# Complexity of Pure and Mixed Qubit Geodesic Paths on Curved Manifolds

Carlo Cafaro1 and Paul M. Alsing2 1SUNY Polytechnic Institute, 12203 Albany,
New York, USA  2Air Force Research Laboratory, Information Directorate, 13441
Rome, New York, USA

###### Abstract

It is known that mixed quantum states are highly entropic states of imperfect
knowledge (i.e., incomplete information) about a quantum system, while pure
quantum states are states of perfect knowledge (i.e., complete information)
with vanishing von Neumann entropy. In this paper, we propose an information
geometric theoretical construct to describe and, to a certain extent,
understand the complex behavior of evolutions of quantum systems in pure and
mixed states. The comparative analysis is probabilistic in nature, it uses a
complexity measure that relies on a temporal averaging procedure along with a
long-time limit, and is limited to analyzing expected geodesic evolutions on
the underlying manifolds. More specifically, we study the complexity of
geodesic paths on the manifolds of single-qubit pure and mixed quantum states
equipped with the Fubini-Study metric and the Sjöqvist metric, respectively.
We analytically show that the evolution of mixed quantum states in the Bloch
ball is more complex than the evolution of pure states on the Bloch sphere. We
also verify that the ranking based on our proposed measure of complexity, a
quantity that represents the asymptotic temporal behavior of an averaged
volume of the region explored on the manifold during the evolution of the
systems, agrees with the geodesic length-based ranking. Finally, focusing on
geodesic lengths and curvature properties in manifolds of mixed quantum
states, we observed a softening of the complexity on the Bures manifold
compared to the Sjöqvist manifold.

###### pacs:

Complexity (89.70.Eg), Entropy (89.70.Cf), Probability Theory (02.50.Cw),
Quantum Computation (03.67.Lx), Quantum Information (03.67.Ac), Riemannian
Geometry (02.40.Ky).

##  I Introduction

We divide the Introduction in three parts to better motivate the selection of
our goals along with their physical relevance. In the first part, we highlight
the use of geometric concepts originally introduced in quantum computing and
later borrowed by high energy physicists to describe and, to a certain extent,
understand black holes behavior. In particular, we emphasize the geometric
characterization of some complexity notions, including gate complexity and
state complexity. In the second part, we outline several distinguishing
features that characterize the physics of systems specified by pure and mixed
quantum states. Neither geometry nor complexity are mentioned in this second
part. In the third part, we finally describe our main objectives.

###  I.1 Geometry in quantum computing and high energy physics

Geometry plays a fundamental role in science pettini00 ; pettini07 , including
quantum computing and high energy physics. In Ref. nielsen06 , Nielsen and
collaborators used methods of Riemannian geometry to propose a way of finding
efficient quantum circuits capable of performing certain computational tasks.
They proposed a geometric measure of quantum algorithm complexity for quantum
circuits constructed with unitary gates. Their formalism led to a geometric
continuous-time version of the discrete gate complexity, a measure of
complexity quantifying how hard it is to build a unitary operator nielsen10 ;
2020 . In such geometric context, finding optimal quantum circuits is
equivalent to finding the shortest path between two points in a certain curved
geometry. Essentially, one introduces a Riemannian metric in the space of
unitary operators acting on a given number of qubits. The metric quantifies
how hard it is to implement a given quantum computational task. Then, the
distance induced by the metric in the space of unitary operators is employed
as a measure of the complexity of the quantum operation. In addition to gate
complexity, one can also introduce in quantum information science the concept
of quantum computational complexity of a state, a measure quantifying how hard
it is to build a unitary transformation that transforms the reference state to
the target state nielsen10 . Geometric concepts (including actions, path
lengths, volumes, and complexity) play a fundamental role in high energy
physics as well. For instance, quantum computational complexity measures of
geometric origin appear to play a fundamental role in encoding properties of
the interiors of black holes leo18 . In Refs. leo14 ; leo15 , it was shown
that the quantum computational complexity of the dual quantum state is
proportional to the spatial volume of the Einstein-Rosen bridge (i.e., a
structure linking two sides of the Penrose diagram of an eternal anti-de
Sitter black hole). In Refs. leo16 ; leo16B , it was argued that the quantum
computational complexity of a holographic state is proportional to the action
of a certain spacetime region termed Wheeler-DeWitt patch. For very insightful
applications of Nielsen’s geometric approach to quantum computational
complexity of states and gates in the single-qubit and multi-qubit scenarios
of special relevance in high energy physics, we refer to Refs. brown19  and
auzzi21 , respectively. The analysis in Ref. brown19  is rather illuminating
because it clearly shows the effects of replacing a non deformed Bloch sphere
equipped with the usual Fubini-Study metric with a deformed Bloch sphere with
a new metric that does not treat all directions in the tangent space in a
similar manner. Indeed, in Nielsen’s geometric approach, the single-qubit
Hilbert space is equipped with a metric that stretches directions that are
hard to move in, assigning them a large distance. Two main consequences of
this new metric can be summarized as follows: First, geodesics are no longer
generated by time-independent Hamiltonians. Second, suitable choices of the
anisotropy penalty factors specifying the new metric can lead to spaces with
negative sectional curvature. This, in turn, is responsible of the chaotic
growth of perturbations (i.e., exponential maximal complexity). For a work
focusing on the connection between a geometric measure of quantum
computational complexity and negative curvature, we refer to Ref. brown17 .
The work in Ref. auzzi21  attracts great interest for several reasons,
including the fact that it addresses the issue of ergodicity of geodesics on
manifolds of negative curvature. This is especially important in view of a
potential application of thermodynamical arguments to complexity evolution. As
previously pointed out, Nielsen’s approach to quantum computation defines a
geometric measure on the space of unitary operators. In Ref. chapman18 ,
instead, the Fubini-Study metric is used to define a geometry on the space of
states to propose a complexity measure assigned to a target state. This
complexity is the minimal distance as measured by the Fubini-Study metric
among all parametrized curves on the space of states that connect the
reference state to the desired target state. Within this approach, the Fubini-
Study metric accounts for the complexity by keeping track of the changes of
the state (by means of applications of unitary operations) throughout the
preparation of the target state. In Ref. ruan21 , a notion of mixed state
complexity is extended to impure quantum states by replacing the Fubini-Study
metric with the Bures metric (or, alternatively, the quantum Fisher
information metric) and, at the same time, extending the nature of quantum
transformations acting on the state to non-unitary operations. Finally,
following what happens for pure states, the complexity of mixed states is
identified with the (Bures) length of the geodesic connecting the reference
and target mixed states. To a certain extent and to the best of our knowledge,
given the novelty of the introduction of the concept of mixed state
complexity, no comparative analysis exists in the literature between
complexity behaviors associated to physical systems specified by pure and
mixed states. We intend to cover this point in this paper.

Knowledge of system | Type of state | Purity | Von Neumann entropy | Temperature | Entanglement
---|---|---|---|---|---
Complete | Pure | Maximal | Minimal | Low | Typical
Partial | Mixed | Not maximal | Not minimal | High | Less typical
Table 1: Schematic description of physical systems in pure and mixed quantum
states in terms of purity, von Neumann entropy, temperature, and entanglement.

###  I.2 Pure and mixed quantum states

In quantum information science, when one has complete knowledge about a
quantum system, one can use a pure state to describe it. However, complete
knowledge is only available in limiting ideal (noiseless) scenarios (i.e.,
isolated/closed quantum systems). In practice, one only has partial knowledge
about a quantum system. Indeed, small errors may happen in the preparation,
evolution, or measurement of the system due to imperfect devices or to
(external) coupling with other degrees of freedom outside of the system that
one is controlling. In these realistic (noisy) cases (i.e., open quantum
systems), quantum systems are described by mixed states. These states are
specified by classical probability distributions over pure states and are used
to represent our probabilistic ignorance of a pure state. The density operator
formalism is a very powerful mathematical tool for incorporating a lack of
complete knowledge about a quantum system. Within this formalism, the
“quantumness” of the system resides in the off-diagonal entries of the density
matrix. These are interference terms between the pure states that specify the
mixture that defines the mixed state. A particular measure of noisiness of a
quantum state is the purity
P$\left(\rho\right)\overset{\text{def}}{=}\mathrm{Tr}\left(\rho^{2}\right)$ of
a density operator $\rho$. The purity of a pure state is equal to one, and the
purity of a mixed state is strictly less than one with
$1/N\leq$P$\left(\rho\right)\leq 1$ for an $N\times N$ density matrix. The
departure of a system from a pure state can also be quantified by means of the
von Neumann entropy
S${}_{vN}\left(\rho\right)\overset{\text{def}}{=}-\mathrm{Tr}\left(\rho\log\rho\right)$.
This quantity specifies the degree of mixing of the state describing a given
finite-dimensional quantum system. For a pure state, the van Neumann entropy
vanishes. Instead, for a maximally mixed state characterized by the complete
absence of off-diagonal entries in the density matrix (thus, describing
something non-interfering and seemingly classical), the von Neumann is maximal
and equals $\log\left(2\right)$ for a qubit system.

To the best of our knowledge, there does not exist any comparative geometric
analysis of the complexity of pure and mixed states in the literature. From an
intuitive standpoint, there are several reasons why one expects mixed states
to be more complex than pure states: 1) Mixed states are generally used to
describe highly entropic systems that can exhibit a temperature higher than
the one specifying systems in a pure state. In statistical mechanics, for
instance, a physical system at thermal equilibrium is described by a thermal
(Gibbs) state huang . The Gibbs state is a mixed state with a well-defined
finite temperature value. However, at zero temperature (i.e.,
$\beta\overset{\text{def}}{=}(k_{B}T)^{-1}\rightarrow\infty$ with $k_{B}$
denoting the Boltzmann constant), the system is in a pure state. In this
limiting case, the density matrix has every element zero except for a single
element on the diagonal. At infinite temperature (i.e., $\beta\rightarrow 0$),
instead, the system is in a maximally mixed state (i.e., a mixture of pure
states with equal statistical weights). For example, consider a spin-$1/2$
particle in a stationary and uniform magnetic field $B_{0}$ along the
$z$-direction. The Hamiltonian of the system can be written as
H$\overset{\text{def}}{=}(\hslash\omega_{0}/2)\sigma_{z}$ with
$\omega_{0}\overset{\text{def}}{=}\left(eB_{0}\right)/m$. Clearly, $e$ and $m$
denote the electric charge and the mass of the electron, respectively.
Moreover, $\hslash\overset{\text{def}}{=}h/(2\pi)$ is the reduced Planck
constant and, finally, $\sigma_{z}$ is the Pauli phase flip operator. At
thermal equilibrium, the density matrix of the system is given by
$\rho_{\mathrm{TE}}\left(\beta\right)\overset{\text{def}}{=}e^{-\beta\mathrm{H}}/\mathrm{Tr}(e^{-\beta\mathrm{H}})$.
A simple calculation shows that $\rho_{\mathrm{TE}}\left(\beta\right)$ becomes
a maximally mixed (or, pure) state as $T$ approaches infinity (or, zero). For
a definition of temperature of arbitrary quantum states, beyond thermal
(Gibbs) states used for physical systems at thermal equilibrium, we refer to
Ref. brunner22 . The temperature quantifies the ability of a quantum system to
cool down or heat up a thermal environment in Ref. brunner22 . Finally, for a
scheme to measure the temperature of individual pure quantum states by means
of quantum interference, we refer to Ref. mitch22 ; 2) Systems in mixed states
are less quantum (or, alternatively, more classical) than systems in pure
states. In Ref. karol98 , it is proven that entanglement, a quintessential
quantum property of physical systems, is typical of pure states, while
separability is connected with quantum mixtures. For intriguing connections of
geometric flavor among purity, separability, and complex behavior in quantum
scattering processes, we refer to Refs. kim11 ; kim12 . In stead, for possible
justifications of why chaoticity viewed as temporal complexity is softer in
quantum systems compared with classical systems, we hint at Refs. caron01 ;
caron04 ; kroger06 ; adom12a ; adom12b . Furthermore, it is known that the
existence of speed limits is not something peculiar to quantum systems
taddei13 ; campo13 ; deffner13 . Indeed, there are speed limits for classical
systems as well oku18 ; campo18 ; kosinski21 . In Ref. kosinski21 , it was
shown that the quantum counterpart of the classical speed limits derived in
Ref. oku18  are obtained by quantum systems specified by density operators
describing states that become more and more mixed as $\hslash$ approaches
zero; 3) Mixed quantum states can undergo a richer variety of transformations
compared to pure states nielsen2000 . In open system dynamics, one needs to
consider general nonunitary quantum evolutions and have the freedom to choose
a variety of distance measures between quantum states. Decoherence and
measurements are examples of noncontrollable and controllable nonunitary
processes, respectively. Quantum channels, for instance, provide us with a
formalism for discussing decoherence, the nonunitary evolution of pure states
into mixed states stefano12 . In conventional formulations of quantum
mechanics, instead, pure states can only be connected in a unitary fashion.
Moreover, the choice of geometric distance measures between pure states is
more restrained than that between impure states; 4) Mixed states evolutions
can exhibit higher speed values than the ones of pure state temporal changes.
In Ref. carlini08 , it is shown that the time optimal mixed state evolution
can be faster than the time optimal pure state evolution. In Ref. deffner13 ,
it is demonstrated that non-Markovian (i.e., memory) effects can speed up
nonunitary quantum evolutions of arbitrarily driven open quantum systems. In
Ref. girolami19 , it is pointed out that finding the optimal unitary for mixed
target states is more challenging than for pure target states; 5) Mixed qubit
states have three local degrees of freedom, while pure qubit states only have
two local degrees of freedom. From a pure geometric perspective, it is
reasonable to expect that mixed states are more complex than pure states
hornedal22 . For instance, unlike what happens in optimal-speed unitary
evolutions of systems in pure states, tight evolutions of closed quantum
systems in mixed states are typically generated by time-varying Hamiltonians
hornedal22 . We refer to Table I for a schematic description of physical
systems in pure and mixed quantum states in terms of purity, von Neumann
entropy, temperature, and entanglement.

Given the lack of a geometric comparative analysis between the complex
behaviors exhibited by physical quantum systems specified by pure and mixed
quantum states and, in addition, given the variety of distinguishing physical
features that characterize the evolution of systems specified by pure and
mixed quantum states, we intend to capture here the complexity of these
evolutions from a geometric standpoint and provide a geometrical picture of
these physical differences.

###  I.3 Our goals

In this paper, we aim to provide a comparative information geometric analysis
of the complexity of geodesic paths of pure and mixed quantum states on the
Bloch sphere and in the Bloch ball, respectively. Our investigation is
partially inspired by the above mentioned geometrically flavored
investigations. Furthermore, it is motivated by our curiosity concerning the
possibility of describing and, to a certain extent, understanding from a
geometric viewpoint the previously mentioned fingerprints of a greater degree
of complexity of mixed quantum states. Finally, it relies on our insights into
the concepts of complexity cafaro18 , geometric formulations of optimal-speed
Hamiltonian evolutions on the Bloch sphere cafaropra20 ; cafaropra22 , and the
role played by the thermodynamic length and divergence (or, alternatively,
action) in studying the complexity of minimum entropy production probability
paths in quantum mechanical evolutions cafaropre20 ; cafaropre22 . The main
questions that we address in this paper can be summarized as follows:

  1. [i]

Can we gain physical insights by identifying the distinguishing features that
characterize the geometry along evolution of pure and mixed quantum states?

  2. [ii]

Expressing the concept of complexity in terms of volumes of explored regions
on curved manifolds, do geodesic paths on manifolds of mixed quantum states
exhibit a higher degree of complexity compared to the complexity of geodesic
paths emerging from the geometry along the evolution of pure quantum states?

  3. [iii]

Does the choice of the metric on the space of mixed quantum states have
crucial observable physical effects on the complexity of the underlying
geodesic paths?

The layout of the rest of the paper is as follows. In Section II, we introduce
our proposed measure of complexity of geodesic paths on curved manifolds. In
Section III, we introduce the geodesic paths on manifolds of pure and mixed
states emerging from the Fubini-Study and the Sjöqvist metrics, respectively.
In Section IV, we study the complexity of the geodesic paths expressed in
terms of temporal averages of volume regions explored by the physical systems
during the quantum evolutions. In Section V, we include several physics
considerations, including comments on the concepts of metric, path length, and
curvature employed in our analysis. These comments also help emphasizing the
physical significance of our proposed complexity measure. In Section VI, we
present our final remarks. Finally, several technical details, including a
comparative analysis between the Sjöqvist and the Bures metrics for mixed
quantum states, appear in Appendix A, B, C, D, E, and F.

##  II Information Geometric Complexity

In this section, we present the notion of information geometric complexity
(IGC) along with the concept of information geometric entropy (IGE). These
quantities will help quantifying how complex are the evolutions of pure and
mixed states. Before introducing formal details, let us emphasize at the
outset that the IGC is essentially the exponential of the IGE. The latter, in
turn, is the logarithm of the volume of the parametric region explored by the
system during its evolution from an initial to a final configuration on the
underlying manifold. The IGE is an indicator of complexity that was initially
proposed in Ref. cafaro07  in the framework of the Information Geometric
Approach to Chaos (IGAC) cafarothesis . For clarity, we mention in this paper
only the necessary information on the IGAC. However, we recommend the
interested reader to consider the compact discussions on the IGAC in Refs.
ali18 ; ali21 .

In what follows, we begin by presenting the IGE in its original classical
setting characterized by probability density functions. Obviously, when
transitioning from classical to quantum settings, parametrized families of
probability distributions are replaced by families of parametrized density
operators.

Assume that $N$-real valued variables $\left(\xi^{1}\text{,...,
}\xi^{N}\right)$ parametrize the points $\left\\{p\left(x\text{;
}\xi\right)\right\\}$ of an $N$-dimensional curved statistical manifold
$\mathcal{M}_{s}$,

$\mathcal{M}_{s}\overset{\text{def}}{=}\left\\{p\left(x\text{;
}\xi\right):\xi\overset{\text{def}}{=}\left(\xi^{1}\text{,...,
}\xi^{N}\right)\in\mathcal{D}_{\boldsymbol{\xi}}^{\mathrm{tot}}\right\\}\text{.}$
(1)

In addition, assume that the microvariables $x$ specifying the probability
distributions $\left\\{p\left(x\text{; }\xi\right)\right\\}$ are elements of
the (continuous) microspace $\mathcal{X}$ while the macrovariables $\xi$
belong to the parameter space $\mathcal{D}_{\boldsymbol{\xi}}^{\mathrm{tot}}$
defined as,

$\mathcal{D}_{\boldsymbol{\xi}}^{\mathrm{tot}}\overset{\text{def}}{=}\left(\mathcal{I}_{\xi^{1}}\otimes\mathcal{I}_{\xi^{2}}\text{...}\otimes\mathcal{I}_{\xi^{N}}\right)\subseteq\mathbb{R}^{N}\text{.}$
(2)

Note that $\mathcal{I}_{\xi^{j}}$ in Eq. (2) is a subset of $\mathbb{R}^{N}$
and characterizes the range of acceptable values for the statistical
macrovariables $\xi^{k}$ with $1\leq k\leq N$. Within the IGAC framework, it
is argued that the IGE is a good measure of temporal complexity of geodesic
paths on $\mathcal{M}_{s}$. The IGE is given by,

$\mathcal{S}_{\mathcal{M}_{s}}\left(\tau\right)\overset{\text{def}}{=}\log\widetilde{vol}\left[\mathcal{D}_{\boldsymbol{\xi}}\left(\tau\right)\right]\text{,}$
(3)

with the average dynamical statistical volume
$\widetilde{vol}\left[\mathcal{D}_{\boldsymbol{\xi}}\left(\tau\right)\right]$
being defined as,

$\widetilde{vol}\left[\mathcal{D}_{\boldsymbol{\xi}}\left(\tau\right)\right]\overset{\text{def}}{=}\frac{1}{\tau}\int_{0}^{\tau}vol\left[\mathcal{D}_{\boldsymbol{\xi}}\left(\tau^{\prime}\right)\right]d\tau^{\prime}\text{.}$
(4)

Note that $\mathcal{D}_{\boldsymbol{\xi}}\left(\tau^{\prime}\right)$ in Eq.
(4) is an $N$-dimensional subspace of
$\mathcal{D}_{\boldsymbol{\xi}}^{\mathrm{tot}}\subseteq\mathbb{R}^{N}$ whose
elements $\left\\{\xi\right\\}$ with
$\xi\overset{\text{def}}{=}\left(\xi^{1}\text{,..., }\xi^{N}\right)$ satisfy
$\xi^{k}\left(\tau_{0}\right)\leq\xi^{k}\leq\xi^{j}\left(\tau_{0}+\tau^{\prime}\right)$
with $\tau_{0}$ denoting the initial value taken by the affine parameter
$\tau^{\prime}$ that characterizes the geodesics on $\mathcal{M}_{s}$ as will
be described in more detail shortly. In Eq. (4), the temporal average
operation is denoted with the tilde symbol. We also emphasize that two
sequential integration procedures define
$\widetilde{vol}\left[\mathcal{D}_{\boldsymbol{\xi}}\left(\tau\right)\right]$
in Eq. (4). The first integration is defined on the explored parameter space
$\mathcal{D}_{\boldsymbol{\xi}}\left(\tau^{\prime}\right)$ and leads to
$vol\left[\mathcal{D}_{\boldsymbol{\xi}}\left(\tau^{\prime}\right)\right]$.
Then, the second integration describes a temporal averaging procedure, is
performed over the duration $\tau$ of the evolution on $\mathcal{M}_{s}$, and
finally yields
$\widetilde{vol}\left[\mathcal{D}_{\boldsymbol{\xi}}\left(\tau\right)\right]$.
The volume
$vol\left[\mathcal{D}_{\boldsymbol{\xi}}\left(\tau^{\prime}\right)\right]$ in
the RHS of Eq. (4) is the volume of an extended region on $\mathcal{M}_{s}$
and is given by,

$vol\left[\mathcal{D}_{\boldsymbol{\xi}}\left(\tau^{\prime}\right)\right]\overset{\text{def}}{=}\int_{\mathcal{D}_{\boldsymbol{\xi}}\left(\tau^{\prime}\right)}\rho\left(\xi^{1}\text{,...,
}\xi^{N}\right)d^{N}\xi\text{.}$ (5)

Since we are limiting our present discussion to the IGE in the context of a
statistical manifold $\mathcal{M}_{s}$ of classical probability distributions,
$\rho\left(\xi^{1}\text{,..., }\xi^{N}\right)$ in Eq. (5) is the so-called
Fisher density and equals the square root of the determinant
$g\left(\xi\right)$ of the Fisher-Rao information metric tensor
$g_{\mu\nu}^{\mathrm{FR}}\left(\xi\right)$,
$g^{\mathrm{FR}}\left(\xi\right)\overset{\text{def}}{=}\det\left[g_{\mu\nu}^{\mathrm{FR}}\left(\xi\right)\right]$.
Therefore, $\rho\left(\xi^{1}\text{,...,
}\xi^{N}\right)\overset{\text{def}}{=}\sqrt{g^{\mathrm{FR}}\left(\xi\right)}$.
Recall that in the continuous microspace setting,
$g_{\mu\nu}^{\mathrm{FR}}\left(\xi\right)$ is defined as

$g_{\mu\nu}^{\mathrm{FR}}\left(\xi\right)\overset{\text{def}}{=}\int
p\left(x|\xi\right)\partial_{\mu}\log p\left(x|\xi\right)\partial_{\nu}\log
p\left(x|\xi\right)dx\text{,}$ (6)

with $\partial_{\mu}\overset{\text{def}}{=}\partial/\partial\xi^{\mu}$. Note
that
$vol\left[\mathcal{D}_{\boldsymbol{\theta}}\left(\tau^{\prime}\right)\right]$
in Eq. (5) assumes a more simple expression for manifolds equipped with metric
tensors specified by factorizable determinants,

$g\left(\xi\right)=g\left(\xi^{1}\text{,...,
}\xi^{N}\right)={\prod\limits_{k=1}^{N}}g_{k}\left(\xi^{k}\right)\text{.}$ (7)

In such a scenario, the IGE in Eq. (3) reduces to

$\mathcal{S}_{\mathcal{M}_{s}}\left(\tau\right)=\log\left\\{\frac{1}{\tau}\int_{0}^{\tau}\left[{\prod\limits_{k=1}^{N}}\left(\int_{\tau_{0}}^{\tau_{0}+\tau^{\prime}}\sqrt{g_{k}\left[\xi^{k}\left(\eta\right)\right]}\frac{d\xi^{k}}{d\eta}d\eta\right)\right]d\tau^{\prime}\right\\}\text{.}$
(8)

We remark the $g\left(\theta\right)$ is not factorizable when the
microvariables $\left\\{x\right\\}$ are correlated. Therefore, in this case,
one is forced to use the general definition of the IGE. We refer to Ref. ali10
for a study on the effects of microscopic correlations on the IGE of Gaussian
statistical models.

In the IGAC theoretical setting, the leading asymptotic behavior of
$\mathcal{S}_{\mathcal{M}_{s}}\left(\tau\right)$ in Eq. (8) characterizes the
complexity of the statistical models being analyzed. To this end, we consider
the leading asymptotic term in the equation for the IGE,

$\mathcal{S}_{\mathcal{M}_{s}}^{\text{{asymptotic}}}\left(\tau\right)\sim\lim_{\tau\rightarrow\infty}\left[\mathcal{S}_{\mathcal{M}_{s}}\left(\tau\right)\right]\text{.}$
(9)

Observe that $\mathcal{D}_{\boldsymbol{\xi}}\left(\tau^{\prime}\right)$
specifies the domain of integration that appears in the expression of
$vol\left[\mathcal{D}_{\boldsymbol{\xi}}\left(\tau^{\prime}\right)\right]$ in
Eq. (5), and is defined as

$\mathcal{D}_{\boldsymbol{\xi}}\left(\tau^{\prime}\right)\overset{\text{def}}{=}\left\\{\xi:\xi^{k}\left(\tau_{0}\right)\leq\xi^{k}\leq\xi^{k}\left(\tau_{0}+\tau^{\prime}\right)\right\\}\text{,}$
(10)

where $\tau_{0}\leq\eta\leq\tau_{0}+\tau^{\prime}$ and $\tau_{0}$ is the
initial value of the affine parameter $\eta$. In Eq. (10),
$\xi^{k}=\xi^{k}\left(\eta\right)$ satisfy the geodesic equations

$\frac{d^{2}\xi^{k}}{d\eta^{2}}+\Gamma_{ij}^{k}\frac{d\xi^{i}}{d\eta}\frac{d\xi^{j}}{d\eta}=0\text{,}$
(11)

with $\Gamma_{ik}^{j}$ in Eq. (11) being the usual Christoffel connection
coefficients,

$\Gamma_{ij}^{k}\overset{\text{def}}{=}\frac{1}{2}g^{kl}\left(\partial_{i}g_{lj}+\partial_{j}g_{il}-\partial_{l}g_{ij}\right)\text{.}$
(12)

Note that the elements of
$\mathcal{D}_{\boldsymbol{\xi}}\left(\tau^{\prime}\right)$ in Eq. (10), an
$N$-dimensional subspace of $\mathcal{D}_{\boldsymbol{\xi}}^{\mathrm{tot}}$,
are $N$-dimensional macrovariables $\left\\{\xi\right\\}$ with components
$\xi^{j}$ bounded by fixed integration limits $\xi^{j}\left(\tau_{0}\right)$
and $\xi^{j}\left(\tau_{0}+\tau^{\prime}\right)$. The temporal functional form
of such limits can be determined by integrating the $N$-coupled nonlinear
second order ODEs in Eq. (11). Having introduced the IGE, we term information
geometric complexity (IGC) the quantity
$\mathcal{C}_{\mathcal{M}_{s}}\left(\tau\right)$ given by

$\mathcal{C}_{\mathcal{M}_{s}}\left(\tau\right)\overset{\text{def}}{=}\widetilde{vol}\left[\mathcal{D}_{\boldsymbol{\xi}}\left(\tau\right)\right]=e^{\mathcal{S}_{\mathcal{M}_{s}}\left(\tau\right)}\text{.}$
(13)

As mentioned earlier, we shall focus on the asymptotic temporal behavior of
the IGC as specified by
$\mathcal{C}_{\mathcal{M}_{s}}^{\text{{asymptotic}}}\left(\tau\right)\overset{\tau\rightarrow\infty}{\sim}e^{\mathcal{S}_{\mathcal{M}_{s}}\left(\tau\right)}$.

The IGC $\mathcal{C}_{\mathcal{M}_{s}}\left(\tau\right)$ can be interpreted by
explaining the meaning of the IGE
$\mathcal{S}_{\mathcal{M}_{s}}\left(\tau\right)$ in Eq. (3). The IGE is an
affine temporal average of the $N$-fold integral of the Fisher density over
geodesics regarded as maximum probability trajectories and, in addition,
measures the number of the explored macrostates in $\mathcal{M}_{s}$. In
particular, the IGE at a given instant is the logarithm of the volume of the
effective parameter space navigated by the system at that specific instant.
The temporal averaging procedure in Eq. (4) is introduced to average out the
conceivably very complicated fine details of the probabilistic dynamical
description of the system on $\mathcal{M}_{s}$. Furthermore, the long-time
limit in Eq. (9) is used to properly specify the selected dynamical indicators
of complexity by neglecting the transient effects which enter the calculation
of the expected value of the volume of the effective parameter space. In
summary, the IGE provides an asymptotic coarse-grained inferential
characterization of the complex dynamics of a system in the presence of
partial knowledge. For further details on the IGC and IGE, we refer to Refs.
cafaro18 ; cafaro17 ; cafaro10 .

Our discussion has followed the original IGAC setting where we assumed to deal
with an underlying continuous microspace yielding a macrospace equipped with a
classical Fisher-Rao information metric in its integral form. However,
shifting to a discrete microspace leading to a macrospace with a Fisher-Rao
information metric expressed in terms of a summation is straightforward,

$g_{\mu\nu}^{\mathrm{FR}}\left(\xi\right)=\sum_{k=1}^{N}\frac{1}{p_{k}\left(\xi\right)}\frac{\partial
p_{k}\left(\xi\right)}{\partial\xi^{\mu}}\frac{\partial
p_{k}\left(\xi\right)}{\partial\xi^{\nu}}\text{.}$ (14)

From Eq. (14), the Fisher-Rao infinitesimal line element
$ds_{\mathrm{FR}}^{2}$ becomes

$ds_{\mathrm{FR}}^{2}=g_{\mu\nu}^{\mathrm{FR}}\left(\xi\right)d\xi^{\mu}d\xi^{\nu}=\sum_{k=1}^{N}\frac{dp_{k}^{2}}{p_{k}}\text{,}$
(15)

where
$dp_{k}\overset{\text{def}}{=}\left(\partial_{\mu}p_{k}\right)d\xi^{\mu}$.
Moreover, the parameters $\left\\{\xi^{k}\right\\}_{1\leq k\leq N}$ were
originally viewed in the IGAC context as statistical macrovariables emerging,
for instance, as suitable expectation values of the microvariables of the
physical system in the presence of partial knowledge. However, due to the fact
that in principle the IGE can be fully constructed from a geometric standpoint
once the infinitesimal line element $ds^{2}$ is known, its extension to
quantum manifolds of density matrices
$\left\\{\rho_{\xi}\left(x\right)\right\\}$ specified by a set of parameters
$\left\\{\xi\right\\}$ with
$\xi\in\mathcal{D}_{\boldsymbol{\xi}}^{\mathrm{tot}}\subseteq\mathbb{R}^{N}$,
including experimentally controllable parameters such as temperature and
magnetic field intensity, is simple as well. For clarity, note that
$\rho_{\xi}\left(x\right)\in\mathcal{M}_{s}^{\left(\text{{quantum}}\right)}$
replaces $p_{\xi}\left(x\right)\overset{\text{def}}{=}p\left(x\text{;
}\xi\right)\in\mathcal{M}_{s}^{\left(\text{{classical}}\right)}$ with
$\mathcal{M}_{s}^{\left(\text{{classical}}\right)}$ equal to $\mathcal{M}_{s}$
in Eq. (1).

Clearly, to provide estimates of the IGE and of the IGC in Eqs. (3) and (13),
respectively, we need to first find the geodesic paths on the manifolds.
Therefore, in the next section, we present the geodesic paths on the manifolds
of pure and mixed states equipped with the Fubini-Study and Sjöqvist metrics,
respectively.

##  III Geodesic Paths

We introduce here the geodesic paths on manifolds of pure and mixed states
equipped with the Fubini-Study and the Sjöqvist metrics, respectively.

###  III.1 Geodesic paths on the Bloch sphere: The Fubini-Study metric

We begin by discussing geodesic paths on manifolds of pure states equipped
with the Fubini-Study metric. In quantum mechanics, it is known that the only
Riemannian metric on the set of rays, up to a constant factor, which is
invariant under all unitary transformations is the angle in Hilbert space
(also known as, the Wootters angle),

$\theta_{\mathrm{Wootters}}\left(\left|\psi_{i}\right\rangle\text{,
}\left|\psi_{f}\right\rangle\right)\overset{\text{def}}{=}\arccos\left[\left|\left\langle\psi_{i}|\psi_{f}\right\rangle\right|\right]\text{,}$
(16)

with $\left|\psi_{i}\right\rangle$ and $\left|\psi_{f}\right\rangle$ being two
pure states. It is also known that a concept of statistical distance can be
defined between different preparations of the same quantum system, or to put
it another way, between different rays in the same Hilbert space wootters81 .
This notion of statistical distance is specified completely by the size of
statistical fluctuations taking place in measurements prepared to discriminate
one state from another. A major finding obtained by Wootters in Ref.
wootters81  was showing that such statistical distance coincides with the
usual distance (i.e., angle) between rays. The infinitesimal line element that
corresponds to the Hilbert space angle is the so-called Fubini-Study metric
$g_{\mu\nu}^{\mathrm{FS}}\left(\xi\right)$, the natural metric on the manifold
of Hilbert space rays. The physical interpretation of this metric in terms of
statistical fluctuations in the outcomes of intrinsically probabilistic
quantum measurements that aim at distinguishing one pure state from another is
a major result obtained in Ref. wootters81 . Before introducing the Fubini-
Study metric $g_{\mu\nu}^{\mathrm{FS}}\left(\xi\right)$ in an explicit manner,
we remark that the extension of Wootters’ reasoning to the problem of
distinguishing mixed quantum states was carried out by Braunstein and Caves in
Ref. braunstein94 . In the case of mixed states, the Bures angle
$\theta_{\mathrm{Bures}}$ and the Bures metric
$g_{\mu\nu}^{\mathrm{Bures}}\left(\xi\right)$ replace the Hilbert space angle
$\theta_{\mathrm{Wootters}}$ and the Fubini-Study metric
$g_{\mu\nu}^{\mathrm{FS}}$, respectively. The Bures angle represents the
length of a geodesic joining two density operators $\rho_{i}$ and $\rho_{f}$
and is given by,

$\theta_{\mathrm{Bures}}\left(\rho_{i}\text{,
}\rho_{f}\right)\overset{\text{def}}{=}\arccos\left[\mathrm{F}_{B}\left(\rho_{i}\text{,
}\rho_{f}\right)\right]\text{.}$ (17)

In Eq. (17), $\mathrm{F}_{B}\left(\rho_{i}\text{, }\rho_{f}\right)$ is the
Bures fidelity defined as

$\mathrm{F}_{B}\left(\rho_{i}\text{,
}\rho_{f}\right)\overset{\text{def}}{=}\left[\mathrm{Tr}\left(\sqrt{\rho_{i}^{1/2}\rho_{f}\rho_{i}^{1/2}}\right)\right]^{2}\text{.}$
(18)

For clarity, we point out that using Eqs. (17) and (18),
$\theta_{\mathrm{Bures}}\left(\rho_{i}\text{, }\rho_{f}\right)$ equals
$\arccos\left(\sqrt{\left\langle\psi_{i}|\rho_{f}|\psi_{i}\right\rangle}\right)$
when
$\rho_{i}\overset{\text{def}}{=}\left|\psi_{i}\right\rangle\left\langle\psi_{i}\right|$.
Furthermore, when both $\rho_{i}$ and $\rho_{f}$ are pure states,
$\theta_{\mathrm{Bures}}\left(\rho_{i}\text{, }\rho_{f}\right)$ reduces to
$\theta_{\mathrm{Wootters}}\left(\left|\psi_{i}\right\rangle\text{,
}\left|\psi_{f}\right\rangle\right)$ in Eq. (16). Finally, for completeness,
we remark here that the Bures distance
$d_{\mathrm{Bures}}\left(\rho_{i}\text{, }\rho_{f}\right)$ is different from
the Bures angle in Eq. (17) and is formally defined as

$d_{\mathrm{Bures}}\left(\rho_{i}\text{,
}\rho_{f}\right)\overset{\text{def}}{=}\sqrt{2\left[1-\sqrt{\mathrm{F}_{B}\left(\rho_{i}\text{,
}\rho_{f}\right)}\right]}\text{.}$ (19)

Returning to the formal introduction of
$g_{\mu\nu}^{\mathrm{FS}}\left(\xi\right)$, consider two neighboring single
qubit pure states $\left|\psi\right\rangle$ and
$\left|\bar{\psi}\right\rangle$ defined as,

$\left|\psi\right\rangle\overset{\text{def}}{=}\sum_{k=0}^{1}\sqrt{p_{k}}e^{i\phi_{k}}\left|e_{k}\right\rangle\text{,
and
}\left|\bar{\psi}\right\rangle\overset{\text{def}}{=}\sum_{k=0}^{1}\sqrt{p_{k}+dp_{k}}e^{i\left(\phi_{k}+d\phi_{k}\right)}\left|e_{k}\right\rangle\text{,}$
(20)

respectively, with $\left\\{\left|e_{k}\right\rangle\right\\}$ being an
orthonormal basis of the Hilbert space of single qubit state vectors. The
infinitesimal line element between $\left|\psi\right\rangle$ and
$\left|\bar{\psi}\right\rangle$ in Eq. (20) is given by the Fubini-Study
metric $ds_{\mathrm{FS}}^{2}$ braunstein94 ,

$ds_{\mathrm{FS}}^{2}\overset{\text{def}}{=}1-\left|\left\langle\bar{\psi}|\psi\right\rangle\right|^{2}=\frac{1}{4}\sum_{k=0}^{1}\frac{dp_{k}^{2}}{p_{k}}+\left[\sum_{k=0}^{1}p_{k}d\phi_{k}^{2}-\left(\sum_{k=0}^{1}p_{k}d\phi_{k}\right)^{2}\right]\text{.}$
(21)

Using the Bloch sphere parametrization of single qubit states,

$\left|\psi\right\rangle=\left|\psi\left(\theta\text{,
}\varphi\right)\right\rangle\overset{\text{def}}{=}\cos\left(\frac{\theta}{2}\right)\left|0\right\rangle+e^{i\varphi}\sin\left(\frac{\theta}{2}\right)\left|1\right\rangle\text{,}$
(22)

where $0\leq\theta\leq\pi$ and $0\leq\varphi<2\pi$, we get by comparing Eqs.
(20) and (22) that

$p_{0}\left(\theta\text{,
}\varphi\right)=\cos^{2}\left(\frac{\theta}{2}\right)\text{,
}p_{1}\left(\theta\text{,
}\varphi\right)=\sin^{2}\left(\frac{\theta}{2}\right)\text{,
}\phi_{0}\left(\theta\text{, }\varphi\right)=0\text{, and
}\phi_{1}\left(\theta\text{, }\varphi\right)=\varphi\text{.}$ (23)

Therefore, substituting Eq. (23) into Eq. (21), the Fubini-Study metric
$ds_{\mathrm{FS}}^{2}$ reduces to

$ds_{\mathrm{FS}}^{2}=g_{\mu\nu}^{\mathrm{FS}}\left(\xi\right)d\xi^{\mu}d\xi^{\nu}=\frac{1}{4}\left[d\theta^{2}+\sin^{2}\left(\theta\right)d\varphi^{2}\right]\text{.}$
(24)

In Eq. (24), $g_{\mu\nu}^{\mathrm{FS}}\left(\xi\right)$ is the Fubini-Study
metric tensor, $1\leq\mu$, $\nu\leq 2$, and $\xi=\left(\xi^{1}\text{,
}\xi^{2}\right)\overset{\text{def}}{=}\left(\theta\text{, }\varphi\right)$.
For completeness, we point out that the Fubini-Study distance between two
antipodal (i.e., orthogonal) states on the Bloch sphere is $\pi/2$. Instead,
the geodesic distance between two antipodal states is $\pi$. Indeed,
$ds_{\mathrm{FS}}^{2}=\left(1/4\right)ds_{\mathrm{BSM}}^{2}$ where
$ds_{\mathrm{BSM}}^{2}$ denotes the Bloch sphere metric (BSM) defined as
uzdin12 ,

$ds_{\mathrm{BSM}}^{2}\overset{\text{def}}{=}d\hat{n}\cdot d\hat{n}\text{.}$
(25)

In Eq. (25), $\hat{n}$ is the unit vector in $\mathbb{R}^{3}$ given by

$\hat{n}\overset{\text{def}}{=}\frac{\left\langle\psi\left(\theta\text{,
}\varphi\right)|\vec{\sigma}|\psi\left(\theta\text{,
}\varphi\right)\right\rangle}{\left\langle\psi\left(\theta\text{,
}\varphi\right)|\psi\left(\theta\text{,
}\varphi\right)\right\rangle}=\left(\sin\left(\theta\right)\cos\left(\varphi\right)\text{,
}\sin\left(\theta\right)\sin\left(\varphi\right)\text{,
}\cos\left(\theta\right)\right)\text{,}$ (26)

with $\vec{\sigma}\overset{\text{def}}{=}\left(\sigma_{x}\text{,
}\sigma_{y}\text{, }\sigma_{z}\right)$ being the Pauli vector operator and
$\left|\psi\left(\theta\text{, }\varphi\right)\right\rangle$ given in Eq.
(22). From Eq. (24), the only nonvanishing Christoffel connection coefficients
are

$\Gamma_{22}^{1}=-\sin\left(\theta\right)\cos\left(\theta\right)\text{, and
}\Gamma_{12}^{2}=\Gamma_{21}^{2}=\frac{\cos\left(\theta\right)}{\sin\left(\theta\right)}\text{.}$
(27)

Therefore, geodesic paths satisfy the geodesic equations in Eq. (11) being
specified by the following system of two coupled second order nonlinear ODEs,

$\ddot{\theta}-\sin\left(\theta\right)\cos\left(\theta\right)\dot{\varphi}^{2}=0\text{,
and
}\ddot{\varphi}+2\frac{\cos\left(\theta\right)}{\sin\left(\theta\right)}\dot{\theta}\dot{\varphi}=0\text{,}$
(28)

where $\dot{\theta}\overset{\text{def}}{=}d\theta/d\eta$ with $\eta$ being an
affine parameter. Integration of Eq. (28) under suitable working conditions
yields geodesic paths given by

$\theta\left(\eta\right)=\cos^{-1}\left[\mathrm{a}_{\mathrm{FS}}\sin\left(\eta\right)\right]\text{,
and
}\varphi\left(\eta\right)=\varphi_{i}+\tan^{-1}\left[\mathrm{c}_{\mathrm{FS}}\tan\left(\eta\right)\right]\text{,}$
(29)

where
$\mathrm{a}_{\mathrm{FS}}^{2}\overset{\text{def}}{=}1-\mathrm{c}_{\mathrm{FS}}^{2}$
and $\mathrm{c}_{\mathrm{FS}}=\mathrm{c}_{\mathrm{FS}}\left(\theta_{i}\text{,
}\dot{\varphi}_{i}\right)\overset{\text{def}}{=}\dot{\varphi}_{i}\sin^{2}\left(\theta_{i}\right)=\mathrm{const}$.
Note that both $\theta\left(\eta\right)$ and $\varphi\left(\eta\right)$ in Eq.
(29) are bounded functions for any $\eta\geq 0$. We remark that the speed of
evolution along these paths is constant and equals
$v_{\mathrm{FS}}\overset{\text{def}}{=}(1/2)\left[\dot{\theta}^{2}+\sin^{2}\left(\theta\right)\dot{\varphi}^{2}\right]^{1/2}$.
For a detailed derivation of the relations in Eq. (29) along with their
extension to arbitrary working conditions, we refer to Appendix A. Having
found the geodesic paths in Eq. (29), we focus now on geodesics on manifolds
of mixed quantum states equipped with the Sjöqvist metric.

###  III.2 Geodesic paths in the Bloch ball: The Sjöqvist metric

W begin by mentioning the motivation underlying the introduction of the
Sjöqvist metric from a practical standpoint in science. From a physical
standpoint, the Sjöqvist metric can be related to measurable quantities in
suitably prepared interferometric measurements. For this reason, it is
sometimes called “interferometric” metric. The metric can be regarded as the
infinitesimal distance $\delta s^{2}\left(\rho\text{,
}\rho+\delta\rho\right)\approx g\left(\dot{\rho}\text{,
}\dot{\rho}\right)\delta t^{2}$ between two neighboring mixed states $\rho$
and $\rho+\delta\rho$ with $\delta\rho=\dot{\rho}\delta t$. The mixed state
$\rho$ encodes the internal degree of freedom of a particle entering a Mach-
Zehnder interferometer with two beam splitters. The mixed state
$\rho^{\prime}\overset{\text{def}}{=}\rho+\delta\rho$ equals $U\rho
U^{\dagger}$ with $U$ being a unitary applied to the particle for a small but
finite time $\delta t$. From an experimental standpoint, the line element
$\delta s^{2}$ is related to the probability P0 of finding the particle in the
$0$-beam (that is, the beam where the unitary transformation $U$ was applied)
after passing the second beam splitter. In particular, up to the leading
nontrivial order in $\delta t$, one finds that P${}_{0}=1-(1/4)\delta s^{2}$.
For more details on a direct experimental access to the Sjöqvist line element,
we refer to Refs. erik20 ; silva21 . In Ref. silva21 , the Sjöqvist metric is
generalized by extending its applicability to degenerate density matrices as
well. Interestingly, studying finite-temperature equilibrium phase
transitions, dramatically different behaviors between the Sjöqvist and the
Bures metrics are noticed in Ref. silva21 . Specifically, the Sjöqvist metric
appears to be more sensitive to the change in parameters than the Bures one.
Indeed, unlike what happens for the Bures metric, the Sjöqvist metric infers
both zero-temperature and finite-temperature phase transitions. We will return
to this point on the difference between the Sjöqvist and Bures metrics later
in our paper.

Resuming the formal introduction of the Sjöqvist metric, consider two rank-$2$
neighboring nondegenerate density operators $\rho\left(t\right)$ and
$\rho\left(t+dt\right)$ connected via a smooth path
$t\mapsto\rho\left(t\right)$ characterizing the evolution of a quantum system.
The nondegeneracy requirement assures that the gauge freedom in the spectral
decomposition of the density operators is represented by the phase of the
eigenvectors. This, in turn, implies there is a one-to-one correspondence
between a rank-$2$ nondegenerate density operator $\rho\left(t\right)$ and the
set of two orthogonal rays
$\left\\{e^{i\phi_{k}\left(t\right)}\left|e_{k}\left(t\right)\right\rangle:0\leq\phi_{k}\left(t\right)<2\pi\right\\}$
that specify the spectral decomposition along the path
$t\mapsto\rho\left(t\right)$. Clearly, if some nonzero eigenvalue of
$\rho\left(t\right)$ was degenerate, the above mentioned correspondence would
not be valid anylonger. The infinitesimal line element between
$\rho\left(t\right)$ and $\rho\left(t+dt\right)$ in the working assumption
that
$\left\\{\sqrt{p_{k}\left(t\right)}e^{i\phi_{k}\left(t\right)}\left|e_{k}\left(t\right)\right\rangle\right\\}_{k=0,1}$
represents the spectral decompositions along the path
$t\mapsto\rho\left(t\right)$ is given by the Sjöqvist metric
$ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$ erik20 ,

$ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\overset{\text{def}}{=}\min\left[d^{2}\left(t\text{,
}t+dt\right)\right]\text{,}$ (30)

with $d^{2}\left(t\text{, }t+dt\right)$ is defined as

$d^{2}\left(t\text{,
}t+dt\right)\overset{\text{def}}{=}\sum_{k=0}^{1}\left\|\sqrt{p_{k}\left(t\right)}e^{i\phi_{k}\left(t\right)}\left|e_{k}\left(t\right)\right\rangle-\sqrt{p_{k}\left(t+dt\right)}e^{i\phi_{k}\left(t+dt\right)}\left|e_{k}\left(t+dt\right)\right\rangle\right\|^{2}\text{.}$
(31)

Following the line of reasoning in Ref. erik20 ,
$ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$ can be recast as

$ds_{\mathrm{Sj\ddot{o}qvist}}^{2}=\frac{1}{4}\sum_{k=0}^{1}\frac{dp_{k}^{2}}{p_{k}}+\sum_{k=0}^{1}p_{k}ds_{k}^{2}\text{,}$
(32)

where $dp_{k}=\dot{p}_{k}dt$ and, recalling Ref. provost80 , $ds_{k}^{2}$ in
Eq. (32) is the Fubini-Study metric along the pure state
$\left|e_{k}\right\rangle$

$ds_{k}^{2}\overset{\text{def}}{=}\left\langle\dot{e}_{k}\left|\left(\hat{1}-\left|e_{k}\right\rangle\left\langle
e_{k}\right|\right)\right|\dot{e}_{k}\right\rangle=\left\langle
de_{k}|de_{k}\right\rangle-\left|\left\langle
e_{k}|de_{k}\right\rangle\right|^{2}\text{,}$ (33)

with $\hat{1}$ being the identity operator on the Hilbert space of single
qubit quantum states. Using the Bloch sphere parametrization of single qubit
mixed states in the Bloch ball, we have

$\rho=\frac{\hat{1}+\vec{r}\cdot\vec{\sigma}}{2}=\frac{1}{2}\left(\begin{array}[c]{cc}1+r\cos\left(\theta\right)&r\sin\left(\theta\right)e^{-i\varphi}\\\
r\sin\left(\theta\right)e^{i\varphi}&1-r\cos\left(\theta\right)\end{array}\right)\text{,}$
(34)

where $\vec{r}$ is the polarization vector given by
$\vec{r}\overset{\text{def}}{=}r\hat{n}$ with $\hat{n}$ defined in Eq. (26).
Note that for mixed quantum states, $0\leq r<1$ and
$\det\left(\rho\right)=\left(1/2\right)\left(1-\vec{r}^{2}\right)\geq 0$
because of the positiveness of $\rho$. For pure quantum states, instead, $r=1$
and $\det\left(\rho\right)=0$. From Eq. (34), we observe that the spectral
decomposition of $\rho$ is given by

$\rho=\sum_{k=0}^{1}p_{k}\left|e_{k}\right\rangle\left\langle
e_{k}\right|\text{.}$ (35)

The two distinct eigenvalues $\left\\{p_{k}\right\\}_{k=0,1}$ are given by,

$p_{0}=p_{0}\left(r\text{, }\theta\text{,
}\varphi\right)\overset{\text{def}}{=}\frac{1+r}{2}\text{, and
}p_{1}=p_{1}\left(r\text{, }\theta\text{,
}\varphi\right)\overset{\text{def}}{=}\frac{1-r}{2}\text{,}$ (36)

respectively. The orthonormal eigenvectors corresponding to $p_{0}$ and
$p_{1}$ in Eq. (36) are

$\left|e_{0}\right\rangle=\left|e_{0}\left(r\text{, }\theta\text{,
}\varphi\right)\right\rangle\overset{\text{def}}{=}\frac{1}{\sqrt{2}}\left(\begin{array}[c]{c}e^{-i\varphi}\sqrt{1+\cos\left(\theta\right)}\\\
\frac{\sin\left(\theta\right)}{\sqrt{1+\cos\left(\theta\right)}}\end{array}\right)=\left(\begin{array}[c]{c}e^{-i\varphi}\cos\left(\frac{\theta}{2}\right)\\\
\sin\left(\frac{\theta}{2}\right)\end{array}\right)\text{,}$ (37)

and,

$\left|e_{1}\right\rangle=\left|e_{1}\left(r\text{, }\theta\text{,
}\varphi\right)\right\rangle\overset{\text{def}}{=}\frac{1}{\sqrt{2}}\left(\begin{array}[c]{c}-e^{-i\varphi}\sqrt{1-\cos\left(\theta\right)}\\\
\frac{\sin\left(\theta\right)}{\sqrt{1-\cos\left(\theta\right)}}\end{array}\right)=\left(\begin{array}[c]{c}-e^{-i\varphi}\sin\left(\frac{\theta}{2}\right)\\\
\cos\left(\frac{\theta}{2}\right)\end{array}\right)\text{,}$ (38)

respectively. Finally, using Eqs. (36), (37), and (38),
$ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$ in Eq. (32) becomes

$ds_{\mathrm{Sj\ddot{o}qvist}}^{2}=g_{\mu\nu}^{\mathrm{Sj\ddot{o}qvist}}\left(\xi\right)d\xi^{\mu}d\xi^{\nu}=\frac{1}{4}\left[\frac{dr^{2}}{1-r^{2}}+d\Omega^{2}\right]\text{,}$
(39)

with
$d\Omega^{2}\overset{\text{def}}{=}d\theta^{2}+\sin^{2}\left(\theta\right)d\varphi^{2}$.
In Eq. (39), $g_{\mu\nu}^{\mathrm{Sj\ddot{o}qvist}}\left(\xi\right)$ is the
Sjöqvist metric tensor, $1\leq\mu$, $\nu\leq 3$, and $\xi=\left(\xi^{1}\text{,
}\xi^{2}\text{, }\xi^{3}\right)\overset{\text{def}}{=}\left(r\text{,
}\theta\text{, }\varphi\right)$. Note when $r$ is constant and equals one,
$ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$ in Eq. (39) reduces to
$ds_{\mathrm{FS}}^{2}$ in Eq. (24). For completeness, we recall that the Bures
metric extends to mixed quantum states the Fubini-Study metric on pure states
bures69 ; uhlmann76 ; hubner92 . Furthermore, as shown in Ref. braunstein94 ,
it is equivalent, up to a proportionality factor of four, to the quantum
Fisher information metric. Interestingly, we remark that the Bures
infinitesimal line element $ds_{\mathrm{Bures}}^{2}$ between $\rho$ and
$\rho+d\rho$ with $\rho$ given in Eq. (34) is given by

$ds_{\mathrm{Bures}}^{2}=g_{\mu\nu}^{\mathrm{Bures}}\left(\xi\right)d\xi^{\mu}d\xi^{\nu}=\frac{1}{4}\left[\frac{dr^{2}}{1-r^{2}}+r^{2}d\Omega^{2}\right]\text{.}$
(40)

For an explicit derivation of Eq. (40), we refer to Appendix B. From Eqs. (39)
and (40), we notice that the angular part of
$ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$ does not exhibit the $r^{2}$-factor which,
instead, appears in $ds_{\mathrm{Bures}}^{2}$. The lack of this factor implies
that the Sjöqvist metric is singular at the origin of the Bloch ball where
$r=0$ and, unlike the Bures metric, is not defined for degenerate density
operators. Finally, we refer to Ref. silva21  for a recent extension of the
Sjöqvist metric for the space of nondegenerate density matrices, to the
degenerate case, i.e., the case in which the eigenspaces have dimension
greater than or equal to one. We will go back to this point on the difference
between the Sjöqvist and Bures metrics later in our paper (see also Appendix C
and Appendix D).

Returning to the Sjöqvist metric analysis, we see from Eq. (39) that the only
nonvanishing Christoffel connection coefficients are

$\Gamma_{11}^{1}=\frac{r}{1-r^{2}}\text{,
}\Gamma_{33}^{2}=-\sin\left(\theta\right)\cos\left(\theta\right)\text{, and
}\Gamma_{23}^{3}=\Gamma_{32}^{3}=\frac{\cos\left(\theta\right)}{\sin\left(\theta\right)}\text{.}$
(41)

Therefore, geodesics satisfy the geodesic equations in Eq. (11) described in
terms of a system of three coupled second order nonlinear ODEs,

$\ddot{r}+\frac{r}{1-r^{2}}\dot{r}^{2}=0\text{,
}\ddot{\theta}-\sin\left(\theta\right)\cos\left(\theta\right)\dot{\varphi}^{2}=0\text{,
and
}\ddot{\varphi}+2\frac{\cos\left(\theta\right)}{\sin\left(\theta\right)}\dot{\theta}\dot{\varphi}=0\text{,}$
(42)

where $\dot{r}\overset{\text{def}}{=}dr/d\eta$ with $\eta$ being an affine
parameter. Interestingly, observe that although the ODE satisfied by the
radial parameter $r$ in Eq. (42) is nonlinear, it is not coupled to the ODEs
describing the evolution of the angular parameters $\theta$ and $\varphi$.
Furthermore, the angular motion is identical to the one that emerges when
employing the Fubini-Study metric. Therefore, we refer to Eq. (29) and to
Appendix A for details on the angular motion. Instead, integration of the
radial equation of motion in Eq. (42) yields,

$r_{\mathrm{Sj\ddot{o}qvist}}\left(\eta\right)=\sin\left[\sin^{-1}\left(r_{i}\right)+\frac{\dot{r}_{i}}{\sqrt{1-r_{i}^{2}}}\eta\right]\text{,}$
(43)

where $r_{i}\overset{\text{def}}{=}r\left(\eta_{i}\right)$,
$\dot{r}_{i}\overset{\text{def}}{=}\dot{r}\left(\eta_{i}\right)$, and
$\eta_{i}$ is set equal to zero. We emphasize that the speed of evolution
along geodesic paths is constant and equals
$v_{\mathrm{Sj\ddot{o}qvist}}\overset{\text{def}}{=}(1/2)\left[\left(1-r^{2}\right)^{-1}\dot{r}^{2}+\dot{\theta}^{2}+\sin^{2}\left(\theta\right)\dot{\varphi}^{2}\right]^{1/2}$.
For an explicit derivation of Eq. (43) along with a discussion on alternative
geodesic parametrizations like the one used in Ref. erik20 , we refer to
Appendix C. Finally, for a discussion on the integration of the geodesic
equations in a Bloch ball equipped with the Bures metric
$g_{\mu\nu}^{\mathrm{Bures}}\left(\xi\right)$, we refer to Appendix D. In
Appendix E, instead, we present a summary of curvature properties of the
manifold of pure states equipped with
$g_{\mu\nu}^{\mathrm{FS}}\left(\xi\right)$ along with those of a manifold of
mixed quantum states endowed with
$g_{\mu\nu}^{\mathrm{Sj\ddot{o}qvist}}\left(\xi\right)$ and
$g_{\mu\nu}^{\mathrm{Bures}}\left(\xi\right)$. More specifically, for each
scenario, we find the expressions of the tensor metric components,
infinitesimal line elements, Christoffel connection coefficients, Ricci tensor
components, Riemann curvature tensor components, scalar curvatures and,
finally, sectional curvatures.

At this point, having found the geodesic paths on curved manifolds equipped
with the Fubini-Study and Sjöqvist metrics, we are ready to use our complexity
quantifiers in Eqs. (3) and (13) to determine how complex evolutions on pure
and mixed states are.

##  IV Complexity of quantum evolution

We study here the complexity of the geodesic paths expressed in terms of
temporal averages of volume regions explored by the physical systems during
the quantum evolutions.

###  IV.1 Actions, lengths, and accessible volumes

Before studying the complexity, let us first comment on the relevance of the
concepts of length and action in the geometric formulation of physical
theories. In the Introduction, we mentioned these concepts play a key role in
the understanding of the physics of black holes leo14 ; leo15 ; leo16 ; leo16B
. However, lengths and actions also play a very important role in the
geometric formulation of thermodynamics orlando22 . In this case, these two
quantities are generally termed thermodynamic length and thermodynamic
divergence, respectively. Indeed, including the theory of fluctuations into
the axioms of equilibrium thermodynamics ruppeiner79 , thermodynamic systems
can be characterized by Riemannian manifolds furnished of a thermodynamic
metric tensor that is identical to the Fisher-Rao information metric crooks07
. Within this geometric setting for thermodynamics, the above mentioned
Riemannian structure allows one to introduce a notion of length for
fluctuations about equilibrium states as well as for thermodynamic processes
proceeding via equilibrium states. In analogy to Wootters’ statistical
distance between probability distributions as presented in Ref. wootters81 ,
the thermodynamic length of a path connecting two points on a manifold of
thermal states can be viewed as a measure of the maximal number of
statistically distinguishable thermodynamic states along the path diosi84 . In
particular, the larger the fluctuations, the closer the points are together.
The thermodynamic divergence of a path, instead, is a measure of the losses in
the process quantified by the total entropy produced along the path. For more
details, we refer to Refs. ruppeiner79 ; weinhold75 ; salamon83 . Having in
mind Wootters’ approach, note that the concepts of action and length are
formally different when studying the geometry along the evolution of states.
For the sake of reasoning, assume that the line element of the Riemannian
space is given by
$ds^{2}\overset{\text{def}}{=}g_{\mu\nu}\left(\xi\right)d\xi^{\mu}d\xi^{\nu}$.
Then, the action A is given by

$\mathrm{A}\overset{\text{def}}{=}\frac{1}{2}m\int_{0}^{\tau}g_{\mu\nu}\left(\xi\right)\dot{\xi}^{\mu}\dot{\xi}^{\nu}d\eta\text{,}$
(44)

with $\dot{\xi}\overset{\text{def}}{=}d\xi/d\eta$. The length $\mathcal{L}$ of
a path $\xi^{\mu}\left(\eta\right)$ with $0\leq\eta\leq\tau$, instead, is
defined as

$\mathcal{L}\overset{\text{def}}{=}\int_{0}^{\tau}\sqrt{g_{\mu\nu}\left(\xi\right)\dot{\xi}^{\mu}\dot{\xi}^{\nu}}d\eta\text{.}$
(45)

However, for particles of mass $m$ moving along geodesics with constant
velocity, both velocity and energy are conserved. In this case, the path
$\mathcal{L}$ and the action $\mathrm{A}$ are linearly related. Indeed, one
has

$\mathrm{A}=\sqrt{\frac{mE}{2}}\mathcal{L}\text{,}$ (46)

where $E\overset{\text{def}}{=}\left(1/2\right)mv^{2}$ and
$v^{2}\overset{\text{def}}{=}g_{\mu\nu}\left(\xi\right)\dot{\xi}^{\mu}\dot{\xi}^{\nu}$
are both constant. Interestingly, we remark that while the length
$\mathcal{L}$ is invariant under reparametrization of the affine parameter
$\eta$, the action $\mathrm{A}$ is not. Before proceeding with the calculation
of lengths in Eq. (45) and volumes of explored regions in Eq. (4) yielding the
complexity of geodesic paths on manifolds of quantum states, we make a couple
of remarks that can help our intuition when considering the Sjöqvist and Bures
cases with pure calculations. First, considering Eqs. (39) and (40) while
performing a change of variables defined by
$r\overset{\text{def}}{=}\sin\left(\alpha_{r}\right)$ with
$0\leq\alpha_{r}\leq\pi/2$, we find that
$4ds_{\mathrm{Sj\ddot{o}qvist}}^{2}=d\alpha_{r}^{2}+d\Omega_{\text{sphere}}^{2}$
and
$4ds_{\mathrm{Bures}}^{2}=d\alpha_{r}^{2}+\sin^{2}\left(\alpha_{r}\right)d\Omega_{\text{sphere}}^{2}$
with
$d\Omega_{\text{sphere}}^{2}\overset{\text{def}}{=}d\theta^{2}+\sin^{2}\left(\theta\right)d\varphi^{2}$.
The structure of the Sjöqvist line element recast in this new form is
reminiscent of the structure of a line element in the usual cylindrical
coordinates $\left(\rho\text{, }\varphi\text{, }z\right)$,
$ds_{\text{cylinder}}^{2}=dz^{2}+d\Omega_{\text{cylinder}}^{2}$ with
$d\Omega_{\text{cylinder}}^{2}\overset{\text{def}}{=}d\rho^{2}+\rho^{2}d\varphi^{2}$,
once one identifies the pair $\left(\alpha_{r}\text{,
}d\Omega_{\text{sphere}}\right)$ with the pair $\left(\rho\text{,
}d\Omega_{\text{cylinder}}\right)$. Therefore, one can imagine associating a
cylinder with a constant (varying) radius to the Sjöqvist (Bures) geometry,
respectively. Note that the varying radius in the Bures case is upper bounded
by the constant value that specifies the radius in the Sjöqvist geometry.
Second, the volumes of the accessible regions of the manifolds in the Sjöqvist
and Bures scenarios are given by,

$V_{\mathrm{Sj\ddot{o}qvist}}^{\left(\text{accessible}\right)}\overset{\text{def}}{=}\frac{1}{8}\int_{0}^{1}\int_{0}^{\pi}\int_{0}^{2\pi}\frac{\sin\left(\theta\right)}{\sqrt{1-r^{2}}}drd\theta
d\varphi=\frac{\pi^{2}}{4}\text{,}$ (47)

and,

$V_{\mathrm{Bures}}^{\left(\text{accessible}\right)}\overset{\text{def}}{=}\frac{1}{8}\int_{0}^{1}\int_{0}^{\pi}\int_{0}^{2\pi}\frac{r^{2}\sin\left(\theta\right)}{\sqrt{1-r^{2}}}drd\theta
d\varphi=\frac{\pi^{2}}{8}\text{,}$ (48)

respectively. Clearly, from Eqs. (47) and (48), we note that
$V_{\mathrm{Bures}}^{\left(\text{accessible}\right)}\leq
V_{\mathrm{Sj\ddot{o}qvist}}^{\left(\text{accessible}\right)}$. This fact, in
turn, is compatible with our intuitive picture proposed in our first remark.
We remark that it is possible that only parts of the accessible geometric
regions are indeed explored during the evolution. In what follows, we shall
finally calculate the lengths and the volumes of the effectively explored
regions of the manifolds in the (pure) Fubini-Study and (mixed) Sjöqvist
cases.

###  IV.2 Evolution on the Bloch sphere

####  IV.2.1 Length

In what follows, we focus on calculating the length of geodesics in the unit
Bloch ball that lay int the $xz$-plane specified by the condition $\varphi=0$.
In the Fubini-Study metric case, we have

$\mathcal{L}_{\mathrm{FS}}\left(\eta_{f}\right)\overset{\text{def}}{=}\frac{1}{2}\int_{0}^{\eta_{f}}\frac{d\theta}{d\eta}d\eta=\frac{1}{2}\dot{\theta}_{i}\eta_{f}\text{,}$
(49)

or, alternatively, in terms of the angular variable $\theta$,

$\mathcal{L}_{\mathrm{FS}}\left(\theta_{f}\right)\overset{\text{def}}{=}\frac{\theta_{f}}{2}\text{.}$
(50)

Note that $\mathcal{L}_{\mathrm{FS}}$ in Eq. (50) denotes the Fubini-Study
distance ($\theta_{f}/2$) and is half the geodesic distance ($\theta_{f}$) on
the Bloch sphere. For completeness, we emphasize that in obtaining Eq. (49) we
exploited the relation $\ddot{\theta}=0$. This relation can be obtained from
Eq. (28) once one imposes the constraint of constant $\varphi$. Moreover, in
getting Eq. (50), we assumed
$\theta_{i}\overset{\text{def}}{=}\theta\left(\eta_{i}\right)=0$, with
$\eta_{i}=0$.

####  IV.2.2 Complexity

From the Fubini-Study metric $ds_{\mathrm{FS}}^{2}$ in Eq. (24), we note that
$g_{\mathrm{FS}}\left(\theta\text{,
}\varphi\right)=\sin^{2}\left(\theta\right)/16$ with $g_{\mathrm{FS}}$
denoting the determinant of the metric tensor $g_{\mu\nu}^{\mathrm{FS}}$.
Therefore, using Eq. (29), the instantaneous explored volume region
$V_{\mathrm{FS}}\left(\eta\right)$ as defined in Eq. (5) becomes

$V_{\mathrm{FS}}\left(\eta\right)=\frac{1}{4}\mathrm{a}_{\mathrm{FS}}\sin\left(\eta\right)\arctan\left[\mathrm{c}_{\mathrm{FS}}\tan\left(\eta\right)\right]\text{.}$
(51)

Recall that
$\mathrm{a}_{\mathrm{FS}}^{2}\overset{\text{def}}{=}1-\mathrm{c}_{\mathrm{FS}}^{2}$
with $\mathrm{c}_{\mathrm{FS}}=\mathrm{c}_{\mathrm{FS}}\left(\theta_{i}\text{,
}\dot{\varphi}_{i}\right)\overset{\text{def}}{=}\dot{\varphi}_{i}\sin^{2}\left(\theta_{i}\right)=\mathrm{const}$.
Then, setting $\theta_{i}=\pi/2$ for illustrative purposes, we note that the
modulus of the volume of the explored region of the manifold of pure states in
Eq. (51) is upper bounded by $\pi/8$,
$\left|V_{\mathrm{FS}}\left(\eta\right)\right|\leq\pi/8$. Therefore, at a time
$\eta$, less than one eight of the accessible region of the manifold is
actually explored since
$V_{\mathrm{FS}}^{\left(\text{accessible}\right)}=\pi$. From Eq. (51), the
average explored region as given in Eq. (4) represents the IGC in Eq. (13) and
turns into

$\mathcal{C}_{\mathrm{FS}}\left(\tau\right)=\frac{1}{4}\mathrm{a}_{\mathrm{FS}}\frac{I_{V_{\mathrm{FS}}}\left(\tau\right)}{\tau}\text{.}$
(52)

The function $I_{V_{\mathrm{FS}}}\left(\tau\right)$ in Eq. (52) is defined as
the integral of $V_{\mathrm{FS}}\left(\eta\right)$ with $0\leq\eta\leq\tau$
and is given by,

$I_{V_{\mathrm{FS}}}\left(\tau\right)\overset{\text{def}}{=}\frac{\mathrm{c}_{\mathrm{FS}}}{\sqrt{\mathrm{c}_{\mathrm{FS}}^{2}-1}}\arctan\left[\sqrt{\mathrm{c}_{\mathrm{FS}}^{2}-1}\sin\left(\tau\right)\right]-\cos\left(\tau\right)\arctan\left[\mathrm{c}_{\mathrm{FS}}\tan\left(\tau\right)\right]\text{.}$
(53)

The above mentioned integral was performed with the help of the Mathematica
software. Furthermore, we remark that $I_{V_{\mathrm{FS}}}\left(\tau\right)$
is a bounded function for any $\tau\geq 0$. The asymptotic temporal expression
of the IGC $\mathcal{C}_{\mathrm{FS}}\left(\tau\right)$ in Eq. (52) will be
compared with the one that we obtain in the case of mixed state evolutions
with distinguishability metric provided by
$g_{\mu\nu}^{\mathrm{Sj\ddot{o}qvist}}\left(\xi\right)$.

###  IV.3 Evolution in the Bloch ball

####  IV.3.1 Length

In what follows, we focus on calculating the length of geodesics in the unit
Bloch ball that lay int the $xz$-plane specified by the condition $\varphi=0$.
In the Sjöqvist metric case, recall that $r^{\prime
2}\left(1-r^{2}\right)^{-1}=\mathrm{const.}$and $\ddot{\theta}=0$. Therefore,
the length $\mathcal{L}_{\mathrm{Sj\ddot{o}qvist}}\left(\eta_{f}\right)$ is
given by

$\mathcal{L}_{\mathrm{Sj\ddot{o}qvist}}\left(\eta_{f}\right)\overset{\text{def}}{=}\frac{1}{2}\int_{0}^{\eta_{f}}\sqrt{1+\frac{r^{\prime
2}}{1-r^{2}}}\frac{d\theta}{d\eta}d\eta=\frac{1}{2}\sqrt{1+\frac{r_{i}^{\prime
2}}{1-r_{i}^{2}}}\dot{\theta}_{i}\eta_{f}\text{,}$ (54)

or, alternatively,

$\mathcal{L}_{\mathrm{Sj\ddot{o}qvist}}\left(\theta_{f}\right)\overset{\text{def}}{=}\frac{1}{2}\sqrt{\theta_{f}^{2}+\left[\sin^{-1}\left(r_{f}\right)-\sin^{-1}\left(r_{i}\right)\right]^{2}}\text{.}$
(55)

Comparing Eqs. (55) and (50), we note that
$\mathcal{L}_{\mathrm{Sj\ddot{o}qvist}}$ reduces to
$\mathcal{L}_{\mathrm{FS}}$ when $r_{f}=r_{i}=1$. For $r_{f}\neq r_{i}$, we
generally have

$\mathcal{L}_{\mathrm{Sj\ddot{o}qvist}}\left(\theta_{f}\right)\geq\mathcal{L}_{\mathrm{FS}}\left(\theta_{f}\right)\text{.}$
(56)

Eq. (56) implies that the length of a geodesic path connecting two arbitrary
points $P_{i}\overset{\text{def}}{=}\left(r_{i}\text{, }\theta_{i}\text{,
}0\right)$ and $P_{f}\overset{\text{def}}{=}\left(r_{f}\text{,
}\theta_{f}\text{, }0\right)$ with $r_{i}\neq r_{f}$ in the Bloch ball laying
in the $xz$-plane (i.e., $\varphi=0$) is longer than the length of two
arbitrary points $\tilde{P}_{i}\overset{\text{def}}{=}\left(\theta_{i}\text{,
}0\right)$ and $\tilde{P}_{f}\overset{\text{def}}{=}\left(\theta_{f}\text{,
}0\right)$ laying on the Bloch sphere with $r_{i}=r_{f}=1$ that intercepts the
$xz$-plane (i.e., $\varphi=0$). Eq. (56) hints to what might happen when
comparing the information geometric complexity of the evolutions of pure and
quantum states as we shall see shortly. Complexities are expressed in terms of
volumes. In parametric spaces of dimension higher than one, relations between
length and volumes are not straightforward. Therefore, we could not easily
take the hint as a “proof” of the higher complexity of the evolution of mixed
quantum states. For this reason, we actually estimate this quantity in an
explicit manner in the following subsection.

####  IV.3.2 Complexity

From the Sjöqvist metric $ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$ in Eq. (39), we
observe that the determinant of the metric tensor
$g_{\mu\nu}^{\mathrm{Sj\ddot{o}qvist}}$ satisfies the relation

$g_{\mathrm{Sj\ddot{o}qvist}}\left(r\text{, }\theta\text{,
}\varphi\right)=\frac{1}{64}\frac{\sin^{2}\left(\theta\right)}{1-r^{2}}\text{.}$
(57)

Therefore, making use of Eqs. (43) and (29), the instantaneous explored volume
region $V_{\mathrm{Sj\ddot{o}qvist}}\left(\eta\right)$ as given in Eq. (5)
turns into

$V_{\mathrm{Sj\ddot{o}qvist}}\left(\eta\right)=\frac{1}{8}\mathrm{a}_{\mathrm{FS}}\frac{\dot{r}_{i}}{\sqrt{1-r_{i}^{2}}}\eta\sin\left(\eta\right)\arctan\left[\mathrm{c}_{\mathrm{FS}}\tan\left(\eta\right)\right]\text{.}$
(58)

Recall that
$\mathrm{a}_{\mathrm{FS}}^{2}\overset{\text{def}}{=}1-\mathrm{c}_{\mathrm{FS}}^{2}$
with $\mathrm{c}_{\mathrm{FS}}=\mathrm{c}_{\mathrm{FS}}\left(\theta_{i}\text{,
}\dot{\varphi}_{i}\right)\overset{\text{def}}{=}\dot{\varphi}_{i}\sin^{2}\left(\theta_{i}\right)=\mathrm{const}$.
Then, putting $\theta_{i}=\pi/2$ for simplicity, we observe that the modulus
$\left|V_{\mathrm{Sj\ddot{o}qvist}}\left(\eta\right)\right|$ of the volume of
the explored region of the manifold of mixed states in Eq. (58) is upper
bounded by a function that grows linearly with $\eta$ with proportionality
coefficient given by $\dot{r}_{i}/\sqrt{1-r_{i}^{2}\text{ }}$ and not by a
constant function as in the case of the evolution of pure quantum states.
Moreover, recalling that
$V_{\mathrm{Sj\ddot{o}qvist}}^{\left(\text{accessible}\right)}=\pi^{2}/4\leq\pi=V_{\mathrm{FS}}^{\left(\text{accessible}\right)}$,
we clearly expect a more complex behavior for sufficiently large values of
$\eta$ in the case of the geometry along evolution of mixed states since

$\frac{V_{\mathrm{Sj\ddot{o}qvist}}^{\left(\text{explored}\right)}\left(\eta\right)}{V_{\mathrm{Sj\ddot{o}qvist}}^{\left(\text{accessible}\right)}}\geq\frac{V_{\mathrm{FS}}^{\left(\text{explored}\right)}\left(\eta\right)}{V_{\mathrm{FS}}^{\left(\text{accessible}\right)}}\text{.}$
(59)

From Eq. (58), the average explored region as given in Eq. (4) denotes the IGC
in Eq. (13) and becomes

$\mathcal{C}_{\mathrm{Sj\ddot{o}qvist}}\left(\tau\right)=\frac{1}{\tau}\int_{0}^{\tau}V_{\mathrm{Sj\ddot{o}qvist}}\left(\eta\right)d\eta\text{,}$
(60)

that is,

$\mathcal{C}_{\mathrm{Sj\ddot{o}qvist}}\left(\tau\right)=\frac{1}{8}\mathrm{a}_{\mathrm{FS}}\frac{\dot{r}_{i}}{\sqrt{1-r_{i}^{2}}}\frac{1}{\tau}\int_{0}^{\tau}\eta\sin\left(\eta\right)\arctan\left[\mathrm{c}_{\mathrm{FS}}\tan\left(\eta\right)\right]d\eta\text{.}$
(61)

Denoting $f\left(\eta\right)\overset{\text{def}}{=}\eta$ and
$\dot{g}\left(\eta\right)\overset{\text{def}}{=}\sin\left(\eta\right)\arctan\left[\mathrm{c}_{\mathrm{FS}}\tan\left(\eta\right)\right]$,
we integrate by parts the integral in Eq. (61) and get

$\int_{0}^{\tau}\eta\sin\left(\eta\right)\arctan\left[\mathrm{c}_{\mathrm{FS}}\tan\left(\eta\right)\right]d\eta=\left[\eta
I_{V_{\mathrm{FS}}}\left(\eta\right)\right]_{\eta=0}^{\eta=\tau}-\int_{0}^{\tau}I_{V_{\mathrm{FS}}}\left(\eta\right)d\eta\text{.}$
(62)

Note that $g\left(\eta\right)=I_{V_{\mathrm{FS}}}\left(\eta\right)$ with
$I_{V_{\mathrm{FS}}}\left(\eta\right)$ given in Eq. (53). Finally,
substituting Eq. (62) into Eq. (61) and considering the asymptotic temporal
behavior of $\mathcal{C}_{\mathrm{Sj\ddot{o}qvist}}\left(\tau\right)$, we
obtain

$\mathcal{C}_{\mathrm{Sj\ddot{o}qvist}}^{\mathrm{asymptotic}}\left(\tau\right)=\frac{1}{8}\mathrm{a}_{\mathrm{FS}}\frac{\dot{r}_{i}}{\sqrt{1-r_{i}^{2}}}I_{V_{\mathrm{FS}}}^{\mathrm{asymptotic}}\left(\tau\right)\text{.}$
(63)

At this point, considering the ratio between
$\mathcal{C}_{\mathrm{Sj\ddot{o}qvist}}^{\mathrm{asymptotic}}\left(\tau\right)$
in Eq. (63) and the asymptotic temporal behavior of
$\mathcal{C}_{\mathrm{FS}}\left(\tau\right)$ in Eq. (52), we get that the
relative asymptotic complexity growth in terms of a ratio exhibits a linear
behavior given by

$\frac{\mathcal{C}_{\mathrm{Sj\ddot{o}qvist}}^{\mathrm{asymptotic}}\left(\tau\right)}{\mathcal{C}_{\mathrm{FS}}^{\mathrm{asymptotic}}\left(\tau\right)}\sim\tau\text{.}$
(64)

In the long-time limit, Eq. (64) expresses the fact that the evolution of
mixed states in the Bloch ball equipped with the Sjöqvist metric explores
averaged volumes of regions larger that the ones inspected during the
evolution of pure states on the Bloch sphere supplied with the Fubini-Study
metric. In particular, there appears to be an asymptotic linear growth of the
ratio between the two IGCs. Finally, in terms of the IGE defined in Eq. (3),
we obtain an asymptotic entropy growth of the relative difference between the
two IGEs given by

$\mathcal{S}_{\mathrm{Sj\ddot{o}qvist}}^{\mathrm{asymptotic}}\left(\tau\right)-\mathcal{S}_{\mathrm{FS}}^{\mathrm{asymptotic}}\left(\tau\right)\sim\log\left(\tau\right)\text{.}$
(65)

Eqs. (65) displays the asymptotic logarithmic discrepancy between the IGE in
the mixed and pure quantum state scenarios. Since the IGC is simply the
exponential of the IGE, we can interpret this entropic deviation as follows.
To a larger IGE there corresponds a larger IGC. Larger IGCs are larger
asymptotic averaged explored volumes. Larger volumes encode, via the metric,
larger fluctuations. The larger the fluctuations, the closer the points (i.e.,
the states) are together. The closer points are together, the greater is the
likelihood of incorrectly distinguishing quantum states during the evolutions
of the quantum system. This, in turn, leads to higher entropic configurations
which are typical of quantum systems in a mixed quantum state. Note that Eqs.
(56), (59), (64), and (65) are non-conflicting and consistent relations in
support of arguments yielding to a higher degree of complexity of evolutions
of mixed states compared to pure quantum states from a geometric perspective.
Indeed, Eq. (56) is an inequality in terms of lengths. Eq. (59) is an
inequality between ratios expressed by means of accessible and instantaneous
explored volumes. Finally, Eqs. (64) and (65) are complexity and entropic
relations that are expressed by means of long-time limits of averaged explored
volumes of regions on and inside the Bloch ball. Interestingly, note that the
asymptotic temporal rates of change of the two IGCs in Eq. (64) scale in a
similar fashion,
$d\mathcal{C}_{\mathrm{Sj\ddot{o}qvist}}^{\mathrm{asymptotic}}/d\tau\sim
d\mathcal{C}_{\mathrm{FS}}^{\mathrm{asymptotic}}/d\tau$. This is a consequence
of a balancing effect that occurs between the asymptotic averaged explored
volumes,
$\bar{V}_{\mathrm{Sj\ddot{o}qvist}}^{\mathrm{asymptotic}}\sim\tau\bar{V}_{\mathrm{FS}}^{\mathrm{asymptotic}}$,
and between the asymptotic temporal rates of change of the averaged explored
volumes,
$d\bar{V}_{FS}^{\mathrm{asymptotic}}/d\tau\sim(1/\tau)d\bar{V}_{\mathrm{Sj\ddot{o}qvist}}^{\mathrm{asymptotic}}/d\tau$.
From a curvature analysis perspective, the manifold of pure states equipped
with the Fubini-Study metric is an isotropic two-dimensional manifold of
constant positive sectional curvature $\mathcal{K}_{\mathrm{FS}}=4$ and
constant scalar curvature $\mathcal{R}_{\mathrm{FS}}=8$. Instead, the manifold
of mixed states equipped with the Sjöqvist metric is an anisotropic three-
dimensional manifold of non-constant but positive sectional curvature and
constant scalar curvature $\mathcal{R}_{\mathrm{Sj\ddot{o}qvist}}=8$. The
positivity of sectional curvatures in both scenarios leads to the presence of
convergence in the geodesic spread analysis on both manifolds. However, given
the anisotropic nature in the mixed quantum states manifold with the Sjöqvist
metric, the study of the geodesic spread equation would be more complicated in
the scenario of mixed states distinguished via the Sjöqvist metric. Our
remarks concerning the asymptotic rates of change of volumes and IGEs are an
indication of this distinct convergent behavior in the pure and mixed quantum
states scenarios. For further details on curvature properties along with a
comparison between the Sjöqvist and Bures manifolds, we refer to Appendix E.

Type of state | Metric | Sectional curvature | Path length | Information geometric complexity
---|---|---|---|---
Pure | Fubini-Study | Constant | Shorter | Lower
Mixed | Sjöqvist | Nonconstant | Longer | Higher
Table 2: Schematic description of geometric properties (i.e., curvature,
length, and complexity) along evolutions of pure and mixed quantum states on
manifolds equipped with the Fubini-Study and Sjöqvist metrics, respectively.

##  V Physical considerations

In this section, we present physical comments on the concepts of metric, path
length, and curvature employed in our investigation. Moreover, we clarify the
physics behind the evolution of quantum states on curved manifolds in terms of
Bloch coordinates. These comments will help highlighting even further the
physical significance of our proposed complexity measure in Eq. (13).

###  V.1 Metric, path length, and curvature

For completeness, we begin by recalling that in quantum mechanics a physical
state is not represented by a normalized state vector
$\left|\psi\left(t\right)\right\rangle\in\mathcal{H}\backslash\left\\{0\right\\}$
but by a ray. A ray is the one-dimensional subspace to which this vector
belongs. Two normalized vectors are equivalent,
$\left|\psi^{\prime}\right\rangle\sim$ $\left|\psi\right\rangle$, if they
belong to the same ray, i.e., if
$\left|\psi^{\prime}\right\rangle=e^{i\phi}\left|\psi\right\rangle$ with
$\phi\in U\left(1\right)$. This equivalence relation specifies equivalence
classes on the sphere $\mathcal{S}^{2N_{\mathcal{H}}+1}$, with
$\dim_{\mathbb{C}}\mathcal{H}=N_{\mathcal{H}}+1$. Finally, the set of
equivalence classes $\mathcal{S}^{2N_{\mathcal{H}}+1}/U(1)$ forms the space of
physical states (rays) which is denoted here by $\mathcal{P}$. The space of
rays is the projective Hilbert space $\mathcal{P}$ which, in turn, is
isomorphic to the complex projective space $\mathbb{C}P^{N}$.

####  V.1.1 Metric

The metric (Eqs. (24), (39), and (40)) on the manifold of quantum states is
fixed once the quantum mechanical fluctuation in energy is specified provost80
. Focusing on pure states, when the uncertainty $\Delta A\left(\eta\right)$ in
the generator of motion $A\left(\eta\right)$ with respect to the parameter
$\eta$ in the projective Hilbert space $\mathcal{P}$ is provided, the metric

$ds_{\mathrm{FS}}^{2}=g_{\mu\nu}\left(\xi\right)\frac{d\xi^{\mu}}{d\eta}\frac{d\xi^{\nu}}{d\eta}d\eta^{2}=\Delta
A^{2}\left(\eta\right)d\eta^{2}$ (66)

is fixed. In particular, assuming $\eta=t$ and
$A\left(\eta\right)=$H$\left(t\right)/\hslash$, we have that
$ds_{\mathrm{FS}}=\left[\Delta E\left(t\right)/\hslash\right]dt$ and the
dispersion $\Delta E\left(t\right)$ of the generator of motion can originate
from a variety of Hamiltonians H$\left(t\right)$. Therefore, the geometry of
the projective Hilbert space, specified by the metric on it, cannot be
modified by the dynamics of the system governed by the Hamiltonian
H$\left(t\right)$ pati91 . This was a major result obtained by Anandan and
Aharonov in Ref. anandan90 . For pure states, the distance function in the
projective Hilbert space $\mathcal{P}$ is the distance between two quantum
states along a given curve in $\mathcal{P}$ as measured by the Fubini-Study
metric defined from the inner product of the representative states in the
$\left(N_{\mathcal{H}}+1\right)$-dimensional Hilbert space $\mathcal{H}$.
Anandan and Aharonov showed that this distance equals the time integral of the
uncertainty of the energy, and does not depend on the particular Hamiltonian
used to move the quantum system along a given curve in $\mathcal{P}$. It is
dependent only on the points in $\mathcal{P}$ to which the quantum states
project. In summary, from a physics standpoint, the metric tensor and its
components on the projective Hilbert space $\mathcal{P}$ are linked to the
dispersion of suitable quantum-mechanical operators (for instance, the
Hamiltonian operator) acting on the underling Hilbert space $\mathcal{H}$.
This connection between metrics and quantum fluctuations is an important
physical consideration to keep in mind throughout our work. This connection
extends to the geometric analysis of quantum mixed states as well zanardi07A ;
zanardi07B ; erik20 .

####  V.1.2 Path length

To explain the physical meaning of the Riemannian distance (Eqs. (49) and
(54)) between two arbitrarily chosen pure quantum states, we follow Wootters
wootters81 . For mixed states, we hint to the work by Braunstein and Caves in
Ref. braunstein94 . Two infinitesimally close points $\xi$ and $\xi+d\xi$
along a path $\xi\left(\eta\right)$ with $\eta_{1}\leq\eta\leq\eta_{2}$ are
statistically distinguishable if $d\xi$ is at least equal to the standard
fluctuation of $\xi$ diosi84 . The line element along the path is
$ds_{\mathrm{FS}}$ with
$ds_{\mathrm{FS}}^{2}=g_{\mu\nu}^{\mathrm{FS}}\left(\xi\right)d\xi^{\mu}d\xi^{\nu}$.
The length of the path $\xi\left(\eta\right)$ with
$\eta_{1}\leq\eta\leq\eta_{2}$ between
$\xi_{1}\overset{\text{def}}{=}\xi\left(\eta_{1}\right)$ and
$\xi_{2}\overset{\text{def}}{=}\xi\left(\eta_{2}\right)$ is defined as

$\mathcal{L}\overset{\text{def}}{=}\int_{\xi_{1}}^{\xi_{2}}\sqrt{ds_{\mathrm{FS}}^{2}}=\int_{\eta_{1}}^{\eta_{2}}\sqrt{\frac{ds_{\mathrm{FS}}^{2}}{d\eta^{2}}}d\eta\text{,}$
(67)

and represents the maximal number $\tilde{N}$ of statistically distinguishable
states along the path. In particular, the geodesic distance between $\xi_{1}$
and $\xi_{2}$ is the path of shortest distance between $\xi_{1}$ and $\xi_{2}$
and is the minimum of $\tilde{N}$. This connection between path length and
number of statistically distinguishable states along the path is a relevant
physical remark to consider throughout our investigation. This viewpoint
extends naturally to the geometric analysis of quantum mixed states as well
braunstein94 .

####  V.1.3 Curvature

What is the physical significance of curvature (Appendix E) in our
investigation? We recall that in the Riemannian geometrization of classical
Newtonian mechanics pettini00 , the curvature $\mathcal{R}$ of the manifold
corresponds, roughly speaking, to the curvature of the potential $V$ expressed
by means of the second derivative of $V$, $\mathcal{R}\sim\partial^{2}V$, with
the Hamiltonian of the system given by H$\left(p\text{,
}q\right)=p^{2}/(2m)+V\left(q\right)$. More generally, in arbitrary
differential geometric settings, the curvature of the manifold determines the
stability (or, alternatively, the instability) of the geodesics via the Jacobi
equation of geodesic spread. This latter curvature interpretation remains
valid in our work. However, we can provide a more specific interpretation for
the concept of curvature in our analysis. As pointed out by Braunstein and
Caves in Ref. caves95 , unlike what happens in general relativity, the
geometry on the space of quantum states does not describe the dynamical
evolution of the physical system. Rather, it places limits on our ability to
discriminate one state from another via measurements. In a sense, the geometry
of quantum states puts the emphasis on the fact that quantum mechanics is
rooted in making statistical inferences based on observed experimental data.
Quantum measurement theory, in turn, is statistical inference in its essence
luo03 . Therefore, given the fact that the problem of distinguishing
neighboring quantum states can be formulated as a parameter estimation problem
braunstein94 , given that quantum mechanics can be regarded as a theory for
making statistical inferences based on observed experimental data luo03 , and,
finally, since the curvature of a manifold is a measure of how difficult is to
do estimations at a given point in statistical science efron75 , it is
reasonable to interpret the curvature of a manifold of quantum states equipped
with suitably defined metric structures as an indicator of how difficult is to
distinguish quantum states by means of parameter estimation at a given point
of the state space. In particular, the higher the curvature, the more
difficult is to do estimation at that point. This is the point of view that we
adopt in this paper. For remarks on a physical interpretation of curvature of
manifolds underlying the information geometry of non-interacting gases
satisfying the Fermi-Dirac and Bose-Einstein statistics, we refer to Ref.
pessoa21 . Finally, for a work on the estimation of the curvature of a quantum
manifold via measurement on a quantum particle constrained to propagate on the
manifold itself, we hint to Ref. paris19 .

###  V.2 Evolution of Bloch coordinates

Having clarified the meaning of metric, path length, and curvature employed in
our work we devote some time explaining the relation between Bloch coordinates
and quantum states on the Bloch sphere and inside the Bloch ball (Appendix A,
C, and D). Let us point out from the start that in our work
$\xi^{\mu}\left(\eta\right)\overset{\text{def}}{=}\left(\xi^{1}\left(\eta\right)\text{,
}\xi^{2}\left(\eta\right)\text{, }\xi^{3}\left(\eta\right)\right)$ in Eq. (66)
is specified by the Bloch coordinates, that is,
$\xi^{\mu}\left(\eta\right)=\left(r\left(\eta\right)\text{,
}\theta\left(\eta\right)\text{, }\varphi\left(\eta\right)\right)$. In this
paper, we focused on the integration of evolution equations for Bloch
coordinates used to parametrize quantum states, either pure or mixed. This
choice was not dictated by mathematical convenience only. Indeed, there is a
clear physical path connecting Bloch coordinates, Bloch vectors, and, finally,
pure and mixed quantum states. For simplicity, we set $\eta=t$ in this
discussion. We observe that from the time evolution of the Bloch coordinates,
both radial and angular, one can generally recover the time evolution of the
density operators for arbitrary quantum states via the time evolution of the
Bloch vector. Conversely, the opposite is also possible. For a detailed study
concerning the time evolution of the Bloch vector of a single two-level atom
that interacts with a single quantized electromagnetic field mode according to
the Jaynes-Cummings model, we refer to Ref. azuma08 . To be explicit here, we
note that the Bloch vector $\vec{p}\left(t\right)$ is defined as

$\vec{p}\left(t\right)\overset{\text{def}}{=}\mathrm{tr}\left[\rho\left(t\right)\vec{\sigma}\right]=\left(r\sin\theta\cos\varphi\text{,
}r\sin\theta\sin\varphi\text{, }r\cos\theta\right)\text{,}$ (68)

with $r=r\left(t\right)$, $\theta=\theta\left(t\right)$, and
$\varphi=\varphi\left(t\right)$. Focusing for simplicity on the case of
unitary quantum evolution, the density operator
$\rho\left(t\right)=(1/2)\left[\mathrm{I}+\vec{p}\left(t\right)\cdot\vec{\sigma}\right]$
in Eq. (68) satisfies the von Neumann equation
$i\hslash\dot{\rho}=\left[\mathrm{H}\left(t\right)\text{,
}\rho\left(t\right)\right]$ with $\dot{\rho}\overset{\text{def}}{=}d\rho/dt$
and $\mathrm{I}$ denoting the identity operator on the single-qubit quantum
state space. Moreover, for a system in a pure state that evolves under a time
independent Hamiltonian H,
$\rho\left(t\right)=\left|\psi\left(t\right)\right\rangle\left\langle\psi\left(t\right)\right|=U\left(t\right)\rho\left(0\right)U^{\dagger}\left(t\right)$
with $U\left(t\right)$ being the unitary time evolution operator given by
$\exp\left[-(i/\hslash)\mathrm{H}t\right]$ and
$\left|\psi\left(t\right)\right\rangle=\cos\left[\theta\left(t\right)/2\right]\left|0\right\rangle+e^{i\varphi\left(t\right)}\sin\left[\theta\left(t\right)/2\right]\left|1\right\rangle$
in the Bloch sphere parametrization. To make very clear this link among Bloch
coordinates, Bloch vectors, pure states, and mixed states, we consider a
simple illustrative example for pure states evolution. Assume the system, a
spin-$1/2$ particle, is initially in the state
$\left|\psi\left(0\right)\right\rangle$ parametrized in terms of
$\left(\theta\left(0\right)\text{,
}\varphi\left(0\right)\right)=\left(\pi/2\text{, }0\right)$. Solving the
Schrödinger’s evolution equation
$i\hslash\partial_{t}\left|\psi\left(t\right)\right\rangle=$H$\left(t\right)\left|\psi\left(t\right)\right\rangle$
with H$\overset{\text{def}}{=}\hslash\omega_{0}\sigma_{z}$, we get

$\rho\left(t\right)=\left|\psi\left(t\right)\right\rangle\left\langle\psi\left(t\right)\right|=\frac{1}{2}\left(\begin{array}[c]{cc}1&e^{-2i\omega_{o}t}\\\
e^{2i\omega_{o}t}&1\end{array}\right)=\frac{1}{2}\left[\mathrm{I}+\vec{p}\left(t\right)\cdot\vec{\sigma}\right]\text{,}$
(69)

with $\vec{p}\left(t\right)=\left(\cos\left(2\omega_{0}t\right)\text{,
}\sin\left(2\omega_{0}t\right)\text{, }0\right)$. Furthermore, the Bloch
angles at time $t$ become $\left(\theta\left(t\right)\text{,
}\varphi\left(t\right)\right)=\left(\pi/2\text{, }2\omega_{0}t\right)$. In
general, the state
$\left|\psi\left(t\right)\right\rangle=U\left(t\right)\left|\psi\left(0\right)\right\rangle$
can be parametrized as
$\left|\psi\left(t\right)\right\rangle=a\left(t\right)\left|0\right\rangle+b\left(t\right)\left|1\right\rangle$
with $\mathbb{C}\ni
a\left(t\right)=\left|a\left(t\right)\right|e^{i\varphi_{a}\left(t\right)}$,
$\mathbb{C}\ni
b\left(t\right)=\left|b\left(t\right)\right|e^{i\varphi_{b}\left(t\right)}$,
$\varphi_{a\text{, }b}\left(t\right)\in\mathbb{R}$, and
$\left|a\left(t\right)\right|^{2}+\left|b\left(t\right)\right|^{2}=1$. With
this pure state parametrization, the Bloch angles $\theta\left(t\right)$ and
$\varphi\left(t\right)$ satisfy the relations,
$\tan\left[\theta\left(t\right)\right]=\left|b\left(t\right)\right|/\left|a\left(t\right)\right|$
and
$\varphi\left(t\right)=\varphi_{b}\left(t\right)-\varphi_{a}\left(t\right)$,
respectively.

Having clarified the physical meaning of geometrical concepts employed in our
analysis, the main take-home message is the following. We have estimated in
this paper the complexity of geodesic paths of both pure and mixed quantum
states by means of a complexity measure (Eq. (13)) expressed in terms of
explored volumes of the suitably metricized curved manifolds that underlay the
dynamics (i.e., the change in Bloch parameters, with changes specified by the
parametric evolution operator). The metric structure on the curved manifolds
of quantum states is fixed by quantum-mechanical fluctuations. Moreover, just
as path lengths can be interpreted in terms of the maximal number of
distinguishable states traversed during the evolution along the path, the
volumes of the parametric space explored in a fixed temporal interval can be
regarded as representing the maximal number of different states visited during
the regional exploration. Clearly, the role played by the infinitesimal
increment $d\xi$ in the path exploration is replaced by the infinitesimal
volume element $dV\overset{\text{def}}{=}\sqrt{g\left(\xi\right)}d^{N}\xi$ in
the regional travel, with
$g\left(\xi\right)\overset{\text{def}}{=}\det\left[g_{\mu\nu}\left(\xi\right)\right]$
and $N$ being the dimensionality of the curved manifold. Essentially, the
Riemannian volume element $dV$ helps gauging the number of distinct states
explored within an infinitesimal volume of a region of the manifold myung00 .
We are ready now for our conclusions.

##  VI Concluding Remarks

We present here a summary of our main findings along with limitations and
possible future directions.

###  VI.1 Summary of results

In this paper, we provided a comparative information geometric analysis of the
complexity of geodesic paths of pure and mixed quantum states on the Bloch
sphere and inside the Bloch ball, respectively. In this geometric setting,
pure and mixed states were chosen to be distinguished by means of the Fubini-
Study (Eq. (24)) and the Sjöqvist metric (Eq. (39)), respectively. After
finding the geodesic paths connecting arbitrary points on (see Appendix A) and
inside the Bloch ball (see Appendix C), we analytically estimated the IGE (Eq.
(3)) and the IGC (Eq. (13)) in both scenarios. The long-time limit of this
pair of entropic measures of complexity of evolution of system in pure (see
Eq. (52)) and mixed (see Eq. (63)) states were compared. We observed a degree
of complexity for the evolution of mixed states with the Sjöqvist geometry
higher than the one specifying the complexity for the evolution of pure states
with the Fubini-Study geometry (see Eqs. (64) and (65)).

The metric structure on the manifold of quantum states is specified by quantum
fluctuations. Path lengths and volumes can be physically interpreted as
indicators of the maximal number of distinguishable states crossed along
trajectories and in volumes of regions of the manifold, respectively. To a
higher count of distinct states passed over in a fixed time interval, there
corresponds a higher degree of complexity of the evolution on the underlying
manifold. Within this physically meaningful geometric description, mixed state
(geodesic) evolutions appear to be generally more complex than pure state
evolutions.

Our main findings can be outlined as follows:

  1. [1]

We proposed a different information geometric way (Eqs. (3) and (13)) to
describe and, to a certain extent, understand the complex behavior of
evolutions of quantum systems in pure and mixed states. The ranking is
probabilistic in nature, it requires a temporal averaging procedure along with
a long-time limit, and is limited to comparing expected geodesic evolutions on
the underlying manifolds.

  2. [2]

We showed (Eqs. (64) and (65)) that the complexity of geodesic paths (Appendix
C) corresponding to the evolution of mixed quantum states in the Bloch ball
equipped with the Sjöqvist metric is higher than the complexity of geodesic
paths (Appendix A) arising from the evolution of pure states on the Bloch
sphere furnished with the Fubini-Study metric.

  3. [3]

We found that the ranking in terms of the information geometric complexity
(64), a quantity that represents the asymptotic temporal behavior of an
averaged volume of the region explored on the manifold during the evolution,
is in agreement with the ranking in terms of lengths (Eq. (56)) and, in
addition, volume ratios in terms of accessible and instantaneous explored
volumes (59). For a schematic summary, we refer to Table II.

  4. [4]

We confirmed that the choice of the metric on the space of mixed states
matters. Specifically, we observed fingerprints of a softening of the
complexity on the Bures manifold (Eq. (125)) compared to the Sjöqvist
manifold. This is in agreement with the presence of longer lengths of geodesic
paths on the Sjöqvist manifold (Eq. (124)). Furthermore, the two manifolds
exhibit different curvature properties. The Bures manifold is isotropic, while
the Sjöqvist manifold is anisotropic (Appendix E). For a schematic outline, we
hint to Table III.

Metric | Sectional curvature | Path length | Information geometric complexity
---|---|---|---
Sjöqvist | Nonconstant | Longer | Stronger
Bures | Constant | Shorter | Weaker
Table 3: Schematic description of distinct features of the Sjöqvist and Bures
metrics in terms of sectional curvatures, path lengths, and information
geometric complexities.

###  VI.2 Outlook

Like most scientific studies, our investigation suffers a few limitations.
From a computational standpoint, our proposed complexity measure requires
volume calculations which are much harder than action or path length
calculations since there are differential equations to be solved. This
particular point is in agreement with what stated in Ref. leo16B . Therefore,
exact analytical solutions are rare and approximate numerical solutions are
unavoidable in more realistic physical scenarios. From a conceptual
perspective, there are at least two weaknesses. First, there is a freedom in
the choice of the metric for mixed quantum states. For further details on the
Sjöqvist metric that concern its extension to the degenerate case along with
its relation to the Bures metric, we refer to Appendix F. Second, we only
limited our work to the study of two-level quantum systems. The ambiguity in
the metric affects the notion of speed which, in turn, is related to the
concepts of length, action, and complexity. The dependence of the ratio of
distance and time on the choice of the metric on the space of mixed quantum
states is in agreement with the considerations carried out in Ref. brody19 .
The restriction to a single qubit, while simple and insightful, cannot be
expected to cover the full richness of a higher-dimensional quantum dynamics
occurring in an exponentially larger Hilbert space. In particular, scaling
laws with respect to the dimensionality of the Hilbert space of suitable
physical quantities cannot be addressed in this limiting scenario. Moreover,
one fundamental quantum phenomenon that escapes a single qubit treatment is
entanglement. This second limitation is similar to the one presented in Ref.
brown19 .

Despite these limitations, we believe our work is relevant also in view of the
fact that it paves the way to further lines of inquiry. For instance, it
naturally triggers the following questions: 1) Using our findings along with
the ones presented in Refs. nielsen06 ; brown19 , can one compare the
complexity of geodesic motion on differently deformed Bloch spheres by adding
anisotropic penalty factors to the Fubini-Study metric?; 2) What happens to
the evolution of mixed states inside a deformed Bloch ball when introducing
anisotropic penalty factors? Is the relative ranking in terms of complexity
between pure and mixed states preserved under any arbitrary deformations of
the Bloch sphere?; 3) Can one compare the complexity of geodesic paths on a
deformed Bloch sphere with the complexity of geodesic paths in a non deformed
Bloch ball? 4) Is there a minimal complexity metric for mixed quantum states?;
5) Using our curvature calculations and the analysis presented in Ref. auzzi21
, how much does one need to deform the Bloch sphere ( that is, introducing
anisotropic penalty factors to get negative sectional curvature) and how high
should the dimensionality of a quantum system be in order to address the issue
of ergodicity and properly apply thermodynamic arguments to complexity
evolution? The introduction of anisotropic penalty factors can be motivated by
experimental considerations. For example, considering a spin-$1/2$ particle in
an external magnetic field, it may be the case that is is easier to apply the
field in some direction rather that in another. In this case, the penalty
factor would be larger where it is more complicated to apply the field.
Incorporating these factors in our analysis would open up to lines of
investigation of relevance in the context of finding optimal Hamiltonian
evolutions, both in terms of efficiency cafaropra20 ; cafaropra22 ; ijqi19 ;
ps19 ; steven20 ; f19 ; e21  and complexity cafaropre20 ; cafaropre22 , in the
presence of physical constraints dictated by experimental limitations.

We hope our work will inspire other scientists and pave the way toward further
investigations in this fascinating research direction. For the time being, we
leave a more in-depth quantitative discussion on these potential extensions
and applications of our theoretical findings to future scientific efforts.

###### Acknowledgements.

C.C. is grateful to the United States Air Force Research Laboratory (AFRL)
Summer Faculty Fellowship Program for providing support for this work. P.M.A.
acknowledges support from the Air Force Office of Scientific Research (AFOSR).
Any opinions, findings and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views
of the Air Force Research Laboratory (AFRL).

## References

  * (1) L. Casetti, M. Pettini, and E. G. D. Cohen, _Geometric approach to Hamiltonian dynamics and statistical mechanics_ , Phys. Rep. 337, 237 (2000).
  * (2) M. Pettini, _Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics_ , Springer-Verlag New York (2007).
  * (3) M. A. Nielsen, M. R. Dowling, and A. C. Doherty, _Quantum computation as geometry_ , Science 311, 1133 (2006).
  * (4) M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information_ , Cambridge University Press (2010).
  * (5) A. Bhattacharyya, P. Nandy, and A. Sinha, _Renormalized circuit complexity_ , Phys. Rev. Lett. 124, 101602 (2020).
  * (6) L. Susskind, _Three lectures on complexity and black holes_ , arXiv:hep-th/1810.11563 (2018).
  * (7) D. Stanford and L. Susskind, _Complexity and shock wave geometries_ , Phys. Rev. D90, 126007 (2014).
  * (8) D. A. Roberts, D. Stanford, and L. Susskind, _Localized shocks_ , J. High Energ. Phys. 2015, 51 (2015).
  * (9) A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, _Holographic complexity equals bulk action?_ , Phys. Rev. Lett. 116, 191301 (2016).
  * (10) A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, _Complexity, action, and black hole_ , Phys. Rev. D93, 086006 (2016).
  * (11) A. R. Brown and L. Susskind, _Complexity geometry of a single qubit_ , Phys. Rev. D100, 046020 (2019).
  * (12) R. Auzzi, S. Baiguera, G. B. De Luca, A. Legramandi, G. Nardelli, and N. Zenoni, _Geometry of quantum complexity_ , Phys. Rev. D103, 106021 (2021).
  * (13) A. R. Brown, L. Susskind, and Y. Zhao, _Quantum complexity and negative curvature_ , Phys. Rev. D95, 045010 (2017).
  * (14) S. Chapman, M. P. Heller, H. Marrachio, and F. Pastawski, _Toward a definition of complexity for quantum field theory states_ , Phys. Rev. Lett. 120, 121602 (2018).
  * (15) S.-M. Ruan, _Circuit Complexity of Mixed States_ , Ph.D. in Physics, University of Waterloo (2021).
  * (16) K. Huang, _Statistical Mechanics_ , John Wiley & Sons, Inc. (1987).
  * (17) P. Lipka-Bartosik, M. Perarnau-Llobet, and N. Brunner, _What is the temperature of a quantum state?_ , arXiv:quanth-ph/2205.0001 (2022)
  * (18) M. T. Mitchinson, A. Purkayastha, M. Brenes, A. Silva, and J. Goold, _Taking the temperature of a pure quantum state_ , Phys. Rev. A105, L030201 (2022).
  * (19) K. Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, _Volume of the set of separable states_ , Phys. Rev. A58, 883 (1998).
  * (20) D.-H. Kim, S. A. Ali, C. Cafaro, and S. Mancini, _Information geometric modeling of scattering induced quantum entanglement_ , Phys. Lett. A375, 2868 (2011).
  * (21) D.-H. Kim, S. A. Ali, C. Cafaro, and S. Mancini, _Information geometry of quantum entangled wave-packets_ , Physica A391, 4517 (2012).
  * (22) L. A. Caron H. Jirari, H. Kröger, X. Q. Luo, G. Melkonyan, and K. J. M. Moriarty, _Quantum chaos at finite temperature_ , Phys. Lett. A288, 145 (2001).
  * (23) L. A. Caron, D. Huard, H. Kroger, G. Melkonyan, K. J. M. Moriarty, and L. P. Nadeau,_Is quantum chaos weaker than classical chaos?_ , Phys. Lett. A322, 60 (2004).
  * (24) H. Kroger, J.-F. Laprise, G. Melkonyan, and R. Zomorrodi,_Quantum chaos versus classical chaos: Why is quantum chaos weaker?”_ , in _The Logistic Map to the Route to Chaos_ , pages 355-367, Eds. M. Ausloos and M. Dirickx, Springer-Verlag Berlin Heidelberg (2006).
  * (25) C. Cafaro, A. Giffin, C. Lupo, and S. Mancini, _Insights into the softening of chaotic statistical models by quantum considerations_ , AIP Conf. Proc. 1443, 366 (2012).
  * (26) C. Cafaro, A. Giffin, C. Lupo, and S. Mancini, _Softening the complexity of entropic motion on curved statistical manifolds_ , Open Systems & Information Dynamics 19, 1250001 (2012).
  * (27) M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, _Quantum speed limits for physical processes_ , Phys. Rev. Lett. 110, 050402 (2013)
  * (28) A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. Huelga, _Quantum speed limits in open quantum systems dynamics_ , Phys. Rev. Lett. 110, 050403 (2013)
  * (29) S. Deffner and E. Lutz, _Quantum speed limit for non-Markovian dynamics_ , Phys. Rev. Lett. 111, 010402 (2013)
  * (30) M. Okuyama and M. Ohzehi, _Quantum speed limit is not quantum_ , Phys. Rev. Lett. 120, 070402 (2018).
  * (31) B. Shanahan, A. Chenu, N. Margolus, and A. del Campo, _Quantum speed limits across the quantum-to-classical transition_ , Phys. Rev. Lett. 120, 070401 (2018).
  * (32) K. Bolonek-Lason, J. Gonera, P. Kosinki, _Classical and quantum speed limits_ , Quantum 5, 482 (2021).
  * (33) M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information_ , Cambridge University Press (2000).
  * (34) C. Cafaro and S. Mancini, _Characterizing the depolarizing quantum channel in terms of Riemannian geometry_ , Int. J. Geom. Meth. Mod. Phys. 9, 1260020 (2012).
  * (35) A. Carlini, A. Hosoya, T. Koike, and Y. Okudaira, _Time optimal quantum evolution of mixed states_ , J. Phys. A: Math. Theor. 41, 045303 (2008).
  * (36) D. Girolami, _How difficult is it to prepare a quantum state?_ , Phys. Rev. Lett. 122, 010505 (2019).
  * (37) N. Hornedal, D. Allan, and O. Sonnerborn, _Extensions of the Mandelstam-Tamm quantum speed limit to systems in mixed states_ , New J. Phys. 24, 055004 (2022).
  * (38) D. Felice, C. Cafaro, and S. Mancini, _Information geometric methods for complexity_ , Chaos 28, 032101 (2018).
  * (39) C. Cafaro, S. Ray, and P. M. Alsing, _Geometric aspects of analog quantum search evolutions_ , Phys. Rev. A102, 052607 (2020).
  * (40) C. Cafaro, S. Ray, and P. M. Alsing, _Optimal-speed unitary quantum time evolutions and propagation of light with maximal degree of coherence_ , Phys. Rev. A105, 052425 (2022).
  * (41) C. Cafaro and P. M. Alsing, _Information geometry aspects of minimum entropy production paths from quantum mechanical evolutions_ , Phys. Rev. E101, 022110 (2020).
  * (42) C. Cafaro, S. Ray, and P. M. Alsing, _Complexity and efficiency of minimum entropy production probability paths from quantum dynamical evolutions_ , Phys. Rev. E105, 034143 (2022).
  * (43) C. Cafaro and S. A. Ali, _Jacobi fields on statistical manifolds of negative curvature_ , Physica D234, 70 (2007).
  * (44) C. Cafaro, _The Information Geometry of Chaos_ , PhD Thesis, State University of New York at Albany, Albany-NY, USA (2008). Available online at arXiv: math-ph/1601.07935 (2016); C. Cafaro, _Information-geometric indicators of chaos in Gaussian models on statistical manifolds of negative Ricci curvature_ , Int. J. Theor. Phys. 47, 2924 (2008); C. Cafaro,_Information geometry, inference methods and chaotic energy levels statistics_ , Mod. Phys. Lett. B22, 1879 (2008); C. Cafaro, _Works on an information geometrodynamical approach to chaos_ , Chaos, Solitons and Fractals 41, 886 (2009).
  * (45) S. A. Ali, C. Cafaro, S. Gassner, and A. Giffin, _An information geometric perspective on the complexity of macroscopic predictions arising from incomplete information_ , Adv. Math. Phys., Volume 2018, Article ID 2048521 (2018).
  * (46) C. Cafaro and S. A. Ali, _Information geometric measures of complexity with applications to classical and quantum physical settings_ , Foundations 1, 45 (2021).
  * (47) S. A. Ali, C. Cafaro, D.-H. Kim, and S. Mancini, _The effect of microscopic correlations on the information geometric complexity of Gaussian statistical models_ , Physica A389, 3117 (2010).
  * (48) S. A. Ali and C. Cafaro, _Theoretical investigations of an information geometric approach to complexity_ , Rev. Math. Phys. 29, 1730002 (2017).__
  * (49) C. Cafaro, A. Giffin, S. A. Ali, and D.-H. Kim, _Reexamination of an information geometric construction of entropic indicators of complexity_ , Appl. Math. Comput. 217, 2944 (2010).
  * (50) W. K. Wootters, _Statistical distance and Hilbert space_ , Phys. Rev. D23, 357 (1981).
  * (51) S. L. Braunstein and C. M. Caves, _Statistical distance and the geometry of quantum states_ , Phys. Rev. Lett. 72, 3439 (1994).
  * (52) R. Uzdin, U. Guenther, S. Rahav, and N. Moiseyev, _Time-dependent Hamiltonians with 100% evolution speed efficiency_ , J. Phys. A: Math. Theor. 45, 415304 (2012).
  * (53) E. Sjöqvist, _Geometry along evolution of mixed quantum states_ , Phys. Rev. Research 2, 013344 (2020).
  * (54) H. Silva, B. Mera, and N. Paunkovic, _Interferometric geometry from symmetry-broken Uhlmann gauge group with applications to topological phase transitions_ , Phys. Rev. B103, 085127 (2021).
  * (55) J. P. Provost and G. Vallee, _Riemannian structure on manifolds of quantum states_ , Commun. Math. Phys. 76, 289 (1980).
  * (56) D. Bures, _An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite_ $\omega^{\ast}$_-algebras_ , Trans. Amer. Math. Soc. 135, 199 (1969).
  * (57) A. Uhlmann, _The “transition probability” in the state space of a_ $\ast$_-algebra_ , Rep. Math. Phys. 9, 273 (1976).
  * (58) M. Hübner, _Explicit computation of the Bures distance for density matrices_ , Phys. Lett. A163, 239 (1992).
  * (59) C. Cafaro, O. Luongo, S. Mancini, and H. Quevedo, _Thermodynamic length, geometric efficiency, and Legendre invariance_ , Physica A590, 126740 (2022).
  * (60) G. Ruppeiner, _Thermodynamics: A Riemannian geometric model_ , Phys. Rev. A20, 1608 (1979).
  * (61) G. E. Crooks,_Measuring thermodynamic length_ , Phys. Rev. Lett. 99, 100602 (2007).
  * (62) L. Diosi, G. Forgacs, B. Lukacs, and H. L. Frisch, _Metricization of thermodynamic-state space and the renormalization group_ , Phys. Rev. A29, 3343 (1984).
  * (63) F. Weinhold, _Metric geometry of equilibrium thermodynamics_ , J. Chem. Phys. 63, 2479 (1975).
  * (64) P. Salamon and R. S. Berry, _Thermodynamic length and dissipated availability_ , Phys. Rev. Lett. 51, 1127 (1983).
  * (65) A. K. Pati, _Relation between “phases” and “distance” in quantum evolution_ , Phys. Lett. A159, 105 (1991).
  * (66) J. Anandan and Y. Aharonov, _Geometry of quantum evolution_ , Phys. Rev. Lett. 65, 1697 (1990).
  * (67) P. Zanardi, P. Giorda, and M. Cozzini, Information-theoretic differential geometry of quantum phase transitions, Phys. Rev. Lett. 99, 100603 (2007).
  * (68) P. Zanardi, L. C. Venuti, and P. Giorda, _Bures metric over thermal state manifolds and quantum criticality_ , Phys. Rev. A76, 062318 (2007).
  * (69) S. L. Braunstein and C. M. Caves, _Geometry of quantum states_. In: B. V. Belavkin, O. Hirota, and R. L. Hudson (eds.), Quantum Communications and Measurement, pp. 21-30. Springer, Boston, MA (1995).
  * (70) S. Luo, _Wigner-Yanase skew information and uncertainty relations_ , Phys. Rev. Lett. 91, 180403 (2003).
  * (71) B. Efron, _Defining the curvature of a statistical problem (with applications to second order efficiency)_ , The Annals of Statistics 3, 1189 (1975).
  * (72) P. Pessoa and C. Cafaro, _Information geometry for Fermi-Dirac and Bose-Einstein quantum statistics_ , Physica A576, 126061 (2021).
  * (73) D. Bonalda, L. Seveso, and M. G. A. Paris, _Quantum sensing of curvature_ , Int. J. Theor. Phys. 58, 2914 (2019).
  * (74) H. Azuma, _Dynamics of the Bloch vector in the thermal Jaynes-Cummings model_ , Phys. Rev. A77, 063820 (2008).
  * (75) J. Myung, V. Balasubramanian, and M. A. Pitt, _Counting probability distributions: Differential geometry and model selection_ , Proceedings of the National Academy of Sciences 97, 11170 (2000).
  * (76) D. C. Brody and B. Longstaff, _Evolution speed of open quantum dynamics_ , Phys. Rev. Research 1, 033127 (2019).
  * (77) C. Cafaro and P. M. Alsing, _Continuous-time quantum search and time-dependent two-level quantum systems_ , Int. J. Quantum Information 17, 1950025 (2019).
  * (78) C. Cafaro and P. M. Alsing, _Theoretical analysis of a nearly optimal analog quantum search_ , Physica Scripta 94, 085103 (2019).
  * (79) C. Cafaro, S. Gassner, and P. M. Alsing, _Information geometric perspective on off-resonance effects in driven two-level quantum systems_ , Quantum Reports 2, 166 (2020).
  * (80) F. Campaioli, W. Sloan, K. Modi, and F. A. Pollock, _Algorithm for solving unconstrained unitary quantum brachistochrone problems_ , Phys. Rev. A100, 062328 (2019).
  * (81) E. O’Connor, G. Guarnieri, and S. Campbell, _Action quantum speed limits_ , Phys. Rev. A103, 022210 (2021).
  * (82) M. Abramowitz and I. A. Stegun, _Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables_ , Dover, New York (1972).
  * (83) H.-J. Sommers and K. Zyczkowski, _Bures volume of the set of mixed states_ , J. Phys. A36, 10083 (2003).
  * (84) P. B. Slater, _Bures metric for certain high-dimensional quantum systems_ , Phys. Lett. A244, 35 (1998).
  * (85) I. Bengtsson and K. Zyczkowski, _Geometry of Quantum States_ , Cambridge University Press (2006).
  * (86) C. Cafaro, D. Felice, and P. M. Alsing, _Quantum Groverian geodesic paths with gravitational and thermal analogies_ , Eur. Phys. J. Plus 135, 900 (2020).
  * (87) C. Cafaro and S. Mancini, _On the complexity of statistical models admitting correlations_ , Physica Scripta 82, 035007 (2010).
  * (88) C. Cafaro and S. Mancini, _Quantifying the complexity of geodesic paths on curved statistical manifolds through information geometric entropies and Jacobi fields_ , Physica D240, 607 (2011).
  * (89) S. Weinberg, _Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity_ , John Wiley and Sons, Inc. (1972).
  * (90) J. M. Lee, _Riemannian Manifolds: An Introduction to Curvature_ , Springer-Verlag New York, Inc. (1997).
  * (91) F. De Felice and C. J. S. Clarke, _Relativity on Curved Manifolds_ , Cambridge University Press (1990).
  * (92) M. L. Boas, _Mathematical Methods in the Physical Sciences_ , John Wiley & Sons, Inc. (2006).
  * (93) A. P. Lightman, W. H. Press, R. H. Price, and S. A. Teukolsky, _Problem Book in Relativity and Gravitation_ , Princeton University Press (1975).
  * (94) J. J. Sakurai and J. Napolitano, _Modern Quantum Mechanics_ , Cambridge University Press (2017).
  * (95) D. Petz and Cs. Sudar, _Extending the Fisher metric to density matrices_ , in _Geometry in Present Days Science_ , eds. O. E. Barndorff-Nielsen and E. B. Vendel, World Scientific, pp. 21-34 (1999).
  * (96) R. Bhatia, _Matrix Analysis_ , Springer-Verlag (1997).
  * (97) M. K. Kwong, _Some results on matrix monotone functions_ , Lin. Algebra Appl. 118, 129 (1989).
  * (98) D. Petz, _Monotone metrics on matrix spaces_ , Lin. Algebra Appl. 244, 81 (1996).
  * (99) T. Furuta, _Concrete examples of operator monotone functions obtained by an elementary method without appealing to Löwner integral representation_ , Lin. Algebra Appl. 429, 972 (2008).
  * (100) P. Gibilisco, F. Hansen, and T. Isola, _On a correspondence between regular and non-regular operator monotone functions_ , Lin. Algebra Appl. 430, 2225 (2009).
  * (101) H. V. da Silva, _Quantum information geometry and applications_ , MS Thesis in Engineering Physics, IT Lisboa (2021).
  * (102) L. P. Hughston, R. Jozsa, and W. K. Wootters,  _A complete classification of quantum ensembles having a given density matrix_ , Phys. Lett. A183, 14 (1993).

##  Appendix A Geodesic paths on the Bloch sphere

In this Appendix, we derive the geodesic paths on the two-sphere. In the first
derivation, we use simple geometric arguments to obtain the equation of a
great circle in spherical coordinates. In the second derivation, we integrate
the geodesic equations to get explicit expressions of geodesic paths
$\theta=\theta\left(\eta\right)$ and $\varphi=\varphi\left(\eta\right)$. Then,
combining the geodesic paths equations, we show that we also get the equation
of a great circle in spherical coordinates that matches our first derivation.

###  A.1 Geometric derivation

Geodesics on the two-sphere lie on great circles. Great circles, in turn, can
be obtained by intersecting a plane passing through the origin in
$\mathbb{R}^{3}$ with the two-sphere. Assume that the equations of the plane
and the two-sphere of unit radius are given by,

$\alpha x+\beta y+\gamma z=0\text{, and }x^{2}+y^{2}+z^{2}=1\text{,}$ (70)

respectively. In Eq. (70), $\alpha$, $\beta$, and $\gamma$ belong to
$\mathbb{R}$. Using spherical coordinates, we set
$x\overset{\text{def}}{=}\sin\left(\theta\right)\cos\left(\varphi\right)$,
$y\overset{\text{def}}{=}\sin\left(\theta\right)\sin\left(\varphi\right)$, and
$z\overset{\text{def}}{=}\cos\left(\theta\right)$. Finally, combining the two
relations in Eq. (70), we get the equation of a great circle in spherical
coordinates

$\cot\left(\theta\right)=\pm a\cos\left(\varphi-\bar{\varphi}\right)\text{.}$
(71)

The constants $a$ and $\bar{\varphi}$ in Eq. (71) are such that
$a^{2}\overset{\text{def}}{=}\left(\alpha^{2}+\beta^{2}\right)/\gamma^{2}$ and
$\tan\left(\bar{\varphi}\right)\overset{\text{def}}{=}\beta/\alpha$,
respectively.

###  A.2 Dynamics derivation

Consider the system of two coupled second order nonlinear ODEs,

$\left\\{\begin{array}[c]{c}\frac{d^{2}\theta}{d\eta^{2}}-\sin\left(\theta\right)\cos\left(\theta\right)\left(\frac{d\varphi}{d\eta}\right)^{2}=0\\\
\frac{d^{2}\varphi}{d\eta^{2}}+2\frac{\cos\left(\theta\right)}{\sin\left(\theta\right)}\frac{d\theta}{d\eta}\frac{d\varphi}{d\eta}=0\end{array}\right.\text{.}$
(72)

Note that the second relation in Eq. (72) is equivalent to,

$\frac{d}{d\eta}\left(\frac{d\varphi}{d\eta}\sin^{2}\left(\theta\right)\right)=0\text{,}$
(73)

that is,

$\left(\frac{d\varphi}{d\eta}\right)^{2}=\frac{\mathrm{c}_{\mathrm{FS}}^{2}}{\sin^{4}\left(\theta\right)}\text{,}$
(74)

with $\mathrm{c}_{\mathrm{FS}}=\mathrm{c}_{\mathrm{FS}}\left(\theta_{i}\text{,
}\dot{\varphi}_{i}\right)\overset{\text{def}}{=}\dot{\varphi}_{i}\sin^{2}\left(\theta_{i}\right)$
being a real constant where
$\theta_{i}\overset{\text{def}}{=}\theta\left(\eta_{i}\right)$ and
$\dot{\varphi}_{i}\overset{\text{def}}{=}(d\varphi/d\eta)_{\eta=\eta_{i}}$
with $\eta_{i}$ set equal to zero. Using Eq. (74), the first relation in Eq.
(72) yields

$\frac{d^{2}\theta}{d\eta^{2}}=\mathrm{c}_{\mathrm{FS}}^{2}\frac{\cos\left(\theta\right)}{\sin^{3}\left(\theta\right)}\text{.}$
(75)

From Eqs. (74) and (75), we note that the original system of coupled ODEs in
Eq. (72) can be uncoupled. However, Eq. (75) is nonlinear and we shall use
some tricks to find a way of integrating it. Imposing the unit-speed
condition,
$\dot{\theta}^{2}+\dot{\varphi}^{2}\sin^{2}\left(\theta\right)=\mathrm{const.}$with
$\mathrm{const.}=1$, we employ Eqs. (74) and (75) to get

$\int
d\eta=\pm\int\frac{\sin\left(\theta\right)}{\sqrt{\sin^{2}\left(\theta\right)-\mathrm{c}_{\mathrm{FS}}^{2}}}d\theta\text{.}$
(76)

Let us perform a change of variables and put
$\epsilon\overset{\text{def}}{=}\cos\left(\theta\right)$. Then, integration of
Eq. (76) yields

$\eta=\eta_{i}\pm\left[\arctan\left(\frac{\epsilon_{i}}{\sqrt{\mathrm{a}_{\mathrm{FS}}^{2}-\epsilon_{i}^{2}}}\right)-\arctan\left(\frac{\epsilon}{\sqrt{\mathrm{a}_{\mathrm{FS}}^{2}-\epsilon^{2}}}\right)\right]\text{,}$
(77)

where
$\mathrm{a}_{\mathrm{FS}}^{2}\overset{\text{def}}{=}1-\mathrm{c}_{\mathrm{FS}}^{2}$.
For simplicity, assume
$\theta_{i}\overset{\text{def}}{=}\theta\left(\eta_{i}\right)=\cos^{-1}\left(\epsilon_{i}\right)=\pi/2$
with $\eta_{i}=0$. Then, $\epsilon_{i}=0$ and manipulation of Eq. (77) yields

$\cos\left(\theta\right)=\pm\mathrm{a}_{\mathrm{FS}}\sin\left(\eta\right)\text{.}$
(78)

We remark that for an arbitrary $\theta_{i}$, the analogue of Eq. (78) squared
is simply given by

$\cos^{2}\left(\theta\right)=\mathrm{a}_{\mathrm{FS}}^{2}\sin^{2}\left[\eta+\arctan\left(\frac{\epsilon_{i}}{\sqrt{\mathrm{a}_{\mathrm{FS}}^{2}-\epsilon_{i}^{2}}}\right)\right]\text{.}$
(79)

As a consistency check, we note that for $\eta_{i}=0$, Eq. (79) correctly
yields

$\cos^{2}\left(\theta_{i}\right)=\mathrm{a}_{\mathrm{FS}}^{2}\sin^{2}\left[\arctan\left(\frac{\epsilon_{i}}{\sqrt{\mathrm{a}_{\mathrm{FS}}^{2}-\epsilon_{i}^{2}}}\right)\right]=\epsilon_{i}^{2}\text{.}$
(80)

We also point out here that we could have set
$\tilde{v}_{\mathrm{FS}}^{2}\overset{\text{def}}{=}\dot{\theta}^{2}+\dot{\varphi}^{2}\sin^{2}\left(\theta\right)=\mathrm{const.}$with
$\mathrm{const.}\neq 1$. Then, Eq. (78) would simply become
$\cos\left(\theta\right)=\pm\mathrm{a}_{\mathrm{FS}}\sin\left(\tilde{v}_{\mathrm{FS}}\eta\right)$
with
$\mathrm{a}_{\mathrm{FS}}^{2}\overset{\text{def}}{=}1-\left(\mathrm{c}_{\mathrm{FS}}/\tilde{v}_{\mathrm{FS}}\right)^{2}$.
Eq. (78) allows us to express $\theta=\theta\left(\eta\right)$. Next, we need
to find the relation $\varphi=\varphi\left(\eta\right)$. Using Eqs. (74) and
(78), we get

$\frac{d\varphi}{d\eta}=\frac{\mathrm{c}_{\mathrm{FS}}}{1-\mathrm{a}_{\mathrm{FS}}^{2}\sin^{2}\left(\eta\right)}\text{.}$
(81)

Integration of Eq. (81) leads to,

$\varphi\left(\eta\right)=\varphi_{i}+\frac{\mathrm{c}_{\mathrm{FS}}}{\sqrt{-1+\mathrm{a}_{\mathrm{FS}}^{2}}}\tanh^{-1}\left[\sqrt{-1+\mathrm{a}_{\mathrm{FS}}^{2}}\tan\left(\eta\right)\right]\text{.}$
(82)

Note that $\sqrt{-1+\mathrm{a}_{\mathrm{FS}}^{2}}=i\mathrm{c}_{\mathrm{FS}}$,
with $i\in\mathbb{C}$ denoting the imaginary unit. Furthermore, recalling from
Ref. irene  that the inverse hyperbolic tangent function of a complex variable
satisfies the relation $\tanh^{-1}\left(z\right)=-i\tan^{-1}\left(iz\right)$,
setting $z\overset{\text{def}}{=}ix$ with $x\in\mathbb{R}$ and
$z\in\mathbb{C}$, we obtain

$-i\tanh^{-1}\left(ix\right)=\arctan\left(x\right)\text{.}$ (83)

Finally, using Eqs. (82) and (83), we get

$\varphi\left(\eta\right)=\varphi_{i}+\tan^{-1}\left[\mathrm{c}_{\mathrm{FS}}\tan\left(\eta\right)\right]\text{,}$
(84)

that is,

$\tan\left[\varphi\left(\eta\right)-\varphi_{i}\right]=\mathrm{c}_{\mathrm{FS}}\tan\left(\eta\right)\text{.}$
(85)

For completeness, we point out that using Eqs. (78) and (85) along with the
following two trigonometric identities,

$\cos^{2}\left(\theta\right)=\frac{\cot^{2}\left(\theta\right)}{1+\cot^{2}\left(\theta\right)}\text{,
and
}\sin^{2}\left(\eta\right)=\frac{\tan^{2}\left(\eta\right)}{1+\tan^{2}\left(\eta\right)}\text{,}$
(86)

we get after some straightforward algebraic manipulations the equation of a
great circle,

$\cot\left(\theta\right)=\pm\sqrt{\frac{1-\mathrm{c}_{\mathrm{FS}}^{2}}{\mathrm{c}_{\mathrm{FS}}^{2}}}\sin\left(\varphi-\varphi_{i}\right)\text{.}$
(87)

Eq. (87) is identical to Eq. (71) once we identify $a$ and $\bar{\varphi}$
with
$\sqrt{\frac{1-\mathrm{c}_{\mathrm{FS}}^{2}}{\mathrm{c}_{\mathrm{FS}}^{2}}}$
and $\varphi_{i}+\pi/2$, respectively.

##  Appendix B The Bures infinitesimal line element

In this Appendix, we derive Eq. (40). Recall that for the single qubit case
the Bures distance between two infinitesimally close density matrices $\rho$
and $\rho+d\rho$ is given by karol03 ,

$ds_{\mathrm{Bures}}^{2}\overset{\text{def}}{=}\frac{1}{2}\sum_{n,m=0}^{1}\frac{\left|\left\langle
e_{m}|d\rho|e_{n}\right\rangle\right|^{2}}{p_{m}+p_{n}}\text{,}$ (88)

where $\left\\{\left|e_{n}\right\rangle\right\\}_{n=0,1}$ is an orthonormal
basis of eigenvectors of $\rho$ with eigenvalues
$\left\\{p_{n}\right\\}_{n=0,1}$. Therefore,
$\rho=p_{0}\left|e_{0}\right\rangle\left\langle
e_{0}\right|+p_{1}\left|e_{1}\right\rangle\left\langle e_{1}\right|$. From the
expression of $\rho$ in Eq. (34), we get

$d\rho=\frac{\partial\rho}{\partial
r}dr+\frac{\partial\rho}{\partial\theta}d\theta+\frac{\partial\rho}{\partial\varphi}d\varphi\text{,}$
(89)

that is,

$d\rho=\frac{1}{2}\left(\begin{array}[c]{cc}\cos\left(\theta\right)dr-r\sin\left(\theta\right)d\theta&e^{-i\varphi}\left[\sin\left(\theta\right)dr+r\cos\left(\theta\right)-ir\sin\left(\theta\right)d\varphi\right]\\\
e^{i\varphi}\left[\sin\left(\theta\right)dr+r\cos\left(\theta\right)+ir\sin\left(\theta\right)d\varphi\right]&-\cos\left(\theta\right)dr+r\sin\left(\theta\right)d\theta\end{array}\right)\text{.}$
(90)

Finally, using Eqs. (36), (37), (38), and (90), $ds_{\mathrm{Bures}}^{2}$ in
Eq. (88) reduces to $ds_{\mathrm{Bures}}^{2}$ in Eq. (40). For further details
on the Bures metric for high-dimensional quantum systems, we refer to Refs.
slater98 ; karol .

##  Appendix C Geodesic paths in the Bloch ball with the Sjöqvist metric

In this Appendix, we study the geodesic paths in the Bloch ball using the
Sjöqvist metric. Recall that the form of the geodesic equation remains
unchanged under affine transformations. An affine transformation of a
parameter $\eta$ is the change of variable
$\eta\rightarrow\eta^{\prime}=\tilde{a}$ $\eta+\tilde{b}$, with $\tilde{a}$
and $\tilde{b}$ in $\mathbb{R}$. Any other transformation will generate extra
terms in the geodesic equation epj . The affine parameter for a geodesic is
unique up to an affine change of variables. Therefore, it is important to keep
in mind that geodesics are curves with a preferred parametrization. In what
follows, we explicitly analyze geodesic paths parametrized in terms of two
distinct affine parametrizations.

###  C.1 The Sjöqvist $\theta$-affine parametrization

In Ref. erik20 , Sjöqvist focused on finding geodesic paths connecting points
in the Bloch ball laying in a plane that contains the origin. In other words,
using spherical coordinates $\left(r\text{, }\theta\text{, }\varphi\right)$
and keeping $\varphi=$const., geodesics were obtained by minimizing $\int$
$ds_{\mathrm{Sj\ddot{o}qvist}}$ over all curves connecting points
$\left(r_{i}\text{, }\theta_{i}\right)$ and $\left(r_{f}\text{,
}\theta_{f}\right)$. More specifically, one obtains the curve
$\left[\theta_{i}\text{, }\theta_{f}\right]\ni\theta\mapsto
r_{\mathrm{Sj\ddot{o}qvist}}\left(\theta\right)\in\left(0\text{, }1\right]$
that minimizes the length
$\mathcal{L}_{\mathrm{Sj\ddot{o}qvist}}\left(\theta_{i}\text{,
}\theta_{f}\right)$ defined as

$\mathcal{L}_{\mathrm{Sj\ddot{o}qvist}}\left(\theta_{i}\text{,
}\theta_{f}\right)\overset{\text{def}}{=}\int_{s_{i}}^{s_{f}}\sqrt{ds_{\mathrm{Sj\ddot{o}qvist}}^{2}}=\frac{1}{2}\int_{\theta_{i}}^{\theta_{f}}\mathrm{L}\left(r^{\prime}\text{,
}r\text{, }\theta\right)d\theta\text{.}$ (91)

In Eq. (91), $r^{\prime}\overset{\text{def}}{=}dr/d\theta$ and
$\mathrm{L}\left(r^{\prime}\text{, }r\text{, }\theta\right)$ is the
Lagrangian-like function given by

$\mathrm{L}\left(r^{\prime}\text{, }r\text{,
}\theta\right)\overset{\text{def}}{=}\sqrt{1+\frac{r^{\prime
2}}{1-r^{2}}}\text{.}$ (92)

From Eq. (92), note that $\mathrm{L}=\mathrm{L}\left(r^{\prime}\text{,
}r\right)$ does not depend explicitly on $\theta$. Therefore,
$\partial\mathrm{L}/\partial\theta=0$. In this case, it happens that the
Euler-Lagrange equation

$\frac{d}{d\theta}\frac{\partial\mathrm{L}\left(r^{\prime}\text{,
}r\right)}{\partial
r^{\prime}}-\frac{\partial\mathrm{L}\left(r^{\prime}\text{,
}r\right)}{\partial r}=0\text{,}$ (93)

reduces to the so-called Beltrami identity,

$\mathrm{L}\left(r^{\prime}\text{,
}r\right)-r^{\prime}\frac{\partial\mathrm{L}\left(r^{\prime}\text{,
}r\right)}{\partial r^{\prime}}=\text{{const}.}$ (94)

Indeed, combining Eq. (93) with the identity

$\frac{d\mathrm{L}\left(r^{\prime}\text{,
}r\right)}{d\theta}=\frac{\partial\mathrm{L}\left(r^{\prime}\text{,
}r\right)}{\partial
r^{\prime}}r^{\prime\prime}+\frac{\partial\mathrm{L}\left(r^{\prime}\text{,
}r\right)}{\partial r}r^{\prime}\text{,}$ (95)

we get

$\frac{d\mathrm{L}\left(r^{\prime}\text{,
}r\right)}{d\theta}=\frac{d}{d\theta}\left(r^{\prime}\frac{\partial\mathrm{L}\left(r^{\prime}\text{,
}r\right)}{\partial r^{\prime}}\right)\text{.}$ (96)

Eq. (96) finally leads to the Beltrami identity in Eq. (94). Using Eqs. (92)
and (94), we get

$\frac{1}{\sqrt{1+\frac{r^{\prime
2}}{1-r^{2}}}}=\text{{const}.}\equiv\mathrm{c}_{\mathrm{S}}\text{,}$ (97)

that is,

$\frac{r^{\prime
2}}{1-r^{2}}=\text{{const}.}\equiv\mathrm{k}\overset{\text{def}}{=}\frac{1-\mathrm{c}_{\mathrm{S}}^{2}}{\mathrm{c}_{\mathrm{S}}^{2}}\text{,}$
(98)

Integrating Eq. (98) and imposing the boundary conditions
$r\left(\theta_{i}\right)=r_{i}$ and $r\left(\theta_{f}\right)=r_{f}$ with
$\theta_{i}=0$, we finally get the geodesic path

$r_{\mathrm{Sj\ddot{o}qvist}}\left(\theta\right)=\sin\left[\sin^{-1}\left(r_{i}\right)+\frac{\sin^{-1}\left(r_{f}\right)-\sin^{-1}\left(r_{i}\right)}{\theta_{f}}\theta\right]\text{.}$
(99)

Evaluating Eq. (97) at $\theta_{i}=0$ and using Eq. (99), we note that the
constant $\mathrm{c}_{\mathrm{S}}$ can be expressed in terms of $r_{i}$,
$r_{f}$, and $\theta_{f}$ as

$\mathrm{c}_{\mathrm{S}}=\mathrm{c}_{\mathrm{S}}\left(r_{i}\text{,
}r_{f}\text{,
}\theta_{f}\right)\overset{\text{def}}{=}\frac{\theta_{f}}{\sqrt{\theta_{f}^{2}+\left[\sin^{-1}\left(r_{f}\right)-\sin^{-1}\left(r_{i}\right)\right]^{2}}}\text{,}$
(100)

with $0\leq\mathrm{c}_{\mathrm{S}}\leq 1$, as correctly expected. For
completeness, we point out that in terms of initial conditions
$r\left(\theta_{i}\right)=r_{i}$ and
$r^{\prime}\left(\theta_{i}\right)=r_{i}^{\prime}$ with $\theta_{i}=0$,
$r_{\mathrm{Sj\ddot{o}qvist}}\left(\theta\right)$ in Eq. (99) can be recast as

$r_{\mathrm{Sj\ddot{o}qvist}}\left(\theta\right)=\sin\left[\sin^{-1}\left(r_{i}\right)+\frac{r_{i}^{\prime}}{\sqrt{1-r_{i}^{2}}}\theta\right]\text{,}$
(101)

where we used
$\cos\left[\sin^{-1}\left(r_{i}\right)\right]=\sqrt{1-r_{i}^{2}}$ in the raw
solution of the form $r\left(\theta\right)=\sin\left(c_{1}\theta+c_{2}\right)$
with real integrations constants $c_{1}$ and $c_{2}$. As a final remark,
observe from Eq. (99) that $\theta$ plays the role of the affine parameter
that characterizes the points on the curve of minimal length connecting the
initial and final points in the Bloch ball.

###  C.2 The canonical $\eta$-affine parametrization

To find the geodesic paths in the Bloch ball suing the Sjöqvist metric and
parametrized in terms of the “canonical” affine parameter $\eta$, we need to