# Jaynes’ principle for quantum Markov processes: Generalized Gibbs - von
Neumann states rule

J Novotný, J Maryška and I Jex  Department of Physics, Faculty of Nuclear
Sciences and Physical Engineering, Czech Technical University in Prague,
Břehová 7, 115 19 Praha 1 - Staré Město, Czech Republic
corresponding author’s<EMAIL_ADDRESS>
###### Abstract

We prove that any asymptotics of a finite-dimensional quantum Markov processes
can be formulated in the form of a generalized Jaynes’ principle in the
discrete as well as in the continuous case. Surprisingly, we find that the
open system dynamics does not require maximization of von Neumann entropy. In
fact, the natural functional to be extremized is the quantum relative entropy
and the resulting asymptotic states or trajectories are always of the
exponential Gibbs-like form. Three versions of the principle are presented for
different settings, each treating different prior knowledge: for asymptotic
trajectories of fully known initial states, for asymptotic trajectories
incompletely determined by known expectation values of some constants of
motion and for stationary states incompletely determined by expectation values
of some integrals of motion. All versions are based on the knowledge of the
underlying dynamics. Hence our principle is primarily rooted in the inherent
physics and it is not solely an information construct. The found principle
coincides with the MaxEnt principle in the special case of unital quantum
Markov processes. We discuss how the generalized principle modifies
fundamental relations of statistical physics.

##  I Introduction

Extremal and variational principles Lanczos ; Basdevant ; Kirchhoff ; Gibbs ;
Gerjvoy ; Rojo ; MaxEnt ; Jaynes1 ; Jaynes2 ; Jaynes3 ; Kapur1 ; Kapur2 ;
Banavar2010  play a central role in the formulation of the laws of Nature and
pervade all branches of science ranging from evolutionary biology to analytic
mechanics or quantum field theory. They represent a common conceptual frame,
in many cases they have a very broad range of applicability and simplify
solution of a given task. They improve our understanding of the studied field
and consequently allow for possible, sometimes quite far reaching, extensions.

A large class of such principles MaxEnt ; Jaynes1 ; Jaynes2 ; Jaynes3 ; Kapur1
; Kapur2 ; Banavar2010  have been found extremely useful if the complexity of
the investigated problem prohibits to reveal its solution by standard
techniques, e.g. by directly solving equations of motion or the solution
contains too much information about the system to be used for its effective
control. Such situation naturally arises in statistical physics, where the
number of mutually interacting subsystems is gargantual. It does not improve
in the quantum domain or for open system dynamics. An open system quantum
dynamics eventually drives macroscopic system into equilibrium or at least a
stationary regime. Due to high complexity of the whole process, also
frequently accompanied with a lack of details of the dynamics and of the
initial state, the analysis of equilibrium mechanisms and the emerging states
constitutes a formidable task. In 1957 Jaynes Jaynes1 ; Jaynes2 ; Jaynes3 , to
circumvent the whole issue, introduced the MaxEnt principle turning the
problem into finding a constrained extreme of entropy. It affirms that among
all states consistent with observed data the one with the maximal entropy best
represents the actual knowledge of the system state. Thus, provided that mean
values $\langle A_{i}\rangle_{\rho}=\operatorname{Tr}\left[\rho A_{i}\right]$
of observables $A_{i}$ (typically conserved quantities) constitute the only
available knowledge about the established equilibrium, its corresponding
state, maximizing the von Neumann entropy
$S(\rho)=-\operatorname{Tr}\left[\rho\ln\rho\right]$ (with Boltzmann constant
$k_{B}=1$), reads Balian

$\rho_{eq}=\frac{1}{Z}\exp\left(-\sum_{i}\gamma_{i}A_{i}\right).$ (1)

$Z=\operatorname{Tr}\\{\exp(-\sum_{i}\gamma_{i}A_{i})\\}$, the normalizing
partition function as a function of independent Lagrange multipliers
$\gamma_{i}$ comprises all relevant information necessary to construct
statistical mechanics characteristics of the equilibrium, i.e. mean values
$\langle A_{i}\rangle_{\rho_{eq}}=-\left(\partial\ln
Z/\partial\gamma_{i}\right)_{\gamma_{j\neq i}}$, entropy $S=\ln
Z+\sum_{i}\gamma_{i}\langle A_{i}\rangle_{\rho_{eq}}$ of the equilibrium state
(1) and the change of entropy $dS=\sum_{i}\gamma_{i}d\langle A_{i}\rangle$ for
a shift of the equilibrium.

The principle was applied to many problems Banavar2010 ; Karmeshu2012 , but
also attracts significant criticism and raises relevant questions at the same
time. To which dynamical systems does the Jaynes’ principle actually apply and
what is its explicit form? Which available information, represented by a set
of observables $A_{i}$, must be taken into considerations? Does the von
Neumann entropy play a unique role or may or should we extremize some other
(entropy like) quantity? For example, the Tsallis entropy is used to build up
non-extensive statistical thermodynamics Tsallis2009 . A minimum relative
entropy principle is frequently used in adaptive strategies Ortega2010 .
However, this requires additional prior information (reference probability
distribution), which must be obtain in a different way and it is not clear, in
general, how the prior distribution should be chosen. Another question is
whether we can generalize the principle also to trajectories, so called the
maximum calibre principle Ghosh , which could enlarge its scope to non-
equilibrium statistical physics. Finally, the main objection urges that such a
principle should be consistent with the underlying dynamics and in principle
we should be able to derive it from microscopic laws. This is the case for
unitary dynamics of finitely dimensional quantum systems for which it was
proven Eisert2011  that its equilibrium states are Gibbs and von Neumann
states described by a maximum entropy constrained with expectation values of
integrals of motion. Our approach and results complement and take a different
angle on problems studied by Burgarth and coauthors Bu1 ; Bu2 .

The main aim of the paper is to provide answers to all above questions and
consistently formulate the Jaynes’ principle for equilibrium and more
generally asymptotic trajectories of all discrete and continuous quantum
Markov trace-preserving processes (QMPs) governing the evolution of finite
dimensional quantum systems. This is achieved exploiting algebraic properties
of attractors and consequently of asymptotic states of QMPs developed in
Albert2014 ; Novotny2018 . We derive three versions of Jayne’s principles
addressing three different settings. The first one concerns the situation when
the initial state and thus all expectation values of observables are known.
The second principle captures problem when the system is known to be
sufficiently near the asymptotic regime and expectation values of only some
constants of motion are known. The third version is designed for situation
typical in statistical physics when quantum system is supposed to be
sufficiently near to some equilibrium state and we know expectation values of
some integrals of motion only.

The structure of the paper is as follows. In section II we introduce dynamics
of discrete and continuous quantum Markov processes, properties of their
asymptotic dynamics relevant for this study, and different exponential forms
of their asymptotic trajectories. Section III) is devoted to formulations and
proofs of all presented Jaynes’ principle. Consequent properties of
corresponding partition function and entropic relations are derived in section
IV. We conclude in section V.

##  II Quantum Markov processes and its asymptotics

In this part we introduce quantum Markov processes and theory for their
asymptotic evolution. Throughout the paper, we assume a quantum system with a
finitely dimensional Hilbert space ${\mathscr{H}}$ of dimension $N$ and
associated Hilbert space of all operators ${\sf B}({\mathscr{H}})$ equipped
with the Hilbert-Schmidt scalar product
$(A,B)=\operatorname{Tr}\\{A^{\dagger}B\\}$ for $A,B\in{\sf
B}({\mathscr{H}})$. The state of the system is described by a density operator
$\rho\in{\sf B}({\mathscr{H}})$ which is positive with trace one.

###  II.1 Quantum Markov dynamics

We focus on the uniform quantum Markovian dynamics, which is memoryless and
its evolution depends solely on the actual state of the system. Two types of
Markovian dynamics are distinguished – quantum Markov chains (QMCHs)
proceeding in discrete steps given by propagator
${\mathcal{T}}_{n}=({\mathcal{T}})^{n}$ for $n\in\mathbb{N}_{0}$ steps
Gudder2008  and continuous quantum Markov dynamical semigroups (QMDSs) with
propagator ${\mathcal{T}}_{t}=\exp({\mathcal{L}}t)$ generated by Lindbladian
${\mathcal{L}}$ with $t\in\mathbb{R}_{0}^{+}$ being continuous time Alicki2007
. Generators ${\mathcal{T}}$ and ${\mathcal{L}}$ act on the space ${\sf
B}({\mathscr{H}})$. Consequently, during both types of Markovian evolution,
the corresponding propagator ${\mathcal{T}}_{s}$ shifts the state of a quantum
system $\rho(t)$ at some time $t$ to the state
$\rho(s+t)={\mathcal{T}}_{s}\rho(t)$ at time $s+t$. A properly defined quantum
evolution requires completely positive trace-preserving propagators (quantum
channels) ${\mathcal{T}}_{s}$. Thus, while the one step generator
${\mathcal{T}}$ admits the decomposition

$\rho(n+1)={\mathcal{T}}(\rho(n))=\sum_{i}K_{j}\rho(n)K_{j}^{\dagger},$ (2)

with Kraus operators $K_{j}$ satisfying $\sum_{j}K_{j}^{\dagger}K_{j}=I$
Chuang2010 , any Lindbladian ${\mathcal{L}}$ must be of the form

${\mathcal{L}}(X)=i[X,H]+\sum_{j}\left(L_{j}XL_{j}^{\dagger}-\frac{1}{2}\left\\{L_{j}^{\dagger}L_{j},X\right\\}\right),$
(3)

with a Hamiltonian $H$ and Lindblad operators $L_{j}$ Alicki2007 . A special
role play unital QMCHs and QMDSs whose evolution preserve the maximally mixed
state $\frac{1}{N}I$, i.e. they must follow ${\mathcal{T}}_{s}(I)=I$, where
$I$ stands for the identity operator.

Analogously we can describe quantum Markov evolution of observables in the
Heisenberg picture, in which mean values of an evolved observable $B(t)$ have
to satisfy

$\operatorname{Tr}\left\\{B(0)\rho(t)\right\\}=\langle
B(0)\rangle_{\rho(t)}=\langle
B(t)\rangle_{\rho(0)}=\operatorname{Tr}\left\\{B(t)\rho(0)\right\\}.$

Thus in the Heisenberg picture, discrete QMCH and continuous QMDS is generated
by the unital adjoint map ${\mathcal{T}}^{\dagger}$ satisfying
${\mathcal{T}}^{\dagger}(I)=I$ and ${\mathcal{L}}^{\dagger}$ respectively.

###  II.2 Asymptotic dynamics of QMPs

In this work we concentrate on the asymptotic dynamics of QMPs in the
Schrödinger and Heisenberg picture Novotny2018 . The former includes
stationary states as well as periodic or quasi-periodic trajectories of a
given QMP. All these asymptotic states live on the subspace $P{\mathscr{H}}$
Novotny2012 ; Amato2023 , i.e. their supports belong to the subspace
$P{\mathscr{H}}$, where so-called ${\mathcal{T}}$-projector $P$ is the
projection onto the support of the maximal invariant state $\sigma$ of the
given quantum Markov process. This invariant state with the maximal support is
called the ${\mathcal{T}}$-state. The ${\mathcal{T}}$-state plays a central
role in investigation of the asymptotic dynamics. In the case of unital QMPs,
the $\mathcal{T}$-state is the maximally mixed state whose support is the
whole Hilbert space ${\mathscr{H}}$ and thus $P=I$. In a general QMP, any
${\mathcal{T}}$-state can be, in principle, constructed as the time-averaged
state

$\sigma=\lim\limits_{t\rightarrow\infty}\frac{1}{t}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{0}^{t}ds{\mathcal{T}}_{s}(\rho(0))$
(4)

of any strictly positive initial state $\rho(0)$. We stress that
$\mathcal{T}$-states are not unique in general. For many QMPs, there is a
freedom in choice of the ${\mathcal{T}}$-state $\sigma$. Similarly, an
asymptotic trajectory $\sigma(t)$ whose all states have its support equal to
$P{\mathscr{H}}$ is called a ${\mathcal{T}}$-trajectory. Note that a
${\mathcal{T}}$-state is a special example of a ${\mathcal{T}}$-trajectory.
Any ${\mathcal{T}}$-trajectory can be constructed as the asymptotic part of
the evolution of some strictly positive initial state.

The asymptotic dynamics of QMPs runs inside the operator subspace $P{\sf
B}({\mathscr{H}})P$ and conveniently can be expressed in terms of its
attractors in the Schrödinger picture. Let us first treat both types of QMPs
separately. We start with QMCHs. Attractors of a QMCH generated by
${\mathcal{T}}$ are all operators from the so-called attractor (or asymptotic)
space ${\sf Atr}({\mathcal{T}})$ defined as

${\sf Atr}({\mathcal{T}})=\bigoplus_{\lambda\in\sigma_{as}}{\sf
Ker}({\mathcal{T}}-\lambda I),$ (5)

where the asymptotic spectrum $\sigma_{as}$ is

$\sigma_{as}=\\{\lambda|\lambda\in\sigma({\mathcal{T}}),|\lambda|=1\\},$

with $\sigma({\mathcal{T}})$ denoting the spectrum of the generator
${\mathcal{T}}$. In other words, the attractor space is the span of
eigenoperators associated with eigenvalues from the asymptotic spectrum. Let
us stress that attractors are, in general, not states (density operators).
These are operators from which we can construct the asymptotic evolution of
the given QMCH. Indeed, equipped with a linearly independent basis
$X_{\lambda,j}$ of individual attractor subspaces corresponding to
$\lambda\in\sigma_{as}$ and their dual basis $X^{\lambda,j}$, i.e.
$\left(X^{\lambda,j},X_{\mu,l}\right)=\delta_{\lambda\mu}\delta_{jl}$, the
asymptotic trajectory of an initial state $\rho(0)$ reads

$\rho(t\gg
1)=\sum_{\lambda\in\sigma_{as},j}\lambda^{t}X_{\lambda,j}\operatorname{Tr}\left[X^{\lambda,j\dagger}\rho(0)\right],$
(6)

where $j$ accounts for possible degeneracies of eigenoperators Novotny2018 .

Similarly, attractors of a QMDS generated by Lindbladian ${\mathcal{L}}$ are
all operators from the so-called attractor (or asymptotic) space ${\sf
Atr}({\mathcal{L}})$ defined in this case as

${\sf Atr}({\mathcal{L}})=\bigoplus_{a\in\sigma_{as}({\mathcal{L}})}{\sf
Ker}({\mathcal{L}}-aI),$ (7)

where the asymptotic spectrum $\sigma_{as}({\mathcal{L}})$ is

$\sigma_{as}({\mathcal{L}})=\\{a|a\in\sigma({\mathcal{L}}),{\sf Re}(a)=0\\},$

with $\sigma({\mathcal{L}})$ denoting spectrum of generator ${\mathcal{L}}$.
Similarly, considering linearly independent basis $X_{a,j}$ and their
corresponding dual basis $X^{a,j}$, the asymptotic trajectory of an initial
state $\rho(0)$ reads Novotny2018

$\rho(t\gg
1)=\sum_{a\in\sigma_{as}({\mathcal{L}}),j}\exp(at)X_{a,j}\operatorname{Tr}\left[X^{a,j\dagger}\rho(0)\right],$
(8)

where $j$ accounts for possible degeneracies of eigenoperators.

Since we intend to treat both types of QMPs simultaneously and in a unified
manner, we formally introduce a one step generator
${\mathcal{T}}={\mathcal{T}}_{1}=\exp({\mathcal{L}})$ of evolution for QMDS
generated by the Lindbladian ${\mathcal{L}}$. To avoid eventual ambiguity of
its operator powers we define
${\mathcal{T}}^{t}=\exp({\mathcal{L}}t)={\mathcal{T}}_{t}$ for
$t\in\mathbb{R}_{+}$. Concerning asymptotic dynamics, its attractor space
${\sf Atr}({\mathcal{T}})$ coincides with the attractor space ${\sf
Atr}({\mathcal{L}})$. However, we must also keep the relationship between any
attractor $X_{\lambda,j}\in{\sf Atr}({\mathcal{T}})$ and its associated
$a\in\sigma_{as}({\mathcal{L}})$ via relation $\lambda=\exp(a)$. In other
words, attractors corresponding to the same $\lambda$ behave differently under
evolution if they are associated with different
$a_{1},a_{2}\in\sigma_{as}({\mathcal{L}})$ satisfying

$\lambda=\exp(a_{1})=\exp(a_{2}).$ (9)

For example, while fixed points of QMCHs are attractors corresponding to
$\lambda=1$, fixed points of QMDSs are attractors corresponding to
$\lambda=\exp(a)=1$ with only $a=0$. In the rest of the paper we will keep the
following notation: we denote attractor spaces of both types of QMPs generated
by one step generator ${\mathcal{T}}$ as ${\sf Atr}({\mathcal{T}})$ and their
asymptotic spectra as $\sigma_{as}$, which allows us to write the asymptotic
dynamics of both types of QMPs as (6).

Analogously we define the attractor space ${\sf Atr}({\mathcal{T}}^{\dagger})$
of the QMP in the Heisenberg picture. Its basis are formed by the dual basis
$X^{\lambda,j}$. In general these two attractor spaces do not coincide, but
there are important mutual relationships between both sets of attractors
Novotny2018  and consequently, as we will see, between asymptotic states and
conserved quantities of the given QMP. Indeed, while the attractor space ${\sf
Atr}({\mathcal{T}})$ contains all asymptotic trajectories, the attractor space
${\sf Atr}({\mathcal{T}}^{\dagger})$ contains all constants of motion
Albert2014 . Moreover the projected part of the attractor space $P{\sf
Atr}({\mathcal{T}}^{\dagger})P$ forms a $C^{*}$-algebra, i.e. with each pair
of operators it contains also their product, their adjoint operators and it is
nonempty since it contains at least the projection $P$. We note that the whole
${\sf Atr}({\mathcal{T}}^{\dagger})$ is also a $C^{*}$-algebra but under the
so-called Choi-Effros product Bhat2022 .

###  II.3 Exponential form of asymptotic states

All these attractor properties listed in the previous part provide us with an
exponential form of all asymptotic states of QMPs. It was shown that any
strictly positive asymptotic state $\rho_{as}$ on $P{\mathscr{H}}$ (6) can be
written as Novotny2018

$\rho_{as}=\exp\left[\ln\sigma-\sum_{j}\gamma_{j}PZ_{j}P\right],$ (10)

where $\\{Z_{j}\\}$ is a hermitian basis of the attractor space ${\sf
Atr}({\mathcal{T}}^{\dagger})$, $\gamma_{j}$ are real parameters and $\sigma$
is any ${\mathcal{T}}$-state. Please note that the state $\sigma$ is strictly
positive on the subspace $P{\mathscr{H}}$ and vanishing on its orthogonal
complement. Thus the logarithm of the state $\sigma$ is given by the logarithm
of its strictly positive part while being zero on its orthogonal part. All the
other asymptotic states, which are not strictly positive, are obtained from
(10) simply by taking the limit $+\infty$ or $-\infty$ for some chosen subset
of parameters $\gamma_{j}$ Novotny2018 .

Consequently, any discrete or continuous asymptotic trajectory (6) can be
written as (10) with time-dependent coefficients. If all these parameters are
finite, it describes an asymptotic trajectory $\omega(t)$ whose each state has
the maximal support $P{\mathscr{H}}$

$\omega(t)=\exp\left[\ln\sigma-\sum_{j}\mu_{j}(t)PZ_{j}P\right].$ (11)

It is a ${\mathcal{T}}$-trajectory. Expressing
$\ln\sigma=\ln\omega(t)+\sum_{j}\mu_{j}(t)PZ_{j}P$ we can write down any
asymptotic trajectory $\rho_{as}(t)$ as

$\displaystyle\rho_{as}(t)$ $\displaystyle=$
$\displaystyle\exp\left[\ln\sigma-\sum_{j}\nu_{j}(t)PZ_{j}P\right]$ (12)
$\displaystyle=$
$\displaystyle\exp\left[\ln\omega(t)-\sum_{j}\gamma_{j}(t)PZ_{j}P\right].$

It has two consequences. First, the role of the ${\mathcal{T}}$-state in the
exponential form of asymptotic states can be replaced by the more general
${\mathcal{T}}$-trajectory. We will find it very useful later when we
formulate one of Jaynes’ principles. Second, if we choose a different
${\mathcal{T}}$-trajectory it only shifts the original values of parameters
$\nu_{j}(t)$ to $\gamma_{j}(t)=\nu_{j}(t)+\mu_{j}(t)$.

We, instead of analyzing the time dependence of parameters $\gamma_{j}(t)$
determining asymptotic trajectories, provide a natural description of
asymptotic trajectories via constants of motion of QMPs, which also allow to
formulate the Jaynes’ principle for all QMPs. A time dependent observable
$C(t)=C(t)^{\dagger}\in{\sf B}({\mathscr{H}})$ is called constant of motion if

$\langle
C(t+s)\rangle_{\rho(t)}=\operatorname{Tr}\left[C(t+s)\rho(t)\right]=\langle
C(s)\rangle_{\rho(0)}$ (13)

for any state trajectory $\rho(t)$, i.e. its expectation value remains
constant along all state trajectories irrespective of the reference time $s$.
In this definition we take into account the fact that, due to open dynamics,
some system degrees of freedom may become suddenly unavailable, leaving the
possibility to define constants of motions freely in this redundant dying
part. This occurs, for example, in the case of discrete QMP that has, apart of
the asymptotic spectrum, generalized eigenvectors corresponding to eigenvalue
$0$ only. Since these can be of different ranks, the set of initial states is
gradually narrowed down by the evolution until it reaches its asymptotic
regime. Outside of this dying part, e.g. for sufficiently long time outside of
the space $P{\mathscr{H}}$, a constant of motion can be defined freely, since
it has no additional contribution to its mean value. Requiring time
homogeneity for constants of motions naturally removes this redundant part.
Indeed, the definition (13) may be rewritten as

$\operatorname{Tr}\left[T_{t}^{\dagger}\left(C(t+s)\right)\rho(0)\right]=\operatorname{Tr}\left[C(s)\rho(0)\right]$

valid for any initial state $\rho(0)$. It directly implies that any constant
of motion $C(t)$ follows

$C(t_{2}-t_{1})={\mathcal{T}}_{t_{1}}^{\dagger}(C(t_{2}))$ (14)

for any $0\leq t_{1}\leq t_{2}$. Thus the constants of motion are hermitian
(observable) trajectories of QMPs in the Heisenberg picture running in the
reversed time direction within the attractor space. It implies the set
${\mathcal{C}}$ of all constants of motion of a given QMP forms a finite-
dimensional real vector space. Indeed, one can easily check that the basis of
all asymptotic trajectories can be simply chosen as
$\\{\lambda^{t}X^{\lambda,j}\\}$ ($\lambda\in\sigma_{as}$). Since attractors
always come in mutually conjugated pairs associated with complex conjugated
eigenvalues, we can choose basis of ${\mathcal{C}}$ as

$\displaystyle C_{\lambda,j}^{(+)}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\lambda^{t}X^{\lambda,j}+\overline{\lambda^{t}}\left(X^{\lambda,j}\right)^{\dagger}\right),$
$\displaystyle C_{\lambda,j}^{(-)}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2i}\left(\lambda^{t}X^{\lambda,j}-\overline{\lambda^{t}}\left(X^{\lambda,j}\right)^{\dagger}\right),$
(15)

for each $\lambda\in\sigma_{as}$ and $j$. Note again that powers of
eigenvalues from asymptotic spectrum of QMDSs are defined via
$\lambda^{t}=\exp(at)$. Consequently, the vector space of all constants of
motion ${\mathcal{C}}$ has the same dimension as the attractor spaces, i.e.

$\mathsf{dim}({\mathcal{C}})=\mathsf{dim}({\sf
Atr}({\mathcal{T}}))=\mathsf{dim}({\sf Atr}({\mathcal{T}}^{\dagger}).$

Of course, the set ${\mathcal{C}}$ contains also real subspace ${\mathcal{I}}$
of all integrals of motion of the QMP. These are time independent constants of
motion, which always contain the identity operator $I$, since we assume only
trace-preserving QMPs keeping $\operatorname{Tr}[\rho(t)]=1$.

Thus, any element $Z_{i}$ of hermitian basis of ${\sf
Atr}({\mathcal{T}}^{\dagger})$ employed in (12) can be expressed in terms of
constants of motion. Let us assume that apart from the identity operator there
are $d$ linearly independent constants of motion and denote elements of the
basis as $\\{I,C_{1}(t),C_{2}(t),\ldots,C_{d}(t)\\}$. Any asymptotic
trajectory can be written as

$\displaystyle\rho_{as}(t)$ $\displaystyle=$
$\displaystyle\exp\left[\ln\sigma(t)-\sum_{j}\gamma_{j}(t)PC_{j}(t)P-\alpha(t)P\right]$
(16) $\displaystyle=$
$\displaystyle\frac{1}{{\mathcal{Z}}}\exp\left[\ln\sigma(t)-\sum_{j}\gamma_{j}(t)PC_{j}(t)P\right],$

where we used $PIP=P$ and the fact that $P$ commutes with all operators of the
form $PXP$. The parameter $\alpha(t)$ is associated with the integral of
motion $I$ and $\mathcal{Z}=exp(\alpha(t))$ is a normalization function. It
depends on parameters $\gamma_{i}(t)$ and ensures
$\operatorname{Tr}[\rho(t)]=1$.

We found the form of all asymptotic trajectories including stationary states
as well, which very clearly resembles states (1). It is based on properties of
attractors in both pictures. However, we stress that states of these
trajectories do not maximize the von Neumann entropy in general and moreover
the form covers also non-stationary asymptotic trajectories. Equipped with the
general form of asymptotic states (16) of QMPs expressed in terms of their
constants of motion, it is our next main task to formulate properly the
Jaynes’ principle for QMPs, i.e. quantum information principle which would
equivalently reproduce all asymptotic trajectories (16) of QMPs.

##  III Formulation of Jaynes’ principles

In order to write down all asymptotic trajectories, we have to know at least
one ${\mathcal{T}}$-trajectory, e.g. one ${\mathcal{T}}$ state. A special role
among them plays the trajectory initiated at the maximal mixed state and its
time-averaged ${\mathcal{T}}$-state. The following lemma gathers their
commutation properties which in turn will be found crucial in formulation of
Jaynes’ principles.

######  Lemma III.1 (The maximally mixed state).

Let $\rho_{I}(t)={\mathcal{T}}_{t}\left(\frac{1}{N}I\right)$ be the trajectory
of a QMP $\mathcal{T}_{t}$ initiated in the maximally mixed state
$\frac{1}{N}I$, $\sigma_{I}(t)=\rho_{I}(t\gg 1)$ its asymptotic part
(asymptotic trajectory) and $\sigma_{I}$ its time-averaged state. Then

  * 1.

$\sigma_{I}(t)$ is a ${\mathcal{T}}$-trajectory and its time average
$\sigma_{I}$ is a ${\mathcal{T}}$-state.

  * 2.

All states of $\rho_{I}(t)$ and thus $\sigma_{I}(t)$ and $\sigma_{I}$ commute
with an arbitrary constant of motion of the given QMP projected by the
corresponding $\mathcal{T}$-projector $P$. Consequently, $\sigma_{I}(t)$ and
$\sigma_{I}$ commute also with all asymptotic states of the given QMP.

###### Proof.

1\. $I\geq\sigma$ for any state $\sigma$ implies that
${\mathcal{T}}_{t}(I)\geq{\mathcal{T}}_{t}(\sigma)$. Thus $\rho_{I}(t)$ has at
any time $t$ the highest rank from all evolved states and thus in the
asymptotic regime it is a ${\mathcal{T}}$-trajectory, denoted as
$\sigma_{I}(t)$. Its time-averaged state $\sigma_{I}$ is by construction a
stationary state of a QMP with the maximal rank, i.e. $\sigma_{I}$ is a
${\mathcal{T}}$-state.

2\. To prove the second statement, let $A_{k}(t)$ be the Kraus operators of
the QMP $\mathcal{T}_{t}$. Applying their commutation properties

$\displaystyle A_{k}(t)PX^{\lambda,i}P$
$\displaystyle=\lambda^{t}PX^{\lambda,i}PA_{k}(t),$ (17) $\displaystyle
PX^{\lambda,i}PA_{k}(t)^{\dagger}$
$\displaystyle=\lambda^{t}A_{k}(t)^{\dagger}PX^{\lambda,i}P$

(see Theorem 6 in Novotny2018 ) to basis of constants of motion chosen in the
form
$\\{\lambda^{t}X^{\lambda,j}+\overline{\lambda}^{t}X^{\lambda,j\dagger},i(\lambda^{t}X^{\lambda,j}-\overline{\lambda}^{t}X^{\lambda,j\dagger})\\}$
we receive general relations

$\displaystyle\mathcal{T}_{s}(PC(t)PB)$ $\displaystyle=$ $\displaystyle
PC(t+s)P\mathcal{T}_{s}(B)$ $\displaystyle\mathcal{T}_{s}(BPC(t)P)$
$\displaystyle=$ $\displaystyle\mathcal{T}_{s}(B)PC(t+s)P$ (18)

for any constant of motion $C(t)$ and operator $B$. Choosing $B=\frac{1}{N}I$
we arrive at the desired relation

$PC(t+s)P\rho_{I}(t)=\rho_{I}(t)PC(t+s)P.$ (19)

Naturally, the same commutation relation is satisfied by states from the
$\mathcal{T}$-trajectory $\sigma_{I}(t)$. Now, since $\sigma_{I}(t+s)$ is also
a $\mathcal{T}$-trajectory for any reference time $s$, any asymptotic
trajectory $\rho(t)$ (16) can be rewritten as

$\rho(t)=\exp\left[\ln\sigma_{I}(t+s)-\sum_{j}\gamma_{j}(t)PC_{j}(t)P-\alpha(t)P\right].$

Combining this form with (19) we finally receive

$\sigma_{I}(t+s)\rho(s)=\rho(s)\sigma_{I}(t+s)$ (20)

for any positive $t$ and $s$, indicating that states from the asymptotic
trajectory of the maximally mixed state commute with all asymptotic states.
$\,\blacksquare$

The final conceptual piece we need in our quest for a generalized Jaynes
principle is a properly chosen functional to be extremized. As we will see,
the quantum relative entropy $S(\rho_{1}|\rho_{2})$ of the state $\rho_{1}$
with respect to the state $\rho_{2}$ defined as

$\displaystyle
S(\rho_{1}|\rho_{2})=\left\\{\begin{array}[]{ll}\operatorname{Tr}\left[\rho_{1}(\ln\rho_{1}-\ln\rho_{2})\right]&\mbox{if
${\sf Supp}(\rho_{1})\subseteq{\sf Supp}(\rho_{2})$}\\\
+\infty&\mbox{otherwise}.\end{array}\right.$

is the needed object. Equipped with the knowledge of asymptotic states (16) we
formulate the first Jaynes’ principle for a fully known initial state evolved
under a quantum Markov process.

######  Theorem 1 (Jaynes’s principle for fully known initial states).

Assume a QMP ${\mathcal{T}}_{t}$ with a ${\mathcal{T}}$-trajectory
$\sigma(t)$. Let $P$ be the ${\mathcal{T}}$-projector and
$\\{I,C_{1}(t),\ldots,C_{d}(t)\\}$ a basis of constants of motion. Let
$\rho(0)$ be the initial state with expectation values $c_{j}=\langle
C_{j}(t)\rangle_{\rho(t)}$. Then individual states

$\rho(t)=\frac{1}{{\mathcal{Z}}}\exp\left[\ln\sigma(t)-\sum_{j}\gamma_{j}PC_{j}(t)P\right],$
(22)

from its asymptotic trajectory (6) minimize, under the given expectation
values, the quantum relative entropy $S(\rho(t)|\sigma(t))$ for any
sufficiently long time $t$. The Lagrange multipliers $\gamma_{j}$ are time
independent constants regardless of a given QMP and are determined by
expectation values $c_{j}$. The partition function $\mathcal{Z}$ is a function
of the Lagrange multipliers $\gamma_{j}$ and ensures
$\operatorname{Tr}[\rho(t)]=1$.

###### Proof.

Let us assume an arbitrary initial state $\rho(0)$. In the following we show
that the Jaynes’ principle formulated in Theorem 1 provides exactly the same
asymptotic trajectory (22). The quantum relative entropy (48) is finite if and
only if $\rho(t)=P\rho(t)P$. Thus, looking for the minimum of
$S(\rho(t)|\sigma(t))$ we can restrict ourselves to states supported on the
subspace $P{\mathscr{H}}$. This modifies known expectation values of constants
of motion into constrains

$\displaystyle\langle C_{j}(t)\rangle_{\rho(t)}$ $\displaystyle=$
$\displaystyle\operatorname{Tr}\\{C_{j}(t)P\rho(t)P\\}=\operatorname{Tr}\\{PC_{j}(t)P\rho(t)\\}=c_{j},$
$\displaystyle\langle I\rangle_{\rho(t)}$ $\displaystyle=$
$\displaystyle\operatorname{Tr}\\{P\rho(t)\\}=1.$ (23)

We seek a constrained minimum of $S(\rho(t)|\sigma(t))$ at any asymptotic time
$t$. Following the standard procedure for constrained extremes we introduce
the Lagrange functional

$\Lambda=\operatorname{Tr}\left[\rho(t)\left(\ln\frac{\rho(t)}{\sigma(t)}+\sum_{j}\gamma_{j}(t)PC_{j}(t)P+\alpha(t)P\right)\right],$

which depends on Lagrange multipliers $\gamma_{j}(t)$ and $\alpha(t)$ mutually
associated with each constrain in (23). Its variation $\delta\Lambda$, i.e.
the first order leading linear contribution,

$\delta\Lambda=\operatorname{Tr}\biggl{[}\delta\rho(t)\biggl{(}\ln\frac{\rho(t)}{\sigma(t)}+(1+\alpha(t))P+\sum_{j}\gamma_{j}(t)PC_{j}(t)P\biggr{)}\biggr{]}$

must be equal to zero for any hermitian variation $\delta\rho(t)$ of the
extremal state $\rho(t)$. We find that the only solution of this condition at
a given asymptotic time $t$ is exactly the state (22). Since the prime role of
${\mathcal{Z}}(t)=\exp[1+\alpha(t)]$ is to normalize the trace of the state
$\rho(t)$ to one, it coincides with the definition given in (22) and we call
it the partition function.

In the following we show that the Lagrange multipliers are time-independent
constants regardless of a given QMP. According to (12), changing a
$\mathcal{T}$-state (trajectory) manifests itself only by shifting the
parameters $\gamma_{i}(t)$ and we are therefore free to choose an arbitrary
$\mathcal{T}$-trajectory. Let us first exploit the $\mathcal{T}$-trajectory
$\sigma_{I}(t)$ and apply (III) to evolution of an arbitrary asymptotic
trajectory $\rho(t)$

$\displaystyle\rho(t+s)=\mathcal{T}_{s}(\rho(t))=$
$\displaystyle\frac{1}{{\mathcal{Z}}(t)}\mathcal{T}_{s}\left(\exp\left[-\sum_{j}\gamma_{j}(t)PC_{j}(t)P\right]\sigma_{I}(t)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{{\mathcal{Z}}(t)}\sum_{N=0}^{\infty}\frac{(-1)^{N}}{N!}\mathcal{T}_{s}\left(\left(\sum_{j}\gamma_{j}(t)PC_{j}(t)P\right)^{N}\sigma_{I}(t)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{{\mathcal{Z}}(t)}\sum_{N=0}^{\infty}\frac{(-1)^{N}}{N!}\left(\sum_{j}\gamma_{j}(t)PC_{j}(t+s)P\right)^{N}\mathcal{T}_{s}(\sigma_{I}(t))$
$\displaystyle=$
$\displaystyle\frac{1}{{\mathcal{Z}}(t)}\exp\left[\ln\sigma_{I}(t+s)-\sum_{j}\gamma_{j}(t)PC_{j}(t+s)P\right],$

which implies $\gamma_{j}(t+s)=\gamma_{j}(t)$,
${\mathcal{Z}}(t+s)={\mathcal{Z}}(t)$ and therefore the values of the Lagrange
multipliers associated with the $\mathcal{T}$-trajectory $\sigma_{I}(t)$ are
fixed by their initial values $\gamma_{j}(t)=\gamma_{j}(0)\equiv\gamma_{j}$,
${\mathcal{Z}}(t)={\mathcal{Z}}(0)\equiv{\mathcal{Z}}$. The same reasoning
applies to Lagrange multipliers for a general $\mathcal{T}$-trajectory
$\sigma(t)$ associated with the $\mathcal{T}$-trajectory $\sigma_{I}(t)$ and
we can finally conclude that the asymptotic trajectory $\rho(t)$ takes the
form (22). $\,\blacksquare$

A natural option is to choose $\sigma(t)$ as some $\mathcal{T}$-state, but as
we will see, by keeping the general $\mathcal{T}$-trajectory in the theorem
will allow us to formulate the Jaynes’ principle for incompletely known
asymptotic states.

Let us focus on important details and consequences concerning the theorem.
First of all, we prove that despite the fact that the expectation values
constitute time dependent constrains, Lagrange multipliers $\gamma_{j}$ are
time-independent. Their physical meaning is prescribed solely by expectation
values of constants of motion of the given QMP and can not be, as expected,
revealed by the Jaynes’ principle itself. Second, the theorem determines all
asymptotic trajectories (6) including those with non-strictly positive states.
Lagrange multipliers can take arbitrary real values and non-strictly positive
asymptotic states emerge when some of these parameters are taken in the limit
$\pm\infty$. It is actually a well known situation in statistical physics,
e.g. in the canonical ensemble, ground states are obtained in the limit
$\beta=1/T$ ($T$ denoting a temperature of the canonical system) going to
$+\infty$ Balian . Third, the Theorem 1 states that expectation values of all
constants of motion are the only information required in the Jaynes’ principle
to determine individual asymptotic trajectory of any initial state. From (6)
with $\sigma(t)=\sigma$, one sees that this asymptotic trajectory corresponds
to a stationary state if and only if all expectation values of non-stationary
constants of motion are zero. Thus, if additional information is provided
stating that the resulting asymptotic state is stationary, the modified
Jaynes’ principle for asymptotically stationary states reduces to minimization
of quantum relative entropy $S(\rho(t)|\sigma)$ only with respect to the
expectation values of integrals of motion, $\sigma(t)=\sigma$ and all Lagrange
multipliers corresponding to non-stationary constants of motion are set to
zero in (10).

This brings us to another interesting problem. In statistical physics,
typically only some expectation values of constants of motion are known and
they constitute our only available information about a given equilibrium
state. Can we adopt the same approach also in this scenario and simply set
Lagrange multipliers associated with unknown (or ignored) expectation values
of constants of motion to zero? The answer is No. This leads to a
contradiction which is easy to demonstrate on the case when we lack knowledge
of all expectation values. Indeed, resetting all Lagrange multipliers in
equivalent representations (22) with different choices of the
${\mathcal{T}}$-state $\sigma$ results in an ambiguity of the asymptotic
state. In fact, it is a very different task which, to be resolved, we must
properly incorporate the given lack of knowledge via evolution into the
description of asymptotic trajectories. We proceed in the following way. We
stress, the initial state of the system is not known. In principle, it could
be any state from trajectories consistent with the given knowledge. Thus also
the initial time is not relevant. As all states evolve eventually into states
living exclusively on the subspace $P{\mathscr{H}}$, we choose the initial
state at $t=0$ being already supported on this subspace. Expectation values of
constants of motion then read

$\langle
C_{i}(0)\rangle_{\rho(0)}=\operatorname{Tr}\left\\{\rho(0)PC_{i}(0)P\right\\}=c_{i}\quad
i\in{\sf K},$ (24)

where the set ${\sf K}$ collects indices of linearly independent constants of
motion whose expectation values are known. The state comprising solely the
given constraints reads

$\rho(0)=\frac{1}{{\mathcal{Z}}}\exp\left(-\sum_{i\in{\sf
K}}\gamma_{i}PC_{i}(0)P\right).$ (25)

However, this is not an asymptotic state yet in general. To get the asymptotic
state with the prescribed expectation values we simply let it evolve towards
its asymptotic trajectory, which is determined by the Jaynes’ principle in the
following form.

######  Theorem 2 (Jaynes’s principle for partially known asymptotic states).

Consider ${\mathcal{T}}_{t}$ be a QMP with $\sigma_{I}(t)$ being the
asymptotic evolution of the maximally mixed state and $P$ the
${\mathcal{T}}$-projector. Let the quantum system be in the asymptotic regime
of the evolution and expectation values of some linearly independent constants
of motion ($c_{j}=\operatorname{Tr}\\{\rho(t)C_{j}(t)\\}$ with $j\in{\sf K}$)
be the only knowledge provided about its asymptotic trajectory. Then states
$\rho(t)$ from the corresponding asymptotic trajectory minimize, under the
given expectation values, quantum relative entropy $S(\rho(t)|\sigma_{I}(t))$
for any sufficiently long time $t$. The asymptotic trajectory takes the form
of generalized Gibbs and von Neumann state

$\rho(t)=\frac{1}{{\mathcal{Z}}}\exp\left(\ln\sigma_{I}(t)-\sum_{j\in{\sf
K}}\gamma_{j}PC_{j}(t)P\right),$ (26)

where the partition function ${\mathcal{Z}}$ of time-independent Lagrange
multipliers $\gamma_{j}$ ensures $\operatorname{Tr}\left[\rho(t)\right]=1$.

###### Proof.

Suppose our knowledge about the initial state is reduced to information about
expectation values of some of constants of motion $C_{j}(t)$ with
$j\in\textsf{K}$, supplemented by the knowledge that the system is known to be
already in the asymptotic regime. At time $t=0$ the state supported on the
subspace $P\mathscr{H}$ with the known expectation values (24) is given by
(25). However, this state is not asymptotic yet. To satisfy the second
requirement we have to let it evolve to its asymptotic trajectory. According
to (III) and Lemma III.1, the evolution of such state reads

$\displaystyle\rho(t)$ $\displaystyle=$
$\displaystyle\mathcal{T}_{t}(\rho(0))=\frac{1}{{\mathcal{Z}}}\mathcal{T}_{t}\left(\frac{1}{N}I\right)\exp\left[-\sum_{j\in\textsf{K}}\gamma_{j}PC_{j}(t)P\right]$
$\displaystyle=$
$\displaystyle\frac{1}{{\mathcal{Z}}}\exp\left[\ln\rho_{I}(t)-\sum_{j\in\textsf{K}}\gamma_{j}PC_{j}(t)P\right].$

After a sufficiently large time $t$, we obtain the asymptotic trajectory
$\rho_{\infty}(t)$

$\rho_{\infty}(t)=\frac{1}{{\mathcal{Z}}}\exp\left[\ln\sigma_{I}(t)-\sum_{j\in\textsf{K}}\gamma_{j}PC_{j}(t)P\right].$

It is the asymptotic trajectory of the given QMP representing the given
knowledge (24). According to Theorem 1, such a quantum trajectory minimizes
the relative quantum entropy $S(\rho(t)|\sigma_{I}(t))$ with respect to the
given knowledge about the asymptotic state. $\,\blacksquare$

We stress that despite the seeming similarity of Theorems 1 and 2 they treat
different situations. While Theorem 1 deals with asymptotic evolution of the
given initial state, Theorem 2 concerns quantum systems being in the
asymptotic regime whose expectation values of some constants of motion are
known. Since we do not know the system’s initial state and consequently the
time when the dynamics of the system enters into its asymptotic regime, the
Jaynes’ principle formulated in Theorem 2 determines the asymptotic
trajectory, but it can not provide the actual position of the quantum system
on this asymptotic trajectory. Another difference lies in treatment of
parameters $\gamma_{j}$. For a specific choice of the
${\mathcal{T}}$-trajectory $\sigma(t)$ in Theorem 1, some of the parameters
$\gamma_{j}$ may be equal to zero and thus the corresponding constants of
motion $C_{j}(t)$ do not appear in the Gibbs form (22). This is much different
to the situation treated by Theorem 2, in which the ${\mathcal{T}}$-trajectory
is uniquely specified. Constants of motion $C_{j}(t)$, $j\notin{\sf K}$, are
completely ignored in the Gibbs and von Neumann state (26) and thus they do
not have assigned any parameters $\gamma_{j}$.

Theorem 2 reveals the unique role of the evolution of the maximally mixed
state among all possible trajectories. At this point let us stress that even
in the case, when only expectation values of integrals of motion are known,
stationarity of the resulting asymptotic trajectory is not guaranteed. It is
ensured only if the maximally mixed state is steady too. But how to treat the
situation when we know that the quantum system has reached a stationary
asymptotic state, or the asymptotic dynamics is fast enough that we observe
its averaged stationary part solely, and additionally we know expectation
values of only some of its integrals of motion? Again, we must integrate both
types of information into the resulting state consistent with evolution of the
given QMP. The following theorem deals with this issue.

######  Theorem 3 (Jaynes’s principle for stationary states with incomplete
knowledge).

Let ${\mathcal{T}}_{t}$ be a QMP and $\sigma_{I}$ be a its time-averaged
trajectory of the maximally mixed state. Suppose the system is in a partially
unknown stationary state $\rho$ of the QMP and expectation values
$c_{j}=\operatorname{Tr}\\{\rho I_{j}\\}$ of some linearly independent
integrals of motion $I_{j}$, $j\in{\sf K}$, be the only available knowledge
about this state. Then the state $\rho$ minimizes, under the given expectation
values, quantum relative entropy $S(\rho|\sigma_{I})$ and takes the form of
generalized Gibbs and von Neumann state

$\rho=\frac{1}{{\mathcal{Z}}}\exp\left(\ln\sigma_{I}-\sum_{j\in{\sf
K}}\gamma_{j}PI_{j}P\right),$ (27)

where the partition function ${\mathcal{Z}}$ of time-independent Lagrange
multipliers $\gamma_{j}$ ensures $\operatorname{Tr}\left[\rho\right]=1$.

###### Proof.

If the system is in some stationary state, it is in the asymptotic regime
already. In view of Theorem 2, the system in the asymptotic regime being
determine solely by expectation values of some integrals of motion evolves as

$\rho(t)=\frac{1}{{\mathcal{Z}}}\exp\left(\ln\sigma_{I}(t)-\sum_{j\in{\sf
K}}\gamma_{j}PI_{j}P\right),$

where $\sigma_{I}(t)$ is asymptotic ${\mathcal{T}}$-trajectory initiated at
the maximally mixed state. All these asymptotic states have the same demanded
expectation values of given integrals of motion, but it is not stationary as
it is required. To get a stationary state we have to make time-average of the
trajectory $\rho(t)$ given above. Averaging can not increase amount of
information imprinted into the state and consequently the resulting stationary
state $\rho$ incorporates the minimum information given by expectation values
of the corresponding integrals of motion. Exploiting the Lemma III.1, we find
that

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\lim\limits_{t\rightarrow\infty}\frac{1}{t}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{0}^{t}ds\frac{1}{{\mathcal{Z}}}\exp\left(\ln\sigma_{I}(s)-\sum_{j\in{\sf
K}}\gamma_{j}PI_{j}P\right)$ (36) $\displaystyle=$
$\displaystyle\left(\underbrace{\lim\limits_{t\rightarrow\infty}\frac{1}{t}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{0}^{t}ds\sigma_{I}(s)}_{\sigma_{I}}\right)\frac{1}{{\mathcal{Z}}}\exp\left(-\sum_{j\in{\sf
K}}\gamma_{j}PI_{j}P\right)$ $\displaystyle=$
$\displaystyle\frac{1}{{\mathcal{Z}}}\exp\left(\ln\sigma_{I}-\sum_{j\in{\sf
K}}\gamma_{j}PI_{j}P\right).$

According to Theorem 1, this state minimizes under the expectation value
constrains $c_{j}=\operatorname{Tr}\\{\rho I_{j}\\}$ with $j\in{\sf K}$ the
quantum relative entropy $S(\rho|\sigma_{I})$. $\,\blacksquare$

The last Theorem 3 treats exactly the setting well known in statistical
physics – equilibrium states characterized by few expectation values of
integrals of motion. We find that all above formulated Jaynes’ principles as
well as the resulting general Gibbs and von Neumann forms of asymptotic states
differs from the MaxEnt principle and associated Gibbs and von Neumann states
(1) used in statistical physics. However, if the Markov evolution preserves
the maximally mixed state (unital QMPs), then projection $P$ is the identity
operator. The trajectory $\sigma(t)$ can be chosen as the fixed maximally
mixed state and may be therefore included into the partition function and we
get finally Gibbs form which for stationary states coincides with (1). From
this perspective, Theorems 1, 2 and 3 prove the MaxEnt principle for
equilibrium states of unital channels and generalizes it to Jaynes’principle
for all asymptotic trajectories of any quantum Markov process.

##  IV Properties of the partition function

Surprisingly, despite the different generalized Gibbs and von Neumann form
(22) of asymptotic states, the partition function keeps its special role in
calculating statistical properties like expectation values of observables,
quantum relative entropy and its change. Indeed, the partition function
${\mathcal{Z}}$ independently of the choice of a $\mathcal{T}$-trajectory
provides the expectation values of constants of motion

$\langle
C_{j}(t)\rangle_{\rho(t)}=-\left(\frac{\partial\ln{\mathcal{Z}}}{\partial\gamma_{j}}\right)_{\gamma_{k\neq
j}}.$ (46)

Proof of this formula exploits the fact that the partition function is a trace
functional of the form $F(X)=\operatorname{Tr}f(X)$ on the space of all
hermitian operators. If $f$ is an analytic function, the variation of the
functional (the first order change in term of $\delta X$) reads $\delta
F_{X}(\delta X)=\operatorname{Tr}\left\\{\delta Xf^{{}^{\prime}}(X)\right\\}$.
With the help of this expression we get for
$X=\ln\sigma(t)-\sum_{k}\gamma_{k}PC_{k}(t)P$

$\displaystyle\left(\frac{\partial\ln{\mathcal{Z}}}{\partial\gamma_{j}}\right)_{\gamma_{k\neq
j}}=$
$\displaystyle\frac{1}{{\mathcal{Z}}}\lim_{\triangle\gamma_{j}\rightarrow
0}\frac{\operatorname{Tr}\left\\{\exp\left(X-\triangle\gamma_{j}PC_{j}(t)P\right)\right\\}-\operatorname{Tr}\left\\{\exp\left(X\right)\right\\}}{\triangle\gamma_{j}}=$
$\displaystyle\frac{1}{{\mathcal{Z}}}\lim_{\triangle\gamma_{j}\rightarrow
0}\frac{\operatorname{Tr}\left\\{-\triangle\gamma_{j}PC_{j}(t)P\exp\left(X\right)\right\\}}{\triangle\gamma_{j}}=-\langle
C_{j}(t)\rangle_{\rho(t)}.$

Different choices of $\mathcal{T}$-trajectories $\sigma_{1}(t)$ and
$\sigma_{2}(t)$ imply different partition functions with correspondingly
shifted Lagrange multipliers. However, as we show (46) is independent of the
particular choice. Indeed, expressing the $\mathcal{T}$-trajectory
$\sigma_{1}(t)$ as
$\sigma_{1}(t)=1/{\mathcal{Z}}_{1}\exp\left[\ln\sigma_{2}(t)-\sum_{j}\omega_{j}PC_{j}(t)P\right]$,
the asymptotic trajectory (22) can be written as

$\displaystyle\rho(t)$ $\displaystyle=$
$\displaystyle\frac{1}{{\mathcal{Z}}}\exp\left(\ln\sigma_{1}(t)-\sum_{j}\gamma_{j}PC_{j}(t)P\right)$
(47) $\displaystyle=$
$\displaystyle\frac{1}{{\mathcal{Z}}^{{}^{\prime}}}\exp\left(\ln\sigma_{2}(t)-\sum_{j}\beta_{j}PC_{j}(t)P\right)$

with ${\mathcal{Z}}^{{}^{\prime}}={\mathcal{Z}}{\mathcal{Z}}_{1}$ and
$\beta_{j}=\gamma_{j}+\omega_{j}$. Therefore, we get

$\frac{\partial\ln{\mathcal{Z}}^{{}^{\prime}}}{\partial\beta_{j}}=\frac{1}{{\mathcal{Z}}^{{}^{\prime}}}\frac{\partial{\mathcal{Z}}^{{}^{\prime}}}{\partial\gamma_{j}}\frac{\partial\gamma_{j}}{\partial\beta_{j}}=\frac{1}{{\mathcal{Z}}{\mathcal{Z}}_{1}}{\mathcal{Z}}_{1}\frac{\partial{\mathcal{Z}}}{\partial\gamma_{j}}=\frac{\partial\ln{\mathcal{Z}}}{\partial\gamma_{j}}.$

Quantum relative entropy of an asymptotic state $\rho(t)$ w.r.t. the given
${\mathcal{T}}$-trajectory $\sigma(t)$ is then easily obtained as

$\displaystyle
S(\rho(t)|\sigma(t))=\operatorname{Tr}\rho(t)\ln\rho(t)-\operatorname{Tr}\rho(t)\ln\sigma(t)=$
$\displaystyle-\operatorname{Tr}\rho(t)\ln{\mathcal{Z}}+\operatorname{Tr}\left[\rho(t)\left(\ln\sigma(t)-\sum_{j}\gamma_{j}PC_{j}(t)P\right)\right]$
$\displaystyle-\operatorname{Tr}\rho(t)\ln\sigma(t)=-\ln{\mathcal{Z}}-\sum_{j}\gamma_{j}\langle
C_{j}(t)\rangle_{\rho(t)}.$ (48)

Similarly due to (46), its change between two infinitesimally closed
asymptotic trajectories given by the differential reads

$\displaystyle
dS(\rho(t)|\sigma(t))=-\sum_{j}\left(\frac{\partial\ln{\mathcal{Z}}}{\partial\gamma_{j}}\right)d\gamma_{j}-\sum_{j}\langle
C_{j}(t)\rangle_{\rho(t)}d\gamma_{j}$
$\displaystyle-\sum_{j}\gamma_{j}d\langle
C_{j}(t)\rangle_{\rho(t)}=-\sum_{j}\gamma_{j}d\langle
C_{j}(t)\rangle_{\rho(t)}.$ (49)

Note that both quantities (48) and (49) are time-independent. Also
$S(\rho(t)|\sigma(t+s))$ is time-independent, since $\omega(t)=\sigma(t+s)$ is
a ${\mathcal{T}}$-trajectory too. We can conclude that in the asymptotic
evolution the quantum relative entropy of two evolving states stays constant
or is infinite.

Finally, let us explore the quantum relative entropy under the interchange
between $\mathcal{T}$-trajectories $\sigma_{1}(t)$ and $\sigma_{2}(t)$.
Applying (48) to (47) gives us gradually

$\displaystyle
S(\rho(t)|\sigma_{2}(t))=-\ln{\mathcal{Z}}^{{}^{\prime}}-\sum_{j}\beta_{j}\langle
C_{j}(t)\rangle_{\rho(t)}=$
$\displaystyle-\ln{\mathcal{Z}}-\ln{\mathcal{Z}}_{1}-\sum_{j}(\gamma_{j}+\omega_{j})\langle
C_{j}(t)\rangle_{\rho(t)}=$ $\displaystyle
S(\rho(t)|\sigma_{1}(t))+S(\sigma_{1}(t)|\sigma_{2}(t))+\sum_{j}\omega_{j}\langle
C_{j}(t)\rangle_{\sigma_{1}(t)-\rho(t)}$

with $\langle
C_{j}(t)\rangle_{\sigma_{1}(t)-\rho(t)}=\operatorname{Tr}\\{C_{j}(t)(\sigma_{1}(t)-\rho(t))\\}$.
Analogously for the same interchange, the differential of quantum relative
entropy reads

$\displaystyle dS(\rho(t)|\sigma_{1}(t))+dS(\sigma_{1}(t)|\sigma_{2}(t))$
$\displaystyle=-\sum_{j}\gamma_{i}d\langle
C_{j}(t)\rangle_{\rho(t)}-\sum_{j}\omega_{i}d\langle
C_{j}(t)\rangle_{\sigma_{1}(t)}$
$\displaystyle=dS(\rho(t)|\sigma_{2}(t))+\sum_{j}\omega_{j}d\langle
C_{j}(t)\rangle_{\rho(t)-\sigma_{1}(t)}.$

This provides a useful relation for thermodynamic finite and infinitesimal
changes performed in the asymptotic regime.

##  V Conclusion

In summary, we have studied asymptotics of quantum systems undergoing a
quantum Markov (finite-dimensional), discrete or continuous, process (QMP) for
different scenarios of available apriori knowledge. We have proven that the
asymptotics of QMPs always takes the exponential Gibbs and von Neumann like
form and hence can be equivalently formulated as a constrained generalized
Jaynes’ problem. In order to obtain the correct dynamically induced asymptotic
state we must minimize the quantum relative entropy with respect to a properly
chosen maximal invariant state or maximal asymptotic trajectory under given
expectation values of some constants of motion (including integrals of motion
problem. In other words, we show which functional has to be chosen as the
proper one when formulating the problem as an extremal one. The results are
formulated in three principles, each treating a different situation. The first
concerns a known initial state and consequently expectation values of all
constants of motion of the given QMP are known. The second deals with a system
being in the asymptotic regime for which only some of the expectation values
of its constants of motion are known. Finally, the third case treats the
situation where we know, in addition, that the system is in a stationary state
and only some expectation values of some integrals of motion are known. The
obtained Jaynes’ principles for QMPs coincide with the standard MaxEnt
principle if and only if the Markov evolution preserves the maximally mixed
state. From this perspective, we have proven the applicability of the MaxEnt
principle to equilibrium states of all unital QMPs and generalized it to
Jaynes’ principle for all asymptotic trajectories of any quantum Markov
process. Thus, for QMPs we are able to bridge the gap between the Jaynes’
principle as a purely formal information approach to open system dynamics
asymptotics and the actual dynamics of the physical process. In particular,
Theorem 1 reveals that asymptotic dynamics of QMPs can be equivalently
captured by extremal principle. In addition, Theorem 2 and 3 tackle the issue
of incorporating partial knowledge about the asymptotic state in accord with
the dynamics of the given QMP.

We found all ingredients needed for the formulation of the extremal principle
for all QMPs - the relevant constants of motion, the T-trajectories as well as
the quantum relative entropy functional. It results in the equivalence between
solving the actual dynamics and the exponential form of the asymptotic
(stationary) state solving the asymptotic dynamics. Our approach allows to
identify the set of privileged observables needed to define the asymptotic
state. The presented method and results are in its spirit very similar to the
one applied by Boltzmann to prove the mechanical origin of entropy increase
Boltzmann ; Huang ; Goldstein ; Lenarcic . The Boltzmann probability
distribution and its evolution equation – the Boltzmann transport equation –
are reincarnated in our formulation to the Markov equation and the collision
invariants correspond to the objects we use to define the asymptotic space. As
mentioned, the proper variables to be used, are dictated by the system
dynamics and can be in many cases found analytically. This is certainly a
benefit in particular when methods how to identify the basis of asymptotic
dynamics can be given Kollar2012 . It would be of interest to apply our method
for situations of several (interacting) subsystems. In particular to study,
what are the essential features of such systems undergoing QMPs, which
determine and control the changes of Lagrange multipliers. Much more elusive
is the determination of the time scale sufficient to reach the asymptotic
regime. The rate in approaching the stationary (equilibrium) regime is
strongly dependent on the concrete form of the dynamics and the initial state
we start with. Some general constraints for relaxation rates may be found in
Chruscinskiprl ; Chruscinskilinalg , a bound for relaxation rates in large
CNOT networks is discussed in Novotny2021 . This question is particularly
interesting for experimentalists, as it defines the natural scale on which the
dynamics of the system takes place and when transient effects become
negligible.

Acknowledgements: J. N., J.M. and I. J. have been supported by the Grant
Agency of the Czech Technical University in Prague, grant No.
SGS22/181/OHK4/3T/14. This publication was funded by the project ”Centre for
Advanced Applied Sciences”, Registry No. CZ.02.1.01/0.0/0.0/16 019/0000778,
supported by the Operational Programme Research, Development and Education,
co-financed by the European Structural and Investment Funds and the state
budget of the Czech Republic. The support for J. N. and I. J. by GAČR of the
Czech Republic under Grant No. 23-07169S is gratefully acknowledged.

Data Availability Statement: No Data associated in the manuscript

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