The fuzzy bit

M. Aldana

Departamento de Ciencias de la Tierra, Universidad Simón Bolívar

Valle de Sartenejas, Baruta. Apartado Postal 89000, Venezuela.

e-mail<EMAIL_ADDRESS>
M. A. Lledó

Departament de Física Teòrica, Universitat de València and

IFIC (CSIC-UVEG)

C/Dr. Moliner, 50, E-46100 Burjassot (València), Spain.

e-mail<EMAIL_ADDRESS>

###### Abstract

In this paper, the formulation of Quantum Mechanics in terms of fuzzy logic
and fuzzy sets is explored. A result by Pykacz, that establishes a
correspondence between (quantum) logics (lattices with certain properties) and
certain families of fuzzy sets, is applied to the Birkhoff-von Neumann logic,
the lattice of projectors of a Hilbert space. Three cases are considered: the
qubit, two qubits entangled and a qutrit ‘nested’ inside the two entangled
qubits. The membership functions of the fuzzy sets are explicitly computed and
all the connectives of the fuzzy sets are interpreted as operations with these
particular membership functions. In this way, a complete picture of the
standard quantum logic in terms of fuzzy sets is obtained for the systems
considered.

##  1 Introduction

In Mackey’s approach to Quantum Mechanics [1], the description of a physical
system is done in terms of certain axioms. We are given two abstract sets: the
set of states of the system denoted by $\mathcal{S}$ and the set of physical
quantities or observables, denoted as $\mathcal{O}$. The results of
measurements on the physical system are assumed to be real numbers, so one
considers $\mathcal{B}(\mathbb{R})$, the family of Borel subsets of the real
line. We are given a function

$\begin{CD}\mathcal{O}\times\mathcal{S}\times\mathcal{B}(\mathbb{R})@>{p}>{}>[0,1]\\\
(A,\rho,E)@>{}>{}>p(A,\rho,E)\end{CD}$

that is interpreted as the probability that the measurement of the observable
$A$ with the system in the state $\rho$ is contained in the Borel subset $E$.
The first six axioms regard properties of the probability function, $p$. These
axioms are natural or have physical interpretation. The first conclusion is
that the physical system is described in terms of its logic $\mathcal{L}$,
together with a convex family of probability measures on $\mathcal{L}$. All
these terms are going to be defined in Section 2 (other sources are, for
example, [2, 3]).

A particular example of logic is the logic of propositions of a classical
system. Propositions are such as ‘the observable $A$ has a value in $E$’. If
$\mathcal{S}$ is the phase space  or space of states of the classical system,
the observables are functions $f:\mathcal{S}\rightarrow\mathbb{R}$ and the
proposition will be true if the system is in a state inside $f^{-1}(E)$. In
this way, physically sensible statements about the system are essentially
subsets of $\mathcal{S}$. The logic of Classical Mechanics is the Boolean
algebra of subsets of a set $\mathcal{S}$. This is the standard relation
between sets and the propositional logic. Let $U$ be a set and consider the
subsets of it. Let $A$ be one such subset and $x$ an element in $U$. Then $A$
can be defined as the collection of elements such that the proposition ‘$x\in
A$’ is true.

So the first six axioms apply to Quantum Mechanics, but also to Classical
Mechanics

The next axiom of Mackey (Axiom VII) is that the logic of a quantum system is,
among all possible logics, the Birkhoff-von Neumann logic of projectors of a
Hilbert space. Quoting Mackey [1], ‘This axiom has a rather different
character from Axioms I-VI. These all have some degree of physical naturalness
and plausibility. Axiom VII seems completely ad hoc.’

In parallel to these logics, the many valued logics of Łukasiewicz [5] and
others, where propositions are not always totally true or false, but have some
degree of truth, were being developed. Intuitively, they seem related to a
probabilistic theory like Quantum Mechanics.

The same relation that the classical, two valued logic, has with set theory,
has the many valued logic with fuzzy sets [6]. A fuzzy subset $\mathcal{A}$ of
a set $\mathcal{U}$, called the universe of discourse, is given by a map
$\mu_{\mathcal{A}}:\mathcal{U}\rightarrow[0,1]$ called the membership
function, that associates to any element of $\mathcal{U}$ a ‘degree of
membership’ with respect to $\mathcal{A}$. The value $\mu_{\mathcal{A}}(x)=1$
means that $x$ belongs totally to $\mathcal{A}$ and $\mu_{\mathcal{A}}(x)=0$
means that $x$ does not belong to $\mathcal{A}$. In between, we can have any
degree of membership. If $\mathcal{A}$ is a ‘crisp’ set, then $\mu_{A}$ is the
characteristic function

$\mu_{\mathcal{A}}=\begin{cases}0\quad\mathrm{if}\,x\notin\mathcal{A},\\\
1\quad\mathrm{if}\,x\in\mathcal{A}.\end{cases}$

The relationship between fuzzy sets and many valued logic was made explicit by
Giles [7].

A theorem by Pykacz [8, 9] makes, finally, the connection between fuzzy sets
and Quantum Mechanics. Essentially, it says that giving a logic (in the sense
used before) is equivalent to giving a family of fuzzy sets with certain
properties. So any logic does have a ‘representation’ in terms of fuzzy sets.
Adding properties to such representation allows to further distinguish between
logics. In particular, one can consider the universe of discourse to be the
set of density matrices in some Hilbert space, and compute the membership
functions in terms of an hermitian operator on such Hilbert space and a Borel
subset of $\mathbb{R}$. Then we recover the Birkhoff-von Neumann logic. This
choice is, then, equivalent to the choice of Mackey in his Axiom VII.

The aim of the present work is to clarify these abstract concepts, by giving
explicitly the membership functions corresponding to the Birkhoff-von Neumann
logic for different systems, all with a finite dimensional Hilbert space. The
paper is organized as follows. In Section 2 we go over the definition of
logic, and of observable and state over that logic. In Section 3 we deepen the
understanding of the axioms of Mackey and the work of Ma̧czyński [10] with his
experimental functions (that later will become the membership functions). In
Section 4 we specify all these concepts for the Birkhoff-von Neumann logic. In
Section 5 we compute the experimental functions for the qubit and in Section 6
we do the same for two qubits entangled, in the formalism of the Bloch matrix
[12, 13]. In Section 7 we exploit the decomposition of the Hilbert space of
the two entangled qubits in terms of eigenvectors of the angular momentum
operator to see how a qutrit, ‘nested’ in the system of two entangled qubits,
arises [14] and compute easily its experimental functions in terms of the
Bloch matrix. At this point we make a detour to study the unitary
transformations of the qutrit in terms of orthogonal symmetric Lie algebras,
inherited from the two entangled qubits [15]. In Section 8 we turn to the
fuzzy set picture. We interpret the connectives of fuzzy sets for the case of
the Birkhoff-von Neumann logic. Logic gates, essential part of quantum
computers, can also be written in terms of the membership functions. We deal
with a few examples and an interesting result is obtained with respect to the
square root of NOT. Finally, in Section 9, we state our conclusions.

##  2 Logics

This section is devoted to give, briefly, the basic definitions of logic,
observable and state, following some standard references [2, 3, 9].

######  Definition 2.1.

A partially ordered set or poset, is a set $L$ together with a binary
relation, ‘$\leq$’, satisfying

  * •

Reflexivity: $a\leq a$ for all $a\in L$.

  * •

Transitivity: for all $a,b,c\in L$ such that $a\leq b$ and $b\leq c$ then
$a\leq c$.

  * •

Antisymmetry: if $a,b\in L$ are such that $a\leq b$ and $b\leq a$ then $a=b$.

The binary relation does not need to be defined for all pairs of elements in
$L$. When it is so, it is a completely ordered set or simply an ordered set.

$\square$

######  Example 2.2.

The set of all the subsets or power set of a three element set $\\{x,y,z\\}$
partially ordered by the inclusion relation is a poset (see Fig. 1).

$\\{x,y,z\\}$$\\{x,y\\}$$\\{y,z\\}$$\\{x\\}$$\\{y\\}$$\\{z\\}$$\emptyset$$\\{x,z\\}$

Figure 1: Partial order in the power set of a three element set.

$\square$

######  Definition 2.3.

If $L$ is a poset, there exists, at most, one element called zero and denoted
as ‘0’, such that $0\leq a\,$ $\,\forall a\in L.$ Also, there exists, at most,
and element called one that will be denoted as ‘1’ such that $a\leq 1$
$\forall a\in L$.

$\square$

In Example 2.2, the empty set $\emptyset$ is the 0 and the set $\\{x,y,z\\}$
is the 1.

######  Definition 2.4.

For $S$ a finite subset of a poset $L$, the meet of $S$ is, if it exists, an
element $\bigwedge S$ that satisfies

  1. 1.

$\bigwedge S\leq s$ for all $s\in S$.

  2. 2.

If $x\in L$ is such that $x\leq s\;$ $\forall s\in S$, then $x\leq\bigwedge
S$.

If $S$ has only two elements $a$ and $b$, then the meet will be denoted as
$a\wedge b$.

For $S$ and $L$ as above, the join of $S$ is, if it exists, an element
$\bigvee S$ that satisfies:

  1. 1.

$s\leq\bigvee S$ for all $s\in S$.

  2. 2.

If $x\in L$ is such that $s\leq x\;$ $\forall s\in S$, then $\bigvee S\leq x$.

If $S$ has only two elements $a$ and $b$ then the join will be denoted as
$a\vee b$.

$\square$

The meet and the join, if they exist, are unique.

######  Definition 2.5.

A poset $L$ such that

  1. 1.

$0$ and $1$ exist, $0\neq 1$,

  2. 2.

for any finite subset $F\subset L$, $\bigwedge S$ and $\bigvee S$ exist,

is a lattice.

$\square$

######  Definition 2.6.

A complementation in a lattice $L$ is a map

$\begin{CD}L@>{\bot}>{}>L\\\ a@>{}>{}>a^{\bot},\end{CD}$

such that:

  1. 1.

$\bot$ is bijective

  2. 2.

$a\leq b\Rightarrow b^{\bot}\leq a^{\bot}$ (order reversing property)

  3. 3.

$(a^{\bot})^{\bot}=a$ (involutivity)

  4. 4.

$a\wedge a^{\bot}=0$ and $a\vee a^{\bot}=1$ (complementarity).

If $x\leq y^{\bot}$, we will say that $x\bot y$.

$\square$

######  Definition 2.7.

When a lattice $L$ has a complementation, we say that $L$ is a complemented
lattice.

$\square$

######  Definition 2.8.

An orthomodular lattice is a complemented lattice such that, for any $x,y\in
L$ with $x\leq y$ then $y=x\vee(y\wedge x^{\bot})$ (orthomodular law).

######  Definition 2.9.

We will say that an orthomodular lattice is $\sigma$-orthocomplete if, for
every countable set $\\{a_{l},a_{2},\dots,a_{n},\dots,\\}$ of pairwise
orthogonal elements, the join $\bigvee_{i}a_{i}$ exists. $\square$

######  Definition 2.10.

We will say that an orthomodular, $\sigma$-orthocomplete lattice is a logic.

$\square$

######  Example 2.11.

Boolean algebra of subsets of a set: classical logic.

Let $L$ be the set of subsets of a certain set $X$ with partial ordering given
by inclusion, join given by the intersection, meet by the union and complement
by the standard complement in $S$. The meet and the join satisfy the
distributive law

$\displaystyle a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c),$ $\displaystyle
a\vee(b\wedge c)=(a\vee b)\wedge(a\vee c),$ (1)

which implies, together with the definition of complement, the orthomodular
property. $L$ is a logic called the classical logic.

$\square$

######  Example 2.12.

Closed subspaces of a Hilbert space: Birkhoff-von Neumann logic [4]. Let
$\mathcal{H}$ be a Hilbert space, possibly of infinite dimension, and consider
$L$ to be the set of its closed subspaces. There is a partial ordering given
by the inclusion; then the meet is the intersection and the join the closure
of the sum of vector spaces. The complement is the orthogonal complement
inside the Hilbert space.

When the closed subspaces are orthogonal, the join is the direct sum of vector
spaces, which is also closed.

One can show that the orthomodular law is satisfied, thus $L$ is a logic
called the standard or Birkhoff-von Neumann logic.

$L$ does not satisfy the distributive law, which is the main difference with
the classical logic. When the Hilbert space is finite dimensional, all the
subspaces are closed and the orthomodular law (Definition 2.8) can be replaced
by the more general modular law:

$x,y,z\in L,\;x\subseteq y,\qquad x\oplus(y\cap z)=(x\oplus z)\cap y.$

$\square$

######  Definition 2.13.

A probability measure in a logic $L$ is a map $\rho:L\rightarrow[0,1]$ such
that

  1. 1.

$\rho(1)=1$

  2. 2.

$\rho(\bigvee_{i}a_{i})=\sum_{i}\rho(a_{i})$ for any sequence of pairwise
orthogonal elements of $L$.

$\square$

Later, we will interpret a probability measure on the Birkhoff-von Neumann
logic as a state of the system. There is an important property that the states
of the system must satisfy:

######  Definition 2.14.

A set of probability measures $\mathcal{S}$ for a quantum logic $L$ is said to
be ordering (full, order determining) if and only if

$\rho(a)\leq\rho(b)\quad\forall\rho\in\mathcal{S}\;\Rightarrow\;a\leq b.$

$\square$

######  Definition 2.15.

Let $L$ be a logic and $\mathcal{B}(\mathbb{R})$ the set of Borel sets of the
real line. A map $A:\mathcal{B}(\mathbb{R})\rightarrow L$ is an $L$-valued
measure if

  1. 1.

$A(\emptyset)=0,\qquad A(\mathbb{R})=1$,

  2. 2.

if $E,F\in\mathcal{B}(\mathbb{R})$ are such that $E\cap F=\emptyset$ then
,$A(E)\bot A(F)$,

  3. 3.

if $E_{i}\in\mathcal{B}(\mathbb{R})$ is a sequence of pairwise disjoint Borel
sets, then $A(E_{1}\cup E_{2}\cup\cdots)=A(E_{1})\vee A(E_{2})\vee\cdots\;\;$
.

$\square$

$L$-valued measures in the Birkhoff-von Neumann logic play the role of the
observables of the quantum theory. Since we can identify a subspace of a
Hilbert space with its orthogonal projection, an $L$-valued measure is, in
that case, a projection valued measure.

######  Definition 2.16.

A family $M_{t},\,t\in T$ of $L$-valued measures is surjective if for each
$f\in L$ there exists $t\in T$ and $E\in\mathcal{B}(\mathbb{R})$ such that
$M_{t}(E)=f$.

$\square$

##  3 The Mackey axioms and the Ma̧czyński
theorem

In [1], Mackey describes Quantum Mechanics as a rigorous mathematical system
in terms of nine axioms that are, as much as possible, plausible and natural
from physical considerations.

One starts with the assumption that, for each quantum system, there exists a
set of states $\mathcal{S}$, a set of observables $\mathcal{O}$ and a function
$p$ such that

$\begin{CD}\mathcal{O}\times\mathcal{S}\times\mathcal{B}(\mathbb{R})@>{p}>{}>[0,1]\\\
(A,\rho,E)@>{}>{}>p(A,\rho,E),\end{CD}$ (2)

were $\mathcal{B}(\mathbb{R})$ denotes the family of Borel subsets of the real
line. The function $p$ has to be interpreted as the probability that the
measurement of the observable $A$, when the system is in the state $\rho$,
leads to a value in the Borel set $E$, so

$\begin{CD}\mathcal{B}(\mathbb{R})@>{\rho_{A}}>{}>[0,1]\\\
E@>{}>{}>p(A,\rho,E),\end{CD}$

is a probability measure for each pair $\rho,A$. We refer to [1] for the
explicit formulation of the nine axioms. The first six axioms describe
properties of the function $p$, based in physical considerations. We are not
detailing those axioms here, as they are stated very clearly in the original
reference [1]. (Also, for a more modern treatment of the subject, see, for
example, [3]). For example, the $\sigma$-orthocompleteness of Definition 2.9
is assumed as Axiom V. We have that $p(A,\rho,E)$ defines a logic in the sense
of Definition 2.10.

Axiom VII is, as we mentioned in the Introduction, ad hoc and doesn’t arise
from any natural, physical consideration. It says that the logic of a quantum
system is the Birkhoff-von Neumann logic of closed subspaces of a Hilbert
space.

Axiom VIII deals with certain observables called questions (see below) and
Axiom IX, with the unitary time evolution of the system.

It is worth to mention here the definition of question by Mackey. A question
is an observable $Q$ such that $\rho_{Q}$ takes only the values 0 or 1. It
corresponds to a proposition that can be answered with ‘yes’ (1) or ‘no’ (0).
Let $A$ be any observable and $E$ any Borel set in $\mathbb{R}$. Then the
question $Q_{A}^{E}$ is the observable that yields 1 whenever a measurement of
$A$ is in $E$ and 0 if the measurement of $A$ is not in $E$.

Let $\rho$ be any state and $Q$ any question. Then the function
$\rho_{Q}(\\{1\\})=s$, $s\in[0,1]$, defines an ordering in the space $L$ of
questions: $Q_{1}\leq Q_{2}$ if and only if
$\rho_{Q_{1}}(\\{1\\})\leq\rho_{Q_{2}}(\\{1\\})$ for all $\rho\in S$. In
particular, it is clear that

$\rho_{Q_{A}^{E}}(\\{1\\})=p(A,\rho,E).$ (3)

In the space of questions, we have an obvious complementation, $Q^{\bot}=1-Q$,
which takes the value 0 when $Q$ takes the value 1 and vice versa. We will say
that two questions are orthogonal if $Q_{1}\leq Q_{2}^{\bot}$, that is,

$\rho_{Q_{1}}(\\{1\\})\leq 1-\rho_{Q_{2}}(\\{1\\})\hbox{ or
}\rho_{Q_{1}}(\\{1\\})+\rho_{Q_{2}}(\\{1\\})\leq 1$

for each $\rho\in S$.

In the language of Ma̧czyński [10], the functions

$f_{A}^{E}(\rho)=p(A,\rho,E)$ (4)

are called experimental functions. Notice that experimental functions are
defined on the equivalence classes of pairs $(A,E)$ under the equivalence
relation

$(A,E)\sim(A^{\prime},E^{\prime})\hbox{ if and only if
}p(A,\rho,E)=p(A^{\prime},\rho,E^{\prime})\qquad\forall\rho\in S.$

The experimental function $f_{A}^{E}(\rho)$ is associated to the experimental
proposition ‘The result of the measurement of $A$ lies in $E$’, which is one
of the questions of Mackey.

In the following, we will consider finite or infinite countable sums of
experimental functions, the infinite case meaning that the series is
convergent. Ma̧czyński introduces two convenient definitions:

######  Definition 3.1.

A finite or countable sequence $f_{1},f_{2},f_{3},\dots$ of experimental
functions is orthogonal if there exists an experimental function $g$ such that

$f_{1}+f_{2}+f_{3}+\dots+g=1.$

######  Definition 3.2.

A finite or countable sequence $f_{1},f_{2},f_{3},\dots$ of experimental
functions is pairwise orthogonal if, for each pair $(i,j)$, $i\neq j$, we have
that $f_{i}+f_{j}\leq 1$. A one element sequence is, by definition,
orthogonal.

$\square$

Clearly, an orthogonal sequence is pairwise orthogonal, but the contrary is
not always true. Nevertheless, if an experimental proposition in an orthogonal
sequence is ‘true’ in some state, that is, its experimental function takes the
value 1 at such state, all the other experimental functions have to take the
value 0, so the rest of the experimental propositions are ‘false’. This is the
same also if the sequence is only pairwise orthogonal. Hence, according to
Ma̧czyński, it makes sense to require that the set of experimental functions
satisfies the postulate that both concepts are equivalent for such set. This
is called by Ma̧czyński the orthogonality postulate. It is equivalent to
Mackey’s fifth axiom and it allowed Ma̧czyński to prove that $L$ is a
complemented lattice. We have the following theorem by Ma̧czyński (version
adapted by Pykacz [9]):

######  Theorem 3.3.

(Ma̧czyński’73 [10]) Let $\mathcal{O}$, $\mathcal{S}$ and $p$ be given as
above, let $L$ be the set of experimental functions defined by $p$ and assume
that they satisfy the orthogonality postulate. Then,

  1. 1.

$L$ is a logic with respect to the natural order of real functions, with
complementation $f^{\bot}=1-f$, $\,f\in L$.

  2. 2.

Each observable $A$ determines a unique $L$-valued measure
$M_{A}:\mathcal{B}(\mathbb{R})\rightarrow L$ defined as

$M_{A}(E)=f_{A}^{E}\,,\qquad\forall E\in\mathcal{B}(\mathbb{R}).$

The family $\\{M_{A},A\in\mathcal{O}\\}$ is surjective

  3. 3.

Each state $\rho\in\mathcal{S}$ determines a unique probability measure
$m_{\rho}:L\rightarrow[0,1]$ defined as

$m_{\rho}(f)=f(\rho)\qquad\forall f\in L.$

The family $\\{m_{\rho},\rho\in\mathcal{S}\\}$ is ordering.

  4. 4.

For each $A\in\mathcal{O}$, $\rho\in\mathcal{S}$ and
$E\in\mathcal{B}(\mathbb{R})$

$p(A,\rho,E)=m_{\rho}\circ M_{A}(E).$

Conversely, if $L$ is a logic admitting an ordering set of probability
measures $\mathcal{S}$ and $\mathcal{O}$ is a surjective set of $L$-valued
measures, then the function defined by

$p(A,\rho,E):=\rho\circ A(E)$

with $\rho\in\mathcal{S}$ and $A\in\mathcal{O}$ is a probability function that
satisfies the orthogonality postulate and whose logic, $L^{f}$, of $p$ is
isomorphic111Two logics are isomorphic if there is a bijection between them
preserving the order relation and the orthocomplementation. to $L$.

$\square$

##  4 Quantum Mechanics

The VIIth axiom of Mackey says that the quantum logic is isomorphic to the
Birkhoff-von Neumann logic $\mathcal{L}$ of closed subspaces (or the
projectors over such subspaces) of a certain separable Hilbert space
$\mathcal{H}$. We saw in Example 2.12 that the partial ordering is given by
the inclusion, the meet is the intersection and the join is the sum. The
complement is the orthogonal complement inside the Hilbert space with inner
product $\langle\,\cdot,\,\cdot\,\rangle$.

We follow closely the argument of Mackey [1]. Let $\psi$ be a unit vector on
the Hilbert space and let $P$ be a projector. It is clear that

$\begin{CD}P@>{}>{}>m_{\psi}(P)=\langle P\psi,\psi\rangle\end{CD}$

is a probability measure on $\mathcal{L}$ representing the pure state $\psi$.
One can prove that a convex linear combination of these probability measures

$m=\sum_{i=1}^{N}a_{i}m_{\psi_{i}},\qquad
a_{i}\in\mathbb{R}^{+},\qquad\sum_{i=1}^{N}a_{i}=1\,$

defines also a probability measure on $\mathcal{L}$, generically corresponding
to a mixed state. Now, by a theorem of Gleason [11], one can also prove that
in dimension bigger than 2, these are all the probability measures if we
assume the reasonable condition that, for each non zero projector, there
exists at least a vector $\phi\in\mathcal{H}$ such that $m_{\phi}(P)\neq 0$.
Observables are defined as $\mathcal{L}$ valued measures, that is, projection
valued measures. By the spectral theorem, there is a one to one correspondence
between projection valued measures and self-adjoint operators. Hence
observables correspond to self-adjoint operators.

We will focuss on finite dimensional Hilbert spaces. General (mixed or pure)
states are described by density matrices, that is, self-adjoint, non negative
matrices with trace 1. If $\rho$ is a density matrix, $A$ a self-adjoint
operator and $E$ a Borel subset of $\mathbb{R}$, the probability function
$p(A,E,\rho)$ is the trace of the corresponding projector times the density
matrix. In any case, $0\leq p(A,E,\rho)\leq 1$, since in the standard
formulation of Quantum Mechanics this quantity is a probability. We will
extract some consequences from this fact.

##  5 The qubit

We want to describe the qubit system with the BvN (Hilbert space or standard)
logic in terms of Ma̧czyński functional representation of Quantum Mechanics.

We then consider the Hilbert space $\mathcal{H}=\mathbb{C}^{2}$. The set of
observables is the set of self-adjoint operators

$\Theta=\left\\{\,A\in M_{2\times
2}(\mathbb{C})\;|\;A=A^{\dagger}\,\right\\}.$

In the basis of the Pauli matrices

$\sigma_{0}=\mathrm{Id}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\quad\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-\mathrm{i}\\\
\mathrm{i}&0\end{pmatrix},\quad\sigma_{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}$

a self-adjoint operator $A$ can be written as
$A=\sum_{\mu=0}^{3}a_{\mu}\sigma_{\mu}\;$, $a_{\mu}\in\mathbb{R}$

$A=\begin{pmatrix}a_{0}+a_{3}&a_{1}-\mathrm{i}a_{2}\\\
a_{1}+\mathrm{i}a_{2}&a_{0}-a_{3}\end{pmatrix},\qquad\det
A=a_{0}^{2}-a_{1}^{2}-a_{2}^{2}-a_{3}^{2}.$ (5)

The set of states $\mathcal{S}$ consists of the density matrices, that is,
$2\times 2$ self-adjoint matrices that are non negative (the eigenvalues are
non negative) and with trace 1:

$\mathcal{S}=\left\\{\,\rho\in M_{2\times
2}(\mathbb{C})\;|\;\rho=\rho^{\dagger},\,\rho\hbox{ is non negative and
}\mathrm{tr}\rho=1\,\right\\}.$

We will write a density matrix as

$\rho=\begin{pmatrix}\rho_{0}+\rho_{3}&\rho_{1}-\mathrm{i}\rho_{2}\\\
\rho_{1}+\mathrm{i}\rho_{2}&\rho_{0}-\rho_{3}\end{pmatrix}=\rho_{0}\mathrm{Id}+\vec{\rho}\cdot\vec{\sigma}\\\
,$ (6)

with $\rho_{\mu}$ real. The trace condition implies that $\rho_{0}=1/2$. From
the characteristic equation

$\det(\rho-\lambda\mathrm{Id})=0$

we obtain

$\rho_{\pm}=\rho_{0}\pm\sqrt{\rho_{1}^{2}+\rho_{2}^{2}+\rho_{3}^{2}},$

and taking into account that $\rho_{\pm}>0$, the non negativity of $\rho$ is
equivalent to $\det\rho\geq 0$, that is,

$\rho_{1}^{2}+\rho_{2}^{2}+\rho_{3}^{2}\leq\frac{1}{4},$ (7)

so they can be represented as a ball of radius 1/2, the Bloch ball (see Fig.2)

$\hat{a}$$\vec{\rho}$$\frac{1}{2}$$\theta$

Figure 2: Bloch ball of states and vector associated to the observable [20].

The pure states correspond to density matrices satisfying

$\mathrm{tr}\rho^{2}=1.$

They form a sphere of radius 1/2, the Bloch sphere. For these states the
equality in (7) holds. Antipodal points represent mutually orthogonal states.
The points inside the Bloch sphere represent mixed states and its center
represents the maximally mixed state.

The eigenvalues of $A$ in (5) are also

$\lambda_{\pm}=a_{0}\pm|\vec{a}|,\qquad\vec{a}=(a_{1},a_{2},a_{3}),$ (8)

and the eigenvectors are given by the conditions

$\displaystyle\vec{v}=\begin{pmatrix}v_{1}\\\ v_{2}\end{pmatrix}$
$\displaystyle\lambda_{+}:\qquad(a_{3}-|\vec{a}|)v_{1}=-(a_{1}-\mathrm{i}a_{2})v_{2}$
$\displaystyle\lambda_{-}:\qquad(a_{1}+\mathrm{i}a_{2})v_{1}=(a_{3}-|\vec{a}|)v_{2}$
(9)

Let $\mathcal{B}(\mathbb{R})$ denote the set of Borel subsets of the real
line. We then have four types of Borel sets with respect to the eigenvalues
$\lambda_{\pm}$:

  * •

Borel sets $E_{0}$ such that $\lambda_{\pm}\notin E_{0}$,

  * •

Borel sets $E_{+}$ such that $\lambda_{+}\in E_{+}$ and $\lambda_{-}\notin
E_{+}$,

  * •

Borel sets $E_{-}$ such that $\lambda_{-}\in E_{-}$ and $\lambda_{+}\notin
E_{-}$,

  * •

Borel sets $E_{\pm}$ such that $\lambda_{+},\lambda_{-}\in E_{\pm}$.

The projector $P_{A}^{E}$ on the subspace of eigenvectors in $\mathcal{H}$
that correspond to eigenvalues contained in $E$ is the same inside of each of
the above four classes. We have:

$\displaystyle P^{0}_{A}=0,\qquad$ $\displaystyle
P^{+}_{A}=\frac{1}{2|\vec{a}|}\begin{pmatrix}|\vec{a}|+a_{3}&a_{1}-\mathrm{i}a_{2}\\\
a_{1}+\mathrm{i}a_{2}&|\vec{a}|-a_{3}\end{pmatrix},$ $\displaystyle
P^{\pm}_{A}=\mathrm{Id},\qquad$ $\displaystyle
P^{-}_{A}=\frac{1}{2|\vec{a}|}\begin{pmatrix}|\vec{a}|-a_{3}&-a_{1}+\mathrm{i}a_{2}\\\
-a_{1}-\mathrm{i}a_{2}&|\vec{a}|+a_{3}\end{pmatrix}.$ (10)

######  Remark 5.1.

If the two eigenvalues coincide, then $|\vec{a}|=0$ so $a_{1}=a_{2}=a_{3}=0$,
the observable is just $a_{0}\mathrm{Id}$ and the projector is the identity.
In that case only $E_{0}$ and $E_{\pm}$ are possible. $\square$

The experimental functions:

$\begin{CD}\mathcal{S}@>{f_{A}^{E}}>{}>[0,1]\\\ \rho
@>{}>{}>f_{A}^{E}(\rho)=p(A,E,\rho),\end{CD}$

represent the probability that the outcome of the measurement of the
observable $A$ in a state $\rho$ is in the Borel subset $E$. This is computed
as the trace of the projector times the density matrix
$\mathrm{tr}(P_{A}^{E}\rho)$:

$\displaystyle f^{0}_{A}(\rho)=0,\qquad$ $\displaystyle
f^{+}_{A}(\rho)=\frac{1}{2}+\frac{1}{|\vec{a}|}(a_{1}\rho_{1}+a_{2}\rho_{2}+a_{3}\rho_{3}),$
$\displaystyle f^{\pm}_{A}(\rho)=1,\qquad$ $\displaystyle
f^{-}_{A}(\rho)=\frac{1}{2}-\frac{1}{|\vec{a}|}(a_{1}\rho_{1}+a_{2}\rho_{2}+a_{3}\rho_{3})\,$

Notice that changing $\vec{a}$ by $-\vec{a}$ we switch from $f^{+}_{A}$ to
$f^{-}_{A}$, so if $A$ is arbitrary it will be enough to consider the
functions, say, $f^{+}_{A}$.

Indeed, we see that the experimental functions, other than 0 and 1, depend
only on the unit vector $\hat{a}=\vec{a}/|\vec{a}|$, with
$\vec{a}=(a_{1},a_{2},a_{3})$. We will denote those functions as

$f_{\hat{a}}(\vec{\rho})=\frac{1}{2}+\hat{a}\cdot\vec{\rho}=\frac{1}{2}+|\vec{\rho}|\cos\theta,$
(11)

where $\vec{\rho}=(\rho_{1},\rho_{2},\rho_{3})$, $|\vec{\rho}|\leq 1/2$ (see
(7)) and $\theta$ is the angle between $\vec{\rho}$ and $\hat{a}$. There is no
dependence on the azimutal angle.

Let us consider the vector $\hat{a}$ fixed. The function $f_{\hat{a}}$ only
depends on $\cos\theta$ and $|\vec{\rho}|$. When $|\vec{\rho}|=1/2$ we have a
pure state and the set of pure states with $\theta$ constant is represented in
the Bloch sphere of Fig. 2 by the blue circles. The green vector represents an
arbitrary state. One can check directly that $0\leq
f_{\hat{a}}(\vec{\rho})\leq 1$.

If we fix $|\vec{\rho}|$, we obtain the functions $f_{\hat{a}}$ in terms of
$\theta$ only. In Fig. 3 we have represented some of these functions for
different values of $|\vec{\rho}|$. The function for pure states
$|\vec{\rho}|=1/2$ corresponds to the blue curve.

Figure 3: Experimental functions of the qubit in terms of the angle between
the Bloch vector and the unitary vector associated to the observable $A$ [20].

The orthogonality condition is

$\displaystyle
f_{\hat{a}}+f_{\hat{a}^{\prime}}=1+(\hat{a}+\hat{a}^{\prime})\cdot\vec{\rho}\leq
1\quad\Rightarrow\quad(\hat{a}+\hat{a}^{\prime})\cdot\vec{\rho}\leq 0,$ (12)

which is only possible, for all $\vec{\rho}$, if $\hat{a}=-\hat{a}^{\prime}$.

We thus conclude that the only types of pairwise orthogonal sequences are:

  1. 1.

$\\{0,1\\}$

  2. 2.

$\\{0,f_{\hat{a}}\\}$

  3. 3.

$\\{f_{\hat{a}},f_{-\hat{a}}\\}$

  4. 4.

$\\{0,f_{\hat{a}},f_{-\hat{a}}\\}$

We recall here that a one element sequence is, by definition, orthogonal and
that (7) implies that $f_{\hat{a}}(\rho)\leq 1$ (as it should be for a
probability). We observe then, that all pairwise orthogonal sequences are also
orthogonal sequences, so the orthogonality postulate is satisfied.

We can restrict the experimental functions to pure states only. Let

$\left\\{|0\rangle,|1\rangle\right\\}$

be the standard basis in $\mathcal{H}$, and let $\Psi\rangle\in\mathcal{H}$.
We choose

$|\Psi\rangle=\Psi_{0}|0\rangle+\Psi_{1}|1\rangle\in\mathcal{H}\qquad\hbox{with}\qquad\langle\Psi|\Psi\rangle=|\Psi_{0}|^{2}+|\Psi_{1}|^{2}=1\,.$

Then

$f_{\hat{a}}^{+}(\Psi_{0},\Psi_{1})=\frac{1}{2}+\frac{1}{|\vec{a}|}\left(a_{1}\mathbf{Re}(\Psi_{0}\bar{\Psi}_{1})-a_{2}\mathbf{Im}(\Psi_{0}\bar{\Psi}_{1})+a_{3}\frac{|\Psi_{0}|^{2}-|\Psi_{1}|^{2}}{2}\right)\,.$

If $\Psi_{0}$ and $\Psi_{1}$ are real, then we can parametrize the state as

$\Psi_{0}=\cos\alpha,\qquad\Psi_{1}=\sin\alpha,\qquad\alpha\in[0,2\pi[$

and the experimental functions take the simple form

$f_{\hat{a}}(\alpha)=\frac{1}{2}+\frac{1}{|\vec{a}|}\left(a_{1}\cos\alpha\sin\alpha+a_{3}\frac{1}{2}(\cos^{2}\alpha-\sin^{2}\alpha)\right)\,.$

If we choose a reference frame in which only $a_{3}=1$ and the rest 0, we get

$f_{\hat{a}}(\alpha)=\cos^{2}\alpha\,.$

(This equation holds even if there is a phase between $\Psi_{0}$ and
$\Psi_{1}$).

##  6 Two qubits entangled.

We have now a Hilbert space that is a tensor product of the Hilbert spaces of
the two qubits, $\mathcal{H}\simeq\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. We
will work with the basis of the matrices $\mathcal{M}_{2\times 2}(\mathbb{C})$
formed as products of the Pauli matrices (Dirac basis):

$\\{\sigma_{\mu}\otimes\sigma_{\nu}\\}_{\mu,\nu=0}^{3}$ (13)

We will use Einstein’s convention, so sum over repeated indices should be
understood. Greek letters will represent indices that run from 0 to 3, while
Latin letters will represent indices that run from 1 to 3. We will use the
notation for the density matrix in terms of the Bloch matrix $[r_{\mu\nu}]$
(see [12])

$\rho=\frac{1}{4}r_{\mu\nu}\sigma_{\mu}\otimes\sigma_{\nu},\qquad[r_{\mu\nu}]=\begin{pmatrix}1&\vec{r}^{\,t}\\\
\vec{s}&R\end{pmatrix},$ (14)

which is real. One has immediately that $\mathrm{tr}\rho=1$ since
$\mathrm{tr}\sigma_{i}=0$ . The vectors $\vec{s}$ and $\vec{r}$ can be
interpreted as the Bloch vectors of the first and second (respectively)
qubits. In fact, tracing out (see property 3 below) the second (first)
subsystem, one obtains

$\rho_{1}=\mathrm{tr}_{2}\rho=\frac{1}{2}\mathrm{Id}+\vec{s}\cdot\vec{\sigma},\qquad\rho_{2}=\mathrm{tr}_{1}\rho=\frac{1}{2}\mathrm{Id}+\vec{r}\cdot\vec{\sigma},$
(15)

which comparing with (6) gives

$\vec{s}=\vec{\rho_{1}},\qquad\vec{r}=\vec{\rho_{2}}.$ (16)

$R=[r_{ij}]$ is called the correlation matrix.

Using the fact that $\mathrm{tr}(\sigma_{\mu}\sigma_{\nu})=2\delta_{\mu\nu}$
and the property 3 below, one has that

$r_{\mu\nu}=\mathrm{tr}(\rho\cdot\sigma_{\mu}\otimes\sigma_{\nu}).$

The eigenvalues of the sum of two operators have no easy expression in terms
of the eigenvalues of each operator, so we will restrict our analysis to
factorizable operators, that is, operators of the form $C=A\otimes B$. We
recall that:

  1. 1.

$C^{\dagger}=A^{\dagger}\otimes B^{\dagger}$.

  2. 2.

$\det(A\otimes B)=(\det A)^{2}(\det B)^{2}$.

  3. 3.

$\mathrm{tr}(A\otimes B)=\mathrm{tr}A\cdot\mathrm{tr}B$

  4. 4.

If $\lambda$ is an eigenvalue of $A$ with eigenvectors $u^{a}$, where $a$ runs
over the multiplicity of $\lambda$, and $\mu$ is an eigenvalue of $B$ with
eigenvectors $v^{b}$, $b$ running on the multiplicity of $\mu$, then
$\lambda\mu$ is an eigenvalue of $A\otimes B$ and $u^{a}\otimes v^{b}$ are
eigenvectors of $\lambda\mu$.

According to (9), we have eigenvalues $\lambda_{\pm}=a_{0}\pm|\vec{a}|$ for
$A$ and $\mu_{\pm}=b_{0}\pm|\vec{b}|$ for $B$, so we have 4 eigenvalues (some
of them may coincide) for $A\otimes B$,

$\alpha_{++}=\lambda_{+}\mu_{+},\quad\alpha_{+-}=\lambda_{+}\mu_{-},\quad\alpha_{-+}=\lambda_{-}\mu_{+},\quad\alpha_{--}=\lambda_{-}\mu_{-}.$

######  Example 6.1.

Let us consider the simple case

$A=B=\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$

Then

$C=\sigma_{3}\otimes\sigma_{3}=\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\
0&0&-1&0\\\ 0&0&0&1\end{pmatrix},$

which has two eigenvalues, $\pm 1$, each one with degeneracy 2. Given the
operator $C$, the decomposition $C=A\otimes B$ is not unique, since, by the
properties of the tensor product, one can rescale $A^{\prime}=rA$,
$B^{\prime}=B/r$ and $C=A^{\prime}\otimes B^{\prime}$ too. It may happen that
there is a natural choice, as for this example. Otherwise we have to choose
one such decomposition. The operators $A\otimes\mathrm{Id}$ and
$\mathrm{Id}\otimes B$ determine, in this case, uniquely a unitary vector of
the standard basis by giving their simultaneous eigenvalues, that is,
$\lambda=\pm 1$ of $A$, $\mu=\pm 1$ of $B$. Whenever there is no possibility
of confusion, we will denote (as usual)

$A\otimes\mathrm{Id}\rightarrow A,\qquad\mathrm{Id}\otimes B\rightarrow B.$

Operators $A$ and $B$ represent local properties of the system, that is,
properties of each qubit.

$\square$

So we will assume that we have chosen one such decomposition $C=A\otimes B$,
since we are limiting our study to decomposable observables. The following
analysis will have an equivalent one for every such decomposition.

It is useful to classify the Borel sets according to the eigenvalues
$\lambda_{\pm},\mu_{\pm}$ that they enclose. There are the following types of
Borel sets (for each type the projector is similar):

  1. 1.

Sets that do not contain any eigenvalue.

  2. 2.

Sets that contain only one eigenvalue.

  3. 3.

Sets that contain only the two eigenvalues of the same subsystem.

  4. 4.

Sets that contain only two eigenvalues, one of each subsystem.

  5. 5.

Sets that contain only three eigenvalues.

  6. 6.

Sets containing the four eigenvalues.

(See Remark 5.1 for the case in which one of the operators has a degenerate
eigenvalue. This would reduce the number of possibilities.)

We compute the experimental functions for each type of Borel set:

  1. 1.

For type 1 we just have $P_{0}\otimes P_{0}=0$.

  2. 2.

Type 2 corresponds to a projector $P_{0}$ in one subsystem times a projector
over the eigenspace of $\lambda_{\pm}$ ($\mu_{\pm}$) in the other. There are
four posible combinations but the result is always 0.

  3. 3.

Type 3 corresponds to $\mathrm{Id}\otimes P_{0}=0,P_{0}\otimes\mathrm{Id}=0.$

  4. 4.

For type 4, let us consider $P_{A\otimes B}^{+,+}=P^{+}_{A}\otimes P^{+}_{B}$,
the projector onto the subspace of eigenvectors in $\mathcal{H}$ that
correspond to the eigenvalues $\lambda_{+},\mu_{+}$, that is, we are
considering a Borel subset $E$ that contains only $\lambda_{+}$ and $\mu_{+}$
where, according to (10), we have:

$P^{+}_{A}=\frac{1}{2|\vec{a}|}\begin{pmatrix}|\vec{a}|+a_{3}&a_{1}-\mathrm{i}a_{2}\\\
a_{1}+\mathrm{i}a_{2}&|\vec{a}|-a_{3}\end{pmatrix},\quad
P^{+}_{B}=\frac{1}{2|\vec{b}|}\begin{pmatrix}|\vec{b}|+b_{3}&b_{1}-\mathrm{i}b_{2}\\\
b_{1}+\mathrm{i}b_{2}&|\vec{b}|-ba_{3}\end{pmatrix}.$

We denote as $\hat{a}$, $\hat{b}$ the respective unit vectors for $\vec{a}$
and $\vec{b}$. The experimental functions on an arbitrary density matrix (14)
can be explicitly computed as:

$f^{+,+}_{\hat{a},\hat{b}}(\rho)=\mathrm{tr}(\rho P_{A\otimes B}^{+,+}).$

After some algebra we get

$f^{+,+}_{\hat{a},\hat{b}}(\vec{s},\vec{r},R)=\frac{1}{4}\left(1+\vec{s}\cdot\hat{a}+\vec{r}\cdot\hat{b}+\hat{a}^{\,t}R\hat{b}\right).$
(17)

This can be read as the probability for the observables $A$ and $B$ getting
the values $\lambda_{+}$ and $\mu_{+}$, respectively. The observable $A\otimes
B$ gets the value $\alpha_{++}=\lambda_{+}\mu_{+}$.

Unlike for the qubit, it is not straightforward to check here that $0\leq
f^{+,+}_{\hat{a},\hat{b}}(\vec{s},\vec{r},R)\leq 1$, but since this is a
probability, according to the Hilbert space machinery, the inequalities must
be satisfied. At the end of the section we will check it for pure states.
These inequalities should be related to the positivity conditions for the
matrix $\rho$, obtained, in the formalism of the Bloch matrix, in [13].

To get the rest of the experimental functions for type 3, we only have to
change in the projectors $\hat{a}$ for $-\hat{a}$ to obtain $\lambda_{-}$ and
$\hat{b}$ for $-\hat{b}$ to obtain $\mu_{-}$.

  5. 5.

Type 5 are projectors of the form $P^{+-,\pm}=\mathrm{Id}\otimes P^{\pm}_{B}$
or $P^{\pm,+-}=P^{\pm}_{A}\otimes\mathrm{Id}$. The experimental function of,
say, $P^{+,+-}=P^{+}_{A}\otimes\mathrm{Id}$ reproduces the result of (11)

$f^{+,+-}_{\hat{a}}(\vec{s})=\frac{1}{2}+\vec{s}\cdot\hat{a}.$ (18)

  6. 6.

Type 6 is just the identity.

A series of experimental functions is orthogonal if its sum is less or equal
than one. This happens, for all density matrices, when we sum probabilities of
events without intersection. For example,

$f^{+,+}_{\hat{a},\hat{b}}(\vec{s},\vec{r},R)+f^{+,-}_{\hat{a},\hat{b}}(\vec{s},\vec{r},R)=\frac{1}{2}+\vec{s}\cdot\hat{a}\leq
1,$ (19)

according to the Hilbert space formalism. We can extract, as a consequence,
the inequalities

$-\frac{1}{2}\leq\vec{s}\cdot\hat{a}\leq\frac{1}{2}.$

In particular, choosing appropriately $\hat{a}$, we get

$-\frac{1}{2}\leq s_{i}\leq\frac{1}{2}.$

The same can be said about the components of $\vec{r}$.

Another case is the following sum

$f^{+,+}_{\hat{a},\hat{b}}(\vec{s},\vec{r},R)+f^{-,-}_{\hat{a},\hat{b}}(\vec{s},\vec{r},R)=\frac{1}{2}+\frac{1}{2}\left({\hat{a}}^{t}R\hat{b}\right)\leq
1,$ (20)

which implies

$-1\leq{\hat{a}}^{t}R\hat{b}\leq 1.$

In particular, choosing the axes $\hat{a}$ and $\hat{b}$ appropriately, we
have that all the entries of the correlation matrix satisfy

$-1\leq R_{ij}\leq 1.$

We can also choose $\hat{a}=(1,0,0)$ and leave $\hat{b}$ arbitrary; then:

${\hat{a}}^{t}R\hat{b}=(R_{11},R_{12},R_{13})\cdot\begin{pmatrix}b_{1}\\\
b_{2}\\\
b_{3}\end{pmatrix}=\left(R_{11}^{2}+R_{12}^{2}+R_{13}^{2}\right)^{1/2}\cos\beta\leq
1,$

where $\beta$ is the angle between the vectors $(R_{11},R_{12},R_{13})$ and
$\hat{b}$. This implies

$R_{11}^{2}+R_{12}^{2}+R_{13}^{2}\leq 1.$

The same reasoning can be done with $\hat{a}=(0,1,0)$ or $\hat{a}=(0,0,1)$. We
then obtain that the vectors corresponding to the rows of $R$,
$(R_{i1},R_{i2},R_{i3})$, satisfy inequalities

$R_{i1}^{2}+R_{i2}^{2}+R_{i3}^{2}\leq 1,\qquad i=1,2,3.$ (21)

If instead we leave $\hat{a}$ arbitrary and choose for $\hat{b}$ the unit
vectors of the three perpendicular axes, we will obtain that the columns of
$R$ satisfy similar inequalities,

$R_{1i}^{2}+R_{2i}^{2}+R_{3i}^{2}\leq 1\qquad i=1,2,3.$ (22)

Again, (21) and (22) are inequalities constraining the space of the density
matrices, independently of any observable.

In [13] the conditions of positivity for the matrix $\rho$ are investigated in
terms of the decomposition that we are using here. One such condition is

$\mathrm{tr}(R^{t}R)+|\vec{s}|^{2}+|\vec{r}|^{2}\leq 3.$

But

$\displaystyle\mathrm{tr}(R^{t}R)=$
$\displaystyle\left(R_{11}^{2}+R_{21}^{2}+R_{31}^{2}\right)+\left(R_{12}^{2}+R_{22}^{2}+R_{32}^{2}\right)+\left(R_{13}^{2}+R_{23}^{2}+R_{33}^{2}\right)=$
$\displaystyle\left(R_{11}^{2}+R_{12}^{2}+R_{13}^{2}\right)+\left(R_{21}^{2}+R_{22}^{2}+R_{23}^{2}\right)+\left(R_{31}^{2}+R_{32}^{2}+R_{33}^{2}\right),$

hence, conditions (21) and (22) imply, in particular, that
$\mathrm{tr}(R^{t}R)\leq 3$, which is consistent with the fact that
$|\vec{s}|^{2}+|\vec{r}|^{2}\geq 0$. For the case where the equalities in (21)
and (22) are fulfilled, one has that $\mathrm{tr}(R^{t}R)=3$, which implies
that $|\vec{s}|=|\vec{r}|=0$ so $\vec{s}=\vec{r}=0$. This condition is known
as the two qubits being locally maximally mixed, i.e., when tracing out one of
the subsystems, the other is left in a maximally mixed state, so its density
matrix is proportional to the identity (see Equation (15)).

#### Pure states.

Let $\\{|u_{1}\rangle,|u_{2}\rangle\\}$ and
$\\{|v_{1}\rangle,|v_{2}\rangle\\}$ be standard basis in $\mathbb{C}^{2}$ and
let $|\psi\rangle\in\mathcal{H}=\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ be a unit
vector representing a pure state:

$|\psi\rangle=\sum_{i,j=1,2}\lambda_{ij}|u_{i}\rangle\otimes|v_{j}\rangle,\qquad\langle\psi|\psi\rangle=\sum_{i,j=1,2}|\lambda_{ij}|^{2}=1,$
(23)

for $\lambda_{ij}\in\mathbb{C}$. As before, we consider the observable
$A\otimes B$ with

$A=\begin{pmatrix}a_{0}+a_{3}&a_{1}-\mathrm{i}a_{2}\\\
a_{1}+\mathrm{i}a_{2}&a_{0}-a_{3}\end{pmatrix},\qquad
B=\begin{pmatrix}b_{0}+b_{3}&b_{1}-\mathrm{i}b_{2}\\\
b_{1}+\mathrm{i}b_{2}&b_{0}-b_{3}\end{pmatrix}.$

We also have the projectors onto the eigenspaces of $\lambda_{+}$,$\,\mu_{+}$
(see (8)), given in terms of the unitary vectors

$\displaystyle\hat{a}=\frac{1}{a}(a_{1},a_{2},a_{3}),\qquad$ $\displaystyle
a^{2}=a_{1}^{2}+a_{2}^{2}+a_{3}^{2},$
$\displaystyle\hat{b}=\frac{1}{b}(b_{1},b_{2},b_{3}),\qquad$ $\displaystyle
b^{2}=b_{1}^{2}+b_{2}^{2}+b_{3}^{2},$

$P_{A}^{+}=\frac{1}{2}\begin{pmatrix}1+\hat{a}_{3}&\hat{a}_{1}-\mathrm{i}\hat{a}_{2}\\\
\hat{a}_{1}+\mathrm{i}\hat{a}_{2}&1-\hat{a}_{3}\end{pmatrix},\qquad
P_{B}^{+}=\frac{1}{2}\begin{pmatrix}1+\hat{b}_{3}&\hat{b}_{1}-\mathrm{i}\hat{b}_{2}\\\
\hat{b}_{1}+\mathrm{i}\hat{b}_{2}&1-\hat{b}_{3}\end{pmatrix}.$

We consider now the case $\hat{a}=(0,0,1)$. Notice that, by a change of the
reference frame, we can always choose our basis in the three dimensional
space, such that this condition is satisfied. Therefore we are not loosing
generality. $B$ is still general. Then we have

$A=\begin{pmatrix}a_{0}+1&0\\\ 0&a_{0}-1\end{pmatrix},\qquad
P_{A}^{+}=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}.$

We compute the probability that a measurement of an observable $A\otimes B$
gives the eigenvalue $\lambda_{+}\mu_{+}$. To facilitate the writing, we make
some definitions:

$\displaystyle\lambda_{1}:=\lambda_{11},\qquad\lambda_{2}:=\lambda_{12}$
$\displaystyle\vec{\lambda}:=\left(\lambda_{1}\lambda^{*}_{2}+\lambda_{2}\lambda^{*}_{1},-\mathrm{i}(\lambda_{1}\lambda^{*}_{2}-\lambda_{2}\lambda^{*}_{1}),|\lambda_{1}|^{2}-|\lambda_{2}|^{2}\right).$

One can prove that
$|\vec{\lambda}|=\sqrt{\lambda_{1}|^{2}+|\lambda_{2}|^{2}}$. Then it results
that

$\displaystyle\tilde{f}_{\hat{a},\hat{b}}^{+,+}=\langle\psi|P_{A}^{+}\otimes
P_{B}^{+}|\psi\rangle=\frac{1}{2}\left(|\lambda_{1}|^{2}+|\lambda_{2}|^{2}+\hat{b}\cdot\vec{\lambda}\right)$

Let $\theta$ be the angle between $\vec{\lambda}$ and $\hat{b}$. Since,
according to (23), $|\lambda_{1}|^{2}+|\lambda_{2}|^{2}\leq 1$,

$\tilde{f}_{\hat{a},\hat{b}}^{+,+}\leq\frac{1}{2}(|\lambda_{1}|^{2}+|\lambda_{2}|^{2})^{1/2}(1+\cos\theta)\leq
1\,.$

The function $\tilde{f}_{\hat{a},\hat{b}}^{+,+}$ is the restriction to pure
states of the function ${f}_{\hat{a},\hat{b}}^{+,+}$ in (17) and we are able
to check directly that it satisfies the inequalities

$0\leq\tilde{f}_{\hat{a},\hat{b}}^{+,+}\leq 1.$

$\square$

##  7 The ‘nested’ qutrit.

In $\mathbb{C}^{2}$ we consider the spin operators

$S_{i}=\frac{1}{2}\sigma_{i},\qquad\qquad(\hbar=1).$

In $\mathcal{H}=\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ we consider the spin
operators $S_{1z}=S_{z}\otimes\mathrm{Id}$ and $S_{2z}=\mathrm{Id}\otimes
S_{z}$. The standard basis is the ordered basis of eigenvectors of these
operators and it is denoted as

$\displaystyle|00\rangle=|0\rangle\otimes|0\rangle,\quad|01\rangle=|0\rangle\otimes|1\rangle,$
$\displaystyle|10\rangle=|1\rangle\otimes|0\rangle,\quad|11\rangle=|1\rangle\otimes|1\rangle,$
(24)

where the indices $0,1$ denote the eigenvalues $+1/2$ and $-1/2$,
respectively.

The total angular momentum $\vec{S}=\vec{S}_{1}+\vec{S}_{2}$, modulus squared,
is given in terms of this basis as

$\vec{S}\cdot\vec{S}=S_{x}^{2}+S_{y}^{2}+S_{z}^{2}=\begin{pmatrix}1&0&0&0\\\
0&1/2&1/2&0\\\ 0&1/2&1/2&0\\\ 0&0&0&1\end{pmatrix},$

with eigenvalues $\lambda=0$ (multiplicity 1) and $\lambda=1$ (multiplicity
3). A basis of eigenvectors is

$\displaystyle|00\rangle,\quad|q_{s}\rangle=\frac{1}{\sqrt{2}}\big{(}|01\rangle+|10\rangle\big{)},\quad|11\rangle,\qquad(\mathrm{eigenvalue}\,1),$
$\displaystyle\mathrm{and}$
$\displaystyle|q_{a}\rangle=\frac{1}{\sqrt{2}}\big{(}|01\rangle-|10\rangle\big{)}\qquad(\mathrm{eigenvalue}\,0).$
(25)

Following [14], we denote this basis as the entangled basis.

As in (14), we express the density matrix of the entangled qubits in the
standard basis as

$\rho^{\mathrm{stn}}=\begin{pmatrix}\rho_{11}&\rho_{12}&\rho_{13}&\rho_{44}\\\
\rho_{12}^{*}&\rho_{22}&\rho_{23}&\rho_{24}\\\
\rho_{13}^{*}&\rho_{23}^{*}&\rho_{33}&\rho_{34}\\\
\rho_{14}^{*}&\rho_{24}^{*}&\rho_{34}^{*}&\rho_{44}\\\
\end{pmatrix}=\frac{1}{4}r_{\mu\nu}\sigma_{\mu}\otimes\sigma_{\nu},\qquad[r_{\mu\nu}]=\begin{pmatrix}1&\vec{r}^{\,t}\\\
\vec{s}&R\end{pmatrix}$

Developing the last term, containing the Bloch matrix, one finds (see [13])

${\rho^{\mathrm{stn}}=\frac{1}{4}}\begin{pmatrix}{\scriptstyle
1+R_{33}+s_{3}+r_{3}}&{\scriptstyle
R_{31}-\mathrm{i}R_{32}+r_{1}-\mathrm{i}r_{2}}&{\scriptstyle
R_{13}-\mathrm{i}R_{23}+s_{1}-\mathrm{i}s_{2}}&{\scriptstyle
R_{11}-\mathrm{i}R_{12}-\mathrm{i}R_{21}-R_{22}}\\\ {\scriptstyle
R_{31}+\mathrm{i}R_{32}+r_{1}+\mathrm{i}r_{2}}&{\scriptstyle
1-R_{33}+s_{3}-r_{3}}&{\scriptstyle
R_{11}+\mathrm{i}R_{12}-\mathrm{i}R_{21}+R_{22}}&{\scriptstyle-
R_{13}+\mathrm{i}R_{23}+s_{1}-\mathrm{i}s_{2}}\\\ {\scriptstyle
R_{13}+\mathrm{i}R_{23}+s_{1}+\mathrm{i}s_{2}}&{\scriptstyle
R_{11}-\mathrm{i}R_{12}+\mathrm{i}R_{21}+R_{22}}&{\scriptstyle
1-R_{33}-s_{3}+r_{3}}&{\scriptstyle-
R_{31}+\mathrm{i}R_{32}+r_{1}-\mathrm{i}r_{2}}\\\ {\scriptstyle
R_{11}+\mathrm{i}R_{12}+\mathrm{i}R_{21}-R_{22}}&{\scriptstyle-
R_{13}-\mathrm{i}R_{23}+s_{1}+\mathrm{i}s_{2}}&{\scriptstyle-
R_{31}-\mathrm{i}R_{32}+r_{1}+\mathrm{i}r_{2}}&{\scriptstyle
1+R_{33}-s_{3}-r_{3}}\end{pmatrix}.$

To change the basis from the standard to the entangled one, we compose the
vertical array of unit vectors with the unitary matrix $A$:

$\begin{pmatrix}|00\rangle\\\ |q_{s}\rangle\\\ |11\rangle\\\
|q_{a}\rangle\end{pmatrix}=A\begin{pmatrix}|00\rangle\\\ |01\rangle\\\
|10\rangle\\\ |11\rangle\end{pmatrix},\quad A=\begin{pmatrix}1&0&0&0\\\
0&1/\sqrt{2}&1/\sqrt{2}&0\\\ 0&0&0&1\\\
0&1/\sqrt{2}&-1/\sqrt{2}&0\end{pmatrix}.$

Then, the density matrix in the entangled basis is
$\rho^{\mathrm{ntgl}}=A^{\dagger}\rho^{\mathrm{stn}}A$. We will denote it as

$\rho^{\mathrm{ntgl}}=\begin{pmatrix}\tau_{11}&\tau_{12}&\tau_{13}&\beta\\\
\tau_{12}^{*}&\tau_{22}&\tau_{23}&\gamma\\\
\tau_{13}^{*}&\tau_{23}^{*}&\tau_{33}&\delta\\\
\beta^{*}&\gamma^{*}&\delta^{*}&\alpha\end{pmatrix},\qquad\tau_{11}+\tau_{22}+\tau_{33}+\alpha=1.$

One could prepare a state with $\alpha=\delta=\gamma=\beta=0$. Then, it
represents a qutrit inside the two entangled qubits. In [14], this ‘nested’
qutrit is studied in the context of the dipolar coupling of the two qubits.
There, a family of Hamiltonians are given, for which time evolution of the
system preserves the constraints defining the qutrit.

Writing the constraints in terms of the Bloch matrix, one can see that the
conditions for the qutrit become simply

$\vec{r}=\vec{s},\qquad R=R^{t}.$ (26)

Hence, the experimental functions (17) are simplified. Notice that we are
giving the experimental functions of a qutrit for observables that are
factorized in terms of two qubits. In [14], only factorizable density matrices
are studied, but with the Bloch matrix method, an arbitrary density matrix is
allowed.

#### Unitary transformations.

It is worthy to stop here to see the form of (special) unitary transformations
for the ‘nested’ qutrit.

In [15], a decomposition of the group $\mathrm{SU}(4)$ (special unitary
transformations of the two qubit system) is given in terms of a structure
oforthogonal symmetric Lie algebra of the Lie algebra of $\mathrm{SU}(4)$,
$\mathfrak{su}(4)$. We give here the definition [16]:

######  Definition 7.1.

Orthogonal symmetric Lie algebra. Let $\mathfrak{l}$ be a Lie algebra over
$\mathbb{R}$ and $s$ an involutive automophism of $\mathfrak{l}$ (that is,
$s^{2}=\mathrm{Id}$ and $s\neq\mathrm{Id}$). Let $\mathfrak{u}$ be the set of
fixed points of $s$ and suppose that $\mathfrak{u}$ is a compactly embedded
subalgebra of $\mathfrak{l}$. Then, the pair $(\mathfrak{l},s)$ is an
orthogonal symmetric Lie algebra.

$\square$

The Lie subalgebra $\mathfrak{u}$ is the eigenvector set of eigenvalue 1 of
$s$. The eigenvector set of eigenvalue -1 is denoted by $\mathfrak{p}$ and it
is, in general, just a vector space. One has

$[\mathfrak{u},\mathfrak{u}]\subset\mathfrak{u},\quad[\mathfrak{u},\mathfrak{p}]\subset\mathfrak{p},\quad[\mathfrak{p},\mathfrak{p}]\subset\mathfrak{u}.$

The Lie algebra $\mathfrak{su}(4)=\\{\hbox{ antihermitian, traceless, $4\times
4$ matrices }\\}$ has different structures of orthogonal symmetric Lie
algebra. The structure that is of interest here is the one whose involutive
automorphism is $s(X)=\bar{X}$. Then

$\displaystyle\mathfrak{u}=\mathfrak{so}(4)=\\{\hbox{ real, antisymmetric
$4\times 4$ matrices }\\},$ $\displaystyle\mathfrak{p}=\\{\hbox{ symmetric,
purely imaginary $4\times 4$ matrices }\\}.$ (27)

A maximal abelian subspace of $\mathfrak{p}$ is the set of $4\times 4$
diagonal matrices. A basis of $\mathfrak{su}(4)$ is given in terms of the
Dirac matrices (13)

$\left\\{\,\frac{\mathrm{i}}{2}\sigma_{i}\otimes\sigma_{0},\quad\frac{\mathrm{i}}{2}\sigma_{0}\otimes\sigma_{i},\quad\frac{\mathrm{i}}{2}\sigma_{i}\otimes\sigma_{j},\quad
i,j=1,2,3\,\right\\}.$ (28)

These correspond to the standard basis of
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ (24), which is a basis of eigenvectors
of

$\sigma_{3}\otimes\sigma_{3}=\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&1\end{pmatrix}.$

Performing a change of basis from the standard one to the Bell basis (see for
example [15])

$\displaystyle|b_{1}\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle+|11\rangle\right),$
$\displaystyle|b_{2}\rangle=\frac{\mathrm{i}}{\sqrt{2}}\left(|10\rangle+|01\rangle\right),$
$\displaystyle|b_{3}\rangle=\frac{1}{\sqrt{2}}\left(|10\rangle-|01\rangle\right),$
$\displaystyle|b_{4}\rangle=\frac{\mathrm{i}}{\sqrt{2}}\left(|00\rangle-|11\rangle\right),$

one can deduce directly the above decomposition. Let

$B=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&0&1\\\ 0&\mathrm{i}&\mathrm{i}&0\\\
0&1&-1&0\\\ \mathrm{i}&0&0&-\mathrm{i}\end{pmatrix}$

and let

$\tau_{\mu\nu}=B\sigma_{\mu\nu}B^{+},$

Then, by an elementary calculation, one obtains

$\mathfrak{u}=\mathrm{span}\left\\{\tau_{01},\tau_{02},\tau_{03},\tau_{10},\tau_{20},\tau_{30}\right\\},$

since they are all real antisymmetric matrices. The rest

$\mathfrak{p}=\mathrm{span}\left\\{\tau_{11},\tau_{12},\tau_{13},\tau_{21},\tau_{22},\tau_{23},\tau_{31},\tau_{32},\tau_{33}\right\\},$

are symmetric, purely imaginary matrices. An abelian subset of $\mathfrak{p}$
is the subset of diagonal matrices

$\mathfrak{a}=\mathrm{span}\\{\tau_{11},\tau_{22},\tau_{33}\\}.$

This coincides with the decomposition given in [15] in the standard basis,
where the commutation relations among the generators are explicitly exhibited.

On the other hand, it is well known the isomorphism

$\mathfrak{so}(4)\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)\cong\mathfrak{su}(2)\oplus\mathfrak{su}(2).$

There is a theorem (see [16], Chapter V, Theorem 6.7) that gives a
decomposition of a group whose Lie algebra has the structure of an orthogonal
symmetric Lie algebra. This theorem, applied to our case, asserts that
$\mathrm{SU}(4)$ admits a decomposition

$\mathrm{SU}(4)\simeq\left(\mathrm{SU}(2)\times\mathrm{SU}(2)\right)\exp{\mathfrak{a}}\left(\mathrm{SU}(2)\times\mathrm{SU}(2)\right).$
(29)

The factor $\exp{\mathfrak{a}}$ is a 3-torus. A unitary transformation can
then be cast in the form

$U=k_{1}U(\alpha,\beta,\gamma)k_{2},\qquad
k_{1},k_{2}\in\mathrm{SU}(2)\times\mathrm{SU}(2)\qquad\hbox{ with }$
$U(\alpha,\beta,\gamma)=\mathrm{e}^{(\alpha\tau_{11}+\beta\tau_{22}+\gamma\tau_{33})}=\mathrm{e}^{\alpha\tau_{11}}e^{\beta\tau_{22}}e^{\gamma\tau_{33}}=U_{1}(\alpha)U_{2}(\beta)U_{3}(\gamma).$

The transformations $k_{1},k_{2}$ are interpreted as local transformations,
that is, transformations involving, separately, the two subsystems of one
qubit. The transformations $U(\alpha,\beta,\gamma)$ are instead genuine non
local transformations. In [13], Appendix C, the general form of such
$U(\alpha,\beta,\gamma)$ is given in terms of the Bloch matrix. For example,
$U(\alpha)$ will transform

$[r_{\mu\nu}]=\begin{pmatrix}1&\vec{r}^{t}\\\ \vec{s}&R\end{pmatrix}$

into

$\vec{r^{\prime}}=\begin{pmatrix}r_{1}\\\ r_{2}\cos\alpha+R_{13}\sin\alpha\\\
r_{3}\cos\alpha-
R_{12}\sin\alpha\end{pmatrix},\qquad\vec{s^{\prime}}=\begin{pmatrix}s_{1}\\\
s_{2}\cos\alpha+R_{31}\sin\alpha\\\ s_{3}\cos\alpha-
R_{21}\sin\alpha\end{pmatrix},$

and

$R^{\prime}=\begin{pmatrix}R_{11}&r_{3}\sin\alpha+R_{12}\cos\alpha&-r_{2}\sin\alpha+R_{13}\cos\alpha\\\
s_{3}\sin\alpha+R_{21}\cos\alpha&R_{22}&R_{23}\\\
-s_{2}\sin\alpha+R_{31}\cos\alpha&R_{32}&R_{33}\end{pmatrix}.$

Such transformations preserve the qutrit condition (26), and so does a general
transformation $U(\alpha,\beta,\gamma)$. Apparently, these transformations
have three parameters, $\alpha,\beta$ and $\gamma$. Nevertheless, one can
check with [13] that, if one assumes the qutrit condition, only differences
among these parameters occur. So we have a two dimensional abelian group $A$
of non local transformations. For completeness, we write here the
transformations. We denote $\theta_{1}=\beta-\alpha$ and
$\theta_{2}=\gamma-\alpha$. Then:

$\displaystyle
r^{\prime}_{1}=r_{1}\cos(\theta_{1}-\theta_{2})+R_{23}\sin(\theta_{1}-\theta_{2}),\qquad$
$\displaystyle r^{\prime}_{2}=r_{2}\cos\theta_{1}-R_{13}\sin\theta_{2},$
$\displaystyle r^{\prime}_{3}=r_{3}\cos\theta_{1}+R_{12}\sin\theta_{1},\qquad$
$\displaystyle R^{\prime}_{12}=R_{12}\cos\theta_{1}+r_{3}\sin\theta_{1},$
$\displaystyle
R^{\prime}_{23}=R_{23}\cos(\theta_{1}-\theta_{2})-r_{1}\sin(\theta_{1}-\theta_{2}),\qquad$
$\displaystyle R^{\prime}_{13}=R_{13}\cos\theta_{2}+r_{2}\sin\theta_{2},$
$\displaystyle R^{\prime}_{11}=R_{11},\qquad\qquad
R^{\prime}_{22}=R_{22},\qquad$ $\displaystyle R^{\prime}_{33}=R_{33}.$

The local transformations are reduced to
$k_{1},k_{2}\in\mathrm{SU}(2)=\mathrm{diag}\left(\mathrm{SU}(2)\times\mathrm{SU}(2)\right)$,
so from (29) we are left with a decomposition of unitary transformations of
the qutrit as

$\mathrm{SU}(2)\times A\times\mathrm{SU}(2).$

This is a subgroup of $\mathrm{SU}(4)$ with 8 parameters. Since the unitary
transformations for the qutrit are $\mathrm{SU}(3)$, the formula above is a
decomposition of $\mathrm{SU}(3)$. The group $A$ is a 2-torus, isomorphic to
the subgroup of $\exp(\mathfrak{a})$ with $\alpha=0$.

At the infinitesimal level, the two (commuting) vectors fields, generators of
the action of $A$ on the set of Bloch matrices, satisfying the qutrit
condition, are

$\displaystyle\frac{\partial}{\partial\theta_{1}}=R_{23}\frac{\partial}{\partial
r_{1}}+R_{12}\frac{\partial}{\partial r_{3}}+r_{3}\frac{\partial}{\partial
R_{12}}-r_{1}\frac{\partial}{\partial R_{23}},$
$\displaystyle\frac{\partial}{\partial\theta_{2}}=-R_{23}\frac{\partial}{\partial
r_{1}}-R_{13}\frac{\partial}{\partial r_{2}}+r_{1}\frac{\partial}{\partial
R_{23}}+r_{2}\frac{\partial}{\partial R_{13}},$

$\blacksquare$

##  8  The fuzzy set picture

Fuzzy sets were introduced by Zadeh [6] in a different context and soon they
were used in many practical applications.

######  Definition 8.1.

A fuzzy set is a pair $\mathcal{A}=(\mathcal{U},\mu_{\mathcal{A}})$ where
$\mathcal{U}$ is a set called the universe of discourse and
$\mu_{\mathcal{A}}$ is a function

$\mu_{\mathcal{A}}:\mathcal{U}\rightarrow[0,1],$ (30)

called the membership function. The number $\mu_{\mathcal{A}}(x)$ indicates
the degree of membership with which the element $x$ belongs to $\mathcal{A}$,
0 meaning no membership and 1 full membership. We will use in indistinctly the
notation $\mathcal{A}$ or $\mu_{\mathcal{A}}$ to refer to a fuzzy set.

$\square$

######  Definition 8.2.

Let $\mathcal{A}$ and $\mathcal{B}$ be two fuzzy sets in the same universe of
discourse $\mathcal{U}$.

  1. 1.

$\mathcal{A}=\mathcal{B}$ if an only if
$\mu_{\mathcal{A}}(x)=\mu_{\mathcal{B}}(x)$ $\;\forall x\in\mathcal{U}$.

  2. 2.

$\mathcal{A}\subseteq\mathcal{B}$ if an only if
$\mu_{\mathcal{A}}(x)\leq\mu_{\mathcal{B}}(x)$ $\;\forall x\in\mathcal{U}$.

  3. 3.

The bold union $\mathcal{A}\sqcup\mathcal{B}$ is defined as

$\mu_{\mathcal{A}\sqcup\mathcal{B}}(x)=\mathrm{min}\\{\mu_{\mathcal{A}}(x)+\mu_{\mathcal{B}}(x),1\\}.$

  4. 4.

The bold intersection $\mathcal{A}\sqcap\mathcal{B}$ is defined as

$\mu_{\mathcal{A}\sqcap\mathcal{B}}(x)=\mathrm{max}\\{\mu_{\mathcal{A}}(x)+\mu_{\mathcal{B}}(x)-1,0\\}.$

  5. 5.

The complementation of $\mathcal{A}$ is

$\mu_{\mathcal{A}^{\prime}}(x)=1-\mu_{\mathcal{A}}(x)\qquad\forall
x\in\mathcal{U}.$

  6. 6.

One defines the empty set $\emptyset$ by $\mu_{\emptyset}(x)=0$, and its
complement is $\mathcal{U}$ itself with $\mu_{\mathcal{U}}(x)=1$

  7. 7.

$\mathcal{A}$ and $\mathcal{B}$ are weakly disjoint if
${\mathcal{A}\sqcap\mathcal{B}}=\emptyset$.

$\square$

The bold union and intersection [7, 17] are not the union and intersection
originally introduced by Zadeh, that is,

$\displaystyle\mu_{\mathcal{A}\cup\mathcal{B}}(x)=\mathrm{max}\\{\mu_{\mathcal{A}}(x),\mu_{\mathcal{B}}(x)\\},$
$\displaystyle\mu_{\mathcal{A}\cap\mathcal{B}}(x)=\mathrm{min}\\{\mu_{\mathcal{A}}(x),\mu_{\mathcal{B}}(x)\\}.$
(31)

The union and intersection (31) do not satisfy the law of excluded middle and
the law of contradiction,

$\mathcal{A}\sqcap{\mathcal{A}^{\prime}}=\emptyset,\qquad\mathcal{A}\sqcup{\mathcal{A}^{\prime}}=\mathcal{U},$

but they are distributive (see Equation (1)); then they are not well suited to
describe a quantum system.

Instead, the bold union and intersection satisfy the law of excluded middle,
the law of contradiction and are not distributive.

On orthogonal sets, $\mathcal{A}\sqcap\mathcal{B}=\emptyset$, one can see that
the bold union is just

$\mu_{\mathcal{A}\sqcup\mathcal{B}}(x)=\mu_{\mathcal{A}}(x)+\mu_{\mathcal{B}}(x).$

Our aim is to interpret them in the context of the Birkhoff-von Neumann
(standard) logic.

In the fuzzy approach to Quantum Mechanics by Pykacz [8, 9], the experimental
functions of Theorem 3.3 are interpreted as membership functions of fuzzy sets
in the universe of discourse formed by all the possible states of the quantum
system. We quote here the theorem of [8, 9] that allows such interpretation.

######  Theorem 8.3.

Any logic $L$ with an ordering set of probability measures $\mathcal{S}$ can
be mapped isomorphically to a logic formed by a family of fuzzy subsets of
$\mathcal{S}$, $\mathcal{L}^{f}(\mathcal{S})$, that satisfies the following
properties:

  1. 1.

$\emptyset$ belongs to the family $\mathcal{L}^{f}(\mathcal{S})$.

  2. 2.

If $\mathcal{A}$ belongs to $\mathcal{L}^{f}(\mathcal{S})$, so does its
complement $\mathcal{A}^{\prime}$.

  3. 3.

Given a countable family $\\{\mathcal{A}_{i}\\}_{i\in I}$ of fuzzy subsets
inside $\mathcal{L}^{f}(\mathcal{S})$ which is pairwise weakly disjoint,
$\mathcal{A}_{i}\sqcap\mathcal{A}_{j}=\emptyset$ for $i\neq j$, then
$\sqcup_{i\in I}\mathcal{A}_{i}$ belongs to $\mathcal{L}^{f}(\mathcal{S})$.

  4. 4.

If $\mathcal{A}\sqcap\mathcal{A}=\emptyset$ then $\mathcal{A}=\emptyset$.

Conversely, any family of fuzzy subsets of a universe $\mathcal{U}$,
satisfying properties 1 - 4 above, is a logic where the partial order is the
inclusion of fuzzy sets and the orthocomplementation is the complementation of
fuzzy sets. The notion of orthogonality coincides with that of weak
disjointness.

The set of ordering measures is given by elements of the universe
$\mathcal{U}$ as

$m_{\rho}(\mathcal{A})=\mu_{\mathcal{A}}(\rho)\qquad\forall\rho\in\mathcal{U}\,.$
(32)

$\square$

The universe of discourse $\mathcal{U}$ is the set of states $\mathcal{S}$. As
stated before, given two membership functions $\mu_{\mathcal{A}}(\rho)$ and
$\mu_{\mathcal{B}}(\rho)$, the bold intersection being the empty set, implies
that the bold union is

$\mu_{\mathcal{A}}\sqcup\mu_{\mathcal{B}}=\mu_{\mathcal{A}}+\mu_{\mathcal{B}}\leq
1,$

and then, the function $m_{\rho}:\mathcal{L}^{f}\rightarrow[0,1]$ above
satisfies the conditions to be a probability measure. The same is true for an
infinite countable succession of $\mathcal{A}_{i}$; this is guaranteed by
property 3 in Theorem 8.3.

In Quantum Mechanics, each state is represented by a density matrix, therefore
the universe of discourse is

$\mathcal{S}=\left\\{\,\rho\in M_{n\times
n}(\mathbb{C})\;|\;\rho=\rho^{\dagger},\,\rho\hbox{ is non negative and
}\mathrm{tr}\rho=1\,\right\\}.$

By stating this we have already introduced the Hilbert space. The membership
functions (4) are determined by an observable $A$ and a Borel set
$E\subset\mathbb{R}$

$f^{A}_{E}(\rho)=p(A,\rho,E)=\mathrm{tr}(\rho P^{A}_{E}),$

where $p(A,\rho,E)$ is the probability map of Mackey (2) and $P_{E}^{A}$ is
the projector on the subspace of $\mathcal{H}$ that is the sum of the
eigenspaces of $A$ in $\mathcal{H}$ with eigenvalue in the Borel set $E$. So
the membership function $\mu_{\mathcal{A}}(\rho)=f_{A}^{E}(\rho)$ contains
information about the observable $A$ and the Borel subset $E$.

###  8.1 The fuzzy bit

For the qubit, the membership (experimental) functions (11) are labelled by a
unit vector $\hat{a}$

$\mu_{\mathcal{A}}=f_{\hat{a}}(\vec{\rho})=\frac{1}{2}+\hat{a}\cdot\vec{\rho}=\frac{1}{2}+|\vec{\rho}|\cos\theta,$

where

$\rho=\frac{1}{2}\mathrm{Id}+\rho_{i}\sigma_{i}=\frac{1}{2}\mathrm{Id}+\vec{\rho}\cdot\vec{\sigma},$

plus the constant functions 0 and 1.

They are functions $f:\mathcal{U}\rightarrow[0,1]$ that, according to Theorem
8.3, can be interpreted as membership functions of fuzzy sets. These functions
are orthogonal if the bold intersection of fuzzy sets is the empty set, so

$\mu_{\mathcal{A}}\sqcap\mu_{\mathcal{B}}=\mathrm{max}\\{\hat{a}\cdot\vec{\rho}+\hat{b}\cdot\vec{\rho},0\\}=0\quad\Leftrightarrow\quad\hat{a}\cdot\vec{\rho}+\hat{b}\cdot\vec{\rho}<0.$

Since this relation has to be satisfied for all $\rho\in\mathcal{S}$ we
recover the result in (12), that is, $\hat{a}=-\hat{b}$.

If the fuzzy sets are orthogonal, then its bold union is

$\mu_{\mathcal{A}}\sqcup\mu_{\mathcal{B}}=\mathrm{min}\\{\mu_{\mathcal{A}}+\mu_{\mathcal{B}},1\\}=1,$

that is, the union is the universe of discourse $S$, which makes sense, since
it is the probability that the eigenvalue of $A$ is +1 or -1, which exhausts
all the possibilities.

We now give some examples that help to clarify the role of the fuzzy sets for
the qubit.

######  Example 8.4.

The gate NOT. We consider a qubit. Let $\\{\,|0\rangle,|1\rangle\,\\}$ be the
ordered basis of $\mathcal{H}\simeq\mathbb{C}^{2}$ of eigenvectors of
$\sigma_{3}$,

$\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$

Let $|\Psi\rangle=v_{0}|0\rangle+v_{1}|1\rangle\in\mathcal{H}$ a vector
representing a pure state, so $|v_{0}|^{2}+|v_{1}|^{2}=1$. There is a
transformation that flips the two basis states (the NOT gate). It is
represented in this basis by the unitary matrix

$U=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}=\sigma_{1},\qquad
U|\Psi\rangle=v_{1}|0\rangle+v_{0}|1\rangle.$

Notice that $\det U=-1$. Its action on a density matrix is

$\rho=\begin{pmatrix}\rho_{00}&\rho_{01}\\\
{\bar{\rho}_{01}}&\rho_{11}\end{pmatrix}\,\longrightarrow\,U\rho
U^{\dagger}=\begin{pmatrix}\rho_{11}&{\bar{\rho}}_{01}\\\
\rho_{01}&\rho_{00}\end{pmatrix}.$

In terms of the sigma matrices we have that

$U\sigma_{1}U^{\dagger}=\sigma_{1},\qquad
U\sigma_{2}U^{\dagger}=-\sigma_{2},\qquad U\sigma_{3}U^{\dagger}=-\sigma_{3},$

so the flip transformation takes $\vec{\rho}=(\rho_{1},\rho_{2},\rho_{3})$ to
$\vec{\rho}{\phantom{{}^{\prime}}}^{\prime}=(\rho_{1},-\rho_{2},-\rho_{3})$.
In the Bloch sphere, this is a rotation of $180^{0}$ around the $x_{1}$ axis.

The probability of finding the system in a state ‘+’ with respect to an
arbitrary axis $\hat{a}$ is

$f_{\hat{a}}^{+}(\rho)=\frac{1}{2}+\hat{a}_{1}\rho_{1}+\hat{a}_{2}\rho_{2}+\hat{a}_{3}\rho_{3}.$

Effecting the NOT transformation changes the probability to

$f_{\hat{a}}^{+}(\rho^{\prime})=\frac{1}{2}+\hat{a}_{1}\rho_{1}-\hat{a}_{2}\rho_{2}-\hat{a}_{3}\rho_{3}.$
(33)

Notice that applying the NOT gate to the fuzzy bit is not the same than
computing its complement
${f_{\hat{a}}^{+}(\rho)}^{\prime}=1-f_{\hat{a}}^{+}(\rho)$ as a fuzzy set.
This has been cause of some confusion in the literature. However, in the
particular case that $\hat{a}$ is the $x_{3}$ axis, $\hat{a}=(0,0,1)$, then
one has that

$f_{\hat{a}}^{+}(\rho^{\prime})=\frac{1}{2}-\rho_{3},$

which is equal to its complement as a fuzzy set

$1-f_{\hat{a}}^{+}(\rho)=\frac{1}{2}-\rho_{3}.$

This probability coincides with
$f_{\hat{a}}^{-}(\rho)=f_{-\hat{a}}^{+}(\rho)$.

$\square$

######  Example 8.5.

The gate $\sqrt{\hbox{NOT}}$. In the same setting than the example above, one
defines the square root of the gate NOT, $\sqrt{\hbox{NOT}}$, as a unitary
transformation such that

$\sqrt{\hbox{NOT}}\cdot\sqrt{\hbox{NOT}}=\hbox{NOT}.$ (34)

This transformation is not unique, but it is usual to choose the following
one:

$U=\frac{1}{2}\begin{pmatrix}1+\mathrm{i}&1-\mathrm{i}\\\
1-\mathrm{i}&1+\mathrm{i}\end{pmatrix}.$

(Notice that $\det U=\mathrm{i}$.) The sigma matrices behave, under this
transformation, as

$U\sigma_{1}U^{\dagger}=\sigma_{1},\qquad
U\sigma_{2}U^{\dagger}=\sigma_{3},\qquad U\sigma_{3}U^{\dagger}=-\sigma_{2},$

so $\vec{\rho}=(\rho_{1},\rho_{2},\rho_{3})$ goes to
$\vec{\rho}{\phantom{{}^{\prime}}}^{\prime}=(\rho_{1},\rho_{3},-\rho_{2})$. In
the Bloch sphere, this is a rotation of $90^{0}$ around the $x_{1}$ axis. It
is clear that (34) is satisfied. The membership functions then change as

$f_{\hat{a}}^{+}(\rho^{\prime})=\frac{1}{2}+\hat{a}_{1}\rho_{1}+\hat{a}_{2}\rho_{3}-\hat{a}_{3}\rho_{2}.$

It is known [18, 19] that the negation in fuzzy logic (orthocomplementation of
fuzzy sets) doesn’t have a square root. Particularly, Lemma 17.1.12 in [18]
reads: “There is no continuous function $f:[0,1]\rightarrow[0,1]$ such that
for any $x\in[0,1]$, $f(f(x))=1-x$”. The authors then provide an example of
square root in which the function $f$ is not continuous.

In our approach, since the NOT gate sending the fuzzy set
$f_{\hat{a}}^{+}(\rho)$ to
${f_{\hat{a}}^{+}(\rho)}^{\prime}=f_{\hat{a}}^{+}(\rho^{\prime})$ does not
coincide with the negation, ${f_{\hat{a}}^{+}(\rho)}^{\prime}\neq
1-f_{\hat{a}}^{+}(\rho)$ in general, a continuous square root of the gate NOT
is allowed.

$\square$

###  8.2 The entangled fuzzy bits

For the entangled fuzzy bits, the membership functions of
$\mathcal{E}=\mathcal{A}\times\mathcal{B}$ are given in terms of the two unit
vectors $\hat{a}$ and $\hat{b}$ (17):

$\mu_{\mathcal{E}}(\vec{s},\vec{r},R)=f^{+,+}_{\hat{a},\hat{b}}(\vec{s},\vec{r},R)=\frac{1}{4}\left(1+\vec{s}\cdot\hat{a}+\vec{r}\cdot\hat{b}+\hat{a}^{\,t}R\hat{b}\right).$

In the two examples (19) and (20) one can check directly that the
orthogonality of the membership functions implies that the bold union is again
a membership function:

$\displaystyle
f^{+,+}_{\hat{a},\hat{b}}(\vec{s},\vec{r},R)+f^{+,-}_{\hat{a},\hat{b}}(\vec{s},\vec{r},R)=f^{+,+-}_{\hat{a},\hat{b}}(\vec{s},\vec{r},R)\leq
1,$ $\displaystyle
f^{+,+}_{\hat{a},\hat{b}}(\vec{s},\vec{r},R)+f^{-,-}_{\hat{a},\hat{b}}(\vec{s},\vec{r},R)\leq
1,$

that is, the sum is again a probability.

######  Example 8.6.

The CNOT gate. The Controlled NOT gate has an input of two qubits. In the
ordered basis of $\mathcal{H}\simeq\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ used
above, the CNOT transformation acts as

$\begin{CD}\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}\\}@>{\hbox{CNOT}}>{}>\\{|00\rangle,|01\rangle,|11\rangle,|10\rangle\\}\end{CD}$

It is represented in this basis by the unitary matrix

$U=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}=\begin{pmatrix}\mathrm{Id}&0\\\
0&\sigma_{1}\end{pmatrix}.$

Notice that $\det U=-1$. We want to see how the membership functions change
with this transformation, so we will compute first the change in the density
matrix. Let us remind the notation (14):

$\rho=\frac{1}{4}r_{\mu\nu}\sigma_{\mu}\otimes\sigma_{\nu},\qquad\hbox{with}\qquad[r_{\mu\nu}]=\begin{pmatrix}1&r_{01}&r_{02}&r_{03}\\\
r_{10}&r_{11}&r_{12}&r_{13}\\\ r_{20}&r_{21}&r_{22}&r_{23}\\\
r_{30}&r_{31}&r_{32}&r_{33}\end{pmatrix}=\begin{pmatrix}1&{\vec{r}\phantom{{}^{\prime}}}^{t}\\\
\vec{s}&R\end{pmatrix}.$

For the transformed matrix

$\rho^{\prime}=U\rho
U^{\dagger}=\frac{1}{4}r^{\prime}_{\mu\nu}\sigma_{\mu}\otimes\sigma_{\nu},\qquad[r^{\prime}_{\mu\nu}]=\begin{pmatrix}1&{\vec{r}\phantom{{}^{\prime}}^{\prime}}^{t}\\\
\vec{s}\phantom{{}^{\prime}}^{\prime}&R^{\prime}\end{pmatrix},$

we obtain [20]:

$\displaystyle{\vec{r}\phantom{{}^{\prime}}^{\prime}}^{t}=(r_{01},r_{32},r_{33}),$
$\displaystyle{\vec{s}\phantom{{}^{\prime}}^{\prime}}^{t}=(r_{11},r_{21},r_{30}),$
$\displaystyle R^{\prime}=\begin{pmatrix}r_{10}&r_{23}&-r_{22}\\\
r_{20}&-r_{13}&r_{12}\\\ r_{31}&r_{02}&r_{03}\end{pmatrix},$ $\displaystyle
r^{\prime}_{00}=r_{00}=1.$

Let us consider $A=B=\sigma_{3}$, so $\hat{a}=\hat{b}=(0,0,1)$. Suppressing
$\hat{a}$ and $\hat{b}$ from the notation in (17), we get

$\displaystyle f^{++}=\frac{1}{4}(1+r_{30}+r_{03}+r_{33}),$
$\displaystyle{f^{\prime}}^{++}=\frac{1}{4}(1+r_{30}+r_{33}+r_{03}),$
$\displaystyle f^{+-}=\frac{1}{4}(1+r_{30}-r_{03}-r_{33}),$
$\displaystyle{f^{\prime}}^{+-}=\frac{1}{4}(1+r_{30}-r_{33}-r_{03}),$
$\displaystyle f^{-+}=\frac{1}{4}(1-r_{30}+r_{03}-r_{33}),$
$\displaystyle{{f^{\prime}}^{-+}}=\frac{1}{4}(1-r_{30}+r_{33}-r_{03}),$
$\displaystyle f^{--}=\frac{1}{4}(1-r_{30}-r_{03}+r_{33}),$
$\displaystyle{f^{\prime}}^{--}=\frac{1}{4}(1-r_{30}-r_{33}+r_{03}).$

As expected, since the first qubit is a control qubit, we have

$\displaystyle f^{++}={f^{\prime}}^{++},$ $\displaystyle
f^{+-}={f^{\prime}}^{+-},$ $\displaystyle f^{-+}={f^{\prime}}^{--},$
$\displaystyle f^{--}={f^{\prime}}^{-+}.$

$\square$

##  9 Conclusions

In this paper the meaning of the relationship between Quantum Mechanics,
logics and fuzzy sets has been clarified. It results that a particular family
of fuzzy sets, with universe of discourse the density matrices in a Hilbert
space, can actually represent faithfully the Birkhoff-von Neumann logic of
projectors of a Hilbert space. This choice is the same than the ‘ad hoc’
choice of Mackey in his Axiom VII. The membership functions of the fuzzy sets
(or, equivalently, the experimental functions of Ma̧czyński) have been
computed for three different systems: the qubit, two entangled qubits and a
qutrit ‘nested’ inside the entangled qubits. A decomposition of SU(3), the
special unitary transformations of the qutrit, inherited from the orthogonal
symmetric Lie algebra structure of $\mathfrak{su}(4)$, has been found. With
the aid of the membership functions, we have proven that a continuous square
root of NOT exists for the qubit.

## Acknowledgements

This work is supported by the Spanish Grants FIS2017- 84440-C2-1-P funded by
MCIN/AEI/10.13039/501100011033 ‘ERDF A way of making Europe’,
PID2020-116567GB-C21 funded by MCIN/AEI/10.13039/501100011033, the Project
PROMETEO/2020/079 (Generalitat Valenciana) and the project CaLIGOLA, Project
ID: 101086123 (HORIZON).

This article is based upon work from COST Action CaLISTA CA21109 supported by
COST (European Cooperation in Science and Technology). www.cost.eu.

M. Aldana wants to thank the Departament de Física Teòrica, Universitat de
València, for its kind hospitality.

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