# Can the Macroscopic Fluctuation Theory be Quantized ?

###### Abstract

The Macroscopic Fluctuation Theory is an effective framework to describe
transports and their fluctuations in classical out-of-equilibrium diffusive
systems. Whether the Macroscopic Fluctuation Theory may be extended to the
quantum realm and which form this extension may take is yet terra incognita
but is a timely question. In this short introductory review, I discuss
possible questions that a quantum version of the Macroscopic Fluctuation
Theory could address and how analysing Quantum Simple Exclusion Processes
yields pieces of answers to these questions.

Denis<EMAIL_ADDRESS>

♣ Laboratoire de Physique de l’Ecole Normale Supérieure, CNRS, ENS &
Université PSL, Sorbonne Université, Université de Paris, 75005 Paris, France.

###### Contents

  1. 1 Introduction
  2. 2 Why, What, and How ?
    1. 2.1 A glimpse on classical Macroscopic Fluctuation Theory
    2. 2.2 What a Quantum Mesoscopic Fluctuation Theory could be aiming at ?
    3. 2.3 How do stochastic effects impact quantum systems ?
    4. 2.4 How can stochastic fluctuations be observed ?
  3. 3 Quantum Simple Exclusion Processes
    1. 3.1 Q-SSEP and Q-ASEP
    2. 3.2 Closed Q-SSEP : Fluctuations at equilibrium
    3. 3.3 Closed Q-SSEP : Where are we ?
    4. 3.4 Open Q-SSEP : Fluctuations out-of-equilibrium
    5. 3.5 Open Q-SSEP : Where are we ?
  4. 4 Conclusion
  5. A Appendix : Does ETH imply ergodicity of quantum trajectories ?

##  1 Introduction

Non-equilibrium phenomena, classical or quantum, are ubiquitous in Nature, but
their understanding is more delicate, and thus yet less profound, than the
equilibrium ones. Last decades has witnessed important conceptual and
practical progresses in this direction, both for classical and quantum
systems. For classical systems, these progresses started with the study of
fluctuating hydrodynamics of many particle systems [1, 2, 3, 4, 5, 6, 7], or
with exact analysis of simple model systems, such as the Symmetric or
Asymetric Simple Exclusion Process (SSEP or ASEP) [8, 9, 10]. They advanced
with the understanding of fluctuation relations [11, 12, 13] and their
interplay with time reversal [14, 15], and culminated with the formulation of
the Macroscopic Fluctuation Theory (MFT) which is an effective theory
describing transports and their fluctuations in diffusive classical systems
[16, 17]. For quantum systems, a good understanding of entanglement dynamics
[18, 19, 20, 21] and quenched dynamics [22, 23, 24] in critical or integrable
models has been obtained. Substantial information on transport has been gained
for integrable many-body systems via the formulation of the Generalized
Hydrodynamics (GHD) [25, 26, 27]. The latter is however well adapted to
describe transport but less fitted to elucidate quantum interference
phenomena. Furthermore, transports in these systems are mainly ballistic.

The questions whether Macroscopic Fluctuation Theory (MFT) may be extended to
deal with quantum systems and which form this extension would take are still
open. If such theory can be formulated, it should aim at describing not only
diffusive transports and their fluctuations but also quantum coherent
phenomena, say the dynamics of quantum interferences or entanglement spreading
and their fluctuations, in out-of-equilibrium extended, diffusive, systems.

Pieces of information on a possible form of such theory has recently been
obtained by studying model systems based on random quantum circuits [28, 29,
30, 31, 32, 33] for which a membrane picture [34, 35, 36, 37, 38] for
entanglement production in many-body systems is emerging.

In parallel, stochastic dynamics in quantum extended systems such as noisy
spin chains were analysed [39, 40, 41, 42]. This leads us to introduce the
Quantum Simple Exclusion Processes. The closed Quantum Symmetric Simple
Exclusion Process (Q-SSEP), was introduced in [43] (under another name), its
open version in [44], and its asymmetric analogue, the Quantum Asymmetric
Simple Exclusion Process (Q-ASEP), was proposed in [45]. Through decoherence,
the average dynamics in such models typically reduces to classical simple
exclusion processes. However, this quasi-classical reduction only applies to
the average dynamics. Fluctuations are beyond this quasi-classical regime and
survive decoherence. They reveal patterns which are possibly generic for
mesoscopic diffusive quantum systems.

In this review, we first introduce basic notions about the MFT and pose a few
questions relevant to its extension to the quantum regime and, second, discuss
possible routes for exploring quantum extensions of the MFT and how these
explorations lead to consider stochastic dynamics in quantum many-body
systems. While classical stochastic dynamics has been a well studied domain
for many decades, stochastic dynamics in quantum extended systems remained
largely unexplored, to the best of our knowledge. We then present a brief
introduction to the Quantum Simple Exclusion Processes and to some of the
recently obtained results concerning their on-going analysis. We have chosen
to select a few claims, summarising these analysis in a necessarily biased
ways, but allowing for a comparison with classical MFT. For each topic, we
also tried to formulate a few questions left opened by these analysis.

As a side comment, we pose a question (or propose a conjecture) about
ergodicity property of monitored extended quantum systems.

##  2 Why, What, and How ?

###  2.1 A glimpse on classical Macroscopic Fluctuation Theory

The Macroscopic Fluctuation Theory [16, 17] is an effective theory describing
current and density fluctuations in classical out-of-equilibrium systems. It
applies to locally diffusive systems satisfying the Fourier-Fick’s law which
asserts that the current is proportional to the density gradient. The
proportionality coefficient is called the diffusion constant. Whenever the
current does not vanish – a situation which reflects transport –, the physical
system is out-of-equilibrium. By the Fourier-Fick’s law, diffusive systems can
be maintained out-of-equilibrium by imposing a non-zero density gradient or by
applying an external field. A standard set-up consists in putting the system
in contact with two reservoirs at different chemical potentials222The
discussion is here presented in terms of particle density and its current, but
one may change the point and discuss about transport of other physical
quantities, say heat or temperature, simply by changing the names.. See Figure
1.

One of the formulation of MFT [16, 17] starts with stochastic differential
equations (SDE) for the density $\mathfrak{n}(x,t)$ and the current
$\mathfrak{j}(x,t)$. In one dimension, they take the following form :

$\displaystyle\partial_{t}\mathfrak{n}(x,t)+\partial_{x}\mathfrak{j}(x,t)=0~{},$
(1a)
$\displaystyle\mathfrak{j}(x,t)=-D(\mathfrak{n})\partial_{x}\mathfrak{n}(x,t)+\varepsilon^{1/2}\,\sigma(\mathfrak{n})^{1/2}\,\xi(x,t)~{},$
(1b)

with $D(\mathfrak{n})$ the diffusion constant, $\sigma(\mathfrak{n})$ the
mobility and $\xi(x,t)$ a Gaussian space-time white noise,
$\mathbb{E}[\xi(x,t)\xi(x^{\prime},t^{\prime})]=\delta(x-x^{\prime})\delta(t-t^{\prime})$.
Here, $\mathbb{E}[\cdot]$ refers to the statistical average and
$\varepsilon:=a_{0}/L$ with $a_{0}$ the short microscopic scale (an ultra-
violet cut-off) and $L$ the linear size of the system. In particular,
$\varepsilon\to 0$ in the large system size limit or in the scaling limit
$a_{0}\to 0$ at $L$ fixed.

The first equation (1a) is a conservation law. The second one (1b) is a noisy
version of the Fourier-Fick’s law
$\mathfrak{j}(x,t)=-D(\mathfrak{n})\partial_{x}\mathfrak{n}(x,t)$. The
mobility quantifies the response of the system to an external field $E$ which,
in the linear approximation, produces an extra current
$\mathfrak{j}_{\mathrm{ext}}=\sigma(\mathfrak{n})\partial_{x}E$. It also codes
for the density fluctuations at equilibrium. Both the diffusion constant and
the mobility may depend non-linearly on the local density. They satisfy the
Einstein relation
$D(\mathfrak{n})=\sigma(\mathfrak{n})\,f^{\prime\prime}(\mathfrak{n})$ with
$f(\mathfrak{n})$ the equilibrium free energy per unit of volume.

The strength of the noise in eq.(1) gets smaller as the system size increases
(or as the ultra-violet cut-off is shrinked), so that analysing eq.(1) is
actually a small noise problem [46]. This reflects the physical fact that
fluctuations in thermodynamical systems are small, sub-leading in the system
size. The MFT describes rare fluctuations in out-of-equilibrium macroscopic
systems.

Through eq.(1), the statistical distribution of the noise induces that of the
current and density profiles. By integrating out the noise, the probability
distribution can (formally) be represented as a path integral with weight,

$\displaystyle\exp\Big{(}-\frac{1}{\varepsilon}\int\\!\\!dxdt\,\frac{\big{(}\mathfrak{j}(x,t)+D(\mathfrak{n}(x,t))\partial_{x}\mathfrak{n}(x,t)\big{)}^{2}}{2\sigma(\mathfrak{n}(x,t))}\Big{)}~{},$
(2)

conditioned by the conservation law
$\partial_{t}\mathfrak{n}(x,t)+\partial_{x}\mathfrak{j}(x,t)=0$. In some
cases, this weight can be deduced from a simple, but elegant, additivity
principle [47]. Configurations extremizing this weight are the most probable.
They correspond to the dominating non-fluctuating solutions of the Fourier-
Fick’s law depending on boundary conditions specific to the physical set-up.

Fluctuations around the dominating configurations are sub-leading in the
system size. They satisfy a large deviation principle. The large prefactor
$\varepsilon^{-1}=L/a_{0}$ in the weight (2) echoes the weakness of the noise
in eq.(1) for macroscopically large systems. It ensures that the large
deviation functions are computable through solutions of extremization problems
(which may nevertheless be difficult to solve) [17, 46, 47].

Figure 1: Classical setup to drive a diffusive systems out-of-equilibrium (the
two blue/red square represent two reservoirs at different chemical potentials
or temperatures).

The MFT also emerged from studies of simple representative toy models. The
iconic exemple is the so-called symmetric simple exclusion process (SSEP). In
its one dimensional version, it describes particles moving along a
chain333Variants of SSEP can be defined on any graph. with the exclusion that
any given site cannot be occupied by more than one particle at each instant.
Particles are injected and extracted at the two ends of the system interval.
In the symmetric case the probability to move toward the left or the right are
equal (they are different in the asymmetric case). Any configuration is
specified by its occupation numbers $\mathfrak{n}_{j}$ at each site $j$, with
$\mathfrak{n}_{j}=0$ for an empty site (represented as ø) and
$\mathfrak{n}_{j}=1$ for a full site (represented as $\bullet$). The evolution
from one configuration to another is a Markovian random process, with
probability transition per unit of time given by the following Markov
transition matrix :

$\displaystyle\mathbb{M}_{{\rm ssep}}(|\text{\o}\text{\o}|)$ $\displaystyle=$
$\displaystyle 0~{},$ $\displaystyle\mathbb{M}_{{\rm
ssep}}(|\\!\bullet\\!\text{\o}|)$ $\displaystyle=$
$\displaystyle-|\\!\bullet\\!\text{\o}|+|\text{\o}\\!\bullet\\!|~{},$ (3)
$\displaystyle\mathbb{M}_{{\rm ssep}}(|\text{\o}\\!\bullet\\!|)$
$\displaystyle=$
$\displaystyle-|\text{\o}\\!\bullet\\!|+|\\!\bullet\\!\text{\o}|~{},$
$\displaystyle\mathbb{M}_{{\rm ssep}}(|\\!\bullet\\!\bullet|)$
$\displaystyle=$ $\displaystyle 0~{}.$

The first and last equations simply express the absence of move either because
there is no particle or because the exclusion freezes the dynamics. The two
middle ones code for symmetric moves to the right or the left when the
configurations allow. The evolution rules are modified at the boundaries to
reflect the rates at which particles are injected and extracted. This system
is known to be integrable and its steady states represented by matrix product
states. There is a vast literature on exact results for SSEP, see e.g. [48,
49, 50, 51].

The large deviation function for density fluctuations in SSEP were computed
from its microscopic definition [52, 53]. Let us consider a protocol as in
Figure 1 in which one fixes the density at the two ends of the interval
$[0,L]$, say $n_{a}$ and $n_{b}$. If $n_{a}\not=n_{b}$ the system is out-of-
equilibrium, with a non-zero mean current proportional to the density
difference $\Delta n:=n_{b}-n_{a}$. In the steady state, the density
statistics satisfies a large deviation principle in the sense that the
distribution of the density profiles behaves as follows444The symbole $\asymp$
means logarithmic equivalence. Here, the limit $\varepsilon\to 0$ is meant as
$a_{0}\to 0$ at fixed $L$.,

$\displaystyle\mathbb{P}\mathrm{rob}[\mathfrak{n}(\cdot)=n(\cdot)]\asymp_{\varepsilon\to
0}e^{-\varepsilon^{-1}\,F[n]}~{},$ (4)

with $F[n]$ the so-called rate function. It plays a role analogue to the free
energy but in non-equilibrium situations, and reduces to it at equilibrium.
Out-of-equilibrium, $F[n]$ becomes a non-linear non-local functional of the
density profile coding for long range correlations.

Equivalently, one can introduce the generating function of connected
correlation functions of the density
$\mathbb{E}\big{[}e^{\varepsilon^{-1}\\!\int\\!dx\,h(x)\mathfrak{n}(x)}\big{]}$,
with $h(x)$ some test function. By expanding in power of $h$, it generates the
density correlation functions. The large deviation principle (4) implies that
it scales as follows,

$\displaystyle\mathbb{E}_{\mathrm{ssep}}\big{[}e^{\varepsilon^{-1}\int\\!dx\,h(x)\mathfrak{n}(x)}\big{]}\asymp_{\varepsilon\to
0}e^{\varepsilon^{-1}\,W[h]}~{},$ (5)

for large system size. The two functions $F[n]$ and $W[h]$ are related by
Legendre transform :
$W[h]=\mathrm{max}_{\\{n(x)\\}}\big{[}\int_{0}^{1}dx\,h(x)n(x)-F[n]\big{]}$.
Formula (5) implies that the cumulants of the density of order $P$ scale like
$\varepsilon^{P-1}$ in the large system size limit.

For SSEP, the large deviation function $W[h]$ is given as the solution of an
extremization problem :

$\displaystyle
W[h]=\int_{0}^{L}\\!\\!\\!dx\big{[}\log\big{(}1+g(x)(e^{h(x)}-1)\big{)}-\log({g^{\prime}(x)}L/{\Delta
n})\big{]}~{},$ (6)

with $g(x)$ solution of the non-linear differential equation,

$\displaystyle\big{(}1+g(x)(e^{h(x)}-1)\big{)}g^{\prime\prime}(x)=g^{\prime}(x)^{2}(e^{h(x)}-1)~{},$
(7)

with boundary conditions $g(0)=n_{a}$ and $g(L)=n_{b}$. This condition is the
Euler-Lagrange equation for $W[h]$ to be extremal with respect to variations
of $g$. Expanding $W[h]$ in power of $h$ yields the first few density
cumulants :

$\displaystyle\mathbb{E}_{\mathrm{ssep}}[\mathfrak{n}(x)]=\bar{n}(x)~{},\quad\mathbb{E}_{\mathrm{ssep}}[\mathfrak{n}(x)\mathfrak{n}(y)]^{c}=-\varepsilon\,(\Delta
n)^{2}\,x(L-y)/L^{2}~{},\quad\mathrm{for}\ x<y,$ (8)

with $\bar{n}(x)$ the linear profile interpolating the two boundary densities,
$\bar{n}(x)=(n_{a}(L-x)+xn_{b})/L$. In particular, the mean current
$\bar{\mathfrak{j}}=-\partial_{x}\bar{n}=-\Delta n/L$, is proportional to the
density gradient. Eq.(8) reveals the presence of long range, but decreasing
with the system size, density correlations.

The SSEP large deviation function (6) was also shown to follow from the MFT
[17]. In its scaling limit, SSEP is mapped onto MFT with
$D_{\mathrm{ssep}}=1$,
$\sigma_{\mathrm{ssep}}(\mathfrak{n})=\mathfrak{n}(1-\mathfrak{n})$, and free
energy
$f_{\mathrm{ssep}}(\mathfrak{n})=\mathfrak{n}\log\mathfrak{n}+(1-\mathfrak{n})\log(1-\mathfrak{n})$.
The MFT has however a wider range of applications as it applies to all
classical, locally diffusive, systems. It is worth stressing that MFT,
formulated as an effective fluctuating hydrodynamics, depends only on two
phenomenological coefficients : the diffusion coefficient $D(\mathfrak{n})$
and the mobility $\sigma(\mathfrak{n})$, which are near equilibrium data.

###  2.2 What a Quantum Mesoscopic Fluctuation Theory could be aiming at ?

As the classical MFT, the Quantum Mesoscopic Fluctuation Theory (Q-MFT) should
describe transports and their fluctuations in out-of-equilibrium systems. To
be applicable to quantum systems, it should also address specifically quantum
effects such as interference, coherence, entanglement, etc.

The Q-MFT is expected to be applicable to locally diffusive but quantum
systems. As a consequence, for the systems to be diffusive, not ballistic,
Q-MFT should be valid at scales above the mean free path. For quantum effects
not to be wash-out by decoherence, it should be applicable at scales below the
decoherence length. This intermediate scale domain is called the mesoscopic
domain.

At a sufficiently coarse-grained level, transports may usually be described by
some hydrodynamic equations, even for genuinely quantum systems as in the
Generalized Hydrodynamics (GHD) [25, 26] adapted to integrable systems. Since
they refer to locally conserved quantities, these equations take the form of
local conservation laws, say of the form
$\partial_{t}\mathfrak{n}+\partial_{x}\mathfrak{j}=0$, completed with
constitutive equations expressing the current $\mathfrak{j}$ in terms of the
density $\mathfrak{n}$ plus possible noise. This is also the way the classical
MFT is formulated. Assuming decoherence, hydrodynamics hypothesises some kind
of local equilibrium so that the system density matrices, reduced to the local
hydrodynamic cells, are diagonal in the basis of the transported conserved
charges. However, the off-diagonal elements survive at mesoscopic scales,
even-though they are sub-leading compared to the diagonal ones. Usual
hydrodynamics describes neither the structure of coherence or entanglement,
nor their statistics or their dynamics. Describing them requires extracting
information on the dynamics of the off-diagonal elements. Phrased differently,
if Q-MFT could be formulated, it is expected to go beyond standard fluctuating
hydrodynamics to include off-diagonal contributions.

At mesoscopic scales, quantum systems interact with extra noisy degrees of
freedom leading to diffusion. For large systems, decoherence is hence at play
so that the average dynamics is quasi-classical. Quantum interference effects
echo fluctuating, non-classical, off-diagonal coherences. To make them
manifest thus requires an understanding of coherent fluctuations, which cancel
out on average and are sub-leading in the system size, and their noisy
dynamics. For large systems, the statistical properties of these fluctuations
can often be described by large deviation functions which code for rare
fluctuations away from the typical value. Of particular interest are the large
deviation functions say for quantum expectations of operators sensitive to
quantum coherences or for entanglement entropies, which are specific to
quantum systems, (see exemples of such large deviation functions in the closed
Q-SSEP in Section 3.3).

In the last few years, impressive progresses on closely related topics has
been achieved in critical or integrable quantum systems. A precise picture of
entanglement and quenched dynamics has been obtained in terms of quasi-
particle dynamics [22]. A good understanding of transport and out-of-
equilibrium phenomena in those systems has been achieved using the Generalized
Hydrodynamics we alluded to above. GHD also lead to a good description of
fluctuation statistics of transport in integrable systems [54] (see [55] for
similar studies in critical systems). However, these understandings are
restricted to predominantly ballistic systems, even-though sub-leading
diffusive effects have been incorporated [56].

A naive way to look for a quantum version of MFT is to quantize equations (1).
Since eq.(1b) is a constraint, expressing the current $\mathfrak{j}$ in terms
of the density $\mathfrak{n}$ plus noise, a direct quantization of eqs.(1) is
however potentially difficult, because, in a quantum theory, a constraint
should be promoted to an operator identity. Following the strategy recently
applied to GHD [57], a possible route could amount to look for fluctuations
above a classical solution
$(\mathfrak{n}_{\mathrm{cl}},\mathfrak{j}_{\mathrm{cl}})$ of the MFT equations
(1), say in form

$\mathfrak{n}=\mathfrak{n}_{\mathrm{cl}}+\partial_{x}\varphi~{},\quad\mathfrak{j}=\mathfrak{j}_{\mathrm{cl}}-\partial_{t}\varphi~{},$

to fulfil the continuity equation (1a), and to quantize the resulting Gaussian
theory for the field $\varphi$. Another route could consist in promoting
eq.(1b) to a dynamical equation. For instance [39], eq.(1b) could be upraised
to a dynamical one by introducing a current friction, say

$\tau_{f}\,\partial_{t}\mathfrak{j}=-D(\mathfrak{n})\partial_{x}\mathfrak{n}-\eta\,\mathfrak{j}+\varepsilon^{1/2}\,\sigma(\mathfrak{n})^{1/2}\,\xi~{},$

with $\eta$ a dimensionless control parameter and $\tau_{f}$ a time scale
parameter, so that the current friction coefficient is $\eta/\tau_{f}$. In the
large friction limit, $\eta\,\mathfrak{j}\gg\tau_{f}\partial_{t}\mathfrak{j}$,
this equation reduces to eq.(1b), but it can now be quantized.

However, both approaches fail to answer questions about the nature of the
noise Q-MFT should incorporate. More importantly, there is also no a priori
guarantee that such direct quantization of the MFT appropriately describes
quantum coherent effects and their fluctuations in diffusive extended systems.
Therefore, we will advocate below another route consisting in analysing simple
microscopic models, say quantum variants of the classical exclusion processes,
and in deducing, if possible, patterns for a Q-MFT applicable to a large class
of mesoscopic diffusive systems.

###  2.3 How do stochastic effects impact quantum systems ?

The previous discussion points towards analysing noisy dynamics in Quantum
Mechanics, in particular in extended systems. There are (at least) three main,
interconnected, ways in which probabilistic phenomena hit Quantum Mechanics :

  * •

Intrinsically, as the results of measurements on quantum systems. Indeed, any
information extracted by monitoring a quantum system is random, as expressed
by the Born’s rule of Quantum Mechanics, and it impacts the quantum system by
the principle of quantum measurement back-action, even for weak measurements.
As a consequence, the evolutions of monitored quantum systems are stochastic.
They are actually described by the so-called quantum trajectories which are
instrumental in controlling quantum systems, see e.g. [58, 59].

  * •

Extrinsically, as noise, providing stochastic models for environments
interacting with quantum systems. Any system interacting with an environment
is subject to a noisy evolution. Noise, classical or quantum, may be viewed as
the collection of the degrees of freedom of the – not precisely known and
controlled – environments. As a consequence, random dynamics arise in Quantum
Mechanics as an efficient way of describing the evolution of systems
interacting with environments or external fields, as in the original model of
bosonic baths introduced by Caldeira and Leggett [60].

  * •

Generically, as models for typical, alias generic, quantum states and
operations. This is for instance the principle underlying the use of random
matrix theory (RMT) in Quantum Mechanics or Quantum Information, as first
advocated by Wigner, see e.g. [61]. More recently, this is also the way random
quantum circuits have been used to study universal behaviours of quantum
chaotic systems. By adding stochasticity, these systems ought to lose their
fine properties pertaining to particularities, such as specific symmetries,
thus allowing the emergence of generic properties.

Analysing quantum simple exclusion processes fits in the two last categories.
The latter are models of stochastic many-body dynamics incorporating diffusive
effects and their fluctuations. Although one may imagine experimental
realisations555One may look for spin chains subjected to strong noisy magnetic
fields, since Q-SSEP describe the effect dynamics of the latter in the limit
of large magnetic fields. , the purpose of these model systems are, as any toy
model, to propose paradigmatic models, which for instance allow exact results
to be derived, and to reveal patterns which are potentially generic for
mesoscopic diffusive quantum systems. Maybe optimistically, the conclusions
then obtained lead to a simple framework, depending on only a few number of
characteristics and parameters, as the MFT does, and applicable to a large
enough universality class of physical systems.

Considering the impact of monitoring on these systems is certainly important
as it has been recently done in quantum circuit models. We will however not
discuss them in this short review.

###  2.4 How can stochastic fluctuations be observed ?

When observing or measuring a quantum system, one gets random outputs whose
distribution is encoded in the system density matrix $\rho$ via Born’s rule :
the probability for an output $o$ is $p_{o}={\rm Tr}(\rho\,\Pi_{o})$, if the
measured observable is $O$ and $\Pi_{o}$ its spectral projector on the
eigenvalue $o$. A way to measure these distributions is to prepare a large
number $\mathcal{N}$ of copies of the system, say $\mathcal{N}\approx 10^{3}$,
$10^{6}$ or bigger, and to repeat the measurement experiment under identical
conditions. The output histogram is then obtained by collecting the
measurement outputs $o_{1},o_{2},\cdots,o_{\mathcal{N}}$ and counting the
number of times $\mathfrak{n}_{o}$ a given output $o$ appears in this list.
Then $\mathfrak{n}_{o}/\mathcal{N}\simeq p_{o}$ for $\mathcal{N}\gg 1$. Given
these data, one has access to the expectation of the observable $O$, or its
higher moments, via $\langle
O^{k}\rangle=\sum_{o}o^{k}\mathfrak{n}_{o}/\mathcal{N}={\rm Tr}(\rho\,O^{k})$.
This is nicely illustrated in Haroche’s photon box experiment [62].

Suppose now that the system evolves in time and that its preparation or its
evolution is noisy, so that the system density matrix is time dependent and
random. To make it explicit (when needed), we shall write $\rho_{t}(\omega)$
with $\omega$ some variables representing possible events and taking values in
some probability space. When iterating measurement processes, two different
setups can be envisioned to get an estimation of the output histograms for an
observable $O$ :
– (i) either one has some control on the noise, so that one can prepare copies
of the systems and let them evolve under identical noisy conditions. Then the
outputs $o_{1},o_{2},\cdots,o_{\mathcal{N}}$ are distributed in a way
depending on the noise sample, say $\omega$, so that their histogram is

$\displaystyle p_{o}(\omega)=\mathfrak{n}_{o}(\omega)/\mathcal{N}\simeq{\rm
Tr}(\rho_{t}(\omega)\,\Pi_{o})~{},\quad\mathrm{for}\ \mathcal{N}\gg 1~{},$ (9)

with $\mathfrak{n}_{o}(\omega)$ the numbers of time the value $o$ appear in
the output lists for all copies of the system666This may alternatively be
formulated in terms of conditioned probability as $p_{o}(\omega)$ is the
probablity to observe the value $o$ conditioned on the noise being in the
state $\omega$. under identical noisy conditions $\omega$.
– (ii) or one does not control the noise so that the latter is sampled
differently at each iteration of the measurement process. Then, if the number
of iterations $\mathcal{N}$ is large enough, and much larger than in the
previous setup, the noisy variables are sampled faithfully according to some
probability measure $p_{\omega}$ so that the output histogram is

$\displaystyle\bar{p}_{o}=\bar{\mathfrak{n}}_{o}/\mathcal{N}\simeq{\rm
Tr}(\bar{\rho}_{t}\,\Pi_{o})~{},\quad\bar{\rho}_{t}:=\sum_{\omega}p_{\omega}\rho_{t}(\omega)~{},\quad\mathcal{N}\gg
1~{}.$ (10)

Phrased differently, by not controlling the noise at the different iterations
of the measurement process, we only have access to the average density matrix,
$\bar{\rho}_{t}=\sum_{\omega}p_{\omega}\rho_{t}(\omega)$, and to the average
quantum expectation values :

$\displaystyle\mathbb{E}[\langle O\rangle_{t}]:={\rm
Tr}(\bar{\rho}_{t}\,O)=\sum_{\omega}p_{\omega}\,{\rm
Tr}(\bar{\rho}_{t}(\omega)\,O)~{}.$ (11)

Under noisy conditions, the histograms
$p_{o}(\omega)=\mathfrak{n}_{o}(\omega)/\mathcal{N}$ may fluctuate from one
noise realisation to the other. A way to quantify the fluctuations of the
quantum expectation values is to evaluate their correlations, say,

$\displaystyle\mathbb{E}[\langle O^{(1)}\rangle_{t}\langle
O^{(2)}\rangle_{t}]:=\sum_{\omega}p_{\omega}\,{\rm
Tr}(\rho_{t}(\omega)\,O^{(1)}){\rm Tr}(\rho_{t}(\omega)\,O^{(2)})~{},$ (12)

for two observables $O^{(1)}$ and $O^{(2)}$. This requires producing the two
histograms $p^{(1)}_{o_{1}}(\omega)$ and $p^{(2)}_{o_{2}}(\omega)$ by
iterating measurements of the two observables $O^{(1)}$ and $O^{(2)}$ under
identical noisy conditions777Actually, this assertion is also true in
classical physics where to estimate the correlations between two quantities
$X$ and $Y$ one has to sample their possible values $x(\omega)$ and
$y(\omega)$ under identical noisy conditions in order to estimate their
correlations $\mathbb{E}[XY]=\sum_{\omega}p_{\omega}x(\omega)y(\omega)$.
However, the consequences of these samplings in quantum mechanics is more
dramatic due to the back-action of the measurement process on the system..

Although controlling the noise may be difficult experimentally and even looks
paradoxical – since, more or less by definition, noise is a name for variables
whose behaviors cannot be controlled – this is not completely unimaginable.
For instance, one may imagine placing copies of the system under the umbrella
of the same noise, say if the randomness arises from an external random
magnetic field888A field frequency spectrum containing a few incommensurable
frequencies would be generate a field whose time evolution is closed enough to
a good realisation of a white-noise.. Or, if the noise is artificially
produced, say by random number generators, one may simply ensure that the
generators are implemented using identical seeds. Or, if the fluctuations
arise from variances in the initial state, one may control this initial state,
etc.

Nevertheless, as for classical systems, having a realistic, generic,
controlled models for the noise and the fluctuations it induces, allows to
estimate the fluctuations due to noisy environments that are not controlled.

Taking the time evolution of the system into account, one may wonder how these
fluctuations evolve in time and whether they survive the large time limit
and/or the large size limit, especially for ergodic systems. A commonly used
formulation of ergodicity, especially in the physics literature on many-body
physics, is to assume that the time averaged system density matrix converges
at large time to some steady density matrix $\rho_{\infty}$ (supposed to be
unique for simplicity)999This notion of ergodicity is not that used in the
mathematical literature, say on quantum billards, which is grounded on semi-
classical structures.:

$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\hskip-5.69054ptds\,{\rm
Tr}(\rho_{s}\,O)={\rm Tr}(\rho_{\infty}\,O)~{}.$ (13)

for some appropriate set of observables $O$. If the system is noisy, one
hypothesises self averaging and extend the above definition to an almost sure
convergence. Defining ergodicity via eq.(13) leads to wonder (say, because of
the measurement back-action) how the time integral $\int_{0}^{T}\\!ds\,{\rm
Tr}(\rho_{s}\,O)$ may be estimated experimentally. However, in the context of
monitored systems and their quantum trajectories [58, 59], such almost sure
convergence is known to hold under quite general hypothesis [63, 64].

At a classical level, if $x\to x_{t}$ is a dynamical flow on some space
$\mathfrak{X}$, ergodocity w.r.t. to some measure $d\nu_{\infty}$ on
$\mathfrak{X}$ is the property that
$\lim_{T\to\infty}T^{-1}\int_{0}^{T}\hskip-2.84526ptds\,f(x_{s})=\int_{\mathfrak{X}}d\nu_{\infty}(x)\,f(x)$,
for appropriate functions $f$. At the quantum level, there are two options :
(i) either one views quantum observables as quantization of functions on some
phase space and density matrices as quantization of measures on that space,
then eq.(13) is a natural analogue of the classical ergodicity (as long as
measurements are not taken into account, see Appendix A); (ii) or one
considers the density matrices $\rho$ as the dynamical variables, then one
would like to apply ergodicity to non-linear functions of $\rho$.

Considering density matrices as the dynamical variable, eq.(13) should better
be called linear ergodicity as it only tests linear functions of the system
density matrix. More generally, to have a hand on fluctuations, one may
consider more general functions and ask that the time averages of say all
polynomials in the density matrix (up to a given degree) converge, that is :

$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\hskip-5.69054ptds\,{\rm
Tr}(\rho_{s}\,O^{(1)})\cdots{\rm
Tr}(\rho_{s}\,O^{(p)})~{}\quad\mathrm{exist}.$ (14)

Non-linear ergodicity would hold if there exists a steady measure
$\mathbb{E}_{\infty}$ on density matrices such that these large time limits
are the moments of the density matrix w.r.t. this measure, that is:

$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\hskip-5.69054ptds\,{\rm
Tr}(\rho_{s}\,O^{(1)})\cdots{\rm
Tr}(\rho_{s}\,O^{(p)})=\mathbb{E}_{\infty}[{\rm Tr}(\rho\,O^{(1)})\cdots{\rm
Tr}(\rho\,O^{(p)})]~{},$ (15)

If the system evolution is noisy, one may require that this property holds
almost surely. Of course, we can also extend this definition by considering
multi-time expectations.

Checking ergodicity experimentally, even at the linear level as in eq.(13), by
producing the appropriate measurement histograms requires double series of
copies $\mathcal{M}\times\mathcal{N}$ of the system: $\mathcal{N}$ copies to
produce the measurement histograms at each given instant, repeated
$\mathcal{M}$ times to sample the time interval from $0$ to $T$. The
difficulties are comparable to those of the direct approach to evaluate the
noisy fluctuations (12). The situation is better if one imagines monitoring
the quantum system. See the Appendix A.

For extended systems, assuming that a steady regime is attained at large time,
we expect the measure $\mathbb{E}_{\infty}$ to be peaked around a mean density
matrix $\bar{\rho}_{\infty}$, with fluctuations $\delta\rho_{t}$ decreasing
with the system size :

$\displaystyle\rho_{t}\simeq\bar{\rho}_{\infty}+\delta\rho_{t}~{},\quad\mathrm{at\
large\ enough\ time},$ (16)

or more precisely, ${\rm Tr}(\rho_{t}\,O)\simeq
Tr(\bar{\rho}_{\infty}\,O)+Tr(\delta\rho_{t}\,O)$, for appropriate observables
$O$. These fluctuations are expected to be small and to scale as
$1/L^{\alpha}$, with $L$ the macroscopic linear size of the system and
$\alpha>0$. The law of large number suggests $\alpha=\frac{1}{2}$. As a
consequence, we expect that time averaging is unnecessary so that the density
matrix, tested appropriately, converges at large time,

$\displaystyle\lim_{t\to\infty}{\rm Tr}(\rho_{t}\,O)={\rm
Tr}(\bar{\rho}_{\infty}\,O)~{},\quad\mathrm{in\ infinite\ volume},$ (17)

for an appropriate set of observables, say local observables. This is a
stronger convergence than that of linear ergodicity (13). This is the expected
behaviour when thermalisation holds but also in some non-equilibrium
situations.

There are fluctuations, in time and from sample to sample, when approaching
the asymptotic state $\bar{\rho}_{\infty}$. See Figure 2. At large time, but
at finite (large) volume, they are expected to reach a steady distribution
coding for correlations between quantum expectation values (at large time),
say

$\mathbb{E}_{\infty}[{\rm Tr}(\rho\,O^{(1)})\cdots{\rm
Tr}(\rho\,O^{(r)})]~{}.$

Deviation from ${\rm Tr}(\bar{\rho}_{\infty}\,O^{(1)})\cdots{\rm
Tr}(\bar{\rho}_{\infty}\,O^{(r)})$ are sub-leading in the system size. At
finite but large enough volume, we may expect that their evolution acquires a
universal status, so that they can be described by some effective, universal,
noisy dynamics. Having good noisy dynamical models then provides a way to have
a hand on those fluctuations and their time evolutions.

Figure 2: Schematic representation of the convergence of quantum expectation
values to their equilibrium values with sub-leading fluctuations $\Delta
O_{t}:={\rm Tr}(O\delta\rho_{t})$ decreasing with the system size.

A natural question is then how could we estimate these fluctuations by looking
at a single time trajectory of the system state. A way to proceed could
consist in slicing the time axis into pieces of time duration of order $\tau$,
with $\tau$ an effective correlation time such that the average over time
scale of order $\tau$ smoothes out the fluctuations, i.e.
$\bar{\rho}_{t}:=\frac{1}{\tau}\int_{t}^{t+\tau}ds\,\rho_{s}$ has a smooth
time evolution. This smoothed mean state $\bar{\rho}_{t}$ at time $t$ can be
evaluated by sampling the time interval $[t,t+\tau]$ at a large number of
times $t_{1},\,t_{2},\cdots,t_{\mathcal{P}}$, that is: ${\rm
Tr}(\bar{\rho}_{t}\,O)=\frac{1}{\mathcal{P}}\sum_{j=1}^{\mathcal{P}}{\rm
Tr}(\rho_{t_{j}}\,O)$. This sampling yields to an estimation of the
fluctuations at time $t$ via

$\frac{1}{\mathcal{P}}\sum_{j=1}^{\mathcal{P}}{\rm
Tr}(\rho_{t_{j}}\,O^{(1)})\cdots{\rm Tr}(\rho_{t_{j}}\,O^{(r)})~{},$

with all $t_{j}\in[t,t+\tau]$ and $\mathcal{P}$ large enough. We could then
declare that the above mentioned effective noisy dynamics is faithful if its
moments coincide with these correlations, that is :

$\mathbb{E}[{\rm Tr}(\rho\,O^{(1)}_{t})\cdots{\rm
Tr}(\rho\,O^{(r)}_{t})]\stackrel{{\scriptstyle?}}{{=}}\frac{1}{\mathcal{P}}\sum_{j=1}^{\mathcal{P}}{\rm
Tr}(\rho_{t_{j}}\,O^{(1)})\cdots{\rm Tr}(\rho_{t_{j}}\,O^{(r)})~{},$

At the linear level, for $t$ large enough, this reproduces the convergence
condition (17) with $\bar{\rho}_{\infty}$ the mean steady state of the
effective dynamics.

##  3 Quantum Simple Exclusion Processes

The quantum simple exclusion processes are models of quantum many-body noisy
dynamics describing fermions hopping stochastically along a one dimensional
chain101010Quantum simple exclusion processes can be defined on any graph..
The chain can be closed with periodic boundary conditions, or open with
injection and extraction of particles at its two ends. The processes can be
symmetric (resp. asymmetric) depending whether the amplitudes to move to the
right or to the left are equal (resp. different). The exclusion constraint is
implemented by the fermionic character of the degrees of freedom. They are
quantum extensions of the classical simple exclusion processes.

###  3.1 Q-SSEP and Q-ASEP

$\bullet$ Closed Q-SSEP.
In the closed symmetric setup, the system dynamics is unitary but noisy. The
system density matrix $\rho_{t}$ evolves unitarily,
$\rho_{t+dt}=e^{-idH_{t}}\rho_{t}\,e^{idH_{t}}$, with $dH_{t}$ the hamiltonian
increment between time $t$ and $t+dt$. For Q-SSEP, the hamiltonian increment
is defined as111111This definition can be generalized by including $cc$ or
$c^{\dagger}c^{\dagger}$ terms not preserving the particle number but still
quadratic. However, transport is associated to locally conserved quantities
and such models might not be well adapted to study transport.

$dH_{t}:=\sqrt{J}\,\sum_{j=1}^{N}\big{(}c^{\dagger}_{j+1}c_{j}\,dW_{t}^{j}+c^{\dagger}_{j}c_{j+1}\,d\overline{W}_{t}^{j}\big{)}~{},$
(18)

where $c_{j}$ and $c_{j}^{\dagger}$ are canonical fermionic operators, one
pair for each site of the chain, with
$\\{c_{j},c_{k}^{\dagger}\\}=\delta_{j;k}$, and $W_{t}^{j}$ and
$\overline{W}_{t}^{j}$ are pairs of complex conjugated Brownian motions, one
pair for each edge along the chain, with zero mean,
$\mathbb{E}[dW^{j}_{t}]=\mathbb{E}[d\overline{W}^{j}_{t}]=0$, and covariance
$\mathbb{E}[dW_{t}^{j}d\overline{W}_{t}^{k}]=\delta^{j;k}\,dt$. Here, $N$ is
the number of sites on the chain and $J$ a bare coupling constant, with the
dimension of a frequency. For the closed Q-SSEP, periodic boundary conditions
are assumed. See Figure 3.

Figure 3: Schematic representation of Q-SSEP or Q-ASEP. Periodic boundary
conditions are imposed in the closed case (no reservoirs). In the open case,
injection/extraction processes at the two ends represent contact with two
reservoirs, here depicted in red or blue.

Upon discretizing the time evolution, this model can be viewed as a random
quantum circuit with Gaussian two-site gates. In this discrete setting, after
$n$ time steps, each of duration $\delta t$, the system density matrix
$\rho_{n}$ is unitary up-dated according to
$\rho_{n}\to\rho_{n+1}=U_{n}\rho_{n}U_{n}^{\dagger}$ with
$U_{n}=V^{1}_{n}V^{0}_{n}$ where the $V_{n}^{e}$’s (with $e=0,1$ for even and
odd) are products of gates acting on adjacent sites (for $N=2M$) :

$\displaystyle V_{n}^{e}$ $\displaystyle=$ $\displaystyle
W^{(n)}_{e+1,e+2}W^{(n)}_{e+3,e+4}\cdots W^{(n)}_{e+N-1,e+N}~{},$ (19)
$\displaystyle W^{(n)}_{j,j+1}$ $\displaystyle=$
$\displaystyle\exp\big{(}{c^{\dagger}_{j+1}c_{j}\,\xi^{(n)}_{j}+c^{\dagger}_{j}c_{j+1}\,\bar{\xi}^{(n)}_{j}}\big{)}~{},$

with $\xi^{(n)}_{j}$ independent identically distributed (i.i.d.) Gaussian
variables with zero mean and variance
$\mathbb{E}[\xi^{(n)}_{j}\bar{\xi}^{(m)}_{k}]=\delta^{n;m}\delta_{j;k}\,J\delta
t$. This discretization yields to simple numerical implementations of the
dynamics. In the following we shall however stick to the time continuous
formulation (which is better adapted to analytical descriptions).

Since $\rho_{t+dt}=e^{-idH_{t}}\rho_{t}\,e^{idH_{t}}$, the equation of motion
for the system density matrix reads (with Itô convention) :

$d\rho_{t}=-i[dH_{t},\rho_{t}]-\frac{1}{2}[dH_{t},[dH_{t},\rho_{t}]]~{}.$ (20)

These equations are classical stochastic differential equations (SDE) but on
quantum density matrices. The double commutator in eq.(20) is an echo of the
fact that the Brownian increments $dW_{t}^{j}$ scale as $\sqrt{dt}$. The
equation of motion for the observables, say $O$, are defined from eq.(20) by
duality: ${\rm Tr}(\rho_{t}\,O)={\rm Tr}(\rho\,O_{t})$. In particular, the
equation of motion for the local particle number operators,
$\hat{n}_{j}:=c_{j}^{\dagger}c_{j}$, reads

$d\hat{n}_{j}=J\,\Delta\hat{n}_{j}\,dt+\sqrt{J}\big{(}d\hat{\mathbb{V}}_{s}^{j}-d\hat{\mathbb{V}}_{s}^{j-1}\big{)}~{},$
(21)

with $\Delta$ the discrete Laplacian,
$\Delta\hat{n}_{j}:=\hat{n}_{j+1}-2\hat{n}_{j}+\hat{n}_{j-1}$, and
$d\hat{\mathbb{V}}_{t}^{j}:=i\big{(}c_{j}^{\dagger}c_{j+1}dW^{j}-\mathrm{h.c.}\big{)}$
noisy operators. Eq.(21) indeed codes for stochastic diffusion for quantum
observables. The Q-SSEP is one of the simplest model of pure, quantum, noisy,
diffusive dynamics.

Since information is lost when averaging over the realizations of the noise,
the average dynamics is not unitary but dissipative. In particular, the
average number operator $\mathbb{E}[\hat{n}_{j}]$ satisfy to pure diffusion:
$\partial_{t}\mathbb{E}[\hat{n}_{j}]=J\,\mathbb{E}[\Delta\hat{n}_{j}]$. By the
Markov property of the Brownian motions, the average dynamics defines a semi-
group generated by a Lindbladian which is obtained by averaging eq.(20) :

$\partial_{t}\bar{\rho}_{t}=\mathcal{L}_{\mathrm{ssep}}(\bar{\rho}_{t})~{},$
(22)

with $\bar{\rho}_{t}:=\mathbb{E}[\rho_{t}]$ the average density matrix. The
Lindbladian $\mathcal{L}_{\mathrm{ssep}}$ is the sum of local Lindbladians,
one for each edge of the chain,
$\mathcal{L}_{\mathrm{ssep}}=J\,\sum_{j=1}^{N}{\mathcal{L}}_{j;j+1}$. Each of
them decomposes as
$\mathcal{L}_{j;j+1}=\overrightarrow{\mathcal{L}}_{j;j+1}+\overleftarrow{\mathcal{L}}_{j;j+1}$,
where

$\displaystyle\overrightarrow{\mathcal{L}}_{j;j+1}(\rho)$
$\displaystyle:=\ell^{-}_{j}\,\rho\,\ell^{+}_{j}-\frac{1}{2}(\ell^{+}_{j}\ell^{-}_{j}\,\rho+\rho\,\ell^{+}_{j}\ell^{-}_{j})~{},$
(23a) $\displaystyle\overleftarrow{\mathcal{L}}_{j;j+1}(\rho)$
$\displaystyle:=\ell^{+}_{j}\,\rho\,\ell^{-}_{j}-\frac{1}{2}(\ell^{-}_{j}\ell^{+}_{j}\,\rho+\rho\,\ell^{-}_{j}\ell^{+}_{j})~{},$
(23b)

with $\ell^{+}_{j}:=c^{\dagger}_{j+1}c_{j}$ and
$\ell^{-}_{j}:=c^{\dagger}_{j}c_{j+1}$. The local Lindbladians
$\overrightarrow{\mathcal{L}}_{j;j+1}$ (resp.
$\overleftarrow{\mathcal{L}}_{j;j+1}$) describe jumps of the fermions to the
right (resp. left). Higher moments of the density matrix, say its quadratic
fluctuations $\mathbb{E}[\rho_{t}\otimes\rho_{t}]$, also satisfy Lindblad
dynamics [65]. One of the motivation for studying Q-SSEP is to get a good
description of quantum diffusive dynamics, beyond the average dynamics,
including fluctuations. See below section 3.2.

The name Q-SSEP is inherited from the fact that the mean dynamics, when
reduced to an appropriate sector, is that of the classical SSEP. At each site
along the chain, the empty $|\bullet\rangle$ and full $|\text{\o}\rangle$
states, with respectively one and zero fermion, form a basis of states and
diagonalize the particule number operator (with eigen-value $1$ or $0$). The
states $|\bm{n}\rangle$ diagonalising all the particle numbers along the chain
are thus indexed by the classical configurations
$\bm{n}=(\nu_{1},\cdots,\nu_{N})$, with $\nu_{j}=0,1,$ the particle number at
site $j$. A density matrix diagonal in this particle number basis specifies a
probability measure on classical configurations since it can be written as
$\rho_{\mathrm{diag}}=\sum_{\bm{n}}Q_{\bm{n}}\,\Pi_{\bm{n}}$, with
$\Pi_{\bm{n}}:=|\bm{n}\rangle\langle\bm{n}|$ and $Q_{\bm{n}}$ a probability
measure on $\bm{n}$ : $\sum_{\bm{n}}Q_{\bm{n}}=1$, $Q_{\bm{n}}\geq 0$. The
mean Q-SSEP dynamics (22) preserves diagonal density matrices and thus defines
a flow – a Markov chain – on probability measures on classical configurations.
This flow is identical to that of the classical SSEP since

$\displaystyle\mathcal{L}_{\mathrm{ssep}}(|\text{\o}\text{\o}\rangle\langle\text{\o}\text{\o}|)$
$\displaystyle=0~{},$
$\displaystyle\mathcal{L}_{\mathrm{ssep}}(|\text{\o}\bullet\rangle\langle\text{\o}\\!\bullet\\!|)$
$\displaystyle=J(-|\text{\o}\bullet\rangle\langle\text{\o}\\!\bullet\\!|+|\\!\bullet\\!\text{\o}\rangle\langle\\!\bullet\text{\o}|)~{},$
$\displaystyle\mathcal{L}_{\mathrm{ssep}}(|\\!\bullet\\!\text{\o}\rangle\langle\bullet\text{\o}|)$
$\displaystyle=J(+|\text{\o}\bullet\rangle\langle\text{\o}\\!\bullet\\!|-|\\!\bullet\\!\text{\o}\rangle\langle\\!\bullet\text{\o}|)~{},$
$\displaystyle\mathcal{L}_{\mathrm{ssep}}(|\\!\bullet\\!\bullet\rangle\langle\\!\bullet\\!\bullet\\!|)$
$\displaystyle=0~{}.$

This can be compared with the Markov transition matrix of classical SSEP in
eqs.(2.1). As a consequence, the generating function of the steady
fluctuations of the classical SSEP occupancies $\mathfrak{n}_{j}$ can be
expressed as a quantum expectation value w.r.t. the steady average Q-SSEP
density matrix :

$\displaystyle\mathbb{E}_{\mathrm{ssep}}[e^{\sum_{j}h_{j}\mathfrak{n}_{j}}]=\mathbb{E}\Big{[}\mathrm{Tr}\big{(}\rho\,e^{\sum_{i}h_{i}\hat{n}_{i}}\big{)}\Big{]}=\mathrm{Tr}\big{(}\bar{\rho}\,e^{\sum_{i}h_{i}\hat{n}_{i}}\big{)}~{},$

with $\hat{n}_{i}=c^{\dagger}_{i}c_{i}$ the quantum number operators,
$\bar{\rho}=\mathbb{E}[\rho]$ the mean Q-SSEP state, averaged w.r.t. the
Q-SSEP steady measure.

The quantum Q-SSEP is however an extension of the classical SSEP as this
correspondance only applies to the average dynamics of diagonal density
matrices. For instance, while only classical configurations are considered in
classical SSEP, cats like states made of superpositions of classical
configurations are allowed in the Q-SSEP, say

$|\cdots\text{\o}\text{\o}\\!\bullet\\!\bullet\text{\o}\\!\bullet\\!\text{\o}\text{\o}\\!\bullet\cdots\rangle+|\cdots\bullet\\!\text{\o}\\!\bullet\\!\text{\o}\text{\o}\\!\bullet\\!\text{\o}\\!\bullet\\!\bullet\cdots\rangle+\cdots~{},$

even-though the states describe identical density profiles at a coarse-grained
level. These states could interfere and Q-SSEP describes their statistics and
dynamics.

$\bullet$ Open Q-SSEP.
The open Q-SSEP is defined similarly except that the chain interval is now
open (no periodic boundary conditions) and the dynamics is modified by
including injection and extraction processes at the two ends of the chain.
These processes are represented by dissipative, deterministic, Lindblad
dynamics. As a consequence, the equations of motion for the system density
matrix read :

$d\rho_{t}=-i[dH_{t},\rho_{t}]-\frac{1}{2}[dH_{t},[dH_{t},\rho_{t}]]+\mathcal{L}_{\mathrm{bdry}}(\rho_{t})dt~{},$
(24)

with $dH_{t}$ as above in eq.(18), and $\mathcal{L}_{\mathrm{bdry}}$ a
boundary Lindbladian acting on sites $1$ and $N$, with
$\mathcal{L}_{\mathrm{bdry}}=\alpha_{1}{\cal L}_{1}^{+}+\beta_{1}{\cal
L}_{1}^{-}+\alpha_{N}{\cal L}_{N}^{+}+\beta_{N}{\cal L}_{N}^{-}$ and

$\displaystyle{\cal L}_{j}^{+}(\star)$ $\displaystyle=c_{j}^{\dagger}\star
c_{j}-\frac{1}{2}(c_{j}c_{j}^{+}\star+\star c_{j}c_{j}^{\dagger})~{},$ (25a)
$\displaystyle{\cal L}_{j}^{-}(\star)$ $\displaystyle=c_{j}\star
c_{j}^{\dagger}-\frac{1}{2}(c_{j}^{\dagger}c_{j}\star+\star
c_{j}^{\dagger}c_{j})~{},$ (25b)

where the parameters $\alpha_{j}$ (resp. $\beta_{j}$) are the injection (resp.
extraction) rates. These boundary processes drive the system out-of-
equilibrium by enforcing a non-trivial current through the system.

General comments about Q-SSEP apply to the open case, if the boundary terms
are taken into account appropriately. In particular, the average dynamics is
also of Lindblad type, with contributions from the boundary Linbladians. On
diagonal density matrices, it reduces to that of the open SSEP, including the
boundary terms. Higher moments also follow a Lindblad dynamics [65]. Despite
the presence of boundary dynamics, the open Q-SSEP preserves gaussianity of
the states and it is thus particularly adapted to exact studies. Duality
property of such model systems were pointed out in [42].

$\bullet$ Closed and open Q-ASEP.
The Q-ASEP is slightly different from the Q-SSEP because the noise the chain
is coupled to is not classical but quantum. The hamiltonian increments
$dH_{t}$ are of the same form as in eq.(18), namely,

$dH_{t}:=\sqrt{J}\,\sum_{j=1}^{N}\big{(}c^{\dagger}_{j+1}c_{j}\,dW_{t}^{j}+c^{\dagger}_{j}c_{j+1}\,d\overline{W}_{t}^{j}\big{)}~{},$
(26)

but where $W_{t}^{j}$ and $\overline{W}_{t}^{j}$ are now quantum Gaussian
processes with zero mean and covariance,

$\mathbb{E}[dW^{j}_{t}\,d\overline{W}^{k}_{t}]=p\,\delta^{j;k}\,dt~{},\quad\mathbb{E}[d\overline{W}^{j}_{t}\,dW^{k}_{t}]=q\,\delta^{j;k}\,dt~{},$
(27)

with $p,\,q$, ($p\geq q$), two real positive parameters. The Q-ASEP correspond
to $p\not=q$, and the Q-TASEP to either $p=0$ or $q=0$. The two processes
$W^{j}_{t}$ and $\overline{W}^{k}_{t}$ are then non-commuting quantum
operators,

$[dW^{j}_{t},d\overline{W}^{k}_{t}]=(p-q)\,\delta^{j;k}\,dt~{}.$ (28)

There are two physical ways to construct this noise. Either one may view them
as the scaling limit of series of two-state probes, say Q-bit or spin half
probes, recursively interacting with the system [66], in a way similar to the
photon box experiment [62]. This construction is parallel to that of the
classical Brownian motion as a scaling limit of random walks. Or, these noise
can be directly constructed in the continuum, as in the theory of quantum
stochastic differential equations [67]. In physical terms, the noise Hilbert
space is the tensor product of noise Hilbert spaces attached to each edge of
the chain,
$\mathcal{H}_{\mathrm{noise}}=\otimes_{j}\mathcal{H}^{(j)}_{\mathrm{noise}}$.
Each of them is a Fock space, over the space of square integrable functions on
the line, on which complex bosonic fields, $\varphi^{j}(s)$ and
$\bar{\varphi}^{j}(s)$, are acting, with canonical commutation relations,

$\big{[}\varphi^{j}(s),\bar{\varphi}^{k}(s^{\prime})\big{]}=\delta^{j;k}\delta(s-s^{\prime})~{}.$
(29)

The noise expectation values are defined as quantum expectation values w.r.t.
a particular state $\Omega_{\mathrm{noise}}$ on the noise Hilbert space.
Averaging over the noise amounts to trace over the noise degrees of freedom,
$\mathbb{E}[(\cdots)]:={\rm Tr}(\Omega_{\mathrm{noise}}(\cdots)]$. The noise
is chosen to be thermal so that
$\mathbb{E}[\bar{\varphi}^{j}(s)\varphi^{k}(s^{\prime})]=\mathsf{n}\,\delta^{j;k}\delta(s-s^{\prime})$
with $\mathsf{n}$ some filling factor. The quantum Brownian motions are then
defined by

$W^{j}_{t}=\lambda\\!\int_{0}^{t}\\!\\!ds\,\varphi^{j}(s)~{},\quad\overline{W}^{j}_{t}=\lambda\\!\int_{0}^{t}\\!\\!ds\,\bar{\varphi}^{j}(s)~{}.$
(30)

Their increments are $dW^{j}_{t}=W^{j}_{t+dt}-W^{j}_{t}$. They satisfy
eqs.(27) with $q=\lambda^{2}\mathsf{n}$ and $p=\lambda^{2}(1+\mathsf{n})$.
Noise expectation values are computed using Wick’s theorem. The Brownian
increments $dW^{j}_{t}$ and the portion of the noise Fock space associated to
the fields $\varphi^{j}(s)$ between time $t$ and $t+dt$ represent the scaling
limit of all probes which have interacted with the system during the time
interval $[t,t+dt)$.

The noisy dynamics of observables, say $O$, generated by the hamiltonian
increments (26) are

$dO_{t}=+i[dH_{t},O]_{t}-\frac{1}{2}[dH_{t},[dH_{t},O]]_{t}~{},$ (31)

up to boundary terms in the case of an open chain. These are quantum
stochastic differential equations as defined in [67]. For $O$ an observable on
the system, its average w.r.t. the noise, $\bar{O}_{t}:=\mathbb{E}[O_{t}]$,
satisfy a Lindblad equation

$\partial_{t}\bar{O}_{t}=\mathcal{L}_{\mathrm{asep}}^{\dagger}(\bar{O})_{t}~{},$
(32)

with dual Lindbladian
$\mathcal{L}_{\mathrm{asep}}^{\dagger}=J\,\sum_{j=1}^{N}\big{(}p\overrightarrow{\mathcal{L}}^{{\dagger}}_{j;j+1}+q\overleftarrow{\mathcal{L}}^{{\dagger}}_{j;j+1}\big{)}$,
with $\overrightarrow{\mathcal{L}}^{{\dagger}}_{j;j+1}$ and
$\overleftarrow{\mathcal{L}}^{{\dagger}}_{j;j+1}$ the duals of the
Lindbladians (23). This average dynamics was actually considered in [68, 69].
Higher moments also satisfy a Lindblad dynamics [65].

Since the two Lindbladians $\overrightarrow{\mathcal{L}}_{j;j+1}$ and
$\overleftarrow{\mathcal{L}}_{j;j+1}$ code respectively for left and right
moves, the total Lindbladian $\mathcal{L}_{\mathrm{asep}}$ codes for
asymmetric moves as in the classical ASEP. The exclusion principle is taken
into account by the fermionic character of the particles. Restricted to
diagonal density matrices, the average of Q-ASEP is that of the classical ASEP
[45]. Q-ASEP is however an extension of the classical ASEP as it describes
off-diagonal fluctuations and coherences.

In the following we restrict our discussion to the closed and open Q-SSEP.

###  3.2 Closed Q-SSEP : Fluctuations at equilibrium

Q-SSEP is particularly adapted to exact analytic results because it is a
quadratic model121212But it is a non-trivial model because of the presence of
noise., for each realization of the noise. In particular it preserves
gaussianity of the system density matrix, again for each realization of the
noise (not in average). The dynamics and statistics of such density matrices
are encoded into that of their matrices of two-point functions $G_{ij}:={\rm
Tr}(\rho_{t}c_{j}^{\dagger}c_{i})$. Hence, analysing Q-SSEP reduces to
studying the statistics and dynamics of its two-point functions. This reduces
the number of degrees of freedoms from $2^{N}$ to $N^{2}$. For the closed
Q-SSEP, the time evolution of the matrix $G$ is also unitary,

$G_{t+dt}=e^{-idh_{t}}\ G_{t}\,e^{idh_{t}}~{},$ (33)

with $dh_{t}$ the so-called one-particle hamiltonian,
$dh_{t}=\sqrt{J}\,\sum_{j=1}^{N}(|j+1\rangle\langle
j|\,dW_{t}^{j}+\mathrm{h.c.}\big{)}$, or equivalently, with $G_{i;j}=\langle
i|G_{t}|j\rangle$ the matrix elements of the matrix $G_{t}$ (we suppress the
index $t$ to lighten the notation),

$\displaystyle dG_{i;j}$
$\displaystyle=-2J\,G_{i;j}dt+J(G_{i+1;i+1}+G_{i-1;i-1})\delta_{i;j}dt$ (34)
$\displaystyle~{}~{}~{}+i\sqrt{J}\big{(}G_{i;j-1}d\overline{W}_{t}^{j-1}+G_{i;j+1}dW_{t}^{j}-G_{i-1;j}dW_{t}^{i-1}-G_{i+1;j}d\overline{W}_{t}^{i}\big{)}~{},$

This is a matrix valued stochastic differential equation (SDE). The matrix $G$
satisfies a closed set of equations. In particular, the mean density
$\mathfrak{n}_{j}:=\mathbb{E}[G_{j;j}]$ evolves diffusively,
$\partial_{t}\mathfrak{n}_{j}=J\,\Delta\mathfrak{n}_{j}$ with $\Delta$ the
discrete Laplacian.

At large time, the matrix $G$ reaches a steady distribution, which we shall
denote $\mathbb{E}_{\infty}[\cdot]$. Since it is unitary, the dynamics (33) is
isospectral, and all quantities $N_{k}:={\rm Tr}(G^{k})$ are constants of
motion fixed by the initial condition $G_{0}$. We set
$N_{k}=\mathfrak{m}_{k}N$. The steady measure depends on this data. To compute
the moments of $G$ in the steady measure is a linear problem specified by
eq.(34). Let us compute the first few.

For the mean, we find

$\mathbb{E}_{\infty}[G_{i;j}]=\mathfrak{m_{1}}\,\delta_{i;j}~{}.$ (35)

This reflects :
(a) decoherence as the off-diagonal elements vanish in average at large time,
(b) equilibrium as the mean density profile is uniform along the chain.
(Recall that $G_{i;i}$ is the quantum expectation value of the particle number
at site $i$).

For the quadratic cumulants, we find (with $i\not=j$) [43],

$\displaystyle\mathbb{E}_{\infty}[|G_{i;i}|^{2}]^{c}=\frac{(\Delta_{2}\mathfrak{m})^{2}}{N+1}~{},\quad\mathbb{E}_{\infty}[G_{i;i}G_{j;j}]^{c}=-\frac{(\Delta_{2}\mathfrak{m})^{2}}{N^{2}-1}~{},$
(36a)
$\displaystyle\mathbb{E}_{\infty}[|G_{i;j}|^{2}]^{c}=\frac{N(\Delta_{2}\mathfrak{m})^{2}}{N^{2}-1}\simeq\frac{(\Delta_{2}\mathfrak{m})^{2}}{N}~{},$
(36b)

with $(\Delta_{2}\mathfrak{m})^{2}:=\mathfrak{m}_{2}-\mathfrak{m}_{1}^{2}$. We
thus learn that fluctuations of the density at coincident points are of order
$1/\sqrt{N}$, as expected for diffusive systems, but at order $1/N$ at non
coincident points. More importantly, we observe that, although vanishing in
average, coherences, represented by the off-diagonal elements of $G$, are of
order $1/\sqrt{N}$ (provided the $\mathfrak{m}_{k}$ scale as $O(N^{0})$). That
is : decoherence is at work in average but there are non-zero fluctuating
coherences, sub-leading with the system size.

Suppose that the system is initially prepared in a factorized state, so that
its initial two point functions are diagonal, ${\rm
Tr}(\rho_{0}c^{\dagger}_{i}c_{j})=\mathfrak{n}_{i}\,\delta_{i;j}$, with
$\mathfrak{n}_{i}$ the initial local particle density at site $i$. Then,
$\mathfrak{m}_{1}=\sum_{i}\mathfrak{n}_{i}/N$ is the mean density, and
$\mathfrak{m}_{2}=\sum_{i}\mathfrak{n}_{i}^{2}/N$, so that
$(\Delta_{2}\mathfrak{m})^{2}$ is the variance of the initial density131313If
the initial state is not factorized, there are possible corrections to
$\mathfrak{m}_{2}$ due to initial coherences.. Eqs.(36) tell us that, despite
the absence of initial coherences, these are progressively, randomly,
generated by the hopping process (18) and they survive the long time limit if
the initial density is inhomogeneous. Phrased differently : fluctuating
coherences are dynamically produced from the inhomogeneities of the density
profile.

Higher cumulants can be recursively computed. Assuming that the initial state
is such that all $\mathfrak{m}_{k}={\rm Tr}(G_{0}^{k})/N$ scale as $O(N^{0})$
for large $N$, all cumulants of order $P$ scale as $1/N^{P-1}$. Let
$W[A]:=\log Z[A]$ with $Z[A]:=\mathbb{E}_{\infty}\big{[}e^{{\rm
Tr}(AG)}\big{]}$, with $A$ a generic test matrix, be their generating
function. We have, order by order in power of $A$ (recall that
$(\Delta_{2}\mathfrak{m})^{2}=\mathfrak{m}_{2}-\mathfrak{m}_{1}^{2}$) :

$W(A)=\mathfrak{m}_{1}\,{\rm
Tr}A+\frac{1}{2}\frac{N(\Delta_{2}\mathfrak{m})^{2}}{N^{2}-1}\,{\rm
Tr}A^{2}-\frac{1}{2}\frac{(\Delta_{2}\mathfrak{m})^{2}}{N^{2}-1}\,({\rm
Tr}A)^{2}+O(|\\!|A|\\!|^{3})~{}.$ (37)

This formula has a simple interpretation : To leading order in the system
size, the matrix of two point functions converges to the non random, uniform,
matrix $G_{\mathrm{eq}}=\mathfrak{m}_{1}\mathbb{I}$, proportional to the
identity, reflecting convergence toward equilibrium. There are sub-leading
fluctuations, so that we can write,

$G\simeq G_{\mathrm{eq}}+\delta G~{},\quad
G_{\mathrm{eq}}=\mathfrak{m}_{1}\mathbb{I}~{},\quad\delta G\simeq
O(1/\sqrt{N})~{},$ (38)

where the fluctuations $\delta G$ decrease with the system size.
Alternatively, the system density matrix converges to the equilibrium Gibbs
state, up to fluctuations,

$\rho\simeq\rho_{\mathrm{eq}}+\delta\rho~{},\quad\rho_{\mathrm{eq}}=\frac{1}{Z_{\mathrm{eq}}}e^{-\mu\,\hat{N}_{\mathrm{tot}}}~{},\quad\delta\rho\simeq
O(1/\sqrt{N})~{},$ (39)

with $\hat{N}_{\mathrm{tot}}=\sum_{j}c^{\dagger}_{j}c_{j}$ the total particle
number, $\mu$ the chemical potential, related to the density via
$\mathfrak{m}_{1}=1/(1+e^{\mu})$ and $Z_{\mathrm{eq}}$ the equilibrium
partition function, $Z_{\mathrm{eq}}=(1+e^{-\mu})^{N}$. Recall that the number
of particles is the only conserved quantity under the dynamics (18) (for any
fixed realization of the noise).

Hence, for large system sizes, the steady measure $\mathbb{E}_{\infty}$ is
peaked around the equilibrium density matrix $\rho_{\mathrm{eq}}$, with
fluctuations $\delta\rho$ sub-leading in the system size. It is interesting to
compare with eq.(16). Q-SSEP gives access to these rare sub-leading
fluctuations and yields to a precise description of their statistics.

###  3.3 Closed Q-SSEP : Where are we ?

$\bullet$ Steady fluctuations and large deviation

The closed Q-SSEP is actually ergodic, in any sector with a fixed number of
particles, in the sense that the steady measure $\mathbb{E}_{\infty}$ is
$SU(N)$ invariant. This follows from the fact that iterated products of the
form

$e^{-idh_{t_{1}}}\,e^{-idh_{t_{2}}}\cdots e^{-idh_{t_{n}}}~{},$

with $dh_{t_{k}}$ the one-particle hamiltonian increment (33), for any
collection of time increments $dt_{k}$, visit densely the group $SU(N)$
(because the matrices $E_{j;j+1}:=|j\rangle\langle j+1|$ and their adjoints
form a system of simple root generators for $su(N)$). The dynamics generated
by successive iterations of the infinitesimal group elements $e^{-idh_{t}}$ is
thus ergodic enough to cover the group $SU(N)$, and this implies that the
generating function, $Z[A]:=\mathbb{E}_{\infty}\big{[}e^{{\rm
Tr}(AG)}\big{]}$, is $SU(N)$ invariant : $Z[UAU^{\dagger}]=Z[A]$ for any $U\in
SU(N)$. As a consequence, it admits an integral representation,

$Z[A]=\int\\!\\!dV\,e^{{\rm Tr}(AV^{\dagger}G_{0}V)}~{},$ (40)

where the integral is taken over the unitary group $U(N)$ with the invariant
Haar measure. This integral is known as the Harish-Chandra-Itzykson-Zuber (HC-
IZ) integral in random matrix theory (RMT) [70, 71].

The proof is simple, using the $SU(N)$ invariance and the iso-spectral
property of the Q-SSEP flow :

$\displaystyle
Z[A]=\int\\!\\!dV\,Z[VAV^{\dagger}]=\mathbb{E}_{\infty}\Big{[}\\!\int\\!\\!dV\,e^{{\rm
Tr}(AV^{\dagger}GV)}\Big{]}=\int\\!\\!dV\,e^{{\rm
Tr}(AV^{\dagger}G_{0}V)}~{}.$

In the first equality we use the $SU(N)$ invariance, in the second we insert
the definition of $Z$ as the generating function for $\mathbb{E}_{\infty}$ and
we permute integration and averaging, and in the last we use the property that
the spectrum of $G$ is conserved by the flow, and thus non random, so that
$\int\\!dVe^{{\rm Tr}(AV^{\dagger}G_{0}V)}$ can be pulled out from the
expectation with respect to $\mathbb{E}_{\infty}$.

Alternatively, the system Hilbert, which is the Fock space of $N$ fermions,
decomposes into the sum of sectors $\Lambda_{M}$ of fixed particle number, say
$M=0,\cdots,N$ :

$\mathcal{H}_{\mathrm{sys}}=\Lambda_{0}\oplus\Lambda_{1}\oplus\cdots\oplus\Lambda_{N}~{},$

Each of the subspaces $\Lambda_{M}$ form a $U(N)$ irreducible representation
(isomorphic to rank $M$ antisymmetric tensors in dimension $N$). Since, for
each realisation of the noise, the Q-SSEP dynamics preserves the total number
of particles, each of those sectors is stable under the Q-SSEP dynamics, and
the invariant measure on each of them is that induced by the $SU(N)$ Haar
measure. This is the measure we would get if we were sampling the states
uniformly, according to the Haar measure, in the corresponding $SU(N)$ orbit.
Thus, the system is ergodic in each sector with a fixed number of particles.

The generating function $Z[A]$, represented as a HC-IZ integral, can be
analysed using tools from RMT. It depends on the spectrum of $G_{0}$ and hence
on $\mathfrak{m}_{k}:={\rm Tr}G_{0}^{k}/N$. Assuming that
$\mathfrak{m}_{k}=O(N^{0})$ for all $k$, then
$w[A]:=\lim_{N\to\infty}\frac{1}{N}\log Z[NA]$ is finite, order by order in
power of $A$, or alternatively,

$\mathbb{E}_{\infty}\big{[}e^{N\,{\rm
Tr}(AG)}\big{]}\asymp_{N\to\infty}e^{N\,w[A]}~{}.$ (41)

From eq.(37), the first few terms are

$w[A]=\mathfrak{m}_{1}\,{\rm
Tr}A+\frac{1}{2}\,(\Delta_{2}\mathfrak{m})^{2}\,{\rm
Tr}A^{2}+\frac{1}{3}(\Delta_{3}\mathfrak{m})^{3}\,{\rm
Tr}A^{3}+O(|\\!|A|\\!|^{4})~{},$ (42)

with $(\Delta_{2}\mathfrak{m})^{2}=\mathfrak{m}_{2}-\mathfrak{m}_{1}^{2}$ and
$(\Delta_{3}\mathfrak{m})^{3}:=\mathfrak{m}_{3}-3\mathfrak{m}_{2}\mathfrak{m}_{1}+2\mathfrak{m}_{1}^{3}$.
This expansion suggests that $w[A]$ can be written as a series expansion in
${\rm Tr}A^{k}$ only : $w[A]=\sum_{k}\frac{1}{k}\mathfrak{f}_{k}{\rm
Tr}A^{k}$, where $\mathfrak{f}_{k}$ stands for the large $N$ limit of
$N^{(k-1)}\mathbb{E}_{\infty}[G_{i_{1}i_{2}}G_{i_{2}i_{3}}\cdots
G_{i_{k}i_{1}}]$ with $i_{1},\cdots,i_{k}$ all distincts. This type of
expectations, for connected products of $G$’s with indices $i_{k}$ placed
along a loop, are going be also important in the study of the open Q-SSEP,
since they are going to be the dominating ones. Intuitively, this indicates
the emergence of a thermodynamic limit in which correlation functions
factorize on dominating connected components.

Eq.(41) equivalently means that the steady probability distribution for $G$
satisfies a large deviation principle (compare with eq.(4)141414To compare
with eq.(4), note that if $a_{0}$ is the mesh of the lattice and $L$ the size
of the system, then $L=Na_{0}$ and $\varepsilon:=a_{0}/L=1/N$. with
$\varepsilon=1/N$),

$\mathbb{P}\mathrm{rob}_{\infty}[G=g]\asymp_{N\to\infty}e^{-N\,I(g)}~{},$ (43)

with rate function $I(g)$, the Legendre transform of $w[A]$. Its series
expansion around its minimum reads :

$I(g)=\frac{{\rm
Tr}(g-\mathfrak{m}_{1}\mathbb{I})^{2}}{2(\Delta_{2}\mathfrak{m})^{2}}-\frac{{\rm
Tr}(g-\mathfrak{m}_{1}\mathbb{I})^{3}}{3(\Delta_{2}\mathfrak{m})^{3}}\frac{(\Delta_{3}\mathfrak{m})^{3}}{(\Delta_{2}\mathfrak{m})^{3}}+O(|\\!|g-\mathfrak{m}_{1}\mathbb{I}|\\!|^{4})~{}.$
(44)

To leading order, $G$ is Gaussian with mean $\mathfrak{m}_{1}\mathbb{I}$ and
variance of order $(\Delta_{2}\mathfrak{m})/N^{1/2}$. But they are higher
cumulants. Clearly, there is a need for a better, more complete, non
perturbative, description of this large deviation function (however see [72]).

$\bullet$ Fluctuation dynamics

The time evolution of fluctuations is governed by the stochastic equation (20)
for the density matrix, or eq.(34) for the two-point functions. For the
average density matrix, $\bar{\rho}_{t}=\mathbb{E}[\rho_{t}]$, it leads to the
Lindblad equation eq.(22). For higher moments, say
$\mathbb{E}[\rho_{t}\otimes\cdots\otimes\rho_{t}]$, made of products of $R$
replicas of the density matrix, the dynamics is also given by a Lindblad
equation,

$\partial_{t}\mathbb{E}[\rho_{t}\otimes\cdots\otimes\rho_{t}]=\mathcal{L}^{(R)}_{\mathrm{ssep}}(\mathbb{E}[\rho_{t}\otimes\cdots\otimes\rho_{t}])~{},$
(45)

with a Lindbladian $\mathcal{L}^{(R)}_{\mathrm{ssep}}$ similar to that for one
replica, as in eq.(23), but with the jump operators $\ell^{\pm}_{j}$ replaced
by their sum over the various replicas,
$\ell^{\pm}_{j}\to\sum_{a}\ell^{\pm}_{a;j}$, where the index $a$ labels the
replicas and the individual operators $\ell^{\pm}_{a;j}$ only acts on the
$a$-th replica.

As shown in [65], eq.(45) can be mapped to a spin chain dynamics, with
underlying $gl(2R)$ symmetry algebra. Namely, the Lindbladian
$\mathcal{L}^{(R)}_{\mathrm{ssep}}$ can be written as follows,

$\mathcal{L}^{(R)}_{\mathrm{ssep}}=J\sum_{j=1}^{N}\Big{(}\sum_{A,B}G_{j+1}^{AB}G_{j}^{BA}-\frac{1}{2}(C_{j+1}+C_{j}+2R)\Big{)}~{},$
(46)

where the $G_{j}^{AB}$’s, with $A=1,\cdots,2R$, form a set of super-operators
(i.e. linear maps acting on operators), acting locally on site $j$, with
commutation relations,

$\big{[}G_{j}^{AB},G_{k}^{CD}\big{]}=\delta_{j,k}(\delta^{BC}G_{j}^{AD}-\delta^{DA}G_{j}^{CB})~{},$
(47)

and $C_{j}+R:=\sum_{A}G_{j}^{AA}$. The super-operators $G_{j}^{AB}$ act on any
operator $X$ by left or right multiplications of $X$ by fermionic creation or
annihilation operators. They form a representation of the Lie algebra $gl(2R)$
on the space of operators localized at site $j$, which is is isomorphic to the
exterior algebra in dimension $2R$. This operator space is also isomorphic to
the direct sum of all fundamental representations of $sl(2R)$. Thus, the
Lindblad dynamics (46) is that of a spin chain with local degree of freedom in
the fundamental representations of $sl(2R)$.

This observation allows to borrow information and techniques from spin chain
studies. In particular, it allows to study steady or low lying states. The
integrability for one replica has been shown in [73] and in some higher
replica sectors. The question whether it is integrable in all sectors, for any
number of replicas, is still open [65]. Lindblad integrable dynamics have
recently been analysed [69, 74, 75, 76, 77, 78, 79, 80, 81] but finding a
series of integrable Lindbladians coding for all replicas dynamics is still an
open question.

It also allows to study the dynamics in the continuous scaling limit. Let
$a_{0}$ be the mesh of lattice, i.e. the size of the elementary edges, so that
the total length of the chain is $L=Na_{0}$. The scaling limit is the limit,

$a_{0}\to 0,\ N\to\infty,\ J\to\infty,\quad\mathrm{with}\ L:=a_{0}N,\
D:=Ja_{0}^{2},\ x:=ja_{0}~{}\ \mathrm{fixed},$ (48)

with $J$ the coupling constant in eq.(18) and $j$ the position of the
insertion point. This amounts to impose a diffusive scaling relation between
time and space. $D$ has the dimension of diffusion constant. We set
$\varepsilon:=a_{0}/L=1/N$, as in MFT.

The discrete diffusion equation for the local particle number has a simple
scaling. Let $\mathfrak{n}(x,t)$ be the scaling limit of the density,
$\mathfrak{n}(x,t)=\mathfrak{n}_{j}(t)$ for
$\mathfrak{n}_{j}(t)=\mathbb{E}[G_{jj}(t)]$. As a consequence of eq.(34), it
evolves diffusively,

$\partial_{t}\mathfrak{n}(x,t)={D}\,\nabla_{x}^{2}\,\mathfrak{n}(x,t)~{},$
(49)

as expected since, in average, the Q-SSEP dynamics is purely diffusive (by
construction). At large time, $\mathfrak{n}(x,t)$ reach its uniform steady
value $\mathfrak{n}=\mathfrak{m}_{1}$.

Let us now look at the dynamics of the quadratic fluctuations in the scaling
limit. We follow the analysis performed in [65] and define the scaling limits
of the density correlations and coherence fluctuations :

$\displaystyle\mathfrak{g}_{+}(x,y,t)$
$\displaystyle:=\lim_{\mathrm{scaling}}\,\mathbb{E}[G_{jj}(t)G_{kk}(t)]~{},$
(50a) $\displaystyle\mathfrak{g}_{-}(x,y,t)$
$\displaystyle:=\lim_{\mathrm{scaling}}\,\mathbb{E}[G_{jk}(t)G_{kj}(t)]~{}.$
(50b)

They admit an $\varepsilon$-expansion:

$\mathfrak{g}_{\sigma}(x,y,t)=\mathfrak{g}^{(0)}_{\sigma}(x,y,t)+\varepsilon\,\mathfrak{g}^{(1)}_{\sigma}(x,y,t)+O(\varepsilon^{2})~{}.$
(51)

They satisfy a hierarchy of linear evolution equations, with a triangular
structure. The leading terms $\mathfrak{g}^{(0)}_{\sigma}(x,y,t)$ evolve
diffusively,

$\partial_{t}\mathfrak{g}^{(0)}_{\sigma}(x,y,t)={D}(\nabla_{x}^{2}+\nabla_{y}^{2})\mathfrak{g}^{(0)}_{\sigma}(x,y,t)~{}.$
(52)

In absence of long range order in the initial state, the off-diagonal
coherences initially vanish at leading order in the scaling limit, while the
density correlations are initially factorized. Since they are diffusively
transported, this holds true at any time, and

$\mathfrak{g}^{(0)}_{+}(x,y,t)=\mathfrak{n}(x,t)\mathfrak{n}(y,t)~{},\quad\mathfrak{g}^{(0)}_{-}(x,y,t)=0~{}.$
(53)

The first relation expresses the fact that density fluctuations are sub-
leading in the system size, so that the density is self-averaging at leading
order as expected (cf. the above discussion of the steady measure). The second
relation echoes that coherences are also sub-leading in the system size as
imposed by decoherence. These properties can be checked on their large time
steady values, since eqs.(35,36) read
$\mathbb{E}_{\infty}[G_{jj}G_{kk}]=\mathfrak{m}_{1}^{2}+\frac{1}{N}(\Delta_{2}\mathfrak{m})^{2}\delta_{jk}$
and
$\mathbb{E}_{\infty}[G_{jk}G_{kj}]=\mathfrak{m}_{1}^{2}\delta_{jk}+\frac{1}{N}(\Delta_{2}\mathfrak{m})^{2}$
with $\mathfrak{m}_{1}=\mathfrak{n}$. Furthermore, if the average densities
are initially uncorrelated151515which is always the case if the system is
initially prepared in a given state, then $\mathfrak{g}^{(1)}_{+}(x,y,t)=0$,
so that the density cumulants are actually of second order in $\varepsilon$.

Remarkably, as a consequence of eq.(34), the time evolution of the (sub-
leading) coherence fluctuations is essentially diffusive but sourced by the
density in-homogeneities [65],

$\displaystyle\partial_{t}\mathfrak{g}^{(1)}_{-}(x,y,t)={D}(\nabla_{x}^{2}+\nabla_{y}^{2})\mathfrak{g}^{(1)}_{-}(x,y,t)+\mathfrak{h}_{+}(x,y,t)~{},$
(54) $\displaystyle\mathrm{with}\
h_{+}(x,y,t)=2\,{D}{L}\,\nabla_{x}\nabla_{y}\big{(}\delta(x-y)\mathfrak{n}(x,t)\mathfrak{n}(y,t)\big{)}~{}.$

Higher order correlations satisfy similar equations.

The validity of the scaling evolution equations (49,52,53,54) over a
significant range of time and space scales has been numerically checked in
[65], for various initial configurations. In particular, these numerical
results show that the source term $\mathfrak{h}_{+}$ in eq.(54) is relevant in
this scaling range (so that the evolution of the coherences is not purely
diffusive but modified by the interaction reflected in local density in-
homogeneities). See Figure 4.

Figure 4: [Left]: Comparison between $g_{-}(L/2,3L/8;t)$ as a function of
$Jt/L^{2}$ on a ring of $L=48$ (blue dashed), $L=96$ (yellow dotted), $L=192$
(gree dot-dashed) sites with domain wall initial conditions and
$\mathfrak{g}^{(1)}_{-}(x,y,t)$ (red solid). The continuum limit result is
approached throughout the depicted time range as $L$ increases. [Right]:
Comparison between $g_{-}(L/2,23L/48;t)$ as a function of $Jt/L^{2}$ on a ring
of $L=96$ (blue dashed) sites with domain wall initial conditions and
$\mathfrak{g}^{(1)}_{-}(x,y,t)$ (red solid). The dotted drown line is the
result of pure diffusion only, without the source term
$\mathfrak{h}_{+}(x,y,t)$. Figures taken from [65].

The structure of the evolution equations (49,52,53,54) is similar to that of
the fundamental MFT equations (1). In both cases, the evolution is mainly
diffusive and non-random at leading order, as dictated by the Fourier-Fick’s
law, but with relevant corrections for fluctuations at sub-leading order in
$\varepsilon$. In MFT these corrections are encoded into the Langevin equation
(1). It would be interesting to know, in the case of Q-SSEP, whether there
exits an effective stochastic process or an effective field theory describing
them. If the answer is positive, this would provide the first step in
constructing the Quantum Mesoscopic Fluctuation tTheory (Q-MFT).

$\bullet$ Entanglement statistics and dynamics

Exact results on the statistics of entanglement can be obtained by borrowing
techniques from RMT. Let us assume that the system is initially prepared in a
state $|\psi_{M}\rangle$ with $M$ particles, with density $\mathfrak{n}=M/N$.
Since purity and particle number are conserved by the Q-SSEP dynamics, it will
remain in a pure state $|\psi_{M}(t)\rangle$ with $M$ particles at any time.
Following [82], let us consider the entanglement of a subsystem
$\mathcal{A}_{\ell}=\\{1,\ldots,\ell\\}$, as measured by the Rényi-$q$
entropies,

$S_{q}(t):=(1-q)^{-1}\log{\rm Tr}\rho_{\ell}(t)^{q}~{},$ (55)

where $\rho_{\ell}$ is the system density matrix reduced to
$\mathcal{A}_{\ell}$ : $\rho_{\ell}(t):={\rm
Tr}_{N\setminus\ell}(|\psi_{M}(t)\rangle\langle\psi_{M}(t)|)$. We assume
$\mathcal{A}_{\ell}$ to be extensive and set $\xi=\ell/N$. The Rényi entropies
are then typically extensive $S_{q}\sim\ell\,s$, with $0\leq s\leq\log 2$. We
are interested in their statistics at large time.

As shown in [82], in the large size limit, ($N$, $\ell$, $M$ large with fixed
ratios $\xi=\ell/N$, $\mathfrak{n}=M/N$), the Rényi entropies satisfy a large
deviation principle, w.r.t. the invariant measure $\mathbb{E}_{\infty}$,

$\mathbb{P}\mathrm{rob}_{\infty}[S_{q}=\ell\,s]\asymp_{N\to\infty}e^{-\ell^{2}\,I_{q}(s)}~{},$
(56)

with a rate function $I_{q}(s)$, which depends on the initial density
$\mathfrak{n}$ and on the fraction of volume $\xi$ occupied by the sub-system.
Compare with eq.(4).

This rate function can be evaluated using RMT techniques (since the invariant
measure on the space of fixed number of particles is that induced by the Haar
measure on $U(N)$). The computation is mapped to an extremization problem. It
takes three different analytic forms in three different regions [82]:

  * •

For low values of the entropy, $0<s<s_{-}$ for some $s_{-}$ depending on $\xi$
and $\mathfrak{n}$, the rate function is dominated by contributions from pure
states, and diverges when $s\to 0^{+}$. In particular,
$I_{q=2}(s)=-\frac{1}{2}\log s+O(s)$ as $s\to 0^{+}$.

  * •

The intermediate region, $s_{-}<s<s_{+}$, contains the minimum $\bar{s}$ of
the rate function, which is the most-probable value of the entropy. In
particular, for $q=2$ it is Gaussian,
$I_{q=2}(s)=\frac{1}{2\gamma}(s-\bar{s})^{2}$ with $\gamma$ a computable
constant.

  * •

For high values of the entropy, $s_{+}<s<s_{\mathrm{max}}=\log 2$ for some
$s_{+}$, the rate function is dominated by contributions from maximally mixed
states, and diverges when $s\to s_{\mathrm{max}}$. In particular,
$I_{q=2}(s)=-\frac{1}{2}\log|s-s_{\mathrm{max}}|+O(s-s_{\mathrm{max}})$ as
$s\to s_{\mathrm{max}}^{-}$.

The transitions between these different regimes are non-analytic, signalling
phase transitions. See Figure 5.

Figure 5: [Left]: Rate function for the Rényi-$2$ entropy, for $m=1/2$, and
different values of $\xi$. For $\xi<m$ and $\xi=m$ respectively, two and three
phases appear, which correspond to different colors. [Right]: Analytic
predictions for the derivative $I^{\prime}_{2}(s)$ (solid lines), against
numerical data from Monte Carlo simulations for $\ell=32$, $L=\ell/\xi$ and
$M=L/2$ (dots). The numerical error is not visible at the scales of the plot.
Figures taken from [82].

Clearly, it is worth understanding the entanglement dynamics and its scaling
limit. The nature of the spreading of entanglement in this class of models is
yet unknown. On one hand, local quantities spread diffusively, with possible
sub-leading relevant corrections, see eqs.(49,53,54). One the other hand, the
numerical analysis of [40] done on noisy spin chains (which are noisy many-
body models, similar to Q-SSEP while not identical) suggests dynamical scaling
exponents compatible with those of the Kadar-Parisi-Zhang (KPZ) class.
Comparing the resulting hydrodynamics description of entanglement spreading
with the membrane picture [34, 35, 36, 37, 38] which recently emerged from
studies of random unitary circuits will also be interesting. This will provide
another step toward the construction of a Q-MFT. We hope to report soon on
this problem [83].

###  3.4 Open Q-SSEP : Fluctuations out-of-equilibrium

To simplify the notation, we now set the system length to unity, $L=1$. Thus,
the lattice mesh is $a_{0}=1/N$ and the small dimensionless scaling parameter
is $\varepsilon:=a_{0}/L=1/N$.

Contrary to the closed version, the open Q-SSEP involves injection and
extraction processes at the two ends of the system interval, as specified in
eq.(24). Due to the boundary processes, this evolution is not unitary.
Nevertheless, for each realisation of the noise, the model is still quadratic
and can be analysed by studying its two-point functions, $G_{i;j}={\rm
Tr}(\rho_{t}c_{j}^{\dagger}c_{i})$. The latter satisfy the following
stochastic differential equation (SDE) :

$\displaystyle dG_{i;j}$
$\displaystyle=-2J\,G_{i;j}dt+J(G_{i+1;i+1}+G_{i-1;i-1})\delta_{i;j}dt$ (57)
$\displaystyle~{}~{}~{}~{}+i\sqrt{J}\big{(}G_{i;j-1}d\overline{W}_{t}^{j-1}+G_{i;j+1}dW_{t}^{j}-G_{i-1;j}dW_{t}^{i-1}-G_{i+1;j}d\overline{W}_{t}^{i}\big{)}$
$\displaystyle~{}~{}~{}~{}~{}+\sum_{p\in\\{1,N\\}}\big{(}\alpha_{p}\delta_{i,p}\delta_{i;j}-\frac{1}{2}(\alpha_{p}+\beta_{p})(\delta_{i,p}+\delta_{j,p})G_{i;j}\big{)}dt~{},$

Compared to eq.(34) valid in the closed set-up, eq.(57) contains boundary
terms reflecting the injection and extraction processes with rates
$\alpha_{1,N}$ and $\beta_{1,N}$, as illustrated in the evolution equation for
the mean occupation numbers, ${n}_{j}:=\mathbb{E}[G_{j;j}]$ :

$\partial_{t}{n}_{j}=J\Delta{n}_{j}+\hskip-8.5359pt\sum_{p\in\\{1,N\\}}\hskip-5.69046pt\delta_{j;p}\big{(}\alpha_{p}(1-{n}_{p})-\beta_{p}{n}_{p}\big{)}~{},$
(58)

with $\Delta$ the discrete Laplacian.

In average, and in the diagonal sector (diagonal w.r.t. the particle number
operators), the open Q-SSEP dynamics reproduce the open classical SSEP with
injection and extraction of particles at its two ends.

As for the closed case, the two-point functions reach a steady distribution at
large time. Since eq.(57) is linear, moments of $G$ can be recursively
computed, at least for the first of them. For the mean, from eq.(58) we get,

$\mathbb{E}_{\infty}[G_{i;j}]=\frac{n_{a}(N+b-j)+n_{b}(j+a-1)}{N+b+a-1}\,\delta_{i;j}~{},$
(59)

with $n_{a}:=\frac{\alpha_{1}}{\alpha_{1}+\beta_{1}}$,
$n_{b}:=\frac{\alpha_{N}}{\alpha_{N}+\beta_{N}}$,
$a:=\frac{J}{\alpha_{1}+\beta_{1}}$, $b:=\frac{J}{\alpha_{N}+\beta_{N}}$. In
the scaling limit, $N\to\infty$ at $x=j/N$ fixed (with $n_{a},n_{b},a,b$
fixed), the average profile becomes,

$n^{*}(x)=n_{a}+x(n_{b}-n_{a})~{}.$ (60)

It interpolates linearly between the two boundary occupations $n_{a}$ and
$n_{b}$. The average current, which is proportional to the gradient of the
density by the Fourier-Fock’s, is thus non-vanishing, provided the two
boundary occupations are un-balanced. The system is hence out-of-equilibrium.

Higher moments of the two-point functions, w.r.t. the steady measure
$\mathbb{E}_{\infty}$, were analysed in [44, 84]. It has been shown that, in
the large size limit, the leading multi-point cumulants of order $P$, say
$\mathbb{E}_{\infty}[G_{i_{1}j_{1}}\cdots G_{i_{P}j_{P}}]^{c}$, come from the
expectation values of cyclic products $G_{i_{1}i_{P}}\cdots
G_{i_{3}i_{2}}G_{i_{2}i_{1}}$. They scale proportionally to $1/N^{P-1}$ in the
large $N$ limit. This may be compared to the analysis of the large deviation
function in the closed case, see eqs.(37,42). Other cumulants of $G_{ji}$’s
are sub-leading at large $N$. For instance, in the limit $N\to\infty$ with
$x=i/N$, $y=j/N$ fixed, one has (for $x<y$),

$\displaystyle\mathbb{E}_{\infty}[G_{ij}G_{ji}]^{c}$
$\displaystyle=\frac{1}{N}(\Delta n)^{2}\,x(1-y)+O(N^{-2})~{},$ (61a)
$\displaystyle\mathbb{E}_{\infty}[G_{ii}^{2}]^{c}$
$\displaystyle=\frac{1}{N}(\Delta n)^{2}\,x(1-x)+O(N^{-2})~{},$ (61b)
$\displaystyle\mathbb{E}_{\infty}[G_{ii}G_{jj}]^{c}$
$\displaystyle=-\frac{1}{N^{2}}(\Delta n)^{2}\,x(1-y)+O(N^{-3})~{},$ (61c)

with $\Delta n:=n_{b}-n_{a}$. As in the closed case, quadratic coherence
fluctuations scale as $O(1/{N})$ in the thermodynamic limit, while the
quadratic density correlations at distinct positions scale as $O(1/N^{2})$ and
hence are sub-leading.

The statistics is however not Gaussian as it can be seen by computing the
leading cumulants with three points (with $x<y<z$):

$\mathbb{E}_{\infty}[G_{ik}G_{kj}G_{ij}]^{c}=\frac{1}{N^{2}}(\Delta
n)^{3}\,x(1-2y)(1-z)+O(N^{-3})~{},$ (62)

in the scaling limit $N\to\infty$ with $x=i/N$, $y=j/N$ and $z=k/N$ fixed.

More generally, moments of cyclic products $G_{i_{1}i_{P}}\cdots
G_{i_{3}i_{2}}G_{i_{2}i_{1}}$ depend on whether the ordering of the points
$x_{k}=i_{k}/N$ along the chain and along the cycles defined by these products
coincide or not. Fixing an order along the chain interval, the different
orderings along these cyclic products are indexed by single cycle permutations
of $P$ elements. The rule for this correspondence is that by turning around
the oriented loop indexed by a permutation $\sigma$ one successively
encounters the points labeled as $i_{1}$, $i_{\sigma(1)}$,
$i_{\sigma^{2}(1)}$, $\cdots$, up to closing the loop back to
$i_{\sigma^{P}(1)}=i_{1}$. We can thus choose $0\leq x_{1}<\cdots<x_{P}\leq
1$, with $x_{k}=i_{k}/N$, and index these cyclic cumulants by single cycle
permutation $\sigma$. The claim is [44, 84] :

$\mathbb{E}[G_{i_{1}i_{\sigma^{P-1}(1)}}\cdots
G_{i_{\sigma^{2}(1)}i_{\sigma(1)}}G_{i_{\sigma(1)}i_{1}}]^{c}=\frac{(\Delta
n)^{P}}{N^{P-1}}\,g^{(P)}_{\sigma}(x_{1},\cdots,x_{P})+O(\frac{1}{N^{P}})~{},$
(63)

For instance, for $x_{1}<x_{2}<x_{3}<x_{4}$, we have :

$\displaystyle\mathbb{E}_{\infty}[G_{i_{1}i_{4}}G_{i_{4}i_{3}}G_{i_{3}i_{2}}G_{i_{2}i_{1}}]^{c}$
$\displaystyle=N^{-3}(\Delta
n)^{4}\,x_{1}(1-3x_{2}-2x_{3}+5x_{2}x_{3})(1-x_{4})+O(N^{-4})~{},$
$\displaystyle\mathbb{E}_{\infty}[G_{i_{1}i_{2}}G_{i_{2}i_{4}}G_{i_{4}i_{3}}G_{i_{3}i_{1}}]^{c}$
$\displaystyle=N^{-3}(\Delta
n)^{4}\,x_{1}(1-3x_{2}-2x_{3}+5x_{2}x_{3})(1-x_{4})+O(N^{-4})~{},$
$\displaystyle\mathbb{E}_{\infty}[G_{i_{1}i_{4}}G_{i_{4}i_{2}}G_{i_{2}i_{3}}G_{i_{3}i_{1}}]^{c}$
$\displaystyle=N^{-3}(\Delta
n)^{4}\,x_{1}(1-4x_{2}-x_{3}+5x_{2}x_{3})(1-x_{4})+O(N^{-4})~{},$

with $x_{k}=i_{k}/N$. A few other cumulants can be directly computed. They are
all polynomials of degree at most one in each of their variables.

Hence, although decoherence is at play in the average open Q-SSEP behaviour,
fluctuating quantum coherences survive at large time, although sub-leading.
Their statistics possesses a rich and intriguing structure. The scalings of
their cumulants with the system size is such they satisfy a large deviation
principle. The open Q-SSEP seems thus to be an appropriate quantum extension
the open classical SSEP.

###  3.5 Open Q-SSEP : Where are we ?

$\bullet$ Steady fluctuations and large deviation

The scaling of these cumulants with $N$ signals the existence of a large
deviation function, with small parameter $\varepsilon=1/N$ as for the MFT, see
(41). But, no explicit, or even implicit, formula for this large deviation
function is presently known. In particular, it is not yet known whether this
large deviation function derives from a minimization principle as its analogue
(6) in MFT does. The answer to this question would be an important step
towards constructing a Q-MFT.

However, a procedure to compute all cyclic cumulants of the two-point
functions, in the steady measure, has been proposed in [84]. It is based on an
interplay between permutations and polynomials and in introducing appropriate
generating functions. Let us just present it in the simplest case of so-called
regular cyclic cumulants for which the order of the points along the cycle and
the chain coincide. Let $\omega_{P}(x_{1},\cdot,x_{P})$ be the corresponding
scaled cumulants,

$\mathbb{E}_{\infty}[G_{i_{1}i_{P}}\cdots
G_{i_{3}i_{2}}G_{i_{2}i_{1}}]=N^{1-P}\,(\Delta
n)^{P}\,\omega_{P}(x_{1},\cdots,x_{P})+O(N^{-P})~{}.$ (64)

with $x_{k}=i_{k}/N$ and $0\leq x_{1}\leq\cdots\leq x_{P}\leq 1$. Let us then
define generating functions $\mathcal{C}_{k}(z)$, made of multiple derivatives
of these cumulants, via the formal power series,

$\mathcal{C}_{k}(z):=\sum_{P\geq 0}z^{P}\
\nabla_{x_{P+1}}\cdots\nabla_{x_{1}}\omega_{P+k+1}(x_{1},\cdots,x_{P+k+1})~{},$
(65)

Since all $\omega_{P+k+1}$ are polynomials of degree at most one in each of
its variables, the term proportional to $z^{P}$ in the function
$\mathcal{C}_{k}(z)$ depends only on the remaining $k$ variables
$x_{P+k+1},\cdots,x_{P+2}$ on which the derivatives are not acting. To
stabilize the notation, we renamed them as $y_{l}:=x_{P+k+1-l}$ for
$l=1,\cdots,k$, so that $\mathcal{C}_{k}(z)$ depend only on
$y_{0},\cdots,y_{k-1}$. Surprisingly, the conditions for the stationarity of
the measure $\mathbb{E}_{\infty}$ can be recasted into the following recursion
relation :

$\mathcal{C}_{k+1}(z)=\Big{[}\big{(}\mathfrak{c}(z)+\frac{y_{k}}{z}\big{)}\mathcal{C}_{k}(z)\Big{]}_{+}~{},$
(66)

where $\big{[}\cdots\big{]}_{+}$ means the part of the Laurent series with
positive degrees and
$\mathfrak{c}(z):=\big{(}\sqrt{1+4z}-1\big{)}/2z=1-z+2z^{2}-5z^{3}+\cdots$ is
the generating function of alternating Catalan numbers. This allows to compute
them recursively, say $\mathcal{C}_{0}(z)=\mathfrak{c}(z)$,
$\mathcal{C}_{1}(z)=\mathfrak{c}(z)^{2}(1-y_{0})$,
$\mathcal{C}_{2}(z)=\big{(}\mathfrak{c}(z)^{3}+y_{1}z^{-1}(\mathfrak{c}(z)^{2}-1)\big{)}(1-y_{0})$,
etc. The cumulants $\omega_{P}$ are then recovered from the functions
$\mathcal{C}_{P-1}(z)$ by evaluating them at $z=0$ and by renaming back the
$y$’s in terms of the $x$ positions :

$\omega_{P}(x_{1},x_{2},\cdots)=x_{1}\,\mathcal{C}_{P-1}(0)|_{(y_{0}=x_{P},y_{1}=x_{P-1},\cdots,y_{P-1}=x_{2})}~{}.$
(67)

For instance,

$\displaystyle\mathcal{C}_{3}(0)$ $\displaystyle=$
$\displaystyle\\!\big{(}5\,y_{1}\,y_{2}-3\,y_{2}-2\,y_{1}+1\big{)}(1-y_{0})~{},$
$\displaystyle\mathcal{C}_{4}(0)$ $\displaystyle=$
$\displaystyle\\!\big{(}-14\,y_{1}\,y_{2}\,y_{3}+9\,y_{2}\,y_{3}+7\,y_{1}\,y_{3}+5\,y_{1}\,y_{2}-4\,y_{3}-3\,y_{2}-2\,y_{1}+1\big{)}(1-y_{0})~{},$
(68) $\displaystyle\mathcal{C}_{5}(0)$ $\displaystyle=$
$\displaystyle\\!\big{(}42\,y_{1}\,y_{2}\,y_{3}\,y_{4}-28\,y_{2}\,y_{3}\,y_{4}-23\,y_{1}\,y_{3}\,y_{4}-19\,y_{1}\,y_{2}\,y_{4}-14\,y_{1}\,y_{2}\,y_{3}+14\,y_{3}\,y_{4}$
$\displaystyle\\!+12\,y_{2}\,y_{4}+9\,y_{1}\,y_{4}+9\,y_{2}\,y_{3}+7\,y_{1}\,y_{3}+5\,y_{1}\,y_{2}-5\,y_{4}-4\,y_{3}-3\,y_{2}-2\,y_{1}+1\big{)}(1-y_{0}).$

A stabilization phenomena occurs such that
$\mathcal{C}_{k}(0)=\mathcal{C}_{k+1}(0)|_{y_{k}=0}$. A similar construction
applies to all scaled cyclic cumulants $g^{(P)}_{\sigma}(x_{1},\cdots,x_{P})$.
See [84] for more details.

However, this construction leaves a few questions open. What is the algebraic
structure underlying it ? Is it related to any known structure in integrable
systems ? Can it be made more explicit to give a better control on all
cumulants for all single cycle permutations ? As pointed out above, the
scalings of these cumulants with the system size is such that it ensures the
(formal) existence of a large deviation function. Can they be resumed to yield
access to this large deviation function ? Does this large deviation function
derives from an extremization problem ? Etc.

$\bullet$ Fluctuation dynamics, entanglement statistics and dynamics

The time evolution of fluctuations is governed by the stochastic equation
(24). Clearly, it can be also recasted as a spin chain dynamics, as in eq.(46)
for the closed case, once applied to any number of replicas. Since the
boundary terms are quadratic in the fermions, they will add boundary terms to
eq.(46) linear in the $gl(2R)$ generators (47).

For one replica, this spin chain has been shown to be integrable [73].
However, as for the closed case, it is still an open question whether it is
integrable for any number of replicas.

The scaling limit of the quadratic fluctuations can be studied as in eq.(51)
for the closed case. Clearly, their bulk evolution equations (54) will be
unchanged by the boundary processes, and the extra boundary terms in the
discrete SDE (57) will lead to extra boundary contact terms completing these
bulk evolution equations.

As for the closed case, it is important to get a good understanding of entropy
statistics and dynamics, either for entanglement entropies or for mutual
information of sub-systems relative to their complements. These questions are
yet unexplored. In particular, describing accurately the steady statistics of
those entropies requires having a good control on the invariant measure.

$\bullet$ Un-reasonable connexions with combinatorics

As it is apparent in the explicit examples, say in eqs.(3.5), all scaled
cumulants $g^{(P)}_{\sigma}$ (w.r.t. the invariant measure
$\mathbb{E}_{\infty}$) are polynomials with integer coefficients. This is
remarkable, but it lacks an a priori explanation.

Since the cumulants involve integer numbers, one may wonder whether they have
a combinatorial meaning. And indeed they do. Let us specialize them to
coincident points, say $x_{k}=-t$, for all $k$. Then, we have,

$\omega_{n+1}|_{\\{x_{k}=-t\\}}=-t\,\Phi_{n}(t)\,(1+t),$

with $\Phi_{n}(t)$ polynomials of degree $n-1$ and positive integer
coefficients. For instance, cf. eq.(3.5),

$\displaystyle\Phi_{2}(t)$ $\displaystyle=$ $\displaystyle 1+2t,$
$\displaystyle\Phi_{3}(t)$ $\displaystyle=$ $\displaystyle 1+5t+5t^{2},$ (69)
$\displaystyle\Phi_{4}(t)$ $\displaystyle=$ $\displaystyle
1+9t+21t^{2}+14t^{3},$ $\displaystyle\Phi_{5}(t)$ $\displaystyle=$
$\displaystyle 1+14t+56t^{2}+84t^{3}+42t^{4}.$

These polynomials are known to count the $(n-k)$ dimensional faces in the
associahedron of order $n$ [85]. This is indeed surprising. There is yet no
good understanding of this connection except that both structures are related
to the moduli spaces of configurations of particles on the line. The
observation that the recursion relation (66) produces the generating functions
of the associahedron has been recently proved [86]. However, the scaled
cumulants $g^{(P)}_{\sigma}$, without evaluating them at coincident points,
yields refinements of the associahedron generating functions (indexed by
single cycle permutations), and it remains an open question to understand
whether their coefficients have natural combinatorial interpretations.

##  4 Conclusion

We have reviewed recent on-going progresses in studies of Quantum Simple
Exclusion Processes and their possible connexions with a putative extension of
the Macroscopic Fluctuation Theory to the quantum regime, called the Quantum
Mesoscopic Fluctuation Theory. These processes have a rich, interesting but
intriguing, structure which is yet to be fully understood. Many questions
about the lessons these quantum processes may teach us, or about their
structure, remain open. Some of these questions have been formulated all along
the review, at the end of each Sections or sub-points they refer to.

##  Appendix A Appendix : Does ETH imply ergodicity of quantum trajectories ?

Checking linear ergodicity, as written in eq.(13),
$\lim_{T\to\infty}{T}^{-1}\int_{0}^{T}\\!ds\,{\rm Tr}(\rho_{s}\,O)={\rm
Tr}(\rho_{\infty}\,O)$, requires information on the system density matrix
$\rho_{s}$ for all time $s\in[0,T]$. If not continuously monitoring the
system, this requires preparing it, observing it, and reconstructing its
state, at sufficiently different times to sample the time interval $[0,T]$.
This may be not exactly what is done is practice…

The situation is better if one is monitoring the quantum system, because
recursively extracting information on the system allows to estimate its state
and hence the linear (13) or non-linear (15) time averages. It is also more
physical because one expects to be able to observe regularly extended systems,
without disturbing them too much, and having ergodicity for generic enough
thermodynamical systems.

Monitoring a quantum system is however at the prize of back-acting on it and,
as a consequence, its time evolution becomes stochastic. The time evolution of
a monitored quantum system is governed by a stochastic differential equation
whose solutions are called quantum trajectories, see e.g. [58, 59]. For
simplicity, for so-called homodyne detections of an observable $O$, the
evolution equation is

$d\rho_{t}=-i[H,\rho_{t}]\,dt+\eta\mathcal{L}_{O}(\rho_{t})\,dt+\sqrt{\eta}\,\big{(}O\rho_{t}+\rho_{t}O-2\rho_{t}{\rm
Tr}(O\rho_{t})\big{)}\,dB_{t}~{},$ (70)

with $\mathcal{L}_{O}(\rho_{t})=-\frac{1}{2}[O,[O,\rho_{t}]]$ and $H$ the
hamiltonian of the system in absence of measurement161616One may also add an
intrinsic Lindbladian if the system is open.. Here, $\eta$ is the rate at
which information is extracted and $B_{t}$ is a Brownian motion related to the
output signal $Y_{t}$ obtained from monitoring the system :

$dY_{t}=2\sqrt{\eta}\,{\rm Tr}(O\rho_{t})+dB_{t}~{}.$ (71)

Reading this signal yields an estimation of the instantaneous values of the
quantum expectation ${\rm Tr}(O\rho_{t})$. One may extend this equation to
cases with multiple monitored observables, say $O^{(a)}$. Monitoring enough
observables allow to estimate the density matrix at any time. Not recording
the output signals of the measurements amounts to average the quantum
trajectories. The average density matrix $\bar{\rho}_{t}$ then satisfies the
following dissipative Lindblad equation :
$\partial_{t}\bar{\rho}_{t}=-i[H,\bar{\rho}_{t}]+\eta\mathcal{L}_{O}(\bar{\rho}_{t})$.

Eq.(70) is a well-posed SDE on density matrices. Its linear ergodicity
property has been established sometime ago [63, 64]. Assuming the uniqueness
of the average steady state $\bar{\rho}_{\infty}$, solution of
$\eta\mathcal{L}_{O}(\bar{\rho}_{\infty})=i[H,\bar{\rho}_{\infty}]$ or its
generalization with multiple observables, then [63, 64]

$\lim_{T\to\infty}{T}^{-1}\int_{0}^{T}\\!ds\,{\rm Tr}(\rho_{s}\,O)={\rm
Tr}(\bar{\rho}_{\infty}\,O)~{},\quad\mathrm{almost\ surely}.$ (72)

The almost sure convergence here means that eq.(72) holds for all output
records, except maybe for a set of measure zero.

The non-linear ergodicity (15) is then a well-posed problem for the processes
(70). Not so many results about the ergodicity of the quantum trajectories are
known, except, of course, when the invariant measure of the SDE (70) is known
to be unique. However, the only known (at least to the author) rigorous
results about constructing this invariant measure are those of [87, 88]. The
latter were proved under assumptions whose physical interpretation is not very
transparent, in particular in the case of many-body extended systems.

On the other hand, the Eigenstate Thermalisation Hypothesis (ETH) [89, 90, 91]
has recently been taken as a sign, if not as a definition, of ergodicity in
many-body physics. ETH asserts that the matrix elements of any say local
observable ${O}$, in the energy eigenbasis, take the following form :

$O_{ij}=\mathcal{D}_{O}({E}_{ij})\delta_{ij}+\sigma(E_{ij})^{-\frac{1}{2}}f_{O}({E}_{ij},\omega_{ij})R^{O}_{ij},$
(73)

with $\sigma(E)$ the density of energy eigenstate and
${E}_{ij}=\frac{1}{2}({E}_{i}+{E}_{j})$ and
$\omega_{ij}=\frac{1}{2}({E}_{i}-{E}_{j})$, with $E_{i}$ the eigen-energies.
Here, $\mathcal{D}_{O}$ and $f_{O}$ are assumed to be smooth functions, fastly
decreasing with $\omega$, and $R^{O}_{ij}$ matrices of order one with
erratically varying elements in the range of energy around ${E}$, with zero
mean and unit variance,

$\overline{R^{O}_{ij}}=0,\quad\overline{R^{O}_{ij}R^{O}_{kl}}=\delta_{jk}\delta_{il}~{}.$

Schematically, this means that the matrix $O_{ij}$ can approximatively be
thought of as a band matrix. Its width is governed by the decay rate of
$f_{O}({E},\omega)$ as a function of $\omega$ and it is estimated to be of the
order of the temperature $1/\beta_{E}$ at the energy $E$. The function
$f_{O}({E},\omega)$ is expected to be approximatively constant on an energy
scale of the order the Thouless energy $\mathcal{E}_{T}$, which is the inverse
of the diffusion time, $\mathcal{E}_{T}:={\hbar D}/{L^{2}}$, with $D$ the
diffusion constant and $L$ the linear size of the system. The “statistics” of
the off-diagonal elements $R^{O}_{ij}$, around an energy $E$, are in practice
defined by sampling its elements in an energy window around $E$. There is of
course some arbitrariness in choosing the size of this window, but the
Thouless energy $\mathcal{E}_{T}$ is a natural choice since
$f_{O}({E},\omega)$ is close to a constant on this scale. Even though
$\mathcal{E}_{T}$ decreases with the volume size, there is still an
exponentially large number of eigen-energy in this window since eigen-energies
are exponentially close (so there is enough energy eigen-states to accurately
do the statistics). It is known that higher moments of the off-diagonal
elements have to be non-trivial [92].

By construction [89, 90, 91], ETH ensures linear ergodicity. Indeed, suppose
that the system is prepared in a pure state in the micro-canonical energy
window $E$ up to $\delta E$, say

$|\psi\rangle=\sum_{i}c_{i}\,|E_{i}\rangle,\quad\sum_{i}|c_{i}|^{2}=1~{},$

with coefficients $c_{i}$, smooth and non-vanishing only in the energy window
$\delta E$ around the energy $E$. The time evolved state is
$|\psi(t)\rangle=\sum_{i}c_{i}\,e^{-iE_{i}t}\,|E_{i}\rangle$ and the quantum
expectation value of an observable $O$ is $\langle
O\rangle_{t}=\langle\psi(t)|O|\psi(t)\rangle$ or $\langle
O\rangle_{t}=\sum_{i,j}O_{ij}\,c_{i}c_{j}^{*}e^{i(E_{j}-E_{i})t}$. It is then
clear that time averaging this expectation value projects it on its diagonal
matrix elements, which are smooth by the ETH, so that [91]

$\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}dt\langle
O\rangle_{t}=\sum_{i}O_{ii}|c_{i}|^{2}\simeq{\rm
Tr}(\rho_{\text{micro}}^{(E)}\,O)~{},$ (74)

with $\rho_{\text{micro}}^{(E)}$ the micro-canonical Gibbs state at energy
$E$. Thus, assuming ETH, linear ergodicity w.r.t. to the micro-canonical
ensemble holds.

Assuming that ETH implies ergodicity of extended many-body systems, as it is
often done in the physics literature, it is hence natural to wonder whether
(or to conjecture that) ETH ensures ergodicity – say as formulated in eq.(15)
– of the quantum trajectories (70), or their generalisation with multiple
monitored observables, for a large enough set of monitored observables.

Acknowledgements:
I thank my collaborators on this topics, Michel Bauer, Fabian Essler, Ludwig
Hruza, Tony Jin, Alexandre Krajenbrink, Marko Medenjak and Lorenzo Piroli, for
enjoyable collaborations and Jean-Bernard Zuber for regular discussions. This
work was in part supported by CNRS, by the ENS and by the ANR project
“ESQuisses”, contract number ANR-20-CE47-0014-01.

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