#  Optimal bounds on the speed of subspace evolution∗†

Sergio Albeverio and Alexander K. Motovilov  Sergio Albeverio, Institute of
Applied Mathematics, and Hausdorff Center of Mathematics, University of Bonn,
Endenicher Allee 60, D-53115 Bonn, Germany<EMAIL_ADDRESS>Alexander K.
Motovilov, Bogoliubov Laboratory of Theoretical Physics, JINR, Joliot-Curie 6,
141980 Dubna, Russia, and Dubna State University, Universitetskaya 19, 141980
Dubna, Russia<EMAIL_ADDRESS>
###### Abstract.

By a quantum speed limit one usually understands an estimate on how fast a
quantum system can evolve between two distinguishable states. The most known
quantum speed limit is given in the form of the celebrated Mandelstam-Tamm
inequality that bounds the speed of the evolution of a state in terms of its
energy dispersion. In contrast to the basic Mandelstam-Tamm inequality, we are
concerned not with a single state but with a (possibly infinite-dimensional)
subspace which is subject to the Schrödinger evolution. By using the concept
of maximal angle between subspaces we derive optimal bounds on the speed of
such a subspace evolution. These bounds may be viewed as further
generalizations of the Mandelstam-Tamm inequality. Our study includes the case
of unbounded Hamiltonians.

###### Key words and phrases:

Mandelstam-Tamm inequality; quantum speed limit; subspace evolution

###### 2000 Mathematics Subject Classification:

Primary 47A15, 47A62; Secondary 47B15, 47B25

∗This work was partially supported by the Heisenberg-Landau Program.

†Published in J. Phys. A: Math. Theor. 55:23 (2022), 235203 [17 pp.]
(https://doi.org/10.1088/1751-8121/ac6bcf).

##  1\. Introduction

By a quantum speed limit one usually calls a lower bound on the time that is
needed for a quantum system to evolve from a given state to a target state or
a target subspace. The history of the subject is already long, being traced
back to the 1945’s pioneering work by Mandelstam and Tamm [1]. The volume of
literature on quantum speed limits and their applications in a variety of
areas is large and by no means we make here an attempt to present a more or
less complete review of all relevant results. Instead, we only inform the
interested reader that comprehensive surveys of the literature on various
quantum speed limits may be found in the recent review articles [2] and [3]
(see also the introductory part of the very recent paper [4]).

We begin with recalling how the main known quantum speed limits look. To this
end, we consider an isolated quantum system described by a Hamiltonian $H$,
which is assumed to be a time-independent self-adjoint operator acting in the
complex Hilbert space $\mathfrak{H}$. Any vector $\phi$ from the unit sphere
in $\mathfrak{H}$ represents a possible pure state of this system. Strictly
speaking, a pure state $\mathcal{S}$ is rather a class of equivalence of norm-
one vectors in $\mathfrak{H}$: the vectors $\phi,\psi\in\mathfrak{H}$ with
$\|\phi\|=\|\psi\|=1$ represent the same pure state if $\psi=u\phi$ for some
$u\in{\mathbb{C}}$ such that $|u|=1$. In an obvious way, the state
$\mathcal{S}$ may be identified with a one-dimensional subspace
$\mathfrak{P}_{\mathcal{S}}$ which is the span of an arbitrarily chosen vector
$\psi$ in $\mathcal{S}$,
$\mathfrak{P}_{\mathcal{S}}:=\bigl{\\{}f=\lambda\psi\,\,\,\big{|}\lambda\in\mathbb{C}\,\bigr{\\}}$.

In what follows we will always suppose that units of measurement are chosen
such that $\hbar=1$. The time evolution of a state vector
$\psi(t)\in\mathfrak{H}$, $t\in\mathbb{R}$, is assumed to be governed by the
Schrödinger equation

$\displaystyle\mathrm{i}\frac{d}{dt}\psi$ $\displaystyle=H\psi,$ (1.1)
$\displaystyle\psi(t)\bigr{|}_{t=0}$ $\displaystyle=\psi_{0},$ (1.2)

where the initial-state vector $\psi_{0}$, $\|\psi_{0}\|=1$, along with all
other vectors on the path $\psi(t)$, $t\in{\mathbb{R}}$, should belong to the
domain $\mathrm{Dom}(H)$ of the Hamiltoninan $H$.

Starting point in the study of quantum speed limits was the following natural
question: How fast can a quantum system with the Hamiltonian $H$ arrive at a
state orthogonal to its initial state $\psi_{0}$?

It is obvious that the answer to this question may be important in various
respects. Perhaps, the very latest motivation stems from quantum information
theory and quantum computing (see, e.g., [2, 3]).

Available answers to the above question have been given in the form of lower
bounds for the orthogonalization time $T_{\perp}$ which is the time necessary
for the system to evolve from the initial state $\psi_{0}$ to a state
$\psi(T_{\perp})$ such that ${\langle}\psi_{0},\psi(T_{\perp}){\rangle}=0$.
(Here and in what follows by ${\langle}\cdot,\cdot{\rangle}$ we denote the
inner product in the Hilbert space $\mathfrak{H}$ assuming that it is linear
in the first entry.)

The oldest among these bounds is the celebrated Mandelstam–Tamm inequality
discovered in the 1945’s paper [1]:

$T_{\perp}\geq\frac{\pi}{2\,\Delta E},$ (1.3)

where $\Delta E$ is the energy dispersion for the initial state $\psi_{0}$,

$\Delta
E=\bigl{(}\|H\psi_{0}\|^{2}-{\langle}H\psi_{0},\psi_{0}{\rangle}^{2}\bigr{)}^{1/2},\quad\psi_{0}\in\mathrm{Dom}(H).$
(1.4)

The second celebrated lower bound for the orthogonalization time, the
Margolus–Levitin inequality [5] has been discovered half a century later, in
1998. This bound reads as

$T_{\perp}\geq\frac{\pi}{2\,\delta E},$ (1.5)

where the quantity

$\delta
E={\langle}H\psi_{0},\psi_{0}{\rangle}-\min\bigl{(}\mathop{\mathrm{spec}}(H)\bigr{)}$
(1.6)

represents the difference between the average energy for the state $\psi_{0}$
and the lower edge of the spectrum of the Hamiltonian $H$ (which is assumed to
be semibounded from below in this case).

The lower bounds (1.3) and (1.5) are not equivalent to each other but both of
them have been proven to be optimal (see, e.g., [2, p. 7] and [3, p. 3923]).

Of course, one notices that by their form the bounds (1.3) and (1.5) resemble
the uncertainty relation for energy and time. These bounds, however, are
related not to the standard deviation in the measuring of the quantity $t$ but
to the well-established time needed for a state of the system to evolve into
an orthogonal state. Thus, in their essence the inequalities (1.3) and (1.5)
are very different from the uncertainty relation.

Next, there is a version of the Mandelstam–Tamm inequality that works for
intermediate time moments $t\in(0,T_{\perp})$. This is the lower estimate
found for the first time in 1973 by Fleming [6]:

$T_{\theta}\geq\frac{\theta}{\Delta E}\,,$ (1.7)

where $\Delta E$ is again given by (1.4) and $T_{\theta}$ denotes the first
time moment when the acute angle

$\angle\bigl{(}\psi_{0},\psi(t)\bigr{)}:=\arccos|{\langle}\psi_{0},\psi(t){\rangle}|$
(1.8)

between the vectors $\psi_{0}$ and $\psi(t)$ reaches a certain value
$\theta\in(0,\pi/2]$.

It is worth to remark that, through the years, the Mandelstam-Tamm bound
(1.3)/(1.7) has been rediscovered several times (for related discussion and
references, see, e.g., [2, p. 5]). The Mandelstam-Tamm bound has also been
extended to the evolution of mixed states [7]. Furthermore, more detailed
estimates for the evolution speed have been established for particular classes
of evolutionary problems (see [2, 3]). Probably, the latest among them is a
speed limit for evolution of thermal states derived in [4]. Mandelstam-Tamm-
type bounds for the orthogonalization time exist even for some non-self-
adjoint (so-called pseudo-Hermitian and, in particular, $PT$-symmetric)
Hamiltonians (see [8, 9] and references therein).

In our recent work [10] we have generalized the Mandelstam-Tamm-Fleming bound
(1.7) to the Schrödinger evolution of a subspace. Like in the vast majority of
publications on quantum speed limits, in [10] we restricted ourselves to the
exclusive consideration of bounded Hamiltonians. However typical quantum-
mechanical Hamiltonians are unbounded operators. In the present work we drop
the requirement of boundedness of $H$ and extend most of the results of [10]
to the subspace evolution governed by arbitrary self-adjoint Hamiltonians.
Assuming that $P_{0}$ is an orthogonal projection in $\mathfrak{H}$ such that
the domain of a (possibly unbounded) self-adjoint operator (Hamiltonian) $H$
is invariant under $P_{0}$, that is,
$P_{0}\,\mathop{\mathrm{Dom}}(H)\subset\mathop{\mathrm{Dom}}(H)$, we study the
subspace path $\mathfrak{P}_{t}=\mathop{\mathrm{Ran}}(P_{t})$,
$t\in{\mathbb{R}}$, formed in the set of all the subspaces of $\mathfrak{H}$
by the ranges of the orthogonal projections $P_{t}={\rm e}^{-{\rm
i}Ht}P_{0}{\rm e}^{{\rm i}Ht}$, $t\in{\mathbb{R}}$.

Our studies of the subspace evolution path $\mathfrak{P}_{t}$,
$t\in{\mathbb{R}}$, are essentially based on the concept of maximal
angle111For discussion of the concept of maximal angle and references see page
3.1. between two subspaces of a Hilbert space. Recall, that the maximal angle
between (arbitrary) subspaces $\mathfrak{Q}$ and $\mathfrak{R}$ of the Hilbert
space $\mathfrak{H}$ is introduced as follows:

$\vartheta(\mathfrak{Q},\mathfrak{R})=\arcsin\|Q-R\|,$ (1.9)

where $Q$ and $R$ are the orthogonal projections in $\mathfrak{H}$ onto
$\mathfrak{Q}$ and $\mathfrak{R}$, respectively. The maximal angle (1.9)
possesses all the properties of a distance, and thus it generates a metric on
the set of all subspaces of $\mathfrak{H}$. By using this metric we establish,
in particular, the following result (see Theorem 3.12 below).

Assume that $T_{\theta}$ is the time moment at which the maximal angle between
the initial subspace $\mathfrak{P}_{0}$ and a subspace in the path
$\mathfrak{P}_{t}$, $t\geq 0$, reaches a certain value $\theta\in(0,\pi/2]$.
Then necessarily

$T_{\theta}\geq\frac{\theta}{\Delta E_{\mathfrak{P}_{0}}}\,,$ (1.10)

where

$\Delta
E_{\mathfrak{P}_{0}}=\sup\limits_{\scriptsize\begin{matrix}\psi\in\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H)\\\
\|\psi\|=1\end{matrix}}\bigl{(}\|H\psi\|^{2}-{\langle}H\psi,\psi{\rangle}^{2}\bigr{)}^{1/2}.$
(1.11)

Clearly, the quantity $\Delta E_{\mathfrak{P}_{0}}$ is nothing but the least
upper bound for the energy dispersion on the states belonging to the initial
subspace $\mathfrak{H}_{0}.$

The Mandelstam-Tamm-Fleming bound (1.7) turns out to be a special case of the
estimate (1.10) for a one-dimensional subspace $\mathfrak{P}_{0}$ spanned by a
particular state $\psi_{0}$.

The plan of the paper is as follows. In Section 2 we collect some facts on the
projection path $P_{t}={\rm e}^{-{\rm i}Ht}P_{0}{\rm e}^{{\rm i}Ht}$,
$t\in{\mathbb{R}}$. In particular, we notice that this path is strongly
continuous on the whole Hilbert space $\mathfrak{H}$. Under the assumption
$P_{0}\,\mathop{\mathrm{Dom}}(H)\subset\mathop{\mathrm{Dom}}(H)$, the path
$P_{t}$, $t\in{\mathbb{R}}$, is, in addition, strongly differentiable in
$t\in{\mathbb{R}}$ on the domain of $H$. In such a case the strong derivative
$\dot{P}_{t}$ is expressed through the commutator of $H$ and $P_{t}$ (see
Theorem 2.1). In Section 3 we work under the additional hypothesis that the
commutator of $H$ and $P_{0}$ is a bounded operator on
$\mathop{\mathrm{Dom}}(H)$ and, hence, its closure is a bounded operator on
the whole space $\mathfrak{H}$. The main result of the section is Theorem 3.6.
It presents the upper bound (3.16) for the maximal angle between the subspaces
$\mathfrak{P}_{s}$ and $\mathfrak{P}_{t}$, $s,t\in{\mathbb{R}}$, in the
subspace path $\mathfrak{P}_{\tau}=\mathop{\mathrm{Ran}}(P_{\tau})$,
$\tau\in{\mathbb{R}}$, through the product of the times difference $|t-s|$ and
the norm of the commutator of $H$ and $P_{0}$. This section also contains the
proof of Theorem 3.12 that we already mentioned. The section is concluded with
a new consideration of the case where $H$ is bounded operator. In such a case
the quantity (1.11) is bounded by half the distance between the upper and
lower edges of the spectrum of $H$. Combining this with (1.10) we obtain our
last lower bound for $T_{\theta}$ (see Theorem 3.18) in this paper. Finally,
in Section 4 we present a summary of the work and point out some open
problems.

Let us add a few words about notations used throughout the paper. By a
subspace we always understand a closed linear subset of a Hilbert space. The
identity operator is denoted by $I$. For a linear operator $L$, by
$\mathop{\mathrm{Dom}}(L)$ we denote its domain and by
$\mathop{\mathrm{Ran}}(L)$, its range. If $Q$ is an orthogonal projection, the
notation $Q^{\perp}$ is always used for the complementary projection,
$Q^{\perp}=I-Q$. By $\mathfrak{M}\oplus\mathfrak{N}$ we understand the
orthogonal sum of two Hilbert spaces (or orthogonal subspaces or simply
orthogonal linear subsets) $\mathfrak{M}$ and $\mathfrak{N}$. By ${\sf
E}_{T}(\sigma)$ we always denote the spectral projection of a self-adjoint
operator $T$ associated with a Borel set $\sigma\subset{\mathbb{R}}$. Notation
$[A,B]$ is used for the commutator of linear operators $A$ and $B$ on
$\mathfrak{H}$. It is assumed that
$\mathop{\mathrm{Dom}}\bigl{(}[A,B]\bigr{)}:=\bigl{\\{}x\in\mathop{\mathrm{Dom}}(A)\cap\mathop{\mathrm{Dom}}(B)\,\big{|}\,\,Ax\in\mathop{\mathrm{Dom}}(B),\,Bx\in\mathop{\mathrm{Dom}}(A)\bigr{\\}}$
and $[A,B]x:=ABx-BAx$ for any
$x\in\mathop{\mathrm{Dom}}\bigl{(}[A,B]\bigr{)}$.

##  2\. Projection path generated by the Schrödinger evolution of a subspace

As it was already underlined, we are concerned not with a single state but
with a multi-dimensional (possibly, even infinite-dimensional) subspace
spanned by the states of the system that are subject to the Schrödinger
evolution. In general, we allow the Hamiltonian $H$ of the system to be an
unbounded self-adjoint operator on the Hilbert space $\mathfrak{H}$ with
domain $\mathop{\mathrm{Dom}}(H)$. In this work we restrict ourselves to the
consideration of a nontrivial subspace $\mathfrak{P}_{0}\subset\mathfrak{H}$,
$\mathfrak{P}_{0}\neq\\{0\\}$, such that

$P_{0}\,\,\mathop{\mathrm{Dom}}(H)\subset\mathop{\mathrm{Dom}}(H),$ (2.1)

where $P_{0}$ stands for the orthogonal projection in $\mathfrak{H}$ onto
$\mathfrak{P}_{0}$. That is, we assume that the linear set
$\mathop{\mathrm{Dom}}(H)$ is invariant under $P_{0}$ in the sense that
$P_{0}f\in\mathop{\mathrm{Dom}}(H)$ for any $f\in\mathop{\mathrm{Dom}}(H)$.
From (2.1) it follows that the set $\mathop{\mathrm{Dom}}(H)$ is also
invariant under the complementary orthogonal projection $P_{0}^{\perp}$,

$P_{0}^{\perp}\,\,\mathop{\mathrm{Dom}}(H)\subset\mathop{\mathrm{Dom}}(H),$
(2.2)

Moreover, any of the hypotheses (2.1) and (2.2) is equivalent to any of the
equalities

$\mathop{\mathrm{Ran}}\left(P_{0}\big{|}_{\mathop{\mathrm{Dom}}(H)}\right)=\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H)\quad\text{and}\quad\mathop{\mathrm{Ran}}\left(P^{\perp}_{0}\big{|}_{\mathop{\mathrm{Dom}}(H)}\right)=\mathfrak{P}^{\perp}_{0}\cap\mathop{\mathrm{Dom}}(H)$
(2.3)

as well as to the combination of them,

$\mathop{\mathrm{Dom}}(H)=\bigl{(}\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H)\bigr{)}\,\,\oplus\,\,\bigl{(}\mathfrak{P}_{0}^{\perp}\cap\mathop{\mathrm{Dom}}(H)\bigr{)}.$
(2.4)

Since $H$ is a self-adjoint operator, its domain $\mathop{\mathrm{Dom}}(H)$ is
dense in $\mathfrak{H}$. By (2.4), this implies that the sets
$\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H)$ and
$\mathfrak{P}_{0}^{\perp}\cap\mathop{\mathrm{Dom}}(H)$ are dense in the
subspaces $\mathfrak{P}_{0}$ and $\mathfrak{P}_{0}^{\perp}$, respectively. In
particular,

$\overline{\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H)}=\mathfrak{P}_{0}.$
(2.5)

From (2.3) it follows that the Hamiltonian $H$ admits the following $2\times
2$ block matrix representation with respect to the decomposition
$\mathfrak{H}=\mathfrak{P}_{0}\oplus\mathfrak{P}_{0}^{\perp}$:

$\displaystyle H$ $\displaystyle=H_{\mathrm{diag}}+H_{\mathrm{off}},\quad
H_{\mathrm{diag}}=\left(\begin{array}[]{cc}H_{\mathfrak{P}_{0}}&0\\\
0&H_{\mathfrak{P}_{0}^{\perp}}\end{array}\right),\quad
H_{\mathrm{off}}=\left(\begin{array}[]{cc}0&B\\\ C&0\end{array}\right),$
(2.10)

where

$\displaystyle H_{\mathfrak{P}_{0}}$
$\displaystyle=P_{0}H|_{\mathfrak{P}_{0}},\quad\mathop{\mathrm{Dom}}(H_{\mathfrak{P}_{0}})=\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H),$
(2.11) $\displaystyle H_{\mathfrak{P}^{\perp}_{0}}$
$\displaystyle=P_{0}^{\perp}H|_{\mathfrak{P}^{\perp}_{0}},\quad\mathop{\mathrm{Dom}}(H_{\mathfrak{P}^{\perp}_{0}})=\mathfrak{P}^{\perp}_{0}\cap\mathop{\mathrm{Dom}}(H),$
(2.12) $\displaystyle B$
$\displaystyle=P_{0}H|_{\mathfrak{P}^{\perp}_{0}},\quad\mathop{\mathrm{Dom}}(B)=\mathop{\mathrm{Dom}}(H_{\mathfrak{P}^{\perp}_{0}}),$
(2.13) $\displaystyle C$
$\displaystyle=P_{0}^{\perp}H|_{\mathfrak{P}_{0}},\quad\mathop{\mathrm{Dom}}(C)=\mathop{\mathrm{Dom}}(H_{\mathfrak{P}_{0}}).$
(2.14)

Notice that, in general, $C\subset B^{*}$ and $B\subset C^{*}$.

Every vector $\psi_{0}\subset\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H)$ is
assumed to evolve into a vector $\psi(t)\in\mathop{\mathrm{Dom}}(H)$, $t>0$,
according to (1.1), (1.2). It is well known that for any
$\psi_{0}\in\mathop{\mathrm{Dom}}(H)$ the Cauchy problem (1.1), (1.2) has a
unique solution $\psi:\mathbb{R}^{+}\to\mathfrak{H}$ in the class of norm
continuously differentiable vector-valued functions from ${\mathbb{R}}^{+}$ to
$\mathfrak{H}$ such that $\psi(t)\in\mathop{\mathrm{Dom}}(H)$ for all $t\geq
0$. For convenience of the reader, we remark that the existence of the
solution to the problem (1.1), (1.2) in the form

$\psi(t)=U(t)\psi_{0},\quad t>0,$ (2.15)

where

$U(t)=\mathrm{e}^{-\mathrm{i}Ht},\quad t\in\mathbb{R},$ (2.16)

follows, e.g., from [11, Theorem VIII.7]. Surely, the exponential (2.16) is
defined by the spectral theorem (see, e.g., [11, Theorem VIII.6]) as

${\rm e}^{-{\rm i}Ht}:=\int_{\mathbb{R}}e^{-{\rm i}\lambda t}{\sf
E}(d\lambda),\quad t\in{\mathbb{R}},$ (2.17)

where ${\sf E}$ stands for the spectral measure on ${\mathbb{R}}$ associated
with the self-adjoint operator $H$. It is worth to notice that the domain of
$H$,

$\mathop{\mathrm{Dom}}(H)=\left\\{f\in\mathfrak{H}\,\bigg{|}\,\,\int_{\mathbb{R}}\lambda^{2}{\langle}{\sf
E}(d\lambda)f,f{\rangle}\right\\},$ (2.18)

is an invariant of $U(t)$,

$\mathop{\mathrm{Ran}}\left(U(t)|_{\mathop{\mathrm{Dom}}(H)}\right)=\mathop{\mathrm{Dom}}(H),\quad\text{for
any $t\in{\mathbb{R}}$}.$ (2.19)

Given $t\in{\mathbb{R}}$, by $\mathfrak{P}_{t}$ we denote the range
$\mathop{\mathrm{Ran}}\left(U(t)P_{0}\right)$ of the product of the unitary
operator (2.16) and the orthogonal projection $P_{0}$ onto the subspace
$\mathfrak{P}_{0}$, that is,

$\mathfrak{P}_{t}:=\mathop{\mathrm{Ran}}\bigl{(}U(t)\big{|}_{\mathfrak{P}_{0}}\bigr{)},\quad
t\in{\mathbb{R}}.$ (2.20)

By (2.15), the subspace $\mathfrak{P}_{t}$, $t>0$, is nothing but the closure
of the span of the vectors $\psi(t)\subset\mathop{\mathrm{Dom}}(H)$
representing the values, at the time moment $t$, of the vector-valued
functions $\psi:\,\,{\mathbb{R}}^{+}\to\mathfrak{H}$ that solve (1.1), (1.2)
for various $\psi_{0}\in\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H)$. So that
we deal with the path $\mathfrak{P}_{t},\,\,t\geq 0,$ in the set of all
subspaces of the Hilbert space $\mathfrak{H}$. Or (and this is the same) with
the path

$P_{t},\quad t\geq
0,\qquad\mathop{\mathrm{Ran}}\bigl{(}P_{t}\bigr{)}=\mathfrak{P}_{t},$ (2.21)

of the orthogonal projections $P_{t}$ in $\mathfrak{H}$ onto the respective
subspaces $\mathfrak{P}_{t}$. Clearly, the orthogonal projections $P_{t}$ onto
the subspaces $\mathfrak{P}_{t}$ defined in (2.20) are explicitly given by

$P_{t}=U(t)P_{0}U(t)^{*}=\mathrm{e}^{-\mathrm{i}Ht}P_{0}\mathrm{e}^{\mathrm{i}Ht},\quad\text{for
any $t\in{\mathbb{R}}$}.$ (2.22)

It is almost obvious that the strong continuity of the unitary group ${\rm
e}^{-{\rm i}Ht}$, $t\in{\mathbb{R}}$, implies the strong continuity of the
path $P_{t}$, $t\in{\mathbb{R}}$, on the whole Hilbert space $\mathfrak{H}$.
Under the assumption (2.1) from (2.19) it immediately follows that the domain
of $H$ is mapped by $P_{t}$ back into the domain of $H$,

$\mathop{\mathrm{Ran}}\left(P_{t}\big{|}_{\mathop{\mathrm{Dom}}(H)}\right)\subset\mathop{\mathrm{Dom}}(H),\quad
t\in{\mathbb{R}}.$ (2.23)

For convenience of the reader we present a proof of the strong
differentiability of the projection family (2.22) under the hypothesis (2.1).

######  Theorem 2.1.

Let $H$ be a (possibly unbounded) self-adjoint operator in the Hilbert space
$\mathfrak{H}$. Assume that $P_{0}$, $P_{0}\neq 0$, is an orthogonal
projection in $\mathfrak{H}$ and that the domain of $H$ is invariant under
$P_{0}$, i.e.,
$\mathop{\mathrm{Ran}}\bigl{(}P_{0}|_{\mathop{\mathrm{Dom}}(H)}\bigr{)}\subset\mathop{\mathrm{Dom}}(H)$.
Then the projection path $P_{t}={\rm e}^{-{\rm i}Ht}P_{0}{\rm e}^{{\rm i}Ht}$,
$t\in{\mathbb{R}}$, is strongly differentiable on $\mathop{\mathrm{Dom}}(H)$
for any $t\in{\mathbb{R}}$, that is, the following limit exists

$\displaystyle\dot{P}_{t}f:=$ $\displaystyle\lim\limits_{\tau\to
0}\left(\frac{P_{t+\tau}-P_{t}}{\tau}f\right),\quad{t\in{\mathbb{R}}},$ (2.24)
$\displaystyle\text{for any }f\in\mathop{\mathrm{Dom}}(H).$

Moreover, the inclusion (2.23) holds and the following equality takes place:

$\displaystyle{\rm i}\dot{P}_{t}f=$ $\displaystyle
HP_{t}f-P_{t}Hf,\quad{t\in{\mathbb{R}}},$ (2.25) $\displaystyle\text{for any
}f\in\mathop{\mathrm{Dom}}(H).$

###### Proof.

As we already noticed, under the hypothesis
$\mathop{\mathrm{Ran}}\bigl{(}P_{0}|_{\mathop{\mathrm{Dom}}(H)}\bigr{)}\subset\mathop{\mathrm{Dom}}(H)$
the inclusion (2.23) follows immediately from (2.19). Assume that
$f\in\mathop{\mathrm{Dom}}(H)$ and $t\in{\mathbb{R}}$. By (2.23) we have
$P_{t}f\in\mathop{\mathrm{Dom}}(H)$. Now take $0\neq\tau\in{\mathbb{R}}$ and
write

$\displaystyle\frac{P_{t+\tau}-P_{t}}{\tau}f$ $\displaystyle=\frac{{\rm
e}^{-{\rm i}H(t+\tau)}P_{0}{\rm e}^{{\rm i}H(t+\tau)}-{\rm e}^{-{\rm
i}Ht}P_{0}{\rm e}^{{\rm i}Ht}}{\tau}f$ $\displaystyle=\frac{{\rm e}^{-{\rm
i}H(t+\tau)}P_{0}{\rm e}^{{\rm i}H(t+\tau)}-{\rm e}^{-{\rm
i}H(t+\tau)}P_{0}{\rm e}^{{\rm i}Ht}+{\rm e}^{-{\rm i}H(t+\tau)}P_{0}{\rm
e}^{{\rm i}Ht}-{\rm e}^{-{\rm i}Ht}P_{0}{\rm e}^{{\rm i}Ht}}{\tau}f$
$\displaystyle=g_{1}(t,\tau,f)+g_{2}(t,\tau,f),$ (2.26)

where the vectors $g_{1}$ and $g_{2}$ are given by

$\displaystyle g_{1}(t,\tau,f)$ $\displaystyle={\rm e}^{-{\rm
i}H(t+\tau)}P_{0}{\rm e}^{{\rm i}Ht}\,\frac{{\rm e}^{{\rm i}H\tau}-I}{\tau}f,$
$\displaystyle g_{2}(t,\tau,f)$ $\displaystyle={\rm e}^{-{\rm i}Ht}\frac{{\rm
e}^{-{\rm i}H\tau}-I}{\tau}P_{0}{\rm e}^{{\rm i}Ht}f.$

For $g_{1}(t,\tau,f)$ we have

$\displaystyle g_{1}(t,\tau,f)$
$\displaystyle=g_{11}(t,\tau,f)+g_{12}(t,\tau,f),$ (2.27)

where

$\displaystyle g_{11}(t,\tau,f)$ $\displaystyle={\rm e}^{-{\rm
i}H(t+\tau)}P_{0}{\rm e}^{{\rm i}Ht}\,\left(\frac{{\rm e}^{{\rm
i}H\tau}-I}{\tau}f-{\rm i}Hf\right),$ $\displaystyle g_{12}(t,\tau,f)$
$\displaystyle={\rm e}^{-{\rm i}H(t+\tau)}P_{0}{\rm e}^{{\rm i}Ht}\,\left({\rm
i}Hf\right).$

The unitarity of ${\rm e}^{-{\rm i}H(t+\tau)}$ and ${\rm e}^{{\rm i}Ht}$
jointly with the fact that the orthogonal projection $P_{0}$ has unit norm,
$\|P_{0}\|=1$, yields

$\displaystyle\|g_{11}(t,\tau,f)\|\leq\left\|{\rm e}^{-{\rm
i}H(t+\tau)}\right\|\left\|P_{0}\right\|\left\|{\rm e}^{{\rm
i}Ht}\right\|\left\|\frac{{\rm e}^{{\rm i}H\tau}-I}{\tau}f-{\rm
i}Hf\right\|\leq\left\|\frac{{\rm e}^{{\rm i}H\tau}-I}{\tau}f-{\rm
i}Hf\right\|.$ (2.28)

Taking into account the strong differentiability of the group ${\rm e}^{{\rm
i}Ht}$ on $\mathop{\mathrm{Dom}}(H)$, from (2.28) one concludes that

$g_{11}(t,\tau,f)\to 0\quad\text{as}\quad\tau\to 0\quad(\text{for any
$t\in{\mathbb{R}}$ and any $f\in\mathop{\mathrm{Dom}}(H)$}).$ (2.29)

At the same time, in view of the strong continuity of the group ${\rm e}^{{\rm
i}Ht}$ on $\mathfrak{H}$,

$g_{12}(t,\tau,f)\mathop{\longrightarrow}\limits_{\tau\to 0}{\rm e}^{-{\rm
i}Ht}P_{0}{\rm e}^{{\rm i}Ht}\,\left({\rm i}Hf\right)={\rm
i}P_{t}Hf\quad(\text{for any $t\in{\mathbb{R}}$ and any
$f\in\mathop{\mathrm{Dom}}(H)$}).$ (2.30)

As for the term $g_{2}(t,\tau,f)$, it is easy to observe that

$\displaystyle g_{2}(t,\tau,f)$
$\displaystyle\mathop{\longrightarrow}\limits_{\tau\to 0}{\rm e}^{-{\rm
i}Ht}(-{\rm i}H)P_{0}{\rm e}^{{\rm i}Ht}f\quad(\text{for any
$t\in{\mathbb{R}}$ and any $f\in\mathop{\mathrm{Dom}}(H)$}).$

This follows again from the strong differentiability of the group ${\rm
e}^{{\rm i}Ht}$ on $\mathop{\mathrm{Dom}}(H)$, taking into account that ${\rm
e}^{{\rm i}Ht}f\in\mathop{\mathrm{Dom}}(H)$ whenever
$f\in\mathop{\mathrm{Dom}}(H)$ and then $P_{0}{\rm e}^{{\rm
i}Ht}f\in\mathop{\mathrm{Dom}}(H)$ by the hypothesis. Now it only remains to
recollect that ${\rm e}^{-{\rm i}Ht}(-{\rm i}H)P_{0}{\rm e}^{{\rm i}Ht}f=-{\rm
i}H{\rm e}^{-{\rm i}Ht}P_{0}{\rm e}^{{\rm i}Ht}f=-{\rm i}HP_{t}f,$ which means
that

$\displaystyle g_{2}(t,\tau,f)$
$\displaystyle\mathop{\longrightarrow}\limits_{\tau\to 0}-{\rm
i}HP_{t}f\quad(\text{for any $t\in{\mathbb{R}}$ and any
$f\in\mathop{\mathrm{Dom}}(H)$}).$ (2.31)

Combining this result with (2.26), (2.27), (2.29), and (2.30) completes the
proof. ∎

######  Remark 2.2.

Equality (2.25) implies that, under the hypothesis of Theorem 2.1, the
projection path (2.22) is a strong solution (on $\mathop{\mathrm{Dom}}(H)$) to
the Cauchy problem

$\displaystyle\mathrm{i}\frac{d}{dt}P_{t}$ $\displaystyle=[H,P_{t}],$ (2.32)
$\displaystyle P_{t}\big{|}_{t=0}$ $\displaystyle=P_{0},$ (2.33)

where $\frac{d}{dt}P_{t}=\dot{P}_{t}$ stands for the strong derivative (2.24)
and

$[H,P_{t}]:=HP_{t}-P_{t}H,\quad\mathop{\mathrm{Dom}}([H,P_{t}])=\mathop{\mathrm{Dom}}(H),$
(2.34)

denotes the commutator of $H$ and $P_{t}$.

The following observation may be helpful in the study of variation of the
subspaces $\mathfrak{P}_{t}$ defined in (2.20).

######  Remark 2.3.

The inclusion (2.23) implies that also
$\mathop{\mathrm{Ran}}\left(P^{\perp}_{t}\big{|}_{\mathop{\mathrm{Dom}}(H)}\right)\subset\mathop{\mathrm{Dom}}(H)$.
This means that, at any moment $t\in{\mathbb{R}}$, the commutator (2.34) may
be written in the form

$[H,P_{t}]=P^{\perp}_{t}HP_{t}-P_{t}HP^{\perp}_{t}.$ (2.35)

From (2.35) it follows that, with respect to the orthogonal decomposition
$\mathfrak{H}=\mathfrak{P}_{t}\oplus\mathfrak{P}^{\perp}_{t}$, the operator
$[H,P_{t}]$ admits representation in the form of the following $2\times 2$
block off-diagonal skew-symmetric operator matrix:

$[H,P_{t}]=\begin{pmatrix}0&-P_{t}H\bigr{|}_{\mathfrak{P}^{\perp}_{t}}\\\
P^{\perp}_{t}H\bigr{|}_{\mathfrak{P}_{t}}&0\end{pmatrix},\quad
t\in{\mathbb{R}},$ (2.36)

which is considered on the domain
$\bigl{(}\mathop{\mathrm{Dom}}(H)\cap\mathfrak{P}_{t}\bigr{)}\oplus\bigl{(}\mathop{\mathrm{Dom}}(H)\cap\mathfrak{P}^{\perp}_{t}\bigr{)}=\mathop{\mathrm{Dom}}(H)$.

######  Remark 2.4.

One more observation is that for any $t\in{\mathbb{R}}$ the commutator (2.34)
of $H$ and $P_{t}$ remains unitary equivalent to its value $[H,P_{0}]$ at
$t=0$, namely,

$[H,P_{t}]=\mathrm{e}^{-\mathrm{i}Ht}[H,P_{0}]\mathrm{e}^{\mathrm{i}Ht},\quad
t\in{\mathbb{R}}.$ (2.37)

##  3\. Bounds for the speed of the subspace evolution

In this section we again assume that $H$ is a self-adjoint operator on the
Hilbert space $\mathfrak{H}$ and that $P_{0}$ is an orthogonal projection on
$\mathfrak{H}$ with $\mathop{\mathrm{Ran}}(P_{0})=\mathfrak{P}_{0}\neq\\{0\\}$
satisfying the condition (2.1).

It is well known that the set of all orthogonal projections in the Hilbert
space $\mathfrak{H}$ (and hence the set of all subspaces of $\mathfrak{H}$) is
a metric space with the distance given by the operator norm,

${\rho}(Q,R):=\|Q-R\|,\qquad{\rho}(\mathfrak{Q},\mathfrak{R}):={\rho}(Q,R),$
(3.1)

where $Q$, $R$ are arbitrary orthogonal projections and $\mathfrak{Q}$,
$\mathfrak{R}$, their respective ranges. It is worth mentioning that

$\|Q-R\|\leq 1$

for any two orthogonal projections $Q$ and $R$ in $\mathfrak{H}$ (see, e.g.,
[12, Section 34]) and, thus, we always have
$\rho(\mathfrak{Q},\mathfrak{R})\leq 1$ for any subspaces $\mathfrak{Q}$ and
$\mathfrak{R}$ in $\mathfrak{H}$.

We start with proving our first quantum speed limit for the subspace variation
based on the metric (3.1). This “naïve” speed limit follows directly from the
equation (2.25) under an additional assumption that the commutator of the
operators $H$ and $P_{0}$ is a bounded operator on its domain
$\mathop{\mathrm{Dom}}\bigl{(}[H,P_{0}]\bigr{)}=\mathop{\mathrm{Dom}}(H)$.

######  Lemma 3.1.

Assume the hypothesis of Theorem 2.1. Assume, in addition, that the commutator
$[H,P_{0}]$ considered on
$\mathop{\mathrm{Dom}}\bigl{(}[H,P_{0}]\bigr{)}=\mathop{\mathrm{Dom}}(H)$ is a
bounded operator, that is,

$V_{{H,P_{0}}}:=\mathop{\sup}\limits_{\scriptsize\begin{matrix}f\in\mathop{\mathrm{Dom}}(H)\\\
\|f\|=1\end{matrix}}\|HP_{0}f-P_{0}Hf\|<\infty,$ (3.2)

and let $P_{t}={\rm e}^{-{\rm i}Ht}P_{0}{\rm e}^{{\rm i}Ht}$,
$t\in{\mathbb{R}}$. Then the closure $\overline{[H,P_{t}]}$ of the commutator
$[H,P_{t}]$, $t\in{\mathbb{R}}$, is a bounded operator on the whole Hilbert
space $\mathfrak{H}$ and

$\bigl{\|}\overline{[H,P_{t}]}\bigr{\|}=V_{{H,P_{0}}}\quad\text{for any
\,}t\in{\mathbb{R}}.$ (3.3)

Furthermore, the following inequality holds

$\|P_{t}-P_{s}\|\leq V_{{H,P_{0}}}\,|t-s|,\quad\text{for any
\,}t,s\in{\mathbb{R}}.$ (3.4)

###### Proof.

Since $H$ is a self-adjoint operator, its domain is dense in $\mathfrak{H}$.
Then the boundedness (3.2) of the commutator $[H,P_{0}]$ on
$\mathop{\mathrm{Dom}}(H)$ implies that its closure $\overline{[H,P_{0}]}$ is
a bounded operator defined on the whole Hilbert space $\mathfrak{H}$.
Continuity of the norm implies
$\bigl{\|}\overline{[H,P_{0}]}\bigr{\|}=\|[H,P_{0}]\|=V_{{H,P_{0}}}$. By
Remark 2.4, for any $t\in{\mathbb{R}}$ the commutator $[H,P_{t}]$ is unitary
equivalent to the commutator $[H,P_{0}]$ and the same concerns their closures.
Hence, $\overline{[H,P_{t}]}$ is a bounded operator on $\mathfrak{H}$ with the
norm coinciding with that of $\overline{[H,P_{0}]}$, and this proves (3.3).

Now assume that $s,t\in{\mathbb{R}}$ and, for definiteness, let $s<t$. Since
$\overline{\mathop{\mathrm{Dom}}(H)}=\mathfrak{H}$, for the norm of
$P_{t}-P_{s}$ we have

$\|P_{t}-P_{s}\|=\mathop{\sup}\limits_{\scriptsize\begin{matrix}f\in\mathfrak{H}\\\
\|f\|=1\end{matrix}}\|P_{t}f-P_{s}f\|=\mathop{\sup}\limits_{\scriptsize\begin{matrix}f\in\mathop{\mathrm{Dom}}(H)\\\
\|f\|=1\end{matrix}}\|P_{t}f-P_{s}f\|.$ (3.5)

Then by Theorem 2.1 (in particular, by the strong differentiability (2.24) of
the path $P_{t}$) one concludes that

$\|P_{t}-P_{s}\|=\mathop{\sup}\limits_{\scriptsize\begin{matrix}f\in\mathop{\mathrm{Dom}}(H)\\\
\|f\|=1\end{matrix}}\left\|\int_{s}^{t}\dot{P}_{\tau}fd\tau\right\|\leq\int_{s}^{t}\mathop{\sup}\limits_{\scriptsize\begin{matrix}f\in\mathop{\mathrm{Dom}}(H)\\\
\|f\|=1\end{matrix}}\left\|\dot{P}_{\tau}f\right\|d\tau.$ (3.6)

Now by taking into account (2.25) and (3.3) one arrives at

$\|P_{t}-P_{s}\|\leq\int_{s}^{t}\mathop{\sup}\limits_{\scriptsize\begin{matrix}f\in\mathop{\mathrm{Dom}}(H)\\\
\|f\|=1\end{matrix}}\|HP_{\tau}f-P_{\tau}Hf\|d\tau=\int_{s}^{t}\bigl{\|}\overline{[H,P_{\tau}]}\bigr{\|}d\tau=V_{{H,P_{0}}}\,(t-s),$
(3.7)

which completes the proof. ∎

Soon we will see that there is a stronger estimate of $\|P_{t}-P_{s}\|$
through $|t-s|$ than the one presented in (3.4). The stronger estimate (see
inequality (3.16) in Theorem 3.6 below) is associated with another natural but
much less known metric on the set of all subspaces of $\mathfrak{H}$. This
metric is associated with the quantity

$\vartheta(\mathfrak{Q},\mathfrak{R}):=\arcsin(\|Q-R\|)$ (3.8)

called the maximal angle between the subspaces $\mathfrak{Q}$ and
$\mathfrak{R}$. The fact that the maximal angle (3.8) is a metric has been
proven in 1993 by L.Brown [13]. An alternative proof of this fact may be found
in [14]. Let us also refer to the discussion of the metric (3.8) in [15].

Notice that, since the maximal angle is a metric, the triangle inequality

$\displaystyle\vartheta(\mathfrak{Q},\mathfrak{R})$
$\displaystyle\leq\vartheta(\mathfrak{Q},\mathfrak{S})+\vartheta(\mathfrak{S},\mathfrak{R})$
(3.9)

holds for any subspaces
$\mathfrak{Q},\mathfrak{R},\mathfrak{S}\subset\mathfrak{H}$.

######  Remark 3.2.

Inequality $x<\arcsin x$, $x\in(0,1]$, implies that always
$\rho(\mathfrak{Q},\mathfrak{R})<\vartheta(\mathfrak{Q},\mathfrak{R})$ if
$\mathfrak{Q}\neq\mathfrak{R}$. Thus, the metric $\vartheta$ is stronger than
the metric $\rho$ in the sense that the bound
$\vartheta(\mathfrak{Q},\mathfrak{R})<c$ for some $c>0$ automatically requires
that also $\rho(\mathfrak{Q},\mathfrak{R})<c$. The converse is not true in
general.

######  Remark 3.3.

One verifies by inspection that the non-negative operator $(Q-R)^{2}$ is block
diagonal with respect to the decomposition
$\mathfrak{H}=\mathfrak{R}\oplus\mathfrak{R}^{\perp}$, more precisely,

$(Q-R)^{2}=RQ^{\perp}R+R^{\perp}QR^{\perp},$ (3.10)

which means that

$\|Q-R\|=\max\bigl{\\{}\|RQ^{\perp}R\|^{1/2},\|R^{\perp}QR^{\perp}\|^{1/2}\bigr{\\}}=\max\bigl{\\{}\|Q^{\perp}R\|,\|R^{\perp}Q\|\bigr{\\}}.$
(3.11)

######  Remark 3.4.

The concept of maximal angle between subspaces can be traced back at least to
Krein, Krasnoselsky, and Milman [16]. Under the assumption that
$(\mathfrak{Q},\mathfrak{R})$ is an ordered pair of subspaces and
$\mathfrak{Q}\neq\\{0\\}$, the notion of the (relative) maximal angle between
$\mathfrak{Q}$ and $\mathfrak{R}$ is applied in [16] to the number
$\varphi(\mathfrak{Q},\mathfrak{R})\in[0,\pi/2]$ defined by

$\sin\varphi(\mathfrak{Q},\mathfrak{R})=\sup\limits_{x\in\mathfrak{Q},\,\|x\|=1}\mathop{\rm
dist}(x,\mathfrak{R}).$ (3.12)

Obviously, equality (3.12) is equivalent to

$\sin\varphi(\mathfrak{Q},\mathfrak{R})=\|Q^{\perp}R\|.$ (3.13)

If both $\mathfrak{Q}\neq\\{0\\}$ and $\mathfrak{R}\neq\\{0\\}$ then (3.11)
implies

$\vartheta(\mathfrak{Q},\mathfrak{R})=\arcsin\bigl{(}\max\bigl{\\{}\|Q^{\perp}R\|,\|R^{\perp}Q\|\\}\bigr{)}=\max\bigl{\\{}\varphi(\mathfrak{Q},\mathfrak{R}),\varphi(\mathfrak{R},\mathfrak{Q})\bigr{\\}}.$
(3.14)

In contrast to $\varphi(\mathfrak{Q},\mathfrak{R})$, the maximal angle
$\vartheta(\mathfrak{Q},\mathfrak{R})$ is always symmetric with respect to the
interchange of the arguments $\mathfrak{Q}$ and $\mathfrak{R}$. Moreover,

$\varphi(\mathfrak{R},\mathfrak{Q})=\varphi(\mathfrak{Q},\mathfrak{R})=\vartheta(\mathfrak{Q},\mathfrak{R})\quad\text{whenever
}\|Q-R\|<1.$ (3.15)

For $\|Q-R\|<1$, this is simply a consequence of the equality
$\|Q^{\perp}R\|=\|R^{\perp}Q\|$, which is easily deduced, e.g., from [17,
Corollary 3.4 (i) and Remark 3.6].

######  Remark 3.5.

The maximal angle between subspaces admits a quantum-mechanical
interpretation. To this end, one may apply the concept of a subspace-state of
a quantum system. Namely, given a subspace $\mathfrak{Q}\subset\mathfrak{H}$,
one says that the system is in the $\mathfrak{Q}$-state if it is in a pure
state described by a (non-specified) normalized vector $x\in\mathfrak{Q}$. By
(3.12) and (3.14) the quantity $\cos^{2}\vartheta(\mathfrak{Q},\mathfrak{R})$
is then treated as a minimum probability for a quantum system which is in a
$\mathfrak{Q}$-state to be found also in an $\mathfrak{R}$-state.

By using the maximal-angle metric (3.8) we obtain the following result.

######  Theorem 3.6.

Assume the hypothesis of Lemma 3.1 and let
$\mathfrak{P}_{\tau}=\mathop{\mathrm{Ran}}\bigl{(}P_{\tau}\bigr{)}$,
$\tau\in{\mathbb{R}}$. Then the following inequality holds:

$\vartheta\bigl{(}\mathfrak{P}_{s},\mathfrak{P}_{t}\bigr{)}\leq
V_{{H,P_{0}}}\,|t-s|\quad\text{for any \,}s,t\in{\mathbb{R}}.$ (3.16)

###### Proof.

Assume, for definiteness, that $s\neq t$ and set

$\tau_{j}=s+j\,\,\frac{t-s}{n},\quad j=0,1,\ldots,n,$ (3.17)

where $n$ is a natural number, $n\in{\mathbb{N}}$. Notice that $\tau_{0}=s$
and $\tau_{n}=t$. Under the assumption that $n\geq 2$, by applying the
triangle inequality (3.9) to the subspaces $\mathfrak{P}_{s}$,
$\mathfrak{P}_{t}$, and intermediate subspaces $\mathfrak{P}_{\tau_{j}}$,
$j=1,2,\ldots,n-1$, one arrives at the following bound for the maximal angle
$\vartheta(\mathfrak{P}_{s},\mathfrak{P}_{t})$ between the subspaces
$\mathfrak{P}_{s}$ and $\mathfrak{P}_{t}$:

$\vartheta(\mathfrak{P}_{0},\mathfrak{P}_{t})\leq\sum_{j=1}^{n}\arcsin\bigl{\|}P_{\tau_{j}}-P_{\tau_{j-1}}\bigr{\|}.$
(3.18)

By (3.17) we have

$|\tau_{j}-\tau_{j-1}|=\frac{|t-s|}{n},\quad j=1,2,\ldots,n.$ (3.19)

Now take $n$ such that $V_{{H,P_{0}}}\frac{|t-s|}{n}\leq 1$. Then combining
(3.18) and (3.19) with the estimate (3.4) in Lemma 3.1 yields

$\displaystyle\vartheta(\mathfrak{P}_{0},\mathfrak{P}_{t})\leq$
$\displaystyle\sum_{j=1}^{n}\arcsin\bigl{(}V_{{H,P_{0}}}\,|\tau_{j}-\tau_{j-1}|\bigr{)}=n\,\arcsin\frac{V_{{H,P_{0}}}|t-s|}{n}.$
(3.20)

Passing in (3.20) to the limit $n\to\infty$ one arrives at (3.16), completing
the proof. ∎

######  Remark 3.7.

Given $s\neq t$ and $V_{P_{0},H}>0$, the bound (3.16) is more tight for
$\|P_{s}-P_{t}\|$ than the bound (3.4) (cf. Remark 3.2).

######  Remark 3.8.

Under the hypothesis of Lemma 3.1, by Remark 2.3 from (3.2) it follows that

$V_{{H,P_{t}}}=\|{P_{t}HP_{t}^{\perp}}\|=\|{P_{t}^{\perp}HP_{t}}\|\quad(=\|\overline{P_{t}HP_{t}^{\perp}}\|=\|\overline{P_{t}^{\perp}HP_{t}}\|)\quad\text{for
any }t\in{\mathbb{R}}$ (3.21)

and, in particular,

$V_{{H,P_{0}}}=\|{P_{0}HP_{0}^{\perp}}\|=\|{P_{0}^{\perp}HP_{0}}\|.$ (3.22)

Therefore, the bound (3.16) may be interpreted in the sense that only the off-
diagonal entries $P_{0}H\bigr{|}_{\mathfrak{P}_{0}^{\perp}}$ and
$P_{0}^{\perp}H\bigr{|}_{\mathfrak{P}_{0}}$ in the block matrix representation
(2.10) of the Hamiltonian $H$ contribute into the variation of the subspace
$\mathfrak{P}_{0}$. If $H$ is block diagonal with respect to the decomposition
$\mathfrak{H}=\mathfrak{P}_{0}\oplus\mathfrak{P}_{0}^{\perp}$ and, hence, the
subspace $\mathfrak{P}_{0}$ is reducing for $H$, it does not change with time
at all. In particular, none of the spectral subspaces of $H$ can be a subject
of the time evolution.

######  Corollary 3.9.

Under the hypothesis of Lemma 3.1, suppose that $T_{\theta}$ is the time when
the maximal angle between the initial subspace $\mathfrak{P}_{0}$ and a
subspace in the path $\mathfrak{P}_{t}$, $t\geq 0$, reaches the value of
$\theta$, $0<\theta\leq\frac{\pi}{2}$, that is,

$\vartheta\bigl{(}\mathfrak{P}_{0},\mathfrak{P}_{T_{\theta}})=\theta.$ (3.23)

Then222For the case where $V_{H,P_{0}}=0$, which implies that
$\mathfrak{P}_{0}$ is a reducing subspace of $H$, one adopts the convention
that $T_{\theta}=\infty$.

$T_{\theta}\geq\frac{\theta}{V_{{H,P_{0}}}}\,.$ (3.24)

######  Example 3.10.

Let the Hamiltonian $H$ describe a two-level quantum system with bound states
$e_{1}$ and $e_{2}$, that is, $\|e_{1}\|=\|e_{2}\|=1$,
${\langle}e_{1},e_{2}{\rangle}=0$, the Hilbert space $\mathfrak{H}$ is the
span of the vectors $e_{1}$ and $e_{2}$, and

$H=E_{1}{\langle}\cdot,e_{1}{\rangle}e_{1}+E_{2}{\langle}\cdot,e_{2}{\rangle}e_{2},$

where the binding energies $E_{1}$ and $E_{2}$ are supposed to be different,
$E_{1}\neq E_{2}$. Assume that $P_{0}$ is the orthogonal projection on the
one-dimensional subspace $\mathfrak{P}_{0}$ spanned by the vector
$e=\frac{1}{\sqrt{2}}(e_{1}+e_{2})$. One immediately observes that

$[H,P_{0}]=\frac{E_{2}-E_{1}}{2}\bigl{(}\langle\cdot,e_{1}\rangle
e_{2}-\langle\cdot,e_{2}\rangle e_{1}\bigr{)}$ (3.25)

and then for the norm of the commutator of $H$ and $P_{0}$ we get

$V_{H,P_{0}}=\bigl{\|}[H,P_{0}]\bigr{\|}=\frac{|E_{2}-E_{1}|}{2}.$ (3.26)

One also verifies by inspection that this norm may be written as

$V_{H,P_{0}}=\bigl{(}\|He\|^{2}-\langle He,e\rangle^{2}\bigr{)}^{1/2}.$ (3.27)

Furthermore, an elementary computation shows that for any
$\tau\in{\mathbb{R}}$ the orthogonal projection $P_{\tau}={\rm e}^{-{\rm
i}H\tau}P_{0}{\rm e}^{{\rm i}H\tau}$ is given by

$P_{\tau}=\frac{1}{2}\left(\langle\cdot,e_{1}\rangle e_{1}+{\rm e}^{-{\rm
i}(E_{2}-E_{1})\tau}\langle\cdot,e_{1}\rangle e_{2}+{\rm e}^{{\rm
i}(E_{2}-E_{1})\tau}\langle\cdot,e_{2}\rangle e_{1}+\langle\cdot,e_{2}\rangle
e_{2}\right).$ (3.28)

Then for any $s,t\in{\mathbb{R}}$ we have

$P_{t}-P_{s}=\frac{1}{2}\left({\rm e}^{-{\rm i}(E_{2}-E_{1})t}-{\rm e}^{-{\rm
i}(E_{2}-E_{1})s}\right)\langle\cdot,e_{1}\rangle e_{2}+\frac{1}{2}\left({\rm
e}^{{\rm i}(E_{2}-E_{1})t}-{\rm e}^{{\rm
i}(E_{2}-E_{1})s}\right)\langle\cdot,e_{2}\rangle e_{1}.$ (3.29)

The eigenvalues of the rank-two operator (3.29) are

$\lambda_{\pm}=\pm\sin\left(\frac{|E_{2}-E_{1}|}{2}|t-s|\right),$ (3.30)

which means in particular that

$\vartheta(\mathfrak{P}_{s},\mathfrak{P}_{t})=\arcsin\|P_{t}-P_{s}\|=\frac{|E_{2}-E_{1}|}{2}|t-s|\,\text{\,
whenever }\frac{|E_{2}-E_{1}|}{2}|t-s|\leq\frac{\pi}{2},$ (3.31)

where, as usually, $\mathfrak{P}_{\tau}=\mathop{\mathrm{Ran}}(P_{\tau})$,
$\tau\in{\mathbb{R}}$.

######  Remark 3.11.

Example 3.10 is used in many papers on quantum speed limits (see, e.g., [2, 3,
8, 9]). In particular, this example proves the sharpness of both the
Mandelstam-Tamm and Margolus-Levitin inequalities (see, e.g., [2, Section
2.4]). Example 3.10 also works for the bounds (3.16) and (3.24). In this
example, due to (3.26) and (3.31) both of these bounds transform into
equalities, which proves that both the bounds (3.16) and (3.24) are optimal.

######  Theorem 3.12.

Assume the hypothesis of Theorem 2.1 and let
$\mathfrak{P}_{\tau}=\mathop{\mathrm{Ran}}\bigl{(}{\rm e}^{-{\rm
i}H\tau}P_{0}{\rm e}^{{\rm i}H\tau}\bigr{)}$, $\tau\in{\mathbb{R}}$. Assume,
in addition, that

$\Delta
E_{\mathfrak{P}_{0}}:=\sup\limits_{\scriptsize\begin{matrix}f\in\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H)\\\
\|f\|=1\end{matrix}}\bigl{(}\|Hf\|^{2}-{\langle}Hf,f{\rangle}^{2}\bigr{)}^{1/2}<\infty.$
(3.32)

Then

$\vartheta\bigl{(}\mathfrak{P}_{s},\mathfrak{P}_{t}\bigr{)}\leq\Delta
E_{\mathfrak{P}_{0}}\,|t-s|\quad\text{for any \,}t,s\in{\mathbb{R}},$ (3.33)

and333For the case where $\Delta E_{\mathfrak{P}_{0}}=0$, which implies that
$\mathfrak{P}_{0}$ is the eigenspace associated with an eigenvalue of $H$, one
adopts the convention that $T_{\theta}=\infty$ (cf. footnote on page 2).

$T_{\theta}\geq\frac{\theta}{\Delta E_{\mathfrak{P}_{0}}}\,,$ (3.34)

where $\theta$, $T_{\theta}$ are the same as in Corollary 3.9.

###### Proof.

We start with the elementary observation that

$\|P_{0}^{\perp}Hf\|=\bigl{(}\|Hf\|^{2}-\|P_{0}Hf\|^{2}\bigr{)}^{1/2}\quad\text{for
any }f\in\mathop{\mathrm{Dom}}(H).$ (3.35)

If, in addition, $f\in\mathfrak{P}_{0}$ and $\|f\|=1$ then automatically
$\|P_{0}y\|\geq|\langle y,f\rangle|$ for any $y\in\mathfrak{H}$. In
particular,

$\|P_{0}Hf\|\geq\langle Hf,f\rangle\quad\text{for any
}f\in\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H),\,\|f\|=1.$ (3.36)

Hence, from (3.35) it follows that

$\|P_{0}^{\perp}Hf\|\leq\bigl{(}\|Hf\|^{2}-\langle
Hf,f\rangle^{2}\bigr{)}^{1/2}\quad\text{for any
}f\in\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H),\,\|f\|=1.$ (3.37)

Meanwhile, by Remark 3.8 (see second equality in (3.22)) we have

$V_{H,P_{0}}=\sup\limits_{\scriptsize\begin{matrix}f\in\mathop{\mathrm{Dom}}(H)\\\
\|f\|=1\end{matrix}}\|P_{0}^{\perp}HP_{0}f\|=\sup\limits_{\scriptsize\begin{matrix}f\in\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H)\\\
\|f\|=1\end{matrix}}\|P_{0}^{\perp}Hf\|,$ (3.38)

by taking into account that, by the hypothesis, the linear set
$\mathop{\mathrm{Dom}}(H)$ is invariant under $P_{0}$ and then
$P_{0}\mathop{\mathrm{Dom}}(H)=\mathfrak{P}_{0}\cap\mathop{\mathrm{Dom}}(H)$
(see equalities (2.1)–(2.3)). By combining (3.38) with (3.32) and (3.37), one
obtains

$V_{H,P_{0}}\leq\Delta E_{\mathfrak{P}_{0}}.$ (3.39)

Then (3.33) follows from the estimate (3.16) in Theorem 3.6, while (3.34) is
implied by the bound (3.24) in Corollary 3.9. The proof is complete. ∎

######  Remark 3.13.

We underline that the above proof is based on the inequality (3.39). Due to
this inequality, the hypothesis of Theorem 3.12 implies the one of Theorem 3.6
and the bound (3.24) implies the bound (3.34). In this sense, Theorem 3.12 is
a weaker version of Theorem 3.6 but the bound (3.34) much closer resembles a
Mandelstam-Tamm-Fleming bound (1.7). (Also notice that in the case of a one-
dimensional subspace $\mathfrak{P}_{0}$ inequality (3.39) turns into equality
and both the bounds (3.24) and (3.34) reduce to the Mandelstam-Tamm-Fleming
bound (1.7) with the energy dispersion (1.4) involving a state vector
$\psi_{0}$ spanning $\mathfrak{P}_{0}$.)

######  Remark 3.14.

Combining equality (3.31) with equalities (3.26) and (3.27) shows that in
Example 3.10 both bounds (3.33) and (3.34) turn into precise equalities. Thus,
Example 3.10 proves that these bounds are sharp.

The following proposition is verified by direct inspection.

######  Proposition 3.15.

Assume that $T$ is a linear operator on a Hilbert space $\mathfrak{H}$ with
domain $\mathop{\mathrm{Dom}}(T)$. Than the following identity holds

$\displaystyle\|Tx\|^{2}-|{\langle}Tx,x{\rangle}|^{2}=\|(T-cI)x\|^{2}-|{\langle}(T-cI)x,x{\rangle}|^{2}$
(3.40) $\displaystyle\hskip 8.5359pt\text{ for any\, $c\in\mathbb{C}$ and
any\, $x\in\mathop{\mathrm{Dom}}(T)$, $\|x\|=1$}.$

The statement below is rather well known. We present and prove it only for
convenience of the reader.

######  Lemma 3.16.

Let $T$ be a bounded self-adjoint operator on the Hilbert space
$\mathfrak{H}$. Then the following inequalities hold:

$0\leq\|Tx\|^{2}-{\langle}Tx,x{\rangle}^{2}\leq\frac{1}{4}(M-m)^{2},\quad\text{for
any\, $x\in\mathfrak{H}$, $\|x\|=1$,}$ (3.41)

where $m=\min\bigl{(}\operatorname{spec}(T)\bigr{)}$ and
$M=\max\bigl{(}\operatorname{spec}(T)\bigr{)}$ are the upper and lower bounds
of the spectrum of $T$, respectively.

###### Proof.

The left inequality in (3.41) is obvious since
$|{\langle}Tx,x{\rangle}|\leq\|Tx\|\|x\|=\|Tx\|$ because of $\|x\|=1$. Now
choose $c=\frac{1}{2}(m+M)$. By Proposition 3.15 one infers that

$\|Tx\|^{2}-{\langle}Tx,x{\rangle}^{2}=\|T_{c}x\|^{2}-|{\langle}T_{c}x,x{\rangle}|^{2}\quad\text{for
any\, $x\in\mathfrak{H}$, $\|x\|=1$},$ (3.42)

where $T_{c}=T-cI$. Notice that $\|T_{c}\|=\frac{1}{2}(M-m)$. Then from (3.42)
it follows that

$\displaystyle\|Tx\|^{2}-{\langle}Tx,x{\rangle}^{2}$
$\displaystyle\leq\|T_{c}x\|^{2}\leq\frac{1}{4}(M-m)^{2},$ (3.43)

proving the right inequality in (3.41) and, thus, completing the whole proof.
∎

######  Remark 3.17.

Combining equalities (3.26) and (3.27) in Example 3.10 proves that the upper
bound in (3.41) is sharp.

Our final bound only concerns the special case of bounded Hamiltonians.

######  Theorem 3.18.

Let $H$ be a bounded self-adjoint operator on the Hilbert space $\mathfrak{H}$
and let

$E_{\rm min}(H)=\min\bigl{(}\mathop{\mathrm{spec}}(H)\bigr{)},\quad E_{\rm
max}(H)=\max\bigl{(}\mathop{\mathrm{spec}}(H)\bigr{)}$ (3.44)

be the upper and lower bounds of the spectrum of $H$, respectively. Let
$P_{t}$, $t\geq 0$, be the projection path (2.22) where $P_{0}$ is an
orthogonal projection in $\mathfrak{H}$. Then

$\vartheta\bigl{(}\mathfrak{P}_{s},\mathfrak{P}_{t}\bigr{)}\leq\frac{E_{\rm
max}(H)-E_{\rm min}(H)}{2}\,|t-s|,\quad\text{for any \,}s,t\in{\mathbb{R}},$
(3.45)

where $\mathfrak{P}_{\tau}=\mathop{\mathrm{Ran}}(P_{\tau})$,
$\tau\in{\mathbb{R}}$. Furthermore, the following lower bound holds444For the
case where $E_{\rm max}(H)-E_{\rm min}(H)=0$, which implies that $H$ is a
multiple of unity, one adopts the convention that $T_{\theta}=\infty$ (cf.
footnotes on pages 2 and 3).:

$T_{\theta}\,\geq\frac{2\theta}{E_{\rm max}(H)-E_{\rm min}(H)}\,,$ (3.46)

where $\theta$ and $T_{\theta}$ are the same as in Corollary 3.9.

###### Proof.

In the case under consideration, the quantity $E_{\mathfrak{P}_{0}}$ from
(3.32) may be rewritten as

$\displaystyle\Delta E_{\mathfrak{P}_{0}}=$
$\displaystyle\sup\limits_{\scriptsize\begin{matrix}f\in\mathfrak{P}_{0}\\\
\|f\|=1\end{matrix}}\bigl{(}\|Hf\|^{2}-{\langle}Hf,f{\rangle}^{2}\bigr{)}^{1/2}\leq\sup\limits_{\scriptsize\begin{matrix}f\in\mathfrak{H}\\\
\|f\|=1\end{matrix}}\bigl{(}\|Hf\|^{2}-{\langle}Hf,f{\rangle}^{2}\bigr{)}^{1/2},$
(3.47)

since the self-adjoint operator $H$ is bounded and
$\mathop{\mathrm{Dom}}(H)=\mathfrak{H}$. Then by Lemma 3.16 one concludes that

$\Delta E_{\mathfrak{P}_{0}}\leq\frac{E_{\rm max}(H)-E_{\rm
min}(H)}{2},\quad\text{for any subspace
\,}\mathfrak{P}_{0}\subset\mathfrak{H}.$ (3.48)

Now inequality (3.33) in Theorem 3.12 implies the estimate (3.45) while the
lower bound (3.46) follows from the lower bound (3.34), and this completes the
proof. ∎

##  4\. Summary and future perspectives

This paper is aimed in establishing sharp lower bounds on the time required
for an initial state subspace $\mathfrak{P}_{0}$ of a quantum system described
by a Hamiltonian $H$ to evolve into another subspace particularly positioned
with respect to the subspace $\mathfrak{P}_{0}$. The operator $H$ is assumed
to be time-independent and self-adjoint; it is allowed to be unbounded. In the
latter case, we require, in addition, that the domain
$\mathop{\mathrm{Dom}}(H)$ of $H$ is invariant under the orthogonal projection
$P_{0}$ on $\mathfrak{P}_{0}$, that is,
$P_{0}\mathop{\mathrm{Dom}}(H)\subset\mathop{\mathrm{Dom}}(H)$. As a measure
of the difference between the subspaces $\mathfrak{P}_{s}$ and
$\mathfrak{P}_{t}$, $s,t\in{\mathbb{R}}$, in the subspace evolution path
generated by $H$ out of $\mathfrak{P}_{0}$ we use the maximal angle
$\vartheta(\mathfrak{P}_{s},\mathfrak{P}_{t})$ between them, defined as in
(1.9).

Our first principal result (see Theorem 3.6) is as follows. Assume that the
commutator of $H$ and $P_{0}$ is a bounded operator on
$\mathop{\mathrm{Dom}}(H)$ and let $V_{H,P_{0}}=\|[H,P_{0}]\|$. Then for any
$s,t\in{\mathbb{R}}$ the maximal angle between the subspaces
$\mathfrak{P}_{s}$ and $\mathfrak{P}_{t}$ satisfies inequality
$\vartheta(\mathfrak{P}_{s},\mathfrak{P}_{t})\leq V_{H,P_{0}}|s-t|$.
Furthermore, we show that this inequality is optimal. Thus, it is the quantity
$V_{H,P_{0}}$ that bounds the maximal possible “angular” speed of variation of
the subspaces in the path $\mathfrak{P}_{t}$, $t\in{\mathbb{R}}$. In
particular, we make the conclusion that $T_{\theta}\geq\theta/V_{{H,P_{0}}}$
where $T_{\theta}$ is the first time moment when the maximal angle between
$\mathfrak{P}_{0}$ and $\mathfrak{P}_{t}$ reaches the value of $\theta$,
$0<\theta\leq\frac{\pi}{2}$ (see Corollary 3.9).

The second principal result of the paper is based on the bound (3.39),
$V_{H,P_{0}}\leq\Delta E_{\mathfrak{P}_{0}}$, where $\Delta
E_{\mathfrak{P}_{0}}$ denotes the maximum energy dispersion (3.32) over the
subspace $\mathfrak{P}_{0}$. By using this bound we derive for $T_{\theta}$
another sharp lower estimate $T_{\theta}\geq\theta/\Delta
E_{\mathfrak{P}_{0}}$ (see Theorem 3.12 and Remark 3.14) which is, in general,
weaker than our previous lower limit $T_{\theta}\geq\theta/V_{{H,P_{0}}}$.
However the estimate $T_{\theta}\geq\theta/\Delta E_{\mathfrak{P}_{0}}$ is
featuring much closer resemblance to a Mandelstam-Tamm-Fleming bound (1.7),
which is a particular case of it.

Surely, it is also interesting to have for $T_{\theta}$ a lower bound written
directly in terms of the spectrum of the Hamiltonian $H$. We have found a
bound of such a kind, namely the bound (3.46) in Theorem 3.18. It should be
underlined, however, that this bound only works in the case where $H$ is a
bounded operator.

We believe that the established bounds on the speed of evolution of a quantum
subspace admit an extension to the case of some unbounded time-dependent self-
adjoint Hamiltonians, and we work on this. It is worth noting that this
extension is not straightforward since, unlike the one-parameter evolution
semigroup entries (2.16), the corresponding propagators do not commute with a
time-dependent Hamiltonian.

Another challenging problem consists in establishing bounds on the evolution
speed for a subspace whose variation is governed by a non-Hermitian
Hamiltonian. Of course, the general non-Hermitian case should be extremely
difficult. Furthermore, this case is not of direct interest for quantum
physics. However, special cases where a non-Hermitian Hamiltonian is similar
to a self-adjoint operator are definitely of interest. This concerns, in
particular, the case of $PT$-symmetric Hamiltonians with real spectrum where
some quantum speed limits have already been established for the evolution of
pure states (see [8], [9], and references therein). One may expect that in
this case quantum speed limits exist for the evolution of subspaces as well.

## References

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  * [9] W. H. Wang, Z. L. Chen, Y. Song, and Y. J. Fan, Optimal time evolution for pseudo-Hermitian Hamiltonians, Theor. Math. Phys. 204 (2020), 1020–1032.
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  * [12] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover Publications, N.Y. (1993).
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#  _problexity_ — an open-source _Python_ library for binary classification
problem complexity assessment

Joanna Komorniczak and Pawel Ksieniewicz  Department of Systems and Computer
Networks,
Faculty of Information and Communication Technology,
Wrocław University of Science and Technology,
Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

###### Abstract

The classification problem’s complexity assessment is an essential element of
many topics in the supervised learning domain. It plays a significant role in
meta-learning – becoming the basis for determining meta-attributes or multi-
criteria optimization – allowing the evaluation of the training set resampling
without needing to rebuild the recognition model. The tools currently
available for the academic community, which would enable the calculation of
problem complexity measures, are available only as libraries of the _C++_ and
_R_ languages. This paper describes the software module that allows for the
estimation of 22 complexity measures for the Python language – compatible with
the _scikit-learn_ programming interface – allowing for the implementation of
research using them in the most popular programming environment of the
_machine learning_ community.

###### keywords:

Problem complexity , Classification , Python

††journal: Neurocomputing

##  1 Motivation and significance

Proper evaluation of the algorithms and methods proposed for solving the
pattern classification task, in accordance with good practices adopted in the
machine learning research environment [1], requires extensive experiments
performed on a large pool of appropriately diverse data sets [2]. In a typical
approach to designing an experimental evaluation procedure, the usefulness of
a selected group of problems is most often assessed by basic reporting of the
sets dimensionality, the number of examples describing them, the number of
problem classes [3] and their prior distribution – in the case of non-
uniformity of their representation [4]. However, it is necessary to remember
that these are only simple measures, briefly describing the problem difficulty
without giving the researcher sufficient insight into the actual complexity of
the task. A true complexity of a problem is contained not only in the basic
characteristics of problem space or class imbalance but also in problem-
specific distribution, understood as its linearity, neighborhood
characteristic, geometrical and topological complexity, and feature dependency
[5].

The classification task is the main issue of supervised machine learning,
finding its applications in almost every branch of life and science, starting
from economics, through epidemiology and medicine, to machine vision and
predictive maintenance systems [6]. Such a variety of undertaken problems
leads to a plethora of various difficulties, expressed in a multiplicity of
class-building clusters, a significant imbalance ratio, or significant class
overlap, to enumerate a few. The considered complexity metrics demonstrate the
potential for assessing the diversity of collected data sets which were
precisely described and spanned over the taxonomy by Lorena et al. [7].

The computations of the problem complexity do not only find their application
in the proper selection of benchmark datasets for the experimental evaluation.
Their application in _meta-learning_ solutions has been particularly popular
in recent years [8], being one of the primary sources of meta-features for
problem identification. Such an approach allows for the induction of task
representations [9] and the automation of the deep neural networks structure
configuration [10]. The measures of problem complexity are also used in
geospatial data – in filtering the predictor noise [11] – or in the difficulty
metrics dedicated to spectral data [12]. Preliminary works assessing the
usefulness of such measures in data stream processing are also present in the
literature [13].

All the above-mentioned examples concern research from the narrow span of the
last few years, in which researchers carried out elements of problem
complexity analysis with the use of two problem-complexity libraries currently
available to the scientific community.

The first library – DCoL – was developed in 2010 by the team of _Universitat
Ramon Llull_ and _Bell Laboratories_ [14] and is an implementation of 14 basic
complexity metrics for the _C++_ language. In 2017 its source code was made
available on the GitHub111https://github.com/nmacia/dcol.

A second library, ECoL, published as supporting software for Lorena et al. [7]
publication, has been developed since September 2016 and is an implementation
of 22 metrics in _R_ language, currently being the most comprehensive solution
for this type available to the scientific community. It reached its current
version (`0.3.0`) in December 2020. Its entire version history is available in
public GitHub repository222https://github.com/lpfgarcia/ECoL/.

It is important to emphasize that in recent years, the _Python_ programming
language has started playing a much more significant role in the development
of machine learning methods than any other experimental environment [15].

This publication presents a _problexity_ library, containing the
implementation of 22 problem complexity measures divided to six categories:
(_i_) feature-based, (_ii_) linearity, (_iii_) neighborhood, (_iv_) network,
(_v_) dimensionality, and (_vi_) class imbalance as well as a
_ComplexityCalculator_ class, introducing additional utilities facilitating
research and enabling simple expansion of the module with additional metrics.
By proposing this library, new methods considering the classification
complexity can be developed, which will further impact the evolution of
machine learning algorithms both in the meta-learning field and in other
research applications from recent years.

The measures of the problem complexity can be used as a substitute criterion
in optimization tasks. We can expect that the quality of the classification
will depend on the problem’s difficulty expressed in measures of complexity.
Testing the classifier’s ability to recognize objects, often using cross-
validation and induction algorithms with computational overhead significantly
larger than measure computation, will be time consuming. We can potentially
speed up the optimization process by problem complexity assessment as an
alternative criterion.

##  2 Software description

This chapter contains the software description of the library. It will present
the package structure, a minimal processing example, and an exemplary analysis
of the results. The chapter will illustrate the implemented measures of
problem complexity and the _ComplecityCalculator_ model.

###  2.1 Software Architecture

The library consists of two main elements – the _measures_ submodule and the
_ComplexityCalculator_ class. Within the measures, six categories are
distinguished:

  * 1.

_Feature-based_ containing $F1$, $F1v$, $F2$, $F3$, and $F4$ measures,

  * 2.

_Linearity_ containing $L1$, $L2$, and $L3$ measures,

  * 3.

_Neighborhood_ with $N1$, $N2$, $N3$, $N4$, $T1$, and $LSC$ measures,

  * 4.

_Network_ containing $density$, $ClsCoef$, and $Hubs$ measures,

  * 5.

_Dimensionality_ containing $T2$, $T3$, and $T4$ measures,

  * 6.

_Class imbalance_ containing $C1$ and $C2$ measures.

The _ComplexityCalculator_ module enables the computation and analysis of data
sets in the context of the difficulty of the classification task. It offers a
set of methods that allows calculating the measure values and presenting the
result as a single score, report, or illustrative graph.

ComplexityCalculator `fit(X, y)` Calculates metrics for given dataset.
`report()` Returns report of problem complexity. `score()` Returns integrated
score of problem complexity. `plot()` Draws `matplotlib` figure illustrating
problem complexity. Measures _f1(), f1v(), f2(), f3(), f4()_ _feature based_
_l1(), l2(), l3()_ _linearity_ _n1(), n2(), n3(), n4(), t1(), lsc()_
_neighborhood_ _density(), clsCoef(), hubs()_ _network_ _t2(), t3(), t4()_
_dimensionality_ _c1(), c2()_ _class imbalance_

Figure 1: Overall schema of the software architecture

###  2.2 Measures

The package divides measures into six categories, introduced in the
publication by Lorena et al. [7]. The measures return $0$ or a value close to
$0$ for simple problems and values close to $1$ for complex problems. The only
measures not limited to 1 are $T2$ and $T3$ from the _dimensionality_
category. Four implemented measures ($L1$, $L2$, $L3$, $N4$) are non-
deterministic; therefore, subsequent calculations on the same data can yield
varying results. In the case of measures from the Linear category, this
behavior results from Linear SVM classifier optimization, whose weights are
initialized randomly. In the case of L3 and N4 measures, the generation of
synthetic instances in a randomized manner makes the calculations non-
deterministic.

####  2.2.1 Feature-based measures

The measures describe the ability of features to separate classes in the
classification problem. They analyze features separately or evaluate how
attributes work together.

  * $F1$

Maximum Fisher’s discriminant ratio.

The measure describes the overlap of feature values in each class. The inverse
of the original formulation, taking into account the most significant
discriminant ratio, is taken into account, the same as in _ECoL_ package.

  * $F1v$

Directional vector maximum Fisher’s discriminant ratio.

The measure computes projection that maximizes class separation by directional
Fisher’s criterion.

  * $F2$

Volume of overlapping region

The measure describes the overlap of the feature values within the classes. It
is determined by the minimum and maximum values of features in each class. The
overlap is then calculated and normalized by the range of values in each
class.

  * $F3$

Maximum individual feature efficiency

The measure describes the efficiency of each feature in the separation of
classes. It considers the maximum value among all features. The equation
proposed by Lorena et al. [7] has been slightly modified to obtain a maximum
complexity value of 1 in case all instances of separate classes overlap.

  * $F4$

Collective feature efficiency

The measure describes the features synergy. The instances separated by the
most discriminant attribute that was not used already are excluded from
further analysis. The process continues until all instances are classified, or
all features are used. The measure is calculated according to the number of
instances in the overlapping region and the total number of samples.

####  2.2.2 Linearity measures

The measures evaluate the level of problem class linear separation. The
measures use Linear Support Vector Machines (svm) classifier.

  * $L1$

Sum of the error distance by linear programming

The measure calculates the distance of incorrectly classified samples from the
svm hyperplane.

  * $L2$

Error rate of linear classifier

The measure is described by the error rate of the Linear svm classifier within
the dataset.

  * $L3$

Non-linearity of linear classifier The measure is described by the
classifier’s error rate on synthesized points of the dataset. The synthetic
points are obtained by linearly interpolating instances of each class. The
class of original examples determines the label of an augmented point, and the
number of artificial points is equal to the original dataset size.

####  2.2.3 Neighborhood measures

The measures analyze the neighborhood of instances in a feature space.
Neighbors of each sample are established based on the distance between problem
instances.

  * $N1$

Fraction of borderline points

The _Minimum Spanning Three_ is generated over input instances in order to
obtain this measure. The value is computed by calculating the number of edges
in the mst between examples of different classes over a total number of
samples.

  * $N2$

Ratio of intra/extra class NN distance

The measure depends on the distances of each problem instance to its nearest
neighbor of the same class and the distance to the nearest neighbor of a
different class. According to the proportions of those values, the final value
is calculated.

  * $N3$

Error rate of NN classifier

The measure is determined by the error rate of the One Nearest Neighbor
Classifier in the Leave One Out evaluation protocol.

  * $N4$

Calculates the Non-linearity of NN classifier (N4) metric

The measure is determined by the error rate of the k-Nearest Neighbor
Classifier on synthetic points, generated by linearly interpolating original
instances. The classifier is fitted on original points and evaluated on
synthetic instances.

  * $T1$

Fraction of hyperspheres covering data

The measure is defined by the number of hyperspheres needed to cover the data
divided by a number of instances. First, a hypersphere is generated for each
problem sample. A sample lies in the center of the hypersphere. Its radius is
dependent on the distance to the instance of another class. The hyperspheres
are eliminated if a different one already covers the center instance. The
elimination starts from the hyperspheres with the largest radius and continues
to the ones with a smaller radius. The hyperspheres that were not eliminated
are taken into account during the calculation of complexity.

  * $LSC$

Local set average cardinality (LSC)

The measure is dependent on the distances between instances and the distances
to the instances’ nearest enemies – the nearest sample of the opposite class.
The number of cases that lie closer to the sample than its closest enemy is
considered during the calculation.

####  2.2.4 Network measures

The measures consider the instances as the vertices of the graph. All measures
of this category generate an epsilon-Nearest Neighbours graph. The epsilon
value is set to 0.15, same as in the _ECoL_ package. The edges are selected
based on the Gower distance between samples, normalized to the range between 0
and 1. The edge is placed between the points if a normalized Gower distance is
smaller than 0.15. Edges between instances of distinct classes are removed.

  * $density$

Density metric

The measure calculates the number of edges in the final graph divided by the
total possible number of edges.

  * $clsCoef$

Clustering Coefficient metric

For the purpose of obtaining this measure, the neighborhood of each vertex is
calculated, i.e., the instances directly connected to it. Then, the number of
edges between the sample’s neighbors is calculated and divided by the maximum
possible number of edges between them. The final measure is calculated based
on the neighborhood of each point in the dataset.

  * $hubs$

Hubs metric

For the purpose of obtaining this measure, the neighborhood of each vertex is
obtained. The measure scores each sample by the number of connections to
neighbors, weighted by the number of connections the neighbors have.

####  2.2.5 Dimensionality measures

The measures analyze the relation between the number of features and the
number of instances in the dataset.

  * $T2$

Average number of features per dimension

For the purpose of obtaining this measure, the number of dimensions describing
the dataset is divided by the number of instances.

  * $T3$

Average number of PCA dimensions per points

To obtain this measure, first, the number of PCA components needed to
represent 95% of data variability is calculated. Then, the value is divided by
the instance number in the dataset.

  * $T4$

Ration of the PCA dimension to the original dimension

To obtain this measure, the number of PCA components needed to represent 95%
of data variability is divided by the original number of dimensions. This
measure describes the proportion of relevant dimensions in the dataset.

####  2.2.6 Class imbalance measures

The Class Imbalance measures evaluate the dataset based on the degree of data
imbalance.

  * $C1$

Entropy of Class Proportions The measure is obtained based on the proportion
of each class’s samples divided by the total number of samples.

  * $C2$

Imbalance Ratio

The measure is obtained based on the proportion of each class’s samples
divided by a number of opposite class samples.

###  2.3 ComplexityCalculator

The library introduces the _ComplexityCalculator_ class to facilitate the use
of measure implementation. Its objects are initialized with a list of measures
and an optional list of (_a_) category colors for visualization purposes, and
(_b_) a dictionary indicating the number of measures in a given category,
which are necessary only in case of non-default collection of measures.

By default, the module will analyze all 22 metrics of the _measures_ module.
Executing the `fit()` method, which takes a set of features $X$ and a set of
labels $y$ as an argument, will calculate the values of the metrics.

The obtained measures’ values can be accessed as a single value using the
`score()` method, optionally taking a vector of weights as a parameter. The
vector length has to correspond to the number of analyzed measures. By
default, each measure has an equal weight, which means that executing the
`score()` method will return the arithmetic mean of measured complexities.

The `report()` method provides a more detailed description of the
classification problem. The method returns a dictionary containing a summary
of each metric value, their arithmetic mean, and other data set
characteristics, such as the number of samples, dimensionality, labels, and
the number of classes with a prior probability.

The values can also be presented in the form of a graph. Executing the plot
method returns a chart that illustrates the value of each measure from
respective categories as well as the default score of the problem.

###  2.4 Minimal processing example

The _problexity_ module is open Python software released under the _GPL-3.0_
license and versioned in the public _Python Package Index_ (PyPI) repository.
Therefore, it can be easily obtained with the _pip_ package installer with the
command:

⬇

> pip install problexity

To enable the possibility to modify the measures provided by `problexity` or
in case of necessity to expand it with functions that it does not yet include,
it is also possible to install the module directly from the source code. If
any modifications are introduced, they propagate to the module currently
available to the environment.

⬇

> git clone https://github.com/w4k2/problexity.git

> cd problexity

> make install

The `problexity` module is imported in the standard Python fashion. At the
same time, for the convenience of implementation, the authors recommend
importing it under the `px` alias:

⬇

1# Importing problexity

2import problexity as px

The library is equipped with the _ComplexityCalculator_ calculator, which
serves as the basic tool for establishing metrics. The following code presents
an example of the generation of a synthetic data set – typical for the
_scikit-learn_ module – and the determination of the value of measures by
fitting the complexity model in accordance with the standard API adopted for
_scikit-learn_ estimators:

⬇

1# Loading benchmark dataset from scikit-learn

2from sklearn.datasets import load_breast_cancer

3X, y = load_breast_cancer(return_X_y=True)

4

5# Initialize CoplexityCalculator with default parametrization

6cc = px.ComplexityCalculator()

7

8# Fit model with data

9cc.fit(X,y)

As the $L1$, $L2$ and $L3$ measures use the recommended _LinearSVC_
implementation from the `svm` module of the _scikit-learn_ package in their
calculations, the warning "_ConvergenceWarning: Liblinear failed to converge,
increase the number of iterations._ " might occur. It is not a problem for the
metric calculation – only indicating the lack of linear problem separability.

The complexity calculator object stores a list of all estimated measures that
can be read by the model’s `complexity` attribute:

⬇

1cc.complexity

⬇ [0.227 0.064 0.000 0.478 0.012 0.225 0.070 0.042 0.043 0.296 0.084 0.025
0.178 0.912 0.741 0.268 0.569 0.053 0.002 0.033 0.047 0.122]

They appear in the list in the same order as the declarations of the used
metrics, which can also be obtained from the hidden method `_metrics()`:

⬇

1cc._metrics()

⬇ [’f1’, ’f1v’, ’f2’, ’f3’, ’f4’, ’l1’, ’l2’, ’l3’, ’n1’, ’n2’, ’n3’, ’n4’,
’t1’, ’lsc’, ’density’, ’clsCoef’, ’hubs’, ’t2’, ’t3’, ’t4’, ’c1’, ’c2’]

The problem difficulty score can also be obtained as a single scalar measure,
which is the arithmetic mean of all measures used in the calculation:

⬇

1cc.score()

⬇ 0.203

The _problexity_ module, in addition to raw data output, also provides two
standard representations of problem analysis. The first is a report in the
form of a dictionary presenting the number of patterns (`n_samples`),
attributes (`n_features`), classes (`classes`), their prior distribution
(`prior_probability`), average metric (`score`) and all member metrics
(`complexities`), which can be obtained using the model’s `report()` method:

⬇

1cc.report()

⬇ { ’n_samples’: 569, ’n_features’: 30, ’n_classes’: 2, ’classes’: array([0,
1]), ’prior_probability’: array([0.373, 0.627]), ’score’: 0.214,
’complexities’: { ’f1’: 0.227, ’f1v’: 0.064, ’f2’: 0.001, ’f3’: 0.478, ’f4’:
0.012, ’l1’: 0.433, ’l2’ : 0.069, ’l3’: 0.049, ’n1’: 0.043, ’n2’: 0.296, ’n3’:
0.084, ’n4’ : 0.039, ’t1’: 0.178, ’t2’: 0.053, ’t3’: 0.002, ’t4’: 0.033, ’c1’
: 0.047, ’c2’: 0.122, ’lsc’: 0.912, ’density’: 0.741, ’clsCoef’: 0.268,
’hubs’: 0.569 } }

The second form of reporting is a graph which, in the polar projection,
collates all metrics, grouped into categories using color codes:

  * red

– feature based measures,

  * orange

– linearity measures,

  * yellow

– neighborhood measures,

  * green

– network measures,

  * teal

– dimensionality measures,

  * blue

– class imbalance measures.

Each problem difficulty category occupies the same graph area, meaning that
contexts that are less numerous in metrics (class imbalance) are not dominated
in this presentation by categories described by many metrics (neighborhood).
The illustration is built with the standard tools of the `matplotlib` module
as a subplot of a figure and can be generated with the following source code:

⬇

1# Import matplotlib

2import matplotlib.pyplot as plt

3

4# Prepare figure

5fig = plt.figure(figsize=(7,7))

6

7# Generate plot describing the dataset

8cc.plot(fig, (1,1,1))

An example of a complexity graph is shown in the Figure 2.

Figure 2: Exemplary complexity graph generated by _problexity_ module

##  3 Comparison with available modules

The juxtaposition in Table 1 presents a comparison of the available libraries
analyzing the difficulty of classification problems. The columns successively
present the _ECoL_ , _DCoL_ , and the _problexity_ libraries. The rows show
the categories of the complexity measures. The values in the cells indicate
the number of available metrics compared to the number described in a
publication by Lorena et al. [7].

Since _DCoL_ was the earliest implemented library of the analyzed ones, it
contains the fewest measures. The _ECoL_ and _problexity_ libraries are based
on the same publication, so the number of measures in each category matches.
The Table also shows the availability of utilities offered by the libraries
and basic information describing them. All of the compared packages offer the
possibility of generating a report containing a summary of the values of
selected measures. In addition, the _problexity_ package includes a method
allowing to represent selected measures as a single value and as well contains
a tool for graphically presenting the measures.

Table 1: Comparison of measures and utilities available in _ECoL_ , _DCoL_ and _problexity_ area |  functionality | _ECoL_ | _DCoL_ | _problexity_
---|---|---|---|---
_Measures_ |  Feature-based | 5/5 | 5/5 | 5/5
|  Linearity | 3/3 | 3/3 | 3/3
|  Neighborhood | 5/5 | 5/5 | 5/5
|  Network | 3/3 | 0/3 | 3/3
|  Dimensionality | 3/3 | 1/3 | 3/3
|  Class imbalance | 2/2 | 0/2 | 2/2
_Utility Module_ |  Score |  |  | ✓
|  Report | ✓ | ✓ | ✓
|  Plot |  |  | ✓
_Basic information_ |  Language | R | C++ | Python
|  Current version | 0.3.0 | — | 0.3.2

##  4 Impact

In recent years, the complexity measures for classification problems have
gained particular interest in the scientific community. Their most common use
is the construction of meta-attributes (features describing sets [16]) to
automate the selection of processing flows typical of the meta-learning topic
[8]. An interesting trend here is, in particular, the construction of abstract
representations of recognition tasks [9], which allow for an initial
generalization of the problem under consideration [17], allowing for a
significant reduction of the time necessary to select the optimal
classification model for a given task [10].

An alternative field of use is the classification of difficult data, with
particular emphasis on multidimensional discrete signals – for example – in
the form of multispectral and geospatial problems [12]. On the one hand, the
measures of complexity allow for agnostic estimation of the correct structure
of the objects in the predictor’s action space [18]. On the other hand, they
are tools useful in filtering its noise [11]. The same agnostic
characteristics show a particular potential in the processing of imbalanced
data [19], allowing the quality of the proposed resampling to be assessed
without having to rebuild the recognition model [20].

As with imbalanced data, complexity measures also find their application in
processing data streams [13]. By demonstrating the relationship between the
quality of the recognition model and some currently available measures, it is
possible to use them as a proxy-classifier, which allows for a significant
reduction in the time of reviewing available solutions when using any
optimization methods [21].

The field of classification problem complexity is still very active, not only
in applications described above but also in the proposals of new measures that
appear each year. The newly proposed measures allow the assessment of other
processing contexts such as category learning [22], rule-based dissociations
[23], or lost points identification [24], to enumerate a few.

##  5 Conclusions

This paper presents the _problexity_ library for _Python_ programming
language. The library contains measures for assessing the complexity of binary
classification problems. Twenty-two evaluation measures of the problem
complexity have been implemented in the following categories: feature-based,
linearity, neighborhood, network, dimensionality, and class imbalance.
Additionally, the library incorporates the _ComplexityCalculator_ module,
which provides additional tools for analyzing classification data sets.

The library was created to fill the gap related to the lack of accessible
measures for assessing the complexity of the classification problem in Python.
Increasing the availability of complexity assessment methods and creating a
tool for exploring them will allow for a more detailed analysis of data sets’
characteristics, potentially impacting the development of new machine learning
methods.

The current version of the package offers a set of measures adapted to binary
classification datasets, which are the most frequent objective of machine
learning applications. The intent of future versions is for the library to
include measures adapted to multiclass problems. As we pointed out in the
impact section, in the field of data complexity evaluation, new measures are
being proposed. The package will be maintained to contain other measures of
classification complexity and possibly extended with regression complexity
evaluation measures. Further works will focus on adapting the library to
analyze data streams in the context of data difficulties. Finally, the
_problexity_ library will continue to be used in research studies of the
classification problem complexity employment.

## Acknowledgements

This work was supported by the Polish National Science Centre under the grant
No. 2019/35/B/ST6/0442 as well as by the statutory funds of the Department of
Systems and Computer Networks, Faculty of Information and Communication
Technology, Wroclaw University of Science and Technology.

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## Required Metadata

## Current executable software version

Table 2: Code metadata (mandatory) Nr. |  Code metadata description |  Please fill in this column
---|---|---
C1 |  Current code version |  0.3.2
C2 |  Permanent link to code/repository used for this code version |  $https://github.com/w4k2/problexity$
C3 |  Legal Code License |  GPL-3.0
C4 |  Code versioning system used |  git
C5 |  Software code languages, tools, and services used |  python
C6 |  Compilation requirements, operating environments & dependencies |
C7 |  If available Link to developer documentation/manual |  $https://problexity.readthedocs.io$
C8 |  Support email for questions |<EMAIL_ADDRESS>