bar{P}\bar{Q})\big)^{-1}$, and hence $\,tr(e_{P_{-1}}e_{Q_{-1}})=\frac{1}{\text{dim}_N L^2(PQ)}$. Therefore, we conclude the following,

\[
\text{dim}_M \big(L^2(\bar{P}\bar{Q})\big)= \frac{[M:Q]}{[P:N]}\text{dim}_N \big(L^2(PQ)\big).
\]

By [4], we see that

\[
\lambda(P,Q)=\frac{tr(e_Pe_Q)}{tr(e_P)}=\frac{tr(e_Q)}{tr(e_{PQ})}=\frac{[P:N]}{\text{dim}_N \big(L^2(PQ)\big)}.
\]

As $[P:N],[Q:N]<4$, two cases may arise. It may happen that $(N\subset P,Q\subset M)$ forms a commuting square. In this case, thanks to <cit.>, we have $H(P|Q)=\log [P:N]$ and the result follows from Jones index rigidity. In the other case, by Theorem $4.1.3$ in [14] (see also [17]) we have $\text{dim}_N\big(L^2(PQ)\big)=[P:N]\big([P:N]-1\big).$ Therefore, $\lambda(P,Q)=\frac{1}{[P:N]-1}$. The rest follows from Corollary  5.2.5 in [14].

Consider a quadrilateral of finite index subfactors

\[
\begin{matrix}
{Q} &\subset &{M} \cr \cup &\ &\cup\cr {N} &\subset &{P}
\end{matrix}
\]

with $N\subset M$ irreducible and suppose that $[M:P]$ and $[M:Q]$ are less than $4$. Then, either

$h(P|Q)\in\big\{\log\big(4~{\cos}^2 (\pi/n)\big)\,:\,n\geq 3\big\}$

$h(P|Q)=h(Q|P)\in\{\log \big({4~{\cos}^2(\pi/2k) -1}\big)\,:\,k\geq 3\}.$

This follows from Corollary $4.4$ in [4] and <Ref>.

§.§ Basic construction of non-degenerate commuting square

\[\begin{matrix}
\mathcal{Q} &\subset & (\mathcal{M}, tr) \cr \cup &\ &\cup\cr \mathcal{N} &\subset & \mathcal{P}
\end{matrix}\]

be a quadruple of finite von Neumann algebras which is a commuting and co-commuting (non-degenerate) square with respect to the $tr$-preserving conditional expectations $E^{\mathcal{M}}_{\mathcal{Q}}, E^{\mathcal{M}}_{\mathcal{P}}$ and $E^{\mathcal{M}}_{\mathcal{N}}$. Following ([40],[26]), we discuss the basic construction for this quadruple. To avoid technicalities, we confine ourselves into the following two cases.

Case I: Each of the elementary inclusions $\mathcal{N\subset P}$, $\mathcal{Q\subset M}$, $\mathcal{N\subset Q}$, and $\mathcal{P\subset M}$ is an inclusion of type $II_1$ factors with finite Jones index.

Case II: Each of the elementary inclusions $\mathcal{N\subset P}$, $\mathcal{Q\subset M}$, $\mathcal{N\subset Q}$, and $\mathcal{P\subset M}$ is a connected inclusion of finite-dimensional $C^*$-algebras (see [15, 26]).

It is known that, in each case, we have a unique Markov trace (for the inclusion $\mathcal{Q\subset M}$) $tr$ on $\mathcal{M}$ (see [26, 15]). Let $e_{\mathcal{Q}}$ be the Jones projection (with respect to $tr$) for the inclusion $\mathcal{Q\subset M}$, and ${\mathcal{Q}}_1:= \langle \mathcal{M}, e_{\mathcal{Q}} \rangle$ be the von Neumann algebra generated in $\mathcal{B}(L^2(\mathcal{M}))$ by $\mathcal{M}$ and $e_{\mathcal{Q}}$, so that $\mathcal{Q\subset M}\subset \mathcal{Q}_1$ is an instance of basic construction. Denote by $\langle \mathcal{P},e^{\mathcal{M}}_{\mathcal{Q}}\rangle$ the von Neumann algebra  generated in $\mathcal{B}(L^2(\mathcal{M}))$ by $\mathcal{P}$ and $e_{\mathcal{Q}}$. Then, $\mathcal{N\subset P}\subset {\mathcal{P}}_1:= \langle \mathcal{P}, e_{\mathcal{Q}}\rangle$ is an instance of basic construction for the inclusion $\mathcal{N\subset P}$ with the Jones projection $e_{\mathcal{Q}}$. It follows that $\mathcal{P}_1\subset \mathcal{Q}_1$, and each of the inclusion $\mathcal{M}\subset {\mathcal{Q}}_1$ and $\mathcal{P}\subset \mathcal{P}_1$ is a subfactor with finite Jones index (resp. connected inclusion of finite-dimensional $C^*$-algebras) in Case I (resp. Case II). Therefore, we have a unique Markov trace $tr_{\mathcal{Q}_1}$ on $\mathcal{Q}_1$ (which restricts to $tr_{\mathcal{M}}$ on $\mathcal{M}$). The construction of the quadruples :

\[
\begin{matrix}
\mathcal{Q} &\subset & \mathcal{M} & \subset & \mathcal{Q}_1\cr
\rotatebox{90}{$\subset$} &\ &\rotatebox{90}{$\subset$} &\ & \rotatebox{90}{$\subset$}\cr
\mathcal{N} &\subset & \mathcal{P} & \subset & \mathcal{P}_1\end{matrix}
\]

is called the basic construction for the non-degenerate commuting square

𝒬    ⊂    ℳ ∪       ∪𝒩    ⊂    𝒫

and the commuting square (with respect to the $tr_{\mathcal{Q}_1}$-preserving conditional expectations $E^{\mathcal{Q}_1}_{\mathcal{M}}, E^{\mathcal{Q}_1}_{\mathcal{P}_1}$ and $E^{\mathcal{Q}_1}_{\mathcal{P}}$)

\[
\begin{matrix}
\mathcal{M} &\subset & \mathcal{Q}_1\cr \cup &\ &\cup\cr \mathcal{P} &\subset & \mathcal{P}_1
\end{matrix}\]

is called the extension of the commuting square (<ref>).

Non-degenerate commuting squares of finite-dimensional $C^*$-algebras and their basic construction play a central role in the abstract subfactor theory (see [41]). They are also instrumental in constructing hyperfinite subfactors with finite Jones index. Below we sketch the construction. We fix a non-degenerate commuting square of finite-dimensional $C^*$-algebras

\[
\begin{matrix}
A_{10} &\subset &A_{11} \cr \cup &\ &\cup\cr A_{00} &\subset & A_{01}
\end{matrix}
\]

with respect to the unique trace $tr$ on $A_{11}$, which is Markov for the inclusion $A_{10}\subset A_{11}$ as in Case II. The basic construction of the above square is again a non-degenerate commuting square, and is denoted by

\[
\begin{matrix}
A_{11} &\subset &A_{12} \cr \cup &\ &\cup\cr A_{01} &\subset & A_{02}\,\,.
\end{matrix}
\]

We can iterate the basic construction and obtain the following ladder of non-degenerate commuting squares

\[
\begin{matrix}
A_{10} &\subset & A_{11} & \subset & A_{12} & \subset &  \cdots \cr
\rotatebox{90}{$\subset $} &\ &\rotatebox{90}{$\subset$} &\ & \rotatebox{90}{$\subset$} \cr
A_{00} &\subset & A_{01} & \subset & A_{02} & \subset & \cdots \,.
\end{matrix}
\]

Then, $tr$ extends to a faithful trace $\tau$ on $\bigcup_k A_{1k}$, and setting $A_{1,\infty}$ (resp.  $A_{0,\infty}$) as the GNS-completion of $\bigcup_k A_{1k}$ (resp. $\bigcup_k A_{0k}$) with respect to $\tau$, we obtain the hyperfinite subfactor $A_{0,\infty}\subset A_{1,\infty}$.  We record here two important well-known facts about this subfactor which will be useful later.

[[37, 15, 26]]
Let $A_{01},A_{10}, A_{0,\infty}$, and $A_{1,\infty}$ be as above. If $\Lambda$ denotes the inclusion matrix for the vertical inclusion $A_{00}\subset A_{10}$, then $[A_{1,\infty}:  A_{0,\infty}]={\lVert \Lambda \rVert}^2.$

[[26]](Ocneanu compactness)
Let $A_{01},A_{10}, A_{0,\infty}$, and $A_{1,\infty}$ be as above. Then,
$\big(A_{0,\infty}\big)^{\prime}\cap A_{1,\infty}=\big(A_{01})^{\prime}\cap A_{10}$

§.§ Some important results

We first recall few results from Section $6$ in [37] concerning the Pimsner-Popa constant and relative entropy for inclusion of finite-dimensional $C^*$-algebras. These will be repeatedly used throughout the article in several places, and hence for the benefit of the reader, we collect them at a place together. We keep the same symbols used in [37] for reader's convenience.

Let $N:=\oplus_{r\in K}N_r$ and $M:=\oplus_{\ell\in L}M_\ell$ be finite-dimensional von Neumann algebras, i,e., the sets of indices $L\mbox{ and }K$ are finite, where $N_r$ is the algebra of $n_r\times n_r$ matrices and $M_\ell$ is the algebra of $m_\ell\times m_\ell$ matrices. Suppose that we have a unital inclusion $N\subset M$. Denote by $\Lambda=(a_{r\ell})_{r\in K\,,\,\ell\in L}$ the inclusion matrix of $N\subset M$ and by $t_\ell$ (respectively $s_r$) the traces of the minimal projections in $M_\ell$ (respectively $N_r$). Thus, if $m=(m_\ell)_{\ell\in L},\,n=(n_r)_{r\in K},\,t=(t_\ell)_{\ell\in L},\,s=(s_r)_{r\in K}$ are column vectors then $\Lambda t=s$ and $\Lambda^Tn=m$. We have the following.

For the unital inclusion $N\subset M$, we have the following formulae,

* $\lambda(M,N)^{-1}=\max_\ell\left(\sum_r\frac{b_{r\ell}s_r}{t_\ell}\right)$, where $b_{r\ell}=\min\{a_{r\ell}\,,n_r\}$.

* $H(M|N)=\sum m_\ell t_\ell\log\left(\frac{m_\ell}{t_\ell}\right)+\sum n_rs_r\log\left(\frac{s_r}{n_r}\right)+\sum_{r,\ell}n_ra_{r\ell}t_\ell\log(c_{r\ell})$,

where $\,c_{r\ell}=\min\{n_r/a_{r\ell}\,,1\}$.

Particular cases of the above theorem are the following (Example $6.5$ in [37]).

Let $N\subset M$ be such that $N$ and $M$ are factors of type $I_n$ and $I_m$ respectively.

* If $m/n>n$, then $\lambda(M,N)=1/m$ and $H(M|N)=\log m$.

* If $m/n\leq n$, then $\lambda(M,N)=(n/m)^2$ and $H(M|N)=2(\log m-\log n)$.

Let $N\subset M$ be such that $M$ is the factor of type $I_m$ and $N=\oplus_{r\in K}N_r$, where each $N_r$ is the type $I_{n_r}$ factor, such that $\sum_{r\in K}n_r=m$. Then, $\lambda(M,N)=\frac{1}{\texttt{\#} K}$ and $H(M|N)=\sum_r\eta(n_r/m)$.

Note that these formulae are very useful in the sense that if we know the finite-dimensional tower for an inclusion $N\subset M$ of $II_1$ factors, then these formulae can (possibly) give us the exact value of $\lambda(M,N)$ and $H(M|N)$, since both behave well under controlled limits.

Now, we prove few results which will be used on several occasions, especially in Sections <ref> and <ref>.

Suppose that for $j=1,2$ we have two symmetric commuting squares $(B_1^j,A_1,B_2^j,A_2)$ of finite-dimensional algebras (see <Ref> in this regard), and we obtain the following ladder of symmetric commuting squares

\[
\begin{matrix}
A_1 &\subset & A_2 &\subset^{\,e_3} & A_3 &\subset^{\,e_4} &\cdots\\
\cup &  & \cup &   & \cup &  & \\
B_1^j &\subset & B_2^j &\subset & B_3^j &\subset &\cdots
\end{matrix}
\]

by iterated basic construction. That is, $A_k=\mathrm{Alg}\{A_{k-1},e_k\}$ for $k\geq 3$, where $e_k$ is the Jones projection, and $B_k^j=\mathrm{Alg}\{B_{k-1}^j,e_k\}$. Let $C_k=\cap_{j=1}^2B_k^j$ for $k\geq 1$ and suppose that $(C_1,A_1,C_2,A_2)$ is a symmetric commuting square. We let $\,D_3=\mathrm{Alg}\{C_2,e_3\}$ and for $k\geq 4,\,D_k=\mathrm{Alg}\{D_{k-1},e_k\}$. Then, $e_k\in C_k$ and $D_k\subset C_k$ for all $k\geq 3$.

It is immediate that $e_k\in C_k$ for all $k\geq 3$ and $D_3\subset C_3$. Suppose that $D_m\subset C_m$ for some $m\geq 4$. Then, $D_{m+1}=\mathrm{Alg}\{D_m,e_{m+1}\}\subset\mathrm{Alg}\{C_m,e_{m+1}\}\subset C_{m+1}$. Therefore, by induction on $k$ we get the result.

Let $\mathcal{N}\subset\mathcal{M}$ be an inclusion of von Neumann algebras and $u\in\mathcal{M}$ be a unitary. Then, in the following diagram

\[
\begin{matrix}
\mathcal{N} &\subset & \mathcal{M}\cr
&\ &\rotatebox{90}{$\subset$}\cr
& & \mathrm{Ad}_u(\mathcal{N})
\end{matrix}
\]

we have $(\mathrm{Ad}_u(\mathcal{N}))^\prime\cap\mathcal{N}=\mathrm{Ad}_u\big(\mathcal{N}^{\,\prime}\cap\mathrm{Ad}_{u^*}(\mathcal{N})\big)$.

Let $\xi\in\mathrm{Ad}_u\big(\mathcal{N}^{\,\prime}\cap\mathrm{Ad}_{u^*}(\mathcal{N})\big)$ and $\xi=uxu^*$ with $x\in\mathcal{N}^{\,\prime}\cap\mathrm{Ad}_{u^*}(\mathcal{N})$. Clearly, $\xi\in\mathcal{N}$ as $x\in\mathrm{Ad}_{u^*}(\mathcal{N})$. For any $unu^*\in\mathrm{Ad}_u(\mathcal{N})$ with $n\in\mathcal{N}$, since $x\mbox{ and }n$ commutes, we get that $\xi unu^*=uxu^*unu^*=uxnu^*=unxu^*=unu^*\xi$. This says that $\xi\in(\mathrm{Ad}_u(\mathcal{N}))^\prime\cap\mathcal{N}$, i,e., the inclusion `$\supseteq$' holds. The reverse inclusion `$\subseteq$' is easy to check.

This lemma can be generalized a bit more, as indicated by the following.

Let $\mathcal{N}\subset\mathcal{M}$ be an inclusion of von Neumann algebras, $\mathcal{S}$ be a subset of $\mathcal{N}$, and $u\in\mathcal{M}$ be a unitary. Then, $\mathrm{Ad}_u(\mathcal{S})^\prime\cap\mathcal{N}=\mathrm{Ad}_u\big(\mathcal{S}^\prime\cap\mbox{Ad}_{u^*}(\mathcal{N})\big)$. Thus, if $\mathcal{N}=\mathcal{M}$, then $\mathrm{Ad}_u(\mathcal{S})^\prime\cap\mathcal{M}=\mathrm{Ad}_u\big(\mathcal{S}^\prime\cap\mathcal{M})$.

Follows along the similar line of <Ref>.

§ COMMUTING CUBE AND TWO SUBFACTORS

The results obtained in this section are generic in nature, and we take help of these on many occasions throughout the article. Here we introduce the notion of `commuting cube'. Similar notion has appeared in different contexts ([3], [30]) in the finite-dimensional situation.

Suppose that the following quadruple

\[
\begin{matrix}
Q &\subset & M\\
\cup & & \cup\\
N &\subset & P\\
\end{matrix}
\]

is a co-commuting square (of either $II_1$ factors or finite-dimensional $C^*$-algebras) with $[P:N]=[M:Q]$ (either Jones index or Watatani index respectively). Then, the quadruple is a commuting square.

Since it is given that

\[
\begin{matrix}
Q &\subset & M\\
\cup & & \cup\\
N &\subset & P\\
\end{matrix}
\]

is a co-commuting square, the dual quadruple

\[
\begin{matrix}
\langle P,e_P\rangle &\subset & \langle M,e_N\rangle\\
\cup & & \cup\\
M &\subset & \langle Q,e_Q\rangle\\
\end{matrix}
\]

is a commuting sqaure by definition. Therefore, $\,E^{\langle M,e_N\rangle}_{\langle P,e_P\rangle}E^{\langle M,e_N\rangle}_{\langle Q,e_Q\rangle}(e_Q)=E^{\langle M,e_N\rangle}_M(e_Q)=E^{\langle Q,e_Q\rangle}_M(e_Q)$. This implies that $E^{\langle M,e_N\rangle}_{\langle P,e_P\rangle}(e_Q)=E^{\langle Q,e_Q\rangle}_M(e_Q)=[M:Q]^{-1}$. Hence, $e_PE^{\langle M,e_N\rangle}_{\langle P,e_P\rangle}(e_Q)=[M:Q]^{-1}e_P$. Since $e_P\in\langle P,e_P\rangle$, we get that $E^{\langle M,e_N\rangle}_{\langle P,e_P\rangle}(e_Pe_Q)=[M:Q]^{-1}e_P$. Therefore, we have the following,

⇒    (e_Pe_Q)=[M:Q]^-1[M:P]^-1

⇒    (e_Pe_Q)=[M:N]^-1    ( [M:Q]=[P:N])

⇒    (e_Pe_Qe_P)=(e_N) .

Now, $N\subset Q$ implies that $e_N\leq e_Q$, and consequently $e_Pe_Ne_P\leq e_Pe_Qe_P$. Since $e_Pe_Ne_P=e_N$, we have $e_N\leq e_Pe_Qe_P$, and hence $\mbox{tr}(e_Pe_Qe_P-e_N)=\mbox{tr}(e_Pe_Q-e_N)\geq 0$. By the faithfulness of $\mbox{tr}$, along with <Ref>, we get that $e_Pe_Qe_P=e_N$. Now, using $e_Ne_P=e_N,\,e_Ne_Q=e_N$ together with $e_Pe_Qe_P=e_N$, it is easy to verify that $(e_P-e_N)(e_Q-e_N)(e_P-e_N)=0$. Since $e_P-e_N$ and $e_Q-e_N$ both becomes projection, we get the following
||(e_P-e_N)(e_Q-e_N)||^2    =    ||(e_P-e_N)(e_Q-e_N)(e_Q-e_N)(e_P-e_N)||

=    ||(e_P-e_N)(e_Q-e_N)(e_P-e_N)||

=    0 .

This gives us that $(e_P-e_N)(e_Q-e_N)=0$, and consequently $e_Pe_Q=e_N$. Therefore, the quadruple $(N\subset P,Q\subset M)$ is a commuting square.

Consider the cube of finite von Neumann algebras described in <Ref>, where $A_1$ is equipped with a faithful normal tracial state, and $C_0=B_0^1\cap B_0^2,\,C_1=B_1^1\cap B_1^2$. In <Ref>, `$B\rightarrow A$' means $B\subset A$.

8 cm!

Cube of finite von Neumann algebras

Suppose that the adjacent faces in <Ref>

\[
\begin{matrix}
B_1^1 &\subset & A_1\\
\cup & &\cup & \\
B_0^1 &\subset & A_0\\
\end{matrix}\qquad\mbox{and}\qquad\begin{matrix}
A_1 &\supset & B_1^2\\
\cup & &\cup & \\
A_0 &\supset & B_0^2\\
\end{matrix}
\]

are commuting squares. Then, the remaining faces

\[
\begin{matrix}
B_0^j &\subset & B_1^j\\
\cup & &\cup & \\
C_0 &\subset & C_1\\
\end{matrix}
\]

for $j=1,2$ and the slice

\[
\begin{matrix}
C_1 &\subset & A_1\\
\cup & &\cup & \\
C_0 &\subset & A_0\\
\end{matrix}
\]

are also commuting squares. Therefore, $C_1\cap B_0^j=C_0$ for $j=1,2,$ and $C_1\cap A_0=C_0$.

Recall that for a quadruple $(N\subset P,Q\subset M)$, if $E^M_Q(P)\subseteq N$ (or $E^M_P(Q)\subseteq N$), then it is a commuting square (<Ref>). To show that the slice $(C_0\subset A_0,C_1\subset A_1)$ is a commuting square, in view of the above, it is enough to show that $E^{A_1}_{A_0}(c_1)\in C_0$ for all $c_1\in C_1$. Since $C_1\subset B^1_1$, by hypothesis we have $E^{A_1}_{A_0}(c_1)\in B^1_0$. Similarly, since $C_1\subset B^2_1$ we get $E^{A_1}_{A_0}(c_1)\in B^2_0$. Hence, $E^{A_1}_{A_0}(c_1)\in C_0$ which finishes the proof.

To show that the remaining faces of the cube are also commuting squares, it is enough to show one of these, and the other follows similarly. So we consider one of the remaining face

\[
\begin{matrix}
B_0^1 &\subset & B_1^1\\
\cup & &\cup & \\
C_0 &\subset & C_1\\
\end{matrix}
\]

and claim that $E^{B_1^1}_{B_0^1}(C_1)\subseteq C_0$. Take any $x\in C_1$ and using the hypothesis that $(B_0^1\subset A_0,B_1^1\subset A_1)$ is a commuting square observe the following,

\[
\]

Since the slice $(C_0\subset A_0,C_1\subset A_1)$ is already a commuting square, we get that $E^{A_1}_{A_0}(x)\in C_0$, which concludes the proof.

A cube of finite von Neumann algebras as in <Ref> is called a commuting cube if both the adjacent faces $(B_0^j\subset A_0,B_1^j\subset A_1)$ for $j=1,2$ are commuting squares.

Pair of spin model subfactors, to be discussed in the next section, provide natural examples of commuting cube. The following remark is very important.

* The floor and the roof in a commuting cube need not be commuting square. We will see counterexamples in Sections <ref> and <ref>.

* If the adjacent faces in a commuting cube are non-degenerate commuting squares, then although the slice $(C_0\subset A_0,C_1\subset A_1)$ is a commuting square, it may fail to become non-degenerate. Same applies to the remaining faces. For example, consider two distinct $2\times 2$ complex Hadamard matrices $u\mbox{ and }v$ of the following form

\[
1 & 1 \\
e^{i\alpha} & -e^{i\alpha} \\
\end{matrix}}\right] \quad\mbox{and}\quad v=\frac{1}{\sqrt{2}}\,\left[{\begin{matrix}
1 & 1 \\
e^{i\beta} & -e^{i\beta} \\
\end{matrix}}\right]
\]

such that $\alpha-\beta\notin\{0,\frac{\pi}{2},-\frac{\pi}{2}\}$. Obtain the cube as illustrated in <Ref>.

8 cm!

A specific commuting cube

In this cube, both the adjacent faces are non-degenerate commuting squares, since $u\mbox{ and }v$ are complex Hadamard matrices. So, it is a commuting cube. Now, it is a straightforward verification that $\mbox{Ad}_u(\Delta_2)\cap\mbox{Ad}_v(\Delta_2)=\bbc$. Hence, the slice in the cube is the quadruple $(\bbc\subset\bbc,\Delta_2\subset M_2)$, which is clearly not non-degenerate, and so are the remaining faces.

Suppose that we have a commuting cube as in <Ref>. If the roof

\[
\begin{matrix}
B_1^2 &\subset & A_1\\
\cup & &\cup & \\
C_1 &\subset & B_1^1\\
\end{matrix}
\]

is a commuting square, then the floor

\[
\begin{matrix}
B_0^2 &\subset & A_0\\
\cup & &\cup & \\
C_0 &\subset & B_0^1\\
\end{matrix}
\]

is also a commuting square.

Let $x\in B_0^1$ and observe the following,
E^A_0_B_0^2(x)    =    E^A_0_B_0^2E^A_1_A_0(x)    (B_0^1⊂A_0)

=    E^A_1_B_0^2(x)

=    E^A_1_B_1^2E^A_1_A_0(x)    ((B_0^2⊂A_0,B_1^2⊂A_1))

=    E^A_1_B_1^2(x)           (B_0^1⊂A_0)

=    E^A_1_B_1^2E^A_1_B_1^1(x)    (x∈B_0^1⊂B^1_1)

=    E^A_1_C_1(x)           ((C_1⊂B_1^1,B_1^2⊂A_1)) .

Since $x\in B_0^1\subset A_0$ and the slice $(C_0\subset A_0,C_1\subset A_1)$ is a commuting square by <Ref>, we get that $E^{A_1}_{C_1}(x)\in C_0$. This shows that $E^{A_0}_{B_0^2}(B_0^1)\subset C_0$, which proves that the floor is also a commuting square.

The converse of the above result does not hold in general. For example, the floor in <Ref> is clearly a commuting square, however the roof is not so, as a non-trivial angle in $\mathrm{Ang}_{M_2}\big(\mbox{Ad}_u(\Delta_2),\mbox{Ad}_v(\Delta_2)\big)$ is present. We postpone the proof until <Ref> (see <Ref>), as it contextually better suits there.

A commuting cube in <Ref> is called a non-degenerate commuting cube if the adjacent faces and the slice are non-degenerate commuting squares.

Therefore, the commuting cube in <Ref> is not a non-degenerate commuting cube. We request the reader to wait till Sections <ref> and <ref> for (non-trivial) examples of non-degenerate commuting cube.

Basic construction of non-degenerate commuting cube : Consider the cube in <Ref> and assume that the adjacent faces $(B_0^j\subset A_0,B_1^j\subset A_1)$, for $j=1,2$, are non-degenerate commuting squares. Suppose that the slice $(C_0\subset A_0,C_1\subset A_1)$ is also non-degenerate (see <Ref>). By <Ref>, the slice is a commuting square, and such a cube is a non-degenerate commuting cube. Let $A_0\subset A_1\subset^{\,e_2} A_2$ be the basic construction, where $e_2\in A_2$ is the Jones' projection. Then, $C_2=\langle C_1,e_2\rangle$ and $B^j_2=\langle B_j^1,e_2\rangle$ are instances of basic constructions for the inclusions $C_0\subset C_1$ and $B^j_0\subset B^j_1$ respectively for $j=1,2$. Moreover, we obtain a new cube as illustrated in <Ref>.

7 cm!

Commuting cube and basic construction

The adjacent faces $(B_1^j\subset A_1,B_2^j\subset A_2)$, for $j=1,2$, and the slice $(C_1\subset A_1,C_2\subset A_2)$ of the upper cube in <Ref> remains non-degenerate commuting squares. Stacking the lower cube and the upper cube, we get a cube whose adjacent faces and the resulting slice $(C_0\subset A_0,C_2\subset A_2)$ are non-degenerate commuting squares. Iterating Jones' basic construction, we (possibly) obtain a quadruple of $II_1$ factors $(N\subset P,Q\subset M)$, where $M=\{\cup_{m\geq 0}\,A_m\}^{\prime\prime},\,P=\{\cup_{m\geq 0}\,B^1_m\}^{\prime\prime},\,Q=\{\cup_{m\geq 0}\,B^2_m\}^{\prime\prime}$, and $N=\{\cup_{m\geq 0}\,C_m\}^{\prime\prime}$. However, there is no guarantee that $P\neq Q$ (in <Ref> we shall see an instance of this). If $P\mbox{ and }Q$ are indeed distinct, with a little more effort as in <Ref> we can conclude that $N\subset P\cap Q$ (note that $P\cap Q$ is not a factor a priori). Moreover, $[M:P]=\big|\big|\Lambda^{A_0}_{B^1_0}\big|\big|^2$ (similarly for $Q$) and $[M:N]=\big|\big|\Lambda^{A_0}_{C_0}\big|\big|^2$. By the Ocneanu compactness, we have $N^{\,\prime}\cap M=C_1^{\,\prime}\cap A_0$. Therefore, basic construction of a non-degenerate commuting cube gives rise to (possibly) quadruple of factors.

Instances of this construction will be seen in <Ref> (<Ref>) and <Ref> (<Ref>). <Ref> has far reaching consequences for `two subfactors' $P,Q\subset M$ obtained as iterated basic construction of a non-degenerate commuting cube of finite-dimensional algebras. It says that if there is a non-trivial `angle' present in the cube we start with, which is definitely computable being in a finite-dimensional situation, then the resulting quadruple $(N\subset P,Q\subset M)$ of $II_1$ factors is far from being a commuting square.

Consider a non-degenerate commuting cube as in <Ref> and define the following operator

\[
\]

for $j=0,1$. Suppose that $S_0^{\,2}=\alpha S_0$ for some $\alpha\in\mathbb{R}_+$. Then, $S_1^{\,2}=\alpha S_1$.

Since the slice is non-degenerate by hypothesis, for any $x\in A_1$, we have $x=\sum a_0c_1$ where $a_0\in A_0$ and $c_1\in C_1$. Then, the following holds :
S_1(x)=(E_B^2_1E_B^1_1E_B^2_1-E_C_1)(x)    =    (E_B^2_1E_B^1_1E_B^2_1-E_C_1)(∑a_0c_1)

=    E_B^2_1E_B^1_1(∑E_B^2_1(a_0)c_1)-∑E_C_1(a_0)c_1

=    ∑ E_B^2_1((E_B^1_1E_B^2_1(a_0))c_1)-∑E_C_1(a_0)c_1

=    ∑(E_B^2_1E_B^1_1E_B^2_1-E_C_1)(a_0) c_1

=    ∑(E_B^2_0E_B^1_0E_B^2_0-E_C_0)(a_0) c_1 ,

where the last line follows since $a_0\in A_0$, and the adjacent faces and the slice are commuting squares. This says that

\[
S_1^{\,2}(x)=\big(E_{B^2_1}E_{B^1_1}E_{B^2_1}-E_{C_1}\big)^2\big(\textstyle\sum a_0c_1\big)=\sum\,\big(E_{B^2_0}E_{B^1_0}E_{B^2_0}-E_{C_1}\big)^2(a_0)\,c_1=\sum\,S_0^{\,2}(a_0)\,c_1\,,
\]

which completes the proof.

Suppose that $(N\subset P,Q\subset M)$ is a quadruple of $II_1$ factors obtained as an iterated basic construction of a non-degenerate commuting cube of finite-dimensional algebras in <Ref>. Consider the following nonnegative matrix

\[
\]

and suppose that $S_0\neq 0$ with $S_0^{\,2}=\alpha S_0$ for some $\alpha\in\mathbb{R}_+$. Then, $(N\subset P,Q\subset M)$ is not a commuting square, and moreover $\mathrm{Ang}_M(P,Q)$ is the singleton set $\{\mathrm{arccos}\sqrt{\alpha}\}$.

Since $S_0\neq 0$, the quadruple $(C_0\subset B^1_0,B^2_0\subset A_0)$ is not a commuting square. Therefore, none of the roofs, including the final one $(N\subset P,Q\subset M)$, in the iterated basic construction (see <Ref>) are commuting squares due to <Ref>. Since basic construction of non-degenerate commuting square is also so, we get that the resulting final slice $(C_0\subset A_0,N\subset M)$ is a non-degenerate commuting square, and so are the resulting final adjacent faces $(B^1_0\subset A_0,P\subset M)$ and $(B^2_0\subset A_0,Q\subset M)$. Therefore, stacking all these non-degenerate commuting cube, we obtain the final commuting cube which is non-degenerate. By <Ref>, we obtain that $S_\infty:=E^M_PE^M_QE^M_P-E^M_N$ (and equivalently, $e_Pe_Qe_P-e_N$) satisfies $S_\infty^{\,2}=\alpha S_\infty$. Since $S_\infty\neq 0$ and $\alpha>0$, result now directly follows from Corollary $3.1$ in [43].

In this paper, the notion of `commuting cube' has been crucially used on several occasions throughout Sections <ref> and <ref> (e.g. $6.11,\,6.12,\,6.13,\,6.14,\,6.23,\,7.2,\,7.38,\,7.45$). We end this section with the following question.

Question : Under the hypothesis of <Ref>, what is the relationship between the spectrum of the angle operators $\Theta(P,Q)$ and $\Theta\big(B^1_0,B^2_0\big)$?

Pair of spin model subfactors

A particular instance of `two subfactors' is the pair of spin model subfactors. Recall that for any $n\times n$ complex Hadamard matrix $u$, we have a commuting and co-commuting (non-degenerate commuting) square

\[
\begin{matrix}
A_{10}= \Delta_n &\subset & M_n=A_{11} \cr \cup &\ & \cup\cr A_{00}= \mathbb{C} &\subset &  u\Delta_n u^*=A_{01}\,\,.
\end{matrix}
\]

The basic construction of $\Delta_n\subset M_n$ is $\Delta_n\otimes M_n$ with the Jones' projection $\mbox{bl-diag}\{E_{11},\ldots,E_{nn}\}$, and that of $\Delta_n\otimes M_n$ is $M_n\otimes M_n$ with the Jones' projection $J_n\otimes I_n$, and so on. Here, $J_n=\frac{1}{n}\sum_{i,j=1}^nE_{ij}$. Iterating basic construction we obtain a subfactor $R_u\subset R$, where $R$ is the hyperfinite type $II_1$ factor. This is called a spin model subfactor. Not much is known about the spin model subfactor, however, the following result is well-known.

The subfactor $R_u\subset R$ is irreducible and $[R:R_u]=n$.

Now, if we take another $n\times n$ complex Hadamard matrix $v$ with $u\neq v$, then we obtain a subfactor $R_v\subset R$. Thus, we obtain (possibly) a pair of spin model (irreducible) subfactors

\[
\begin{matrix}
R_u &\subset & R \cr
& & \cup\cr
&  & R_v
\end{matrix}\]

Two problems arise here. Firstly, it may happen that $R_u=R_v$ even if $u\neq v$, in which case we fail to obtain a `pair' of subfactors. Secondly, for $R_u\neq R_v$ it is not immediate that $R_u\cap R_v$ is a factor, that is, we do not have a quadruple of factors a priori. To overcome the first difficulty, that is to obtain “two subfactors" of $R$, one has to first characterize $R_u=R_v$ in terms of $u\mbox{ and }v$, and then proceed. We have a complete answer for this.

Recall that a matrix in $GL_n(\bbc)$ is said to be monomial (Generalized permutation matrix) if each row and column has exactly one non-zero entry. Let $N$ denote the set of all monomial matrices. It is known that $N$ is a subgroup of $GL_n(\bbc)$ and $N=\mathcal{N}_{GL_n}(\Delta_n)$, the normalizer of $\Delta_n$ in $GL_n(\bbc)$. Consider the subgroup $\mathbb{G}:=N\cap U(n)$ of $GL_n(\bbc)$, where $U(n)$ is the set of all unitary matrices.

An equivalence relation : For two unitary matrices $A$ and $B$ in $M_n$, consider the equivalence relation $A\sim B$ if $B=AC$, where $C\in\mathbb{G}$. Recall that two complex Hadamard matrices $H_1,H_2$ are called Hadamard equivalent, denoted by $H_1\simeq H_2$, if $H_2=D_1P_1H_1P_2D_2$, where $D_1,D_2$ are unitary diagonal matrices and $P_1,P_2$ are permutation matrices. Hence, the equivalence relation `$\sim$' is a sub-equivalence relation of the Hadamard equivalence `$\simeq$'. This is because for any $C\in N$, there exists an invertible $D\in\Delta_n$ and a permutation matrix $P$ such that $C=PD$. Hence, $C\in\mathbb{G}$ forces $D$ to become a unitary diagonal matrix. It turns out that it is this finer equivalence relation between the complex Hadamard matrices $u$ and $v$ of order $n\times n$ that completely characterize $R_u\neq R_v$. More precisely, we have the following result.

* For distinct $n\times n$ complex Hadamard matrices $u\mbox{ and }v$, the pair of spin model subfactors $R_u\mbox{ and }R_v$ of the hyperfinite type $II_1$ factor $R$ are distinct $(R_u\neq R_v)$ if and only if $\,u\nsim v$.

* If two $n\times n$ complex Hadamard matrices $u\mbox{ and }v$ are Hadamard inequivalent, then the corresponding spin model subfactors $R_u\mbox{ and }R_v$ of $R$ are always distinct $(R_u\neq R_v)$.

The proof of <Ref> requires a bit of work. Although in part $(i)$, the `only if' part is easy, it is the `if' part that requires a bit of work. We will be using the Pimsner-Popa probabilistic constant as a major tool to prove it.

Let $\Delta_n$ denote the diagonal subalgebra of $M_n$ and $\,U\in M_n$ be a unitary matrix. Then, $\Delta_n$ is a Masa (maximal abelian self-adjoint subalgebra) in $M_n$, and in fact any Masa in $M_n$ is of the form $U\Delta_n U^*$ for some unitary $U\in M_n$. Consider the following unital inclusion of subalgebras
\[\begin{matrix}
\Delta_n &\subset & M_n\\
& & \cup\\
& & U\Delta_n U^*\\
\end{matrix}\]

such that $U$ is not the identity matrix. We are going to determine the value of $\,\lambda(\Delta_n\,,\,U\Delta_n U^*)$. First note that $\lambda(M_n,U\Delta_nU^*)=\lambda(M_n,\Delta_n)=\frac{1}{n}$ (example in $6.5$, Page $94$ in [37]). That is, $E_{U\Delta_nU^*}(x)\geq \frac{1}{n}x$ for all $x\in(M_n)_+$, and hence $\,\lambda(\Delta_n\,,\,U\Delta_n U^*)\geq \frac{1}{n}$. We shall show that the best constant is determined in terms of the Hamming numbers of the rows of $U$.

The unique trace preserving conditional expectation $\,E:M_n\longrightarrow\Delta_n$ is given by $A\longmapsto (a_{ii})_{ii}$ for any $A=(a_{ij})_{1\leq i,j\leq n}\in M_n$. Here, the notation $(\,.\,)_{ii}$ denotes a diagonal matrix in $M_n$. We are going to use the following well-known facts.

Weyl's inequality for eigenvalues : Let $A\mbox{ and }B$ be $n\times n$ self-adjoint matrices and $\widetilde{A}=A+B$. Suppose that the eigenvalues of $A$ are ordered as $\lambda_1(A)\leq\lambda_2(A)\leq\ldots\leq\lambda_n(A)$, and similarly for the matrices $\widetilde{A}\mbox{ and }B$. Then, one has the following,

\[\lambda_k(A)+\lambda_1(B)\leq\lambda_k(\widetilde{A})\leq\lambda_k(A)+\lambda_n(B)\,.\]

Matrix determinant lemma : Suppose $A$ is an invertible $n\times n$ matrix and $\mathbf{u},\mathbf{v}$ are two vectors in $\bbc^n$ written in column matrices. Then,

\[
\det\big(A+\mathbf{u}\mathbf{v}^{\textsf{T}}\big)=\big(1+\mathbf{v}^{\textsf {T}}A^{-1}\mathbf{u}\big)\,\det(A)\,.
\]

By the definition of $\,\lambda$, to find $\lambda(\Delta_n,U\Delta_n U^*)$, we need to find the best constant $t\in[0,1]$ in the following inequality,

\[
U(U^*xU)_{kk}U^*\geq tx\,,\quad\forall\,\,x\in(\Delta_n)_+\,.
\]

This is because $E^{M_n}_{U\Delta_nU^*}=\mbox{Ad}_U\circ E^{M_n}_{\Delta_n}\circ \mbox{Ad}_{U^*}$.

For $1\leq i\leq n$, consider the minimal projections $p_i:=E_{ii}$ in $M_n$. Then,
λ(Δ , UΔU^*)=min_1≤i≤n sup {t≥0:(U^*p_iU)_kk≥t U^*p_iU} .

First observe that $\Delta_n$ is generated by the minimal projections $p_i=E_{ii}$ in $M_n$, and $U(U^*p_iU)_{kk}U^*\geq tp_i$ holds if and only if $(U^*p_iU)_{kk}\geq t\,U^*p_iU$ holds. Let us denote

\[
\widetilde{\lambda}_i:=\sup_{t\geq 0}\,\big\{(U^*p_iU)_{kk}\geq t\,U^*p_iU\big\}\,,
\]

for $\,i\in\{1,\ldots,n\}$. It is obvious from the definition of $\lambda$ that $\lambda(\Delta_n\,,\,U\Delta_nU^*)\leq\widetilde{\lambda}_i$ for each $i$. Thus, we get that

\[
\lambda(\Delta\,,\,U\Delta U^*)\leq\min_{1\leq i\leq n}\,\widetilde{\lambda}_i\,.
\]

Conversely, for any $x\in(\Delta_n)_+$ we have $x=\sum_{i=1}^n\,\alpha_ip_i$ with all $\alpha_i\geq 0$. Thus, for any $t\in[0,1],\,(U^*xU)_{kk}\geq t\,U^*xU$ holds if and only if the following inequality
∑_i=1^n α_i(U^*p_iU)_kk    ≥    ∑_i=1^n α_it U^*p_iU

holds, and the best constant $t$ satisfying the above inequality gives us $\lambda(\Delta\,,\,U\Delta U^*)$. Now, if we set $t_0=\mbox{min}_{1\leq i\leq n}\,\widetilde{\lambda}_i$, then by the definition of $\widetilde{\lambda}_i$ we get that

\[
(U^*p_iU)_{kk}\geq t_0\,U^*p_iU
\]

for all $i$, and consequently <Ref> is satisfied for $t=t_0$. By the definition of $\lambda$, we get that

\[
\lambda(\Delta\,,\,U\Delta U^*)\geq t_0=\min_{1\leq i\leq n}\,\widetilde{\lambda}_i\,,
\]

which completes the proof.

Let $\sigma=-t$ and for any fixed $i\in\{1,\ldots,n\}$ consider the following one-parameter family of $n\times n$ matrices

\[
\{A_\sigma:=(U^*p_iU)_{kk}+\sigma\,U^*p_iU\,:\,\sigma\in[-1,0]\}\,.
\]

Since $U^*p_iU$ is a rank one matrix, any member of this family is a matrix of the form $D_\sigma+\sigma\,\mathbf{u}\mathbf{u}^*$, where $\mathbf{u}\in\mathbb{C}^n$ is a vector written as a column matrix and $\mathbf{u}^*$ denotes the complex conjugate of the vector $\mathbf{u}$ written as a row matrix. Thus, any matrix $A_\sigma$ is a rank-$1$ perturbation of a diagonal matrix $D_\sigma$. Throughout this subsection, we denote $U^*=(u_{ij}),\,1\leq i,j\leq n$. We are interested in the following inequality

\[(U^*p_iU)_{kk}+\sigma U^*p_iU\geq 0\,,\]

that is,
(|u_1i|^2,…,|u_ni|^2)+σ [

u_1i     …    u_ni

≥0 .

Except possibly the minimum eigenvalue of the following self-adjoint matrix

\[
\mbox{diag}\big(|u_{1i}|^2,\ldots,|u_{ni}|^2\big)+\sigma\,\left[{\begin{matrix}
\vdots\\
\end{matrix}}\right]\begin{matrix}\left[{\begin{matrix}
\overline{u_{1i}} & \ldots & \overline{u_{ni}}\\
\end{matrix}}\right]\\[1.0ex] \\ \\ \end{matrix}\]

all the other $(n-1)$ eigenvalues, counted with multiplicities, are non-negative.

We write the matrix $A_\sigma$ as $P+D$ where $D$ is the diagonal matrix $\mbox{diag}\big(|u_{1i}|^2,\ldots,|u_{ni}|^2\big)$, and $P$ is the rank one perturbation $A_\sigma-D$. Since $\frac{1}{\sigma}P$ is a rank-$1$ projection, we have the spectrum of $P$ is given by $\sigma(P)=\{0,\sigma\}$. We write the eigenvalues in increasing order as the following,

\[
\lambda_1=\sigma\leq \lambda_2=\lambda_3=\ldots=\lambda_n=0\,.
\]

Similarly, we write the eigenvalues of $D$ in increasing order as the following

\[
\]

where we take $\mu_k=|u_{ki}|^2$ after possible renaming. Now, let the eigenvalues of $A_\sigma$ be $\widetilde{\lambda}_1\leq\ldots\widetilde{\lambda}_n$. By the Weyl's inequality, we get the following

\[
\lambda_k+\mu_1\leq\widetilde{\lambda}_k\leq\lambda_k+\mu_n\,.
\]

Putting $k=2$, we get that $\widetilde{\lambda}_2\geq\lambda_2+\mu_1=\mu_1\geq 0$. Thus, $\widetilde{\lambda}_k\geq 0$ for all $k=2,\ldots,n$, and this completes the proof.

If all the entries of the unitary matrix $U$ are non-zero, then $\lambda\big(\Delta\,,\,U\Delta U^*\big)=\frac{1}{n}\,$.

Fix any $i\in\{1,\ldots,n\}$, and let $\mathbf{v}^T=\big(\overline{u_{1i}},\ldots,\overline{u_{ni}}\big)$ and $\mathbf{u}^T=\big(\sigma\,u_{1i},\ldots,\sigma\,u_{ni}\big)\,.$ Since, $u_{ij}\neq 0$ is given for all $i,j\in\{1,\ldots,n\}$, using the matrix determinant lemma, we get the following,
((|u_1i|^2,…,|u_ni|^2)+σ [

u_1i     …    u_ni

=    (1+v^T (1/|u_1i|^2,…,1/|u_ni|^2) u)  ((|u_1i|^2, … ,|u_ni|^2))

=    ∏_j=1^n |u_ji|^2(1+nσ) .

The quantity $\prod_{j=1}^n\,|u_{ji}|^2(1+n\sigma)$ is non-negative if and only if $(1+n\sigma)$ is non-negative. The matrix in <Ref> is positive semi definite if and only if all its eigenvalues are non-negative. Since, all the $(n-1)$ eigenvalues, counted with multiplicities, are non-negative by <Ref>, we get that  the best constant $\sigma\in[-1,0]$ in <Ref> is determined by the inequality $1+n\sigma\geq 0$, since determinant is product of the eigenvalues. That is, the best constant $\sigma$ is given by $\sigma=-1/n\,,$ and consequently the best constant $t$ in the Pimsner-Popa constant is given by $t=1/n$. Since this happens for any $1\leq i\leq n$, by Lemma <ref> we conclude the proof.

Given a nonzero vector $\mathbf{u}\in\bbc^n$, the Hamming number is given by,

\[
h(\mathbf{u}):=\mbox{number of non-zero entries in }\mathbf{u}\,.
\]

If $\Delta_n$ and $U\Delta_n U^*$ are two Masas in $M_n$, where $U\in U(n)$, then
the Pimsner-Popa constant is given by the following,

\[
\lambda(\Delta_n\,,\,U\Delta_n U^*)=\min_{1\leq i\leq n}\,\big(\mbox{$h\left(U^*\right)_i$}\big)^{-1}
\]

where $\left(U^*\right)_i$ is the $i$-th column of $U^*$.

Consider the $i$-th column of $U^*$ for $1\leq i\leq n$. Let $k_i$ be the number of zero elements in the set $\{u_{1i},\ldots,u_{ni}\}$. That is, $k_i$ many eigenvalues, counted with multiplicities, of the diagonal matrix $\,\mbox{diag}\big(|u_{1i}|^2,\ldots,|u_{ni}|^2\big)$ are $0$. Let $u_{j_1i}=\ldots=u_{j_{k_i}i}=0$. Then, all the $j_\ell$-th row for $1\leq\ell\leq k_i$ of the following $n\times n$ matrix

\[
\left[{\begin{matrix}
\vdots\\
\end{matrix}}\right]\begin{matrix}\left[{\begin{matrix}
\overline{u_{1i}} & \ldots & \overline{u_{ni}}\\
\end{matrix}}\right]\\[1.0ex] \\ \\ \end{matrix}
\]

are identically zero. Since $\overline{u_{j_1i}}=\ldots=\overline{u_{j_{k_i}i}}=0$, all the $j_\ell$-th column for $1\leq\ell\leq k$ of the above matrix are also identically zero. To obtain the Pimsner-Popa constant, we are interested in the best constant $\sigma\in[-1,0]$ for which Eqn. $\ref{best constant matrix}$ holds. That is, all the eigenvalues of the following matrix

\[
\vdots\\
\end{matrix}}\right]\begin{matrix}\left[{\begin{matrix}
\overline{u_{1i}} & \ldots & \overline{u_{ni}}\\
\end{matrix}}\right]\\[1.0ex] \\ \\ \end{matrix}
\]

must be non-negative. Since swapping two rows or columns only changes the sign of the determinant, the characteristic equation of the matrix $A_\sigma$ becomes the following,
0    =    (A_σ-xI_n)

=    ((-x,…,-x,B_σ))

where $B_\sigma$ is a $(n-k_i)\times (n-k_i)$ matrix. This says that positive semi-definiteness of $A_\sigma$ is completely determined by positive semi-definiteness of $B_\sigma$. Now, observe that the matrix $B_\sigma$ is as in the earlier situation of <Ref>, where all the entries of the following matrix

\[
\mbox{diag}\big(\,|u_{j_{k_i+1}i}|^2,\ldots,|u_{j_ni}|^2\big)
\]

are non-zero. Hence, the best constant $\sigma$ in <Ref> for the matrix $B_\sigma$ is given by $\sigma=-\frac{1}{n-k_i}$. The number $n-k_i$ is exactly the number of nonzero entries in the $i$-th column of $U^*$. By Lemma <ref>, the proof is now completed.

If $U$ is a unitary matrix in $M_n$ with $U=(u_{ij})$, then $\sum_{i,j=1}^n\eta(|u_{ij}|^2)\leq\log\big(\min_{1\leq i\leq n}\,h(U)_i\big)^n$.

Let $U=(u_{ij})$ be a unitary matrix in $M_n$ and consider the pair of Masas $(\Delta_n,U\Delta_n U^*)$ in $M_n$. By Cor. $3.2$ in [9], $h(\Delta_n|U\Delta_n U^*)=\frac{1}{n}\sum_{i,j=1}^n\eta(|u_{ij}|^2)$. By <Ref>, we have $h(\Delta_n|U\Delta_n U^*)\leq-\log\lambda(\Delta_n,U\Delta_n U^*)$ and the claim now follows from <Ref>.

The pair $\big(\Delta_n,U\Delta_nU^*\big)$ in the type $I_n$ factor $M_n$ is an orthogonal pair in the sense of Popa $[39]$ only if the unitary $U$ has no zero entry.

Let $\big(\Delta_n,U\Delta_nU^*\big)$ be an orthogonal pair. Then, $(\bbc\subset \Delta_n,U\Delta_n U^*\subset M_n)$ is a commuting square. By <Ref>, $h(\Delta_n|U\Delta_n U^*)=H(\Delta_n|U\Delta_n U^*)=H(\Delta_n|\bbc)=\log n$ (apply Theorem $6.2$ in [37]). Now by Cor. $3.3$ in [9], we know that $\big(\Delta_n,U\Delta_nU^*\big)$ is an orthogonal pair in $M_n$ if and only if $h(\Delta_n|U\Delta_n U^*)=\log n$. Since $h(\Delta_n|U\Delta_n U^*)\leq-\log\lambda(\Delta_n,U\Delta_n U^*)$ by <Ref>, we have the following inequalities

\[
\log n=h(\Delta_n|U\Delta_n U^*)\leq\log\big(\min_{1\leq i\leq n}\,h(U)_i\big)\leq\log n
\]

by <Ref>. Hence, all the entries of the unitary $U$ must be non-zero.

Proof of <Ref>:  For part $(i)$, we prove the contrapositive, that is, $R_u=R_v$ if and only if $u\sim v$. First observe that $R_u=R_v$ if and only if $\lambda(R_u,R_v)=1$. This is because $\lambda(R_u,R_v)=1$ implies that $R_u\subseteq R_v$ by <Ref>. Since $[R:R_u]=[R:R_v]=n$, we get that $R_u=R_v$. The converse direction is obvious.

Now, assume that $R_u=R_v$. Then $\lambda(R_u,R_v)=1$, and consequently $\lambda(u\Delta_2u^*,v\Delta_2v^*)=1$ because $\lambda(R_u,R_v)$ is the limit of the decreasing sequence of $\lambda$ at each step of the tower of basic constructions (<Ref>) and $0\leq\lambda\leq 1$. By <Ref> we get that $u^*v$ must be a permutation matrix, that is $u\sim v$. Thus, we conclude that $R_u=R_v$ implies $u\sim v$. Conversely, suppose that $u\sim v$. Then $v=uPD$, where $P$ is a permutation matrix in $M_n$ and $D\in U(n)$ is a diagonal unitary matrix. Then $\mbox{Ad}_{u^*v}(\Delta_n)\subseteq\Delta_n$, and consequently $\mbox{Ad}_{v}(\Delta_n)\subseteq\mbox{Ad}_u(\Delta_n)$. Immediately, we get that $R_v\subseteq R_u$. Since $[R:R_u]=[R:R_v]=n$, we get that $[R_u:R_v]=1$, and hence $R_u=R_v$. This completes part $(i)$, and part $(ii)$ immediately follows from part $(i)$, since the equivalence relation `$\sim$' is finer than the Hadamard equivalence relation.

Notice that the `only if' part in <Ref> is very easy, and it is the `if' part that requires work. <Ref> and <Ref> are the crucial ingredients in proving the `if' part.

The Pimsner-Popa constant for a pair of type   factors

The results in this section is a computational tool and key ingredient to compute the Pimsner-Popa constant in Sections <ref> and <ref>. Here, we determine the Pimsner-Popa constant for the following situation namely,

\[\begin{matrix}
M_n &\subset & M_n\oplus M_n\\
& & \cup\\
& & UM_nU^*\\
\end{matrix}\]

where $U\in M_n\oplus M_n$ is a unitary matrix.

Henceforth, we write $M_n\oplus M_n$ as $\Delta_2\otimes M_n$, where $\Delta_2$ denotes the diagonal subalgebra (Masa) of $M_2$. The embedding $M_n\xhookrightarrow{} \Delta_2\otimes M_n$ is the diagonal embedding $x\mapsto I_2\otimes x$. Since $U$ is a unitary in $\Delta_2\otimes M_n$, we have $U=\mbox{bl-diag}\{U_1,U_2\}$ with $U_1,\,U_2\in M_n$ unitary matrices. The unique trace preserving conditional expectation $E:\Delta_2\otimes M_n\longrightarrow M_n$ is given by $\tau\otimes\mbox{id}$, where $\tau:\Delta_2\longrightarrow\bbc$ is the normalized trace induced from $M_2$. For any $x\in (M_n)_+$, we have
E_UM_nU^*(x)    =    UE_M_n(U^*(1⊗x)U)U^*

=    UE_M_n({U_1^*xU_1,U_2^*xU_2})U^*

=    1/2U(U_1^*xU_1+U_2^*xU_2)U^*

Hence, for $0\leq t\leq 1$ the inquality $E_{UM_nU^*}(x)\geq tx$ is equivalent to the following inequality,
U_1^*xU_1     0

0     U_2^*xU_2

U_2^*xU_2     0

0     U_1^*xU_1

]≥0 .

We have $E_{UM_nU^*}(x)\geq\frac{1}{2}x$ for all $x\in (M_n)_+$, that is, $\lambda\big(M_n\,,\,UM_nU^*\big)\geq\frac{1}{2}$.

Follows immediately from <Ref>.

For $n\geq 3$ and any unitary $U\in M_n\oplus M_n$, the pair $\big(M_n\,,UM_nU^*\big)$ is never an orthogonal pair in $M_n\oplus M_n$ in the sense of Popa $[39]$.

On contrary assume that $\big(M_n\,,\,UM_nU^*\big)$ is an orthogonal pair. Then the quadruple $(\bbc\subset M_n\,,\,UM_nU^*\subset M_n\oplus M_n)$ is a commuting square. By <Ref>, we get that $h\big(M_n|UM_nU^*\big)=H\big(M_n|UM_nU^*\big)=H(M_n|\bbc)=\log\,n$ (see Section $6$ in [37]). By <Ref>, we have $\lambda\big(M_n\,,\,UM_nU^*\big)^{-1}\leq 2$, and hence $\log n=h\big(M_n|UM_nU^*\big)\leq\log\,2$ using <Ref>. This contradicts the fact that $n\geq 3$.

If $\,U_1^*U_2$ is not a diagonal matrix in $M_n$, then $\,\lambda\big(M_n\,,\,UM_nU^*\big)=\frac{1}{2}\,.$

Suppose that

\[
\frac{1}{2}\,\left[{\begin{matrix}
U_1^*xU_1 & 0\\
0 & U_2^*xU_2\\
\end{matrix}}\right]+\frac{1}{2}\,\left[{\begin{matrix}
U_2^*xU_2 & 0\\
0 & U_1^*xU_1\\
\end{matrix}}\right]\geq \big(\frac{1}{2}+\varepsilon\big)\,\left[{\begin{matrix}
U_1^*xU_1 & 0\\
0 & U_2^*xU_2\\
\end{matrix}}\right]
\]

for all $x\in (M_n)_+$ and $\varepsilon>0$. That is, we have the following,
U_2^*xU_2     0

0     U_1^*xU_1

]    ≥    2ε [
U_1^*xU_1     0

0     U_2^*xU_2

]  ∀   x∈(M_n)_+

Let $V=U_1^*U_2$, a unitary matrix in $M_n$, and choose $x_j=U_1E_{jj}U_1^*$ for $1\leq j\leq n$. Each $x_j\in (M_n)_+$ and we have for each $1\leq j\leq n$ the following,

\[
\left[{\begin{matrix}
V^*E_{jj}V & 0\\
0 & E_{jj}\\
\end{matrix}}\right]\geq 2\varepsilon\left[{\begin{matrix}
E_{jj} & 0\\
0 & V^*E_{jj}V\\
\end{matrix}}\right]
\]

in $\Delta_2\otimes M_n$. Hence, for any $i\neq j$ in $\{1,\ldots,n\}$ we have

\[\left[{\begin{matrix}
0 & 0\\
0 & E_{1i}\\
\end{matrix}}\right]\left[{\begin{matrix}
V^*E_{jj}V & 0\\
0 & E_{jj}\\
\end{matrix}}\right]\left[{\begin{matrix}
0 & 0\\
0 & E_{i1}\\
\end{matrix}}\right]\geq 2\varepsilon\left[{\begin{matrix}
0 & 0\\
0 & E_{1i}\\
\end{matrix}}\right]\left[{\begin{matrix}
E_{jj} & 0\\
0 & V^*E_{jj}V\\
\end{matrix}}\right]\left[{\begin{matrix}
0 & 0\\
0 & E_{i1}\\
\end{matrix}}\right]\]

as in a $C^*$-algebra $a\geq b$ implies that $x^*ax\geq x^*bx$ for any $x$. Since $\,\varepsilon>0$, letting $V=(v_{pq})_{1\leq p,q\leq n}$ we get the following,
0    =    E_1iV^*E_jjVE_i1

=    E_1i(∑_p,q v_qpE_pq)E_jj(∑_r,s v_rsE_rs)E_i1

=    E_1i(∑_p,s v_jpv_jsE_ps)E_i1

=    v_jiv_jiE_11 .

This says that $\,v_{ji}=0$ for any $j\neq i$ i,e. $V$ is a diagonal matrix in $M_n$. This is a contradiction, and we get that $\varepsilon=0$.

One has $\lambda\big(M_n\,,\,UM_nU^*\big)=1$ if and only if $U_1^*U_2=\alpha I_n$ for some $\alpha\in\mathbb{S}^1$.

If $\,U_1^*U_2=\alpha I_n$, then $UM_nU^*=I_2\otimes U_1M_nU_1^*=I_2\otimes M_n$, and hence $\lambda\big(M_n\,,\,UM_nU^*\big)=1$. Conversely, $\lambda\big(M_n\,,\,UM_nU^*\big)=1$ implies that $H(M_n|UM_nU^*)=0$ and consequently, $M_n\subseteq UM_nU^*$, that is, $U^*M_nU\subseteq M_n$. Therefore, $U_1^*AU_1=U_2^*AU_2$ for all $A\in M_n$ and hence $U_1U_2^*=\beta I_n$ for some $\beta\in\bbc$, which completes the proof.

For the inclusion of algebras $M_n\subset\Delta_2\otimes M_n\supset UM_nU^*$, where $\,U\in\Delta_2\otimes M_n$ is a unitary matrix given by $\,U=\mbox{bl-diag}\{U_1,U_2\}$ with unitary matrices $U_1,U_2\in M_n$, one has the following,
λ(M_n , UM_nU^*)=
1        U_1^*U_2 , 1/2        U_1^*U_2 .

Follows from Lemma (<ref>, <ref>).

Note that there is one remaining situation when $U_1^*U_2$ is a diagonal unitary matrix. However, this situation does not arise in our case in Sections <ref> and <ref>, and hence we leave it to the interested readers.

Subfactors arising from   complex Hadamard matrices

Recall that $2\times 2$ complex Hadamard matrices form a single family up to Hadamard equivalence, and any element of this family is given by the following
e^iα_1     e^i (α_1+α_3)

e^iα_2     -e^i(α_2+α_3)

where $\alpha_j\in[0,2\pi),\,j=1,2,3$. Also recall the equivalence relation `$\sim$' from <Ref>, <Ref>. Consider any two $2\times 2$ complex Hadamard matrices $u$ and $v$ such that $u\nsim v$ and obtain the (distinct) irreducible spin model subfactors each with index 2 :

\[\begin{matrix}
R_u &\subset & R\\
& & \cup\\
& & R_v
\end{matrix}
\]

Note that $u\nsim v$ if and only if $v\neq uPD$, where $D$ is a diagonal unitary matrix and $P$ is a permutation matrix. In this $2\times 2$ situation, we have $PD$ is a unitary matrix of the form $\left[{\begin{matrix}
\alpha & 0\\
0 & \beta\\
\end{matrix}}\right]$ or $\left[{\begin{matrix}
0 & \alpha\\
\beta & 0\\
\end{matrix}}\right],$ where $\alpha,\beta\in\mathbb{S}^1$. Given any $u=F^{\alpha_1}_{\alpha_2}(\alpha_3)$ and $v=F^{\beta_1}_{\beta_2}(\beta_3)$ (<Ref>), consider the following matrices

\[
\widetilde{u}=\frac{1}{\sqrt{2}}\begin{bmatrix}
1 &  1\\
e^{i(\alpha_2-\alpha_1)} & -e^{i(\alpha_2-\alpha_1)}
\end{bmatrix}\quad\mbox{and}\quad\widetilde{v}=\frac{1}{\sqrt{2}}\begin{bmatrix}
1 &  1\\
e^{i(\beta_2-\beta_1)} & -e^{i(\beta_2-\beta_1)}
\end{bmatrix}
\]

associated to them. Since $u\sim\widetilde{u}$, by <Ref> we have $R_u=R_{\widetilde{u}}$, and similarly for $v$. Hence, the quadruples of von Neumann algebras $(R_u\cap R_v\subset R_u,R_v\subset R)$ and $(R_{\widetilde{u}}\cap R_{\widetilde{v}}\subset R_{\widetilde{u}},R_{\widetilde{v}}\subset R)$ are the same. Therefore, without loss of generality, throughout the <Ref> we will assume that

1     1

e^iα     -e^iα

1     1

e^iβ     -e^iβ

for $\alpha,\beta\in[0,2\pi)$. Note that $\beta=\alpha\pm\pi$ implies that $v=u\sigma_1$, which gives $u\sim v$. Therefore, in our situation we have $\alpha,\beta\in[0,2\pi)$ with $\beta\neq\alpha,\alpha\pm\pi$.

In this section, we first compute the Pimsner-Popa constant $\,\lambda(R_u,R_v)$. Recall that as mentioned before, it is not obvious whether intersection of factors is again a factor. We prove that $R_u\cap R_v$ is a $II_1$ subfactor of $R$ with $[R:R_u\cap R_v]=4$. Moreover, we show that $R_u\cap R_v\subset R$ is a vertex model subfactor. Finally, we compute the Sano-Watatani angle $\mbox{Ang}_R(R_u,R_v)$ and relative entropy $h(R_u|R_v)$ between the subfactors $R_u\mbox{ and }R_v$, and obtain legitimate bounds for the relative entropy $H(R_u|R_v)$. We also characterize the quadruple of $II_1$ factors $(R_u\cap R_v\subset R_u,R_v\subset R)$ in terms of a bi-unitary matrix in $M_4$.

Note that

\[
\mathbb{C}\subset \Delta_2\subset M_2\subset \Delta_2 \otimes M_2\subset M_2\otimes M_2\subset \Delta_2\otimes M_2\otimes M_2\subset M_2\otimes M_2\otimes M_2\subset\cdots
\]

is a tower of Jones' basic construction, and thus $R$ is the closure in the SOT topology of the union of these subalgebras. Following <Ref>, the ladder of basic constructions of the commuting square

\[
\begin{matrix}
u\Delta_2 u^* &\subset & M_2 \cr \cup &\ & \cup\cr \mathbb{C} &\subset &  \Delta_2
\end{matrix}
\]

is depicted in <Ref> (note that in our convention $u_0=u$), where the unitary matrices $u_j$ are given in <Ref>.

$R$ $R$
d1$\Delta_2\otimes M_2\otimes M_2$
d2$M_2\otimes M_2$
d3$\Delta_2\otimes M_2$
f_1$u_3(M_2\otimes M_2)u_3^*$
f_2$u_2(\Delta_2 \otimes M_2)u_2^*$
e_1$v_3(M_2\otimes M_2)v_3^*$
e_2$v_2(\Delta_2 \otimes M_2)v_2^*$

A pair of $2\times 2$ Hadamard matrices and basic constructions

Although the basic construction (in the general $n\times n$ case) is well-known (see [26, 34, 29], for instance), our basic construction is slightly different than the one in the existing literature as it will be more handy in the subsequent computations. We remark that $R_u$ is the closure in the SOT topology of the union of the left vertical algebras as in <Ref>, and similarly $R_v$ is that of the right vertical algebras.

To fix notations, we let $u=(u_{ij})$ be a $2\times 2$ complex Hadamard matrix and consider the following matrices

\[
\eta=\left[{\begin{matrix}
\frac{u_{12}}{u_{11}} & 0\\
0 & \frac{u_{22}}{u_{21}}\\
\end{matrix}}\right]\quad \mbox{and}\quad \xi=\frac{1}{\sqrt{2}}\,\left[{\begin{matrix}
1 & \frac{u_{12}}{u_{11}}\\
1 & \frac{u_{22}}{u_{21}}\\
\end{matrix}}\right]\,.
\]

Since $u$ is unitary, it is easy to check that $\,\eta\mbox{ and }\xi$ are unitary matrices. Let $I_2^{(k)}$ denote the unit element $I_2\otimes\ldots\otimes I_2$ in $(M_2)^{\otimes\,k}$. We have the following tower of basic construction (with the convention that $u_0=u$).

The tower of the basic construction for $\mathbb{C}\subset u\Delta_2u^*$ is given by

\[
\mathbb{C}\subset u_0\Delta_2u^*_0\subset u_1M_2u^*_1\subset u_2(\Delta_2\otimes M_2)u^*_2\subset u_3(M_2\otimes M_2)u^*_3\subset\cdots
\]

where each $u_i$ is a unitary matrix given by the following prescription :

* for $k\in\bbn$, we have $\,u_{2k}=u_{2k-1}\left(\xi_k\otimes I_2^{(k)}\right)$, where
ξ_k    =
ξ^* ;       k=1, F_2 ;       k≥2,

and $\,F_2=\frac{1}{\sqrt{2}}\,
\left[{\begin{matrix}
1 & 1\\
1 & -1\\
\end{matrix}}\right]$ is the Fourier matrix in $M_2$.

* for $k\in\bbn\cup\{0\}$, we have $\,u_{2k+1}=(I_2\otimes u_{2k})\left(E_{11}\otimes I_2^{(k+1)}+E_{22}\otimes\eta_k\right)$, where
η_k    =
η ;       k=0, σ_3⊗I_2^(k) ;       k≥1.

and $\sigma_3$ is Pauli spin matrix.

For the proof of this theorem we request the interested reader to visit the Appendix. We draw a flowchart in <Ref> to display our plan of actions in this section.

rounded rectangle,
minimum width=2.5cm,
minimum height=1cm] (block1)  Start with $2\times 2$ complex Hadamard

matrices $u\mbox{ and }v$ such that $u\nsim v$;

trapezium left angle = 65,
trapezium right angle = 115,
trapezium stretches,
below=of block1,
minimum width=3.5cm,
minimum height=1cm] (block2)  Obtain the pair of spin model subfactors $R_u\mbox{ and }R_v\,$;

below=of block2,
minimum width=3.5cm,
minimum height=1cm] (block3)  Computation of $\lambda(R_u,R_v)\,$;

below=of block3,
minimum width=3.5cm,
minimum height=1cm] (block40)  Identifying the finite-dimensional grid for $R_u\cap R_v\,$;

below=of block40,
minimum width=3.5cm,
minimum height=1cm] (block4)  Construction of the subfactor $R_{u,v}\subset R\,$;

below right=of block4,
minimum width=3.5cm,
minimum height=1cm] (block6)  Factoriality of $R_u\cap R_v\,$;

rounded rectangle,
below=of block6,
minimum width=3.5cm,
minimum height=1cm] (block50) Characterization of

$\big(R_u\cap R_v\subset R_u,R_v\subset R\big)$

being commuting square;

below=of block4,
minimum width=3.5cm,
minimum height=1cm] (block7)  Relative commutant $(R_u\cap R_v)^\prime\cap R\,$;

below=of block7,
minimum width=2.5cm,
minimum height=1cm] (block8)  Characteriztion of

$\,(R_u\cap R_v\subset R_u,R_v\subset R)\,$;

rounded rectangle,
below=of block8,
minimum width=2.5cm,
minimum height=1cm] (block80)  Relative entropy and angle

between $R_u\mbox{ and }R_v\,$;

[-latex] (block1) edge (block2)
(block2) edge (block3)
(block3) edge (block40)
(block40) edge (block4)
(block6) edge (block50)
(block6) edge (block7)
(block7) edge (block8)
(block8) edge (block80);

[-latex] (block4) -| (block6)
node[pos=0.5,fill=white,inner sep=0];

Flowchart for our plan of actions in <Ref>

§.§ Computation of the Pimsner-Popa constant

Let $u\mbox{ and }v$ be two complex Hadamard matrices of the form described in <Ref>. In this subsection, we compute $\lambda(R_u,R_v)$. Before we begin, we pause for a moment to discuss the following example. Consider the following diagram

\[
\begin{matrix}
u\Delta_2u^* &\subset & M_2 &\supset &v\Delta_2v^*\\
\cup & &\cup & &\cup\\
\bbc &\subset & \Delta_2 &\supset &\bbc
\end{matrix}
\]

where $u$ and $v$ are any two distinct $2\times 2$ complex Hadamard matrices. The quadruple $(\bbc\subset\Delta_2,u\Delta_2u^*\subset M_2)$ (resp., $u$ replaced by $v$) is a symmetric commuting square, since $u$ (resp., $v$) is a complex Hadamard matrix. Now, $\lambda(\bbc,\bbc)=1$ but $\lambda(u\Delta_2u^*,v\Delta_2v^*)=\lambda(\Delta_2,u^*v\Delta_2v^*u)=\min_{1\leq i\leq 2}\,\big(\mbox{$h\left(v^*u\right)_i$}\big)^{-1}$ by <Ref>. If we choose $u=\frac{1}{\sqrt{2}}\left(\begin{smallmatrix}
1 & 1\\
i & -i
\end{smallmatrix}\right)$ and $v=\frac{1}{\sqrt{2}}\left(\begin{smallmatrix}
1 & 1\\
1 & -1
\end{smallmatrix}\right)$, we see that $\lambda(u\Delta_2u^*,v\Delta_2v^*)=\frac{1}{2}\neq\lambda(\bbc,\bbc)$. This example illustrates that finding the value of $\lambda(R_u,R_v)$ is not obvious.

Notation :  $A(u_4,v_4):=v_4^*u_4(\sigma_3\otimes I_4)u_4^*v_4(\sigma_3\otimes I_4)$ in $M_8$, and $B(u_2,v_2):=v_2^*u_2(\sigma_3\otimes I_2)u_2^*v_2$ in $M_4$.

Let $k\geq 2$ be a natural number. The following matrix

\[
v_{2k}^*u_{2k}\left(\sigma_3\otimes I_2^{(k)}\right)u_{2k}^*v_{2k}\left(\sigma_3\otimes I_2^{(k)}\right)
\]

in $M_2^{(k+1)}$ is not diagonal if the following matrix

$($i,e. for $k=2)$ in $M_2^{(3)}$ is not diagonal.

Recall the tower of basic construction in Theorem <ref> for any $2\times 2$ complex Hadamard matrix. For two $2\times 2$ complex Hadamard matrices $u\mbox{ and }v$, we denote by $\xi_k(u)\mbox{ and }\eta_k(u)$ the corresponding matrices $\xi_k\mbox{ and }\eta_k$ in the tower of basic construction for the unitary $u$, and similarly for $v$. Note that we are interested in $k\geq 3$, since the statement is obviously true for $k=2$, and we have $\xi_{n+1}(u)=\xi_{n+1}(v)=F_2\mbox{ and }\eta_n(u)=\eta_n(v)=\sigma_3\otimes I_2^{(n)}$ for any $n\in\bbn$ by <Ref>. The matrices $F_2\mbox{ and }\sigma_3$ both are self-adjoint. For $k\geq 3$, we have the following,

=    (F_2⊗I_2^(k))v_2k-1^*u_2k-1(F_2⊗I_2^(k))(σ_3⊗I_2^(k))(F_2⊗I_2^(k))u_2k-1^*v_2k-1(F_2⊗I_2^(k))(σ_3⊗I_2^(k))

=    (F_2⊗I_2^(k)) {I_2^(k),η_k-1(v)}^* (I_2⊗v_2k-2^*u_2k-2) {I_2^(k),η_k-1(u)}(σ_1⊗I_2^(k))

{I_2^(k),η_k-1(u)}^*(I_2⊗u_2k-2^*v_2k-2) {I_2^(k),η_k-1(v)}(1/√(2)(I_2-iσ_2)⊗I_2^(k)) .

For $k\geq 3$, we have $\eta_{k-1}(u)=\eta_{k-1}(v)=\sigma_3\otimes I_2^{(k-1)}$. Hence,

=    (F_2⊗I_2^(k)) {I_2^(k),σ_3⊗I_2^(k-1)}[
x_2k-2(σ_3⊗I_2^(k-1))     x_2k-2(σ_3⊗I_2^(k-1))

x_2k-2     -x_2k-2

with $\,x_{2k-2}=v_{2k-2}^*u_{2k-2}\left(\sigma_3\otimes I_2^{(k-1)}\right)u_{2k-2}^*v_{2k-2}$ in $M_2\otimes M_2^{(k-1)}$. Therefore, we get that

=    [
{x_2k-2 , σ_3⊗I_2^(k-1)}            [x_2k-2 , σ_3⊗I_2^(k-1)]

[x_2k-2 , σ_3⊗I_2^(k-1)]             {x_2k-2 , σ_3⊗I_2^(k-1)}

where $[\,,\,]$ and $\{\,,\,\}$ denotes the commutator and the anti-commutator respectively. Now, suppose that for $k=2$, the following matrix

in $M_8$ is not diagonal. Assume that the statement is true up to $(k-1)$-th step. Now for the $k$-th step, first by the induction hypothesis we get that $x_{2k-2}\left(\sigma_3\otimes I_2^{(k-1)}\right)$ is not a diagonal matrix. From <Ref>, we see that if the commutator $[x_{2k-2},\sigma_3\otimes I_2^{(k-1)}]\neq 0$, then we are through and if $\,[x_{2k-2},\sigma_3\otimes I_2^{(k-1)}]=0$, then the anticommutator $\{x_{2k-2},\sigma_3\otimes I_2^{(k-1)}\}$ becomes $2x_{2k-2}\left(\sigma_3\otimes I_2^{(k-1)}\right)$, which is not a diagonal matrix by the induction hypothesis. Thus, if the matrix in <Ref> is not diagonal, then the matrix in <Ref> is also not diagonal, which completes the proof.

The matrix $A(u_4,v_4)$ is diagonal if and only if the following self-adjoint matrix

\[
B(u_2,v_2)=v_2^*u_2(\sigma_3\otimes I_2)u_2^*v_2
\]

is diagonal in $M_4$.

Using Theorem <ref>, we obtain the following,
u_4^*v_4    =    [
u_2^*     (σ_3⊗I_2)u_2^*

u_2^*     -(σ_3⊗I_2)u_2^*

v_2     v_2

v_2(σ_3⊗I_2)     -v_2(σ_3⊗I_2)

=    [
u_2^*v_2+(σ_3⊗I_2)u_2^*v_2(σ_3⊗I_2)      u_2^*v_2-(σ_3⊗I_2)u_2^*v_2(σ_3⊗I_2)

u_2^*v_2-(σ_3⊗I_2)u_2^*v_2(σ_3⊗I_2)      u_2^*v_2+(σ_3⊗I_2)u_2^*v_2(σ_3⊗I_2)
] .

Hence, we get that
=    [
u_2^*v_2+(σ_3⊗I_2)u_2^*v_2(σ_3⊗I_2)     -u_2^*v_2+(σ_3⊗I_2)u_2^*v_2(σ_3⊗I_2)

-u_2^*v_2+(σ_3⊗I_2)u_2^*v_2(σ_3⊗I_2)      u_2^*v_2+(σ_3⊗I_2)u_2^*v_2(σ_3⊗I_2)
] .

A(u_4,v_4)    =    v_4^*u_4(σ_3⊗I_4)u_4^*v_4(σ_3⊗I_4)

=    [
v_2^*u_2+(σ_3⊗I_2)v_2^*u_2(σ_3⊗I_2)      v_2^*u_2-(σ_3⊗I_2)v_2^*u_2(σ_3⊗I_2)

v_2^*u_2-(σ_3⊗I_2)v_2^*u_2(σ_3⊗I_2)      v_2^*u_2+(σ_3⊗I_2)v_2^*u_2(σ_3⊗I_2)]

u_2^*v_2+(σ_3⊗I_2)u_2^*v_2(σ_3⊗I_2)      -u_2^*v_2+(σ_3⊗I_2)u_2^*v_2(σ_3⊗I_2)

-u_2^*v_2+(σ_3⊗I_2)u_2^*v_2(σ_3⊗I_2)      u_2^*v_2+(σ_3⊗I_2)u_2^*v_2(σ_3⊗I_2)]

=    2[
A(u_4,v_4)_11       A(u_4,v_4)_12

A(u_4,v_4)_21       A(u_4,v_4)_22

\[
A(u_4,v_4)_{11}=A(u_4,v_4)_{22}=v_2^*u_2(\sigma_3\otimes I_2)u_2^*v_2(\sigma_3\otimes I_2)+(\sigma_3\otimes I_2)v_2^*u_2(\sigma_3\otimes I_2)u_2^*v_2\,,
\]

\[
A(u_4,v_4)_{12}=A(u_4,v_4)_{21}=v_2^*u_2(\sigma_3\otimes I_2)u_2^*v_2(\sigma_3\otimes I_2)-(\sigma_3\otimes I_2)v_2^*u_2(\sigma_3\otimes I_2)u_2^*v_2\,.
\]

Hence, $A(u_4,v_4)=I_2\otimes A(u_4,v_4)_{11}+\sigma_1\otimes A(u_4,v_4)_{12}$. This says that the matrix $A(u_4,v_4)$ is diagonal if and only if $A(u_4,v_4)_{11}$ is diagonal and $A(u_4,v_4)_{12}=0$. That is, $A(u_4,v_4)$ is diagonal if and only if the self-adjoint matrix $v_2^*u_2(\sigma_3\otimes I_2)u_2^*v_2$ commutes with $\sigma_3\otimes I_2$ and $v_2^*u_2(\sigma_3\otimes I_2)u_2^*v_2(\sigma_3\otimes I_2)$ is diagonal, which is further equivalent to $B(u_2,v_2)=v_2^*u_2(\sigma_3\otimes I_2)u_2^*v_2$ is diagonal.

The matrix $B(u_2,v_2)=v_2^*u_2(\sigma_3\otimes I_2)u_2^*v_2$ is not diagonal.

Using <Ref>, a tedious but straightforward computation gives us the following,
v_2^*u_2(σ_3⊗I_2)u_2^*v_2    =    1/2[
σ_3v^*uσ_3u^*v         v^*uσ_3u^*vσ_3

-σ_3v^*uσ_3u^*v          v^*uσ_3u^*vσ_3

1     1

1     -1

We denote $C=\sigma_3v^*u\sigma_3u^*v\in M_2$. Observe that the matrix $B(u_2,v_2)$ is diagonal if and only if the following holds,
C-C^*=0        C+C^* .

Now, another straightforward computation shows that

\[
\cos(\alpha-\beta) & i\sin(\alpha-\beta)\\