# Deviation bounds and concentration inequalities for quantum noises

Tristan Benoist  Institut de Mathématiques de Toulouse, UMR5219, Université de
Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France  Lisa Hänggli  Cambyse
Rouzé  Department of Mathematics, Technische Universität München, 85748
Garching, Germany  Munich Center for Quantum Science and Technology (MCQST),
München, Germany

###### Abstract

We provide a stochastic interpretation of non-commutative Dirichlet forms in
the context of quantum filtering. For stochastic processes motivated by
quantum optics experiments, we derive an optimal finite time deviation bound
expressed in terms of the non-commutative Dirichlet form. Introducing and
developing new non-commutative functional inequalities, we deduce
concentration inequalities for these processes. Examples satisfying our bounds
include tensor products of quantum Markov semigroups as well as Gibbs samplers
above a threshold temperature.

##  1 Introduction

Distinguishing unknown states of a given system constitutes one of the most
fundamental tasks in experimental science, information theory and statistics –
in the classical, as well as in the quantum realm. Classically, the simplest
formulation of the problem is as follows: An experimentalist is given
independent samples $Y_{1}=y_{1},\dots,Y_{n}=y_{n}$ from some unknown
distribution $\pi$ taking values in a finite sample space $\Omega$ and is
required to learn the distribution $\pi$. A natural estimate for the sought
after distribution is then given by the empirical measure

$\displaystyle L_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{Y_{i}}\,,$ (1.1)

where $\delta_{a}$ refers to the Dirac distribution at $a\in\mathbb{R}$.
Noting that $L_{n}$ is a random element of the set ${\mathcal{M}}_{1}(\Omega)$
of probability measures endowed with the topology of the total variation
distance, we may quantify its aptitude for the intended task in terms of the
probability of $L_{n}$ being close to the desired distribution $\pi$. More
precisely, Sanov’s theorem describes the asymptotic efficiency of the
empirical measure as an estimator of the measure $\pi$: denoting by
$\mathcal{B}(\mu,R)$ the open ball centered at
$\mu\in{\mathcal{M}}_{1}(\Omega)$ of radius $R$, the theorem states that for
any $R>0$,

$\lim_{n\to\infty}\frac{1}{n}\ln\mathbb{P}(L_{n}\in\mathcal{B}(\pi,R)^{c})=-\inf_{\nu\in\mathcal{B}(\pi,R)^{c}}D(\nu\|\pi)\,.$
(1.2)

Here $D(\mu\|\nu)$ denotes the relative entropy between two probability
measures $\mu<\\!<\nu$, which is defined as
$D(\mu\|\nu)=\sum_{i\in\Omega}\mu(i)\ln({\mu(i)}/{\nu(i)})$. In words, Sanov’s
theorem tells us that the probability of the empirical measure being at least
distance $R$ away from the true measure $\pi$ decreases exponentially, with an
asymptotic rate given by the right-hand side of Equation 1.2.

Recall that in the above first formulation of the problem of distinguishing
unknown states, we assumed access to arbitrary samples drawn from the unknown
distribution $\pi$ – a fact employed in the theorem of Sanov. However, in
practice, the distribution $\pi$ is often generated from a random process
(e.g. , a Gibbs sampler [48]). In particular, $\pi$ may be the (unique)
stationary measure of a time-continuous Markov process $t\mapsto X_{t}$ on
$\Omega$, which corresponds to the continuous-time Markov semigroup $t\mapsto
e^{tL}$ on $L^{\infty}(\Omega)$ generated by $L$. In this case, the measure
$\pi$ is only reached as $t\to\infty$ and the experimentalist does not have
access to the random variables $Y_{j}$ for $t<\infty$. It is thus crucial to
find a dynamical strategy to learn $\pi$ from measurements performed at finite
times. One way to achieve this goal is to continuously measure the sample
paths corresponding to instances of the underlying process [27]. In that case,
the occupation time

$\displaystyle L_{t}=\frac{1}{t}\int_{0}^{t}\delta_{X_{s}}\mathrm{d}s\,,$
(1.3)

is a candidate for the estimator of the target measure $\pi$. Indeed, by the
Donsker-Varadan theorem [24], we have that for any Borel set
$B\subset{\mathcal{M}}_{1}(\Omega)$,

$\displaystyle-\inf_{\mu\in\operatorname{int}(B)}I(\mu)\leq\liminf_{t\to\infty}\frac{1}{t}\ln\mathbb{P}(L_{t}\in
B)\leq\limsup_{t\to\infty}\frac{1}{t}\ln\mathbb{P}(L_{t}\in
B)\leq-\inf_{\mu\in\operatorname{cl}(B)}I(\mu)\,,$ (1.4)

with $\operatorname{int}(B)$ the interior of $B$, $\operatorname{cl}(B)$ the
closure of $B$, and

$I(\nu):=\sup_{u>0}\left[-\sum_{i\in\Omega}\nu(i)\frac{(Lu)(i)}{u(i)}\right]\,,$

where the supremum runs over functions $u:\Omega\to(0,\infty)$. Since the
function $I(\nu)$ only vanishes for $\nu=\pi$ being the unique stationary
measure of the Markov chain, Eq. (1.4) guarantees that the probability of the
occupation time $L_{t}$ lying in a set not containing $\pi$ decays
exponentially in $t$ with a rate determined by the function $I(\cdot)$. In
other words, the probability of $L_{t}$ not constituting a good estimator for
the target measure $\pi$ decays exponentially in time. Moreover, in the case
of a reversible generator $L$, i.e. , $\pi(i)L_{ij}=\pi(j)L_{ji}$, we have
that $I(\nu)=\mathcal{E}_{L}(f)$, where $f=\sqrt{{d\nu}/{d\pi}}$ and

$\displaystyle{\cal
E}_{L}(g):=-\mathbb{E}_{\pi}[g\,L(g)]\equiv-\sum_{i,j\in\Omega}\pi(i)\,g(i)\,L_{ij}\,g(j)\
,\quad g:\Omega\to\mathbb{R}\ ,$ (1.5)

is the so-called Dirichlet form associated with the generator $L$. More
generally, Deuschel and Stroock proved in [26, Theorem 5.3.10] (see also [63]
for an extension to unbounded functions $f$ when $\Omega$ is a general Polish
space) that for any function $f:\Omega\to\mathbb{R}$, all $r\geq 0$, and all
initial distributions $\nu\in{\mathcal{M}}_{1}(\Omega)$, it holds

$\displaystyle\lim_{t\to\infty}\frac{1}{t}\,\log\mathbb{P}_{\nu}\,\Big{(}\,\frac{1}{t}\int_{0}^{t}\,f(X_{s})\,ds-\mathbb{E}_{\pi}[f]>r\Big{)}=-I_{f}\big{(}\mathbb{E}_{\pi}[f]+r\big{)}\,,$
(1.6)

where $I_{f}$ is the lower semi-continuous regularization of the function

$\displaystyle J_{f}:\mathbb{R}\to\overline{\mathbb{R}},\quad
x\mapsto\inf_{g:\Omega\to\mathbb{R}}\Big{\\{}-\mathbb{E}_{\pi}[\,gL(g)\,]\,\Big{|}\,\mathbb{E}_{\pi}[g^{2}]=1,\,\mathbb{E}_{\pi}[\,fg^{2}]=x\Big{\\}}\,,$

with the convention that $\inf\emptyset=\infty$.

Although they correspond to two different experimental setups, the spatial and
temporal empirical averages $L_{n}$ and $L_{t}$ are intimately related through
a functional inequality known as the logarithmic Sobolev inequality (LSI): The
generator $L$ is said to satisfy the latter if there exists a positive
constant $\alpha$ such that for any probability measure $\nu<\\!<\pi$, and
$f=\sqrt{{d\nu}/{d\pi}}$,

$\displaystyle\alpha\,D(\nu\|\pi)\leq\mathcal{E}_{L}(f)\,.$ (1.7)

The largest constant $\alpha$ achieving this bound is called the logarithmic
Sobolev constant of $L$ and is denoted by $\alpha_{2}(L)$. As explained above,
each side of (1.7) corresponds to a rate function of a large deviation
principle associated with two distinct strategies used to gain information on
$\pi$. Thus, the LSI bears the statistical interpretation of providing a
comparison between the asymptotic efficiency of the spatial and temporal
empirical averages: up to the multiplicative constant $\alpha_{2}(L)$, the
temporal strategy performs at least as well as the spatial strategy. From a
more practical point of view, the logarithmic Sobolev inequality can also be
used to derive finite time concentration bounds on the tail probability in
(1.6) through a so-called transporation-information inequality [64, 37]:
assuming that $\Omega$ is a metric space with metric
$d:\Omega\times\Omega\to\mathbb{R}_{+}$, the latter relates the Wasserstein
distance between the measure $\pi$ and any other probability measure
$\nu<\\!<\pi$ to the Dirichlet form evaluated at $f=\sqrt{d\nu/d\pi}$:

$\displaystyle W_{1}(\nu,\pi)\leq\,\sqrt{2C\,{\cal E}_{L}(f)}\,.$ (1.8)

Here $C>0$ is some constant independent of $\nu$, and
$W_{1}(\nu,\mu):=\max_{\|f\|_{\operatorname{Lip}}\leq
1}\mathbb{E}_{\mu}[f]-\mathbb{E}_{\nu}[f]$ is the Wasserstein distance between
$\nu$ and $\pi$. The inequality (1.8) implies concentration bounds for the
estimator $L_{t}$ of the following form [37]: For any Lipschitz function
$f:\Omega\to\mathbb{R}$, any initial measure $\nu<\\!<\pi$, and all $r>0$,

$\displaystyle\mathbb{P}_{\nu}\Big{(}\,\frac{1}{t}\int_{0}^{t}\,f(X_{s})\,ds-\mathbb{E}_{\pi}[f]>r\Big{)}\leq\,\Big{\|}\frac{d\nu}{d\pi}\Big{\|}_{L^{2}(\pi)}\,\exp\Big{(}-\frac{tr^{2}}{2C\,\|f\|_{\operatorname{Lip}}^{2}}\Big{)}\,.$
(1.9)

In a typical quantum optics experiment [61], the collected data is represented
by some quantum noises. In absence of a system perturbing the electromagnetic
field, say an atom, the measured time resolved signal has either the law of a
Brownian motion or of a ‘zero intensity’ Poisson process. In the presence of
the atom, the Brownian motion gains a drift depending on the atom’s state.
Similarly, the Poisson process gains a positive intensity which depends also
on the atom’s state. In this article, we focus our study precisely on these
time resolved signal processes. We therefore do not derive bounds on
quantities generalizing expressions such as $\int_{0}^{t}f(X_{s})\mathrm{d}s$
as introduced before, but rather consider random processes $X_{t}$ which
either satisfy a stochastic differential equation of the form
$\mathrm{d}X_{t}=a_{t}\mathrm{d}t+\mathrm{d}W_{t}$, where $t\mapsto a_{t}$ is
predictable and $t\mapsto W_{t}$ a Brownian motion, or such as $X_{t}$ is a
Poisson process with stochastic predictable intensity. These choices have two
motivations: first they correspond to the actual measurement processes
occurring in current experiments and second it is for these processes that we
can obtain deviation bounds from non-commutative generalizations of Dirichlet
forms.

##### Quantum setup

Let us present our framework in more details. Without conditioning on the
measured data, the evolution of the state of the atom is modeled by a quantum
Markov semigroup (QMS), that is a semigroup $t\mapsto e^{t{\cal L}}$ of
completely positive, unital maps acting on the algebra ${\cal B}({\cal H})$ of
linear operators on a finite dimensional Hilbert space ${\cal H}$. These are
generalizations of Markov semigroups, changing the vector space from positive
vectors to positive semi-definite matrices. A state of the atom is then lifted
from a $\ell^{1}$ normalized positive vector in $\mathbb{R}^{d}$ to a trace
one positive semi-definite matrix in ${\cal B}({\cal H})\equiv
M_{d}(\mathbb{C})$:

$\mathcal{D}({\cal H})=\\{\rho\in M_{d}(\mathbb{C}):\rho\geq 0,\mathop{\rm
Tr}\nolimits\rho=1\\}.$

We assume that $t\mapsto e^{t{\cal L}}$ is primitive, which means that it
possesses a unique full-rank invariant state $\sigma$. As recalled below (cf.
Theorem 2), the generator ${\cal L}$ admits a Lindblad form: there exist
$H\in{\cal B}_{\operatorname{sa}}({\cal H})$, $k\in\mathbb{N}$ and
$\mathbf{L}:{\cal H}\to{\cal H}\otimes\mathbb{C}^{k}$ such that

$\displaystyle\forall X\in{\cal B}({\cal
H}),\quad\mathcal{L}(X)=i[H,X]+\mathbf{L}^{*}(X\otimes{\rm{id}}_{\mathbb{C}^{k}})\mathbf{L}-\frac{1}{2}(\mathbf{L}^{*}.\mathbf{L}\,X+X\,\mathbf{L}^{*}.\mathbf{L})\,.$
(1.10)

In analogy with the classical setting, the Dirichlet form associated to the
generator ${\cal L}$ is defined by

$\displaystyle{\cal E}_{\cal L}(X):=\langle X,\,{\cal
L}(X)\rangle_{\sigma}\,,\qquad X\in{\cal B}({\cal H})\,,$ (1.11)

where $(X,Y)\mapsto\langle X,Y\rangle_{\sigma}:=\mathop{\rm
Tr}\nolimits[\sigma^{\frac{1}{2}}X\sigma^{\frac{1}{2}}Y]$ is the so-called KMS
inner product corresponding to the state $\sigma$, of associated norm
$\|.\|_{L^{2}(\sigma)}$.

The connection between the QMS and the measured signal is made through a
dilation of the former. The semigroup $t\mapsto e^{t{\cal L}}$ admits a
unitary dilation $t\mapsto U_{t}$ modelling the continuous time dynamics
resulting from the interaction of the quantum system and its environment. The
family of unitaries $t\mapsto U_{t}$ satisfies a quantum stochastic
differential equation (QSDE) recalled in Section 2.1. The processes modeling
the time resolved measurement are derived from the quantum noises describing
the environment in such QSDE. The operators $\mathbf{L}$ encode the
interaction between the atom and the environment and prescribe what are the
induced drifts and intensities of the signals.

We are interested in estimating the expectation values, with respect to the
state $\sigma$, of linear combinations of $L_{j}+L_{j}^{*}$ and
$L_{j}^{*}L_{j}$, with $L_{j}$ the components of the vector $\mathbf{L}$. For
example,

$\displaystyle\langle O(u)\rangle_{\sigma}:=\mathop{\rm
Tr}\nolimits[\,\sigma\,O(u)]\,,\qquad
O(u):=\sum_{j=1}^{k}u_{j}\,(L_{j}+L_{j}^{*})\,,$ (1.12)

where $u$ is a normalized vector in $\mathbb{R}^{k}$. These expectations are
accessed through the above mentioned quantum noise measurements. Considering
the Brownian motion with an added drift, the time averaged measured signal,
and therefore our estimator, is given by

$\displaystyle L_{t}(u)=\frac{1}{t}\,\int_{0}^{t}\mathop{\rm
Tr}\nolimits[O(u)\varrho_{s-}]\mathrm{d}s\,+\frac{W_{t}}{t}\,,$ (1.13)

where $t\mapsto W_{t}$ is a standard Brownian motion. Here
$t\mapsto\varrho_{t}$ is a stochastic process on the system’s states initiated
at the deterministic state $\rho\in{\cal D}({\cal H})$, whose average
evolution equals $\mathbb{E}[\varrho_{t}]=e^{t{\cal L}^{*}}(\rho)$. The fact
that the drift is not deterministic but merely an adapted process is a
prescription from quantum physics. The average estimator converges to the mean
of $O(u)$ in the stationary state $\sigma$ since

$\mathbb{E}[L_{t}(u)]=\frac{1}{t}\,\mathop{\rm
Tr}\nolimits\big{[}\,O(u)\int_{0}^{t}\mathrm{e}^{s\mathcal{L}^{*}}(\rho)\,\mathrm{d}s\big{]}\underset{t\to\infty}{\longrightarrow}\,\langle
O(u)\rangle_{\sigma}\,.$ (1.14)

Similarly, for Poisson processes the estimator is
$R_{t}(u)=\frac{1}{t}N_{t}(u)$, with $N_{t}(u)$ a Poisson process with
intensity $t\mapsto\mathop{\rm
Tr}\nolimits[(u.\mathbf{L})^{*}u.\mathbf{L}\varrho_{t}]$. Hence
$\mathbb{E}[R_{t}(u)]$ converges to $\mathop{\rm
Tr}\nolimits[(u.\mathbf{L})^{*}u.\mathbf{L}\sigma]$.

In [47] it was proved that the estimators $L_{t}$ and $R_{t}$ converge also
almost surely. Our main goal is to refine this result; in particular, we seek
for deviation bounds on the measurement signals in the presence of an atom.
Our main tools are Dirichlet forms and functional inequalities.

##### Summary of main results

Sanov’s theorem admits a quantum generalization [11], and Deuschel-Stroock’s
large deviation theorem was also extended to the quantum discrete and
continuous time settings [42, 59] – however, the link to the quantum Dirichlet
form was not made in these articles. The first goal of this paper is to fill
this missing gap. More precisely, we aim at quantifying the probability that
the estimator $L_{t}(u)$ deviates from the average $\langle
O(u)\rangle_{\sigma}$. We first prove in Theorem 6 the following upper bound:
for any $r>0$,

$\displaystyle\mathbb{P}(L_{t}(u)-\langle O(u)\rangle_{\sigma}\geq
r)\leq\|\sigma^{-\frac{1}{2}}\rho\sigma^{-\frac{1}{2}}\|_{L^{2}(\sigma)}\exp\Big{(}-t\inf_{\|X\|_{L^{2}(\sigma)}=1}{\cal
E}_{\cal L}(X)+\frac{1}{2}\,(r+\langle
O(u)\rangle_{\sigma}-f_{u}(X))^{2}\Big{)}$ (1.15)

for some explicit function $f_{u}:{\cal B}({\cal H})\to\mathbb{R}$. Theorem 6
is a non commutative version of the commutative finite time bound from [64,
Theorem 1]. As for the commutative bound, it is optimal, in the sense that,
when the generator ${\cal L}$ is symmetric with respect to the KMS inner
product $\langle.,.\rangle_{\sigma}$, the exponential rate is the rate
function of the large deviation principle verified by the estimator $L_{t}(u)$
(see Theorem 7 and [26, Theorem 4.2.58] for the commutative version): for any
Borel set $B\subset\mathbb{R}$,

$\displaystyle-\inf_{{s}\in\operatorname{int}(B)}I_{u}({s})\leq\liminf_{t\to\infty}\frac{1}{t}\log\mathbb{P}(L_{t}(u)\in
B)\leq\limsup_{t\to\infty}\frac{1}{t}\log\mathbb{P}(L_{t}(u)\in
B)\leq-\inf_{{s}\in\operatorname{cl}(B)}I_{u}({s})\,,$ (1.16)

with $\operatorname{int}(B)$ the interior of $B$, $\operatorname{cl}(B)$ the
closure of $B$, and where

$\displaystyle I_{u}(s):=\inf_{\|X\|_{L^{2}(\sigma)=1}}\,{\cal E}_{\cal
L}(X)+\frac{1}{2}\,(s-f_{u}(X))^{2}\,.$ (1.17)

In analogy with the classical setting, the generator ${\cal L}$ is said to
satisfy a quantum logarithmic Sobolev inequality (qLSI) if there exists a
constant $\alpha>0$ such that, for any $\rho\in{\cal D}({\cal H})$ [53, 8],

$\displaystyle\alpha\,D(\rho\|\sigma)\,\leq\,{\cal E}_{\cal
L}(\sigma^{-\frac{1}{4}}\,\sqrt{\rho}\,\sigma^{-\frac{1}{4}})\,.$ (1.18)

Here, we leave the problem of finding a statistical interpretation of Equation
1.18 open. Instead, we explore in Section 5 the use of qLSI as well as other
functional inequalities in deriving finite time concentration bounds on the
tail probability of the random variable $L_{t}(u)-\langle
O(u)\rangle_{\sigma}$. To serve this purpose, we consider a quantum
generalization of the Wasserstein distance of order $1$ [57]: assuming that
the generator ${\cal L}$ is symmetric with respect to the so-called GNS inner
product $(X,Y)\mapsto\mathop{\rm Tr}\nolimits[\sigma X^{*}Y]$, there exist
parameters $\omega_{i}\in\mathbb{R}$, $i\in[k]$, such that [28] for all
$i\in[k]$, $\sigma\,L_{i}\,\sigma^{-1}=e^{-\omega_{i}}L_{i}$. Then, the
quantum Wasserstein distance associated to the generator ${\cal L}$ between
two quantum states $\rho_{1}$ and $\rho_{2}$ is defined as

$\displaystyle W_{1,{\cal
L}}(\rho_{1},\rho_{2}):=\sup_{\begin{subarray}{c}X=X^{*}\\\
\|X\|_{\operatorname{Lip}}\leq 1\end{subarray}}\,\mathop{\rm
Tr}\nolimits[\rho_{1}X]-\mathop{\rm Tr}\nolimits[\rho_{2}X]\,,$ (1.19)

where the non-commutative Lipschitz constant of an observable $X=X^{*}$ is
defined as

$\|X\|_{\operatorname{Lip}}:=\Big{(}\sum_{j\in[k]}(\mathrm{e}^{-\omega_{j}/2}+\mathrm{e}^{\omega_{j}/2})\,\|[L_{j},X]\|_{\infty}^{2}\Big{)}^{1/2}\,.$

Extending the commutative framework of [37], we say that the generator ${\cal
L}$ satisfies a non-commutative transportation-information inequality of
constant $C>0$ if for any state $\rho\in{\cal D}({\cal H})$,

$\displaystyle W_{1,{\cal L}}(\rho,\sigma)\leq\sqrt{2C\,{\cal E}_{\cal
L}(\sigma^{-\frac{1}{4}}\,\sqrt{\rho}\,\sigma^{-\frac{1}{4}})}\,.$ (1.20)

Building on previous results from [16, 57, 19], we prove in Section 5.2 that
(1.18) implies the transportation-information inequality (1.20) with constant
$C=\alpha^{-2}8^{-1}$. The latter is then proven to imply a Gaussian
concentration inequality for the trajectory of the following form in Section
5.3: there exist an explicit $u$ dependent constant $C_{u}$ such that for all
$t,r>0$ and any initial state $\rho\in{\cal D}({\cal H})$,

$\displaystyle\mathbb{P}\Big{(}L_{t}(u)-\langle
O(u)\rangle_{\sigma}>r\Big{)}\leq\|\Gamma_{\sigma}^{-1}(\rho)\|_{\mathbb{L}_{2}(\sigma)}\,\exp\left(-\frac{t\,r^{2}}{C_{u}\|\Delta_{\sigma}^{\frac{1}{4}}(L^{*}_{u})+\Delta_{\sigma}^{-\frac{1}{4}}(L_{u})\|_{\operatorname{Lip}}^{2}}\right)\,.$

This bound is to our knowledge the first Gaussian concentration bound derived
for the estimators $L_{t}(u)$ in the non commutative setting.

##### Layout of the paper

Section 2 introduces the concepts that we are going to use in the rest of the
article, namely quantum stochastic differential equations (Section 2.1),
quantum trajectories (Section 2.2) and quantum Markov semigroups (Section
2.3). In Section 3, we provide our main upper bound (Theorem 6) and use it to
derive a large deviation principle in the case of a reversible semigroup
(Theorem 7). The proof of Theorem 6 is postponed to Section 4. In Section 5,
we provide a self-contained introduction to classical and quantum
transportation cost metrics (Section 5.1) and related inequalities (Section
5.2), and apply the latter to the derivation of concentration inequalities in
Theorem 8. We end this article with a few examples illustrating our bounds,
including generalized depolarizing channels and tensor products of quantum
Markov semigroups, in Section 6.

##  2 Notations and preliminaries

The set of $k\times k$ matrices with complex entries is denoted by
$M_{k}(\mathbb{C})$, the unit ball in $\mathbb{C}^{k}$ by
$S^{k-1}(\mathbb{C})$, and the one in $\mathbb{R}^{k}$ by
$S^{k-1}(\mathbb{R})$. Let ${\cal H}$ be a separable Hilbert space. We denote
by $\mathcal{B}({\cal H})$ the space of linear operators on ${\cal H}$, by
$\mathcal{B}_{\operatorname{sa}}({\cal H})$ the subspace of self-adjoint
operators on ${\cal H}$, and by $\mathcal{B}_{+}({\cal H})$ the cone of
positive semidefinite operators. The adjoint of an operator $Y$ is written as
$Y^{*}$. The operator norm on ${\cal B}({\cal H})$ is denoted by
$\|.\|_{\infty}$, whereas Schatten norms are denoted by $\|.\|_{p}$ for $p\geq
1$. The identity operator on a vector space ${\cal A}$ is denoted by
${\rm{id}}_{\cal A}$, dropping the index ${\cal A}$ when the corresponding
space is obvious from the context. We denote by $\mathcal{D}({\cal H})$ the
set of positive semidefinite, trace one operators on ${\cal H}$, also called
_density operators_ , and by ${\cal D}_{>}({\cal H})$ the subset of faithful
density operators. In the following, we will often identify a density operator
$\rho\in\mathcal{D}({\cal H})$ and the _state_ it defines, that is, the
positive linear functional $\mathcal{B}({\cal H})\ni X\mapsto\mathop{\rm
Tr}\nolimits[\rho\,X]$. Given a state $\rho\in{\cal D}({\cal H})$, a self-
adjoint operator $O$ on ${\cal H}$ and a real number $r\in\mathbb{R}$, we
denote the probability that a measurement of $O$ in the state $\rho$ yields a
value $\lambda>r$ by

$\displaystyle\mathbb{P}_{\rho}\big{(}O>r\big{)}:=\mathop{\rm
Tr}\nolimits\big{[}\rho\,\mathds{1}_{]r,\infty[}\,(O)\big{]}\,,$ (2.1)

where $\mathds{1}_{]r,\infty[}$ denotes the characteristic function of the
interval $]r,\infty[$.

###  2.1 Non-commutative noises and quantum stochastic calculus

In this section we give a concise introduction to quantum noises and quantum
stochastic calculus without proofs. For a complete introduction and proofs of
the results stated here, we refer the reader to the pioneering work of Hudson
and Parthasarathy [40] (see also [56] and references therein), but also to the
book of Holevo [39], and the introduction to quantum filtering by Bouten _et
al._ [13].

####  2.1.1 Non-commutative noises

In non-commutative probability, a measurable space is replaced by a unital
$*$-algebra of operators. The elements of the algebra replace random
variables. Expectations are replaced by normalized positive linear forms
called states. A standard non-commutative equivalent of the (commutative)
canonical Brownian motion and Poisson process are defined by elements of a
$*$-algebra of operators on a symmetric Fock space. The Wiener and Poisson
measures are translated to the vacuum state of this Fock space. We now
properly define these objects.

Fix a natural number $k$. Let $\Gamma_{s}^{(0)}=\mathbb{C}$, and for any
natural number $n$, let

$\Gamma_{s}^{(n)}=S_{n}L^{2}(\mathbb{R}_{+};\mathbb{C}^{k})^{\otimes n},$

where $L^{2}(\mathbb{R}_{+};\mathbb{C}^{k})$ denotes the space of square
integrable functions of the half-line with values in $\mathbb{C}^{k}$, and
$S_{n}$ is the orthogonal projection onto the symmetric subspace of the tensor
product. For example, $S_{2}f\otimes g=\frac{1}{2}(f\otimes g+g\otimes f)$.
The symmetric Fock space, $\Gamma_{s}$, is then defined as the completion of
the direct sum of the Hilbert spaces $\Gamma_{s}^{(n)}$:

$\Gamma_{s}=\overline{\bigoplus_{n\in\mathbb{N}_{0}}\Gamma_{s}^{(n)}}.$

Since we work only with symmetric Fock spaces, we will omit the index $s$ from
now on. We will mostly work in reference to the vacuum state
$\Omega\in\Gamma$, defined by $\Omega=1\oplus_{n\in\mathbb{N}}0$.

On the Fock space introduced above we now define creation and annihilation
operators: For any $f\in L^{2}(\mathbb{R}_{+};\mathbb{C}^{k})$, the creation
operator $a^{*}(f)$ is defined by

$a^{*}(f)\Psi=\sqrt{n+1}S_{n+1}\,f\otimes\Psi,\quad\forall\Psi\in\Gamma^{(n)}.$

The annihilation operator $a(f)$ is defined by

$a(f)\Psi=\sqrt{n}\langle
f|\otimes{\rm{id}}_{\Gamma_{s}^{(n-1)}}\Psi,\quad\forall\Psi\in\Gamma^{(n)},$

where $\langle f|$ is the dual of $f$ in $L^{2}\equiv
L^{2}(\mathbb{R}_{+};\mathbb{C}^{k})$. These operators are extended by
linearity to the space of finitely many particle vectors:
$\Gamma^{\mathrm{fin}}=\\{\Psi\in\Gamma:\exists N\in\mathbb{N}\mbox{ s.t.
}\Psi\in\left(\bigoplus_{n=0}^{N}\Gamma^{(n)}\right)\oplus\left(\bigoplus_{n>N}0_{\Gamma^{(n)}}\right)\\}.$
That way they are densely defined on $\Gamma$ since
$\Gamma=\overline{\Gamma^{\mathrm{fin}}}$. Computing the dual of $a^{*}(f)$
shows that $a^{*}(f)=a(f)^{*}$. It also follows from the definition that the
creation and annihilation operators verify the canonical commutation
relations,

$\displaystyle[a(f),a^{*}(g)]=\langle f,g\rangle\,{\rm{id}}\ ,$ (2.2)

where $\langle.,.\rangle$ denotes the inner product in
$L^{2}(\mathbb{R}_{+};\mathbb{C}^{k})$. We now introduce the second
quantization operator $\mathrm{d}\Gamma$ that appears in the non-commutative
generalization of the Poisson process. For any bounded operator
$h:L^{2}(\mathbb{R}_{+};\mathbb{C}^{k})\to
L^{2}(\mathbb{R}_{+};\mathbb{C}^{k})$, let

$\mathrm{d}\Gamma(h)=\bigoplus_{n\in\mathbb{N}_{0}}\mathrm{d}\Gamma_{n}(h),\quad\mbox{with}\quad\mathrm{d}\Gamma_{n}(h)=\sum_{j=1}^{n}{\rm{id}}_{L^{2}}^{\otimes(j-1)}\otimes
h\otimes{\rm{id}}_{L^{2}}^{\otimes(n-j)}$

and domain $\Gamma^{\mathrm{fin}}$. A standard example of such an operator is
the particle number operator: for a fixed $p\in\mathbb{N}_{0}$ and
$\Psi\in\Gamma^{(p)}\oplus 0\subset\Gamma$, we have
$\mathrm{d}\Gamma({\rm{id}}_{L^{2}})\Psi=p\Psi$. Note that $\mathrm{d}\Gamma$
is a morphism of the vector space of bounded operators on $L^{2}$:
$\mathrm{d}\Gamma(h_{1}+h_{2})=\mathrm{d}\Gamma(h_{1})+\mathrm{d}\Gamma(h_{2})$.

The creation and annihilation operators lead to the definition of the non-
commutative replacement for the Brownian motion, which is based on the
operator-valued functions

$A_{u}:t\mapsto A_{u}(t)=a(\mathds{1}_{[0,t[}\,u),\quad\mbox{and}\quad
A_{u}^{*}:t\mapsto A_{u}^{*}(t)=a^{*}(\mathds{1}_{[0,t[}\,u),$

called the annihilation and creation processes respectively, with $u$ a unit
vector in $\mathbb{C}^{k}$ and $\mathds{1}_{[0,t[}$ the characteristic
function of the interval $[0,t[$. Note that by Equation 2.2,

$\displaystyle[A_{u}^{*}(t),A_{v}(s)]=\langle
v,u\rangle\min(s,t)\,{\rm{id}}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ while }\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
[A_{u}(t),A_{v}(s)]=0\,,$ (2.3)

where, by a slight abuse of notations, we used the same brackets
$\langle.,.\rangle$ to denote the canonical inner product on $\mathbb{C}^{k}$.
To simplify the notations, for an orthonormal basis
$\mathbf{u}=\\{u_{1},\dotsc,u_{k}\\}$ of $\mathbb{C}^{k}$ we write

$\mathbf{A}_{\mathbf{u}}(t)=(A_{u_{1}}(t),\dotsc,A_{u_{k}}(t))\quad\mbox{and}\quad\mathbf{A}_{\mathbf{u}}^{*}(t)=(A_{u_{1}}^{*}(t),\dotsc,A_{u_{k}}^{*}(t))\,.$

Using the second quantization operator we define gauge processes that relate
to classical Poisson processes. For any $r\in M_{k}(\mathbb{C})$, let

$\Lambda_{r}:t\in\mathbb{R}_{+}\mapsto\mathrm{d}\Gamma(\operatorname{mult}(\mathds{1}_{[0,t[}r)),$

with $\operatorname{mult}(g)$ the operator on
$L^{2}(\mathbb{R}_{+};\mathbb{C}^{k})$ of point-wise multiplication by the
$k\times k$ matrix valued bounded function $g$. For a unit vector
$u\in\mathbb{C}^{k}$ we denote $\Lambda_{u}=\Lambda_{|u\rangle\langle u|}$,
and for $\mathbf{u}$ an orthonormal basis of $\mathbb{C}^{k}$, we write

$\Lambda_{\mathbf{u}\mathbf{u^{*}}}(t)=(\Lambda_{|u_{i}\rangle\langle
u_{j}|}(t))_{i,j=1}^{k}$

and
$\mathbf{\Lambda}_{\mathbf{u}}(t)=(\Lambda_{u_{1}}(t),\dotsc,\Lambda_{u_{k}}(t))$.

Using these creation, annihilation and gauge processes, we can reconstruct the
classical Brownian motion and Poisson processes. First note that, for
$\mathbf{u}$ an orthonormal basis of $\mathbb{C}^{k}$, the set
$\\{A_{u_{j}}(t)+A_{u_{j}}^{*}(t):t\in\mathbb{R}_{+},j\in\\{1,\dotsc,k\\}\\}$
is formed of commuting self-adjoint operators by Equation 2.3. Hence, by the
spectral theorem, the latter can be jointly represented as multiplication
operators on an appropriate Hilbert space. Moreover, these multiplication
operators in fact coincide with multiplication operators by $k$ independent
Brownian motions. Similarly, the processes
$t\mapsto\Lambda_{u_{i}}(t)+\sqrt{\lambda_{i}}A_{u_{i}}(t)+\sqrt{\lambda_{i}}A_{u_{i}}^{*}(t)+\lambda_{i}t$
can be jointly represented as $k$ independent Poisson processes of respective
intensity $\lambda_{i}$. Let us now state this fact rigorously:

######  Theorem 1.

Let $q\in\\{0,\dotsc,k\\}$. Let $B_{1},\dotsc,B_{q}$ be $q$ independent
Brownian motions and $N_{q+1},\dotsc,N_{k}$ be $p$ independent Poisson
processes of respective intensities $\lambda_{q+1},\dotsc,\lambda_{k}$. Assume
the Brownian motions and the Poisson processes are independent. Let
$\mathcal{W}$ be the Hilbert space of square integrable, complex functions of
$(B_{1},\dotsc,B_{q},N_{q+1},\dotsc,N_{k})$. Then, for any orthonormal basis
$\mathbf{u}$ of $\mathbb{C}^{k}$, there exists a unitary transformation
$J:\Gamma\to\mathcal{W}$, such that for any $t\in\mathbb{R}_{+}$,
$i\in\\{1,\dotsc,q\\}$, and $j\in\\{q+1,\dotsc,k\\}$,

$A_{u_{i}}(t)+A_{u_{i}}^{*}(t)=J^{*}M(B_{i}(t))J\quad\mbox{and}\quad\Lambda_{u_{j}}(t)+\sqrt{\lambda_{j}}(A_{u_{j}}(t)+A_{u_{j}}^{*}(t))+\lambda_{j}t=J^{*}M(N_{j}(t))J,$

with $M(x)$ the operator of multiplication by $x\in\mathbb{R}$. Moreover,
$J\Omega=\mathbf{1}$.

In [39, Section 3 and 4], Holevo provides a proof of Theorem 1 relying on
chaos expansions to construct explicitly the unitary transformation $J$.
Another proof, using Stone’s theorem, is outlined by Bouten _et al._ in [13,
Section 4.1 and 4.2]. Note that the proof can also be carried out using
Gelfand’s representation theorem for commutative C∗-algebras.

As $\mathbf{A}_{\bf u}+\mathbf{A}_{\bf u}^{*}$ is isomorphic to the
multiplication by $\mathbf{B}_{\bf u}$, we will only work in the Fock space
$\Gamma$ from now on, and denote $\mathbf{B}_{\bf u}=\mathbf{A}_{\bf
u}+\mathbf{A}_{\bf u}^{*}$. Similarly, we denote
$N_{u_{j}}=\Lambda_{u_{j}}(t)+\sqrt{\lambda_{j}}(A_{u_{j}}+A_{u_{j}}^{*})+\lambda_{j}t$.
The unitary transformation between the Hilbert spaces $\Gamma$ and
$\mathcal{W}$ will always be implicit.

It is however essential to keep in mind that for $u,v\in S^{k-1}(\mathbb{C})$
non-orthogonal, $[(A_{u}(t)+A_{u}^{*}(t)),(A_{v}(t)+A_{v}^{*}(t)]\neq 0$ and
$[(\Lambda_{u}(t)+A_{u}(t)+A_{u}^{*}(t)+t),(\Lambda_{v}(t)+A_{v}(t)+A_{v}^{*}(t)+t)]\neq
0$.

Summarizing, using non-commutative noises, we constructed a $q$ dimensional
Brownian motion $\mathbf{B}$ and $p$ independent Poisson processes
$\mathbf{N}$ as multiplication operators on the symmetric Fock space $\Gamma$.
In the next subsection, we start from these processes and introduce the so-
called non-commutative Girsanov transform to unravel the family of stochastic
processes we are concerned with in this paper.

####  2.1.2 Non-commutative Girsanov transforms

For classical stochastic processes, the drift of a diffusion or the intensity
of a point process can be modified through a so-called Girsanov
transformation: Let $(\mathbf{X},\mathbf{N})$ be a couple of processes where
is $\bf X$ the solution of
$d\mathbf{X}_{t}=\mathbf{a_{t}}\mathrm{d}t+\mathrm{d}\mathbf{W}_{t}$ with
$\mathbf{W}$ a multidimensional Brownian motion and $\mathbf{N}$ is a vector
of Poisson processes with corresponding vector of stochasic intensities
$\mathbf{\lambda}_{t}$. Let $\mathbb{P}_{T}$ denote the law of this couple of
processes up to time $T$. Let $\mathbb{Q}_{T}$ be a law such that, up to time
$T$, $(\mathbf{X},\mathbf{N})$ is a couple of independent processes with
$\mathbf{X}$ a multidimensional Brownian motion and $\mathbf{N}$ a vector of
independent Poisson processes of unit intensity also independent of
$\mathbf{X}$. Then, by Girsanovs theorem, there exists a
$\mathbb{Q}$-martingale $Z$, such that
$\mathrm{d}\mathbb{P}_{T}=Z_{T}\mathrm{d}\mathbb{Q}_{T}$. More precisely, $Z$
is a solution of

$Z_{t}^{-1}\mathrm{d}Z_{t}=\mathbf{a}_{t}.\mathrm{d}\mathbf{X}_{t}+(\mathbf{\lambda}_{t}-\mathbf{1}).(\mathrm{d}\mathbf{N}_{t}-\mathrm{d}t),\quad
Z_{0}=1$

where $\mathbf{x}.\mathbf{y}$ is the dot product between $\mathbf{x}$ and
$\mathbf{y}$.

We will now explore a non-commutative construction of such a law
transformation. In the next subsection we provide a martingale $Z$ that
performs said non-commutative law transformation, but requires the
introduction of an auxiliary process.

The transforms we discuss are essentially motivated by quantum physics. For
example, the noises $\mathbf{A}$ and $\mathbf{\Lambda}$ can model an
electromagnetic field. The transform then defines the state of the field
(_i.e._ , over the $*$-algebra of operators generated by $\mathbf{A}$ and
$\mathbf{\Lambda}$) after its interaction with an atom. For a physical
discussion of these models we refer the reader to [61]. The transformation we
consider can be obtained as limits of quantum physics axiomatic Hamiltonian
dynamics (see [3, 25]).

Our non-commutative transforms are defined using quantum stochastic calculus.
The latter is a well established theory that makes sense of integrals like
$\int_{0}^{t}\mathbf{F}_{s}.\mathrm{d}\mathbf{A}_{\mathbf{u}}(s)+\int_{0}^{t}\mathbf{G}_{s}.\mathrm{d}\mathbf{\Lambda}_{\mathbf{uu}^{*}}(s)$,
for $s\mapsto\mathbf{F}_{s}$ and $s\mapsto\mathbf{G}_{s}$ appropriate adapted
operator valued functions. They are adapted in the sense that for any
$t\in\mathbb{R}_{+}$, through the natural isomophism ${\cal
H}\otimes\Gamma\equiv{\cal
H}\otimes\Gamma_{s}(L^{2}([0,t[))\otimes\Gamma_{s}(L^{2}([t,\infty[))$,
$\mathbf{F}_{t}$ is mapped to
$\widehat{\mathbf{F}}_{t}\otimes{\rm{id}}_{\Gamma_{s}(L^{2}([t,\infty[))}$
with $\widehat{\mathbf{F}}_{t}$ an operator on ${\cal
H}\otimes\Gamma_{s}(L^{2}([0,t[))$, mimicking the classical stochastic
calculus definition. We do not define such integrals further here; for an
introduction to this subject we refer the reader to Parthasarathy’s book [56].
We use the definition of quantum stochastic integrals used there. For readers
used to classical stochastic calculus, the difference can be summarized by a
non-commutative version of Itô rules: for $u,v\in S^{k-1}(\mathbb{C})$ and
$r_{1},r_{2}\in M_{k}(\mathbb{C})$,

$\displaystyle\mathrm{d}A_{u}(t)\mathrm{d}A_{v}^{*}(t)=\langle
u,v\rangle\mathrm{d}t\ ,$
$\displaystyle\quad\mathrm{d}\Lambda_{r_{1}}(t)\mathrm{d}A_{u}^{*}(t)=\mathrm{d}A_{r_{1}u}^{*}(t)\
,$ (2.4)
$\displaystyle\mathrm{d}A_{u}(t)\mathrm{d}\Lambda_{r_{1}}(t)=\mathrm{d}A_{r_{1}^{\mathsf{T}}u}(t)\
,$
$\displaystyle\quad\mathrm{d}\Lambda_{r_{1}}(t)\mathrm{d}\Lambda_{r_{2}}(t)=\mathrm{d}\Lambda_{r_{1}r_{2}}(t)\
,$

and all other products of infinitesimal increments being $0$. Setting
$\mathbb{E}_{0}$ as the vacuum state

$\mathbb{E}_{0}:\mathrm{Poly}(a(f),a^{*}(f),\mathrm{d}\Gamma(h))_{f\in
L^{2},h\in\mathcal{B}(L^{2})}\to\mathbb{C},A\mapsto\langle\Omega,A\Omega\rangle,$

we have
$\mathbb{E}_{0}(\mathrm{d}A_{u}(t))=\mathbb{E}_{0}(\mathrm{d}A_{u}^{*}(t))=\mathbb{E}_{0}(\mathrm{d}\Lambda_{r}(t))=0$
for any $u\in\mathbb{C}^{k}$ and $r\in M_{k}(\mathbb{C})$. Anything we do in
the present article involving quantum stochastic calculus relies only on these
computational rules. They can thus be taken as axiomatic starting points.

To give an intuition for the stochastic integrals we will use, let us mention
that for any bounded function $f\in L^{2}(\mathbb{R}_{+})$, $u\in
S^{k-1}(\mathbb{C})$ and $r\in M_{k}(\mathbb{C})$,

$\int_{0}^{t}\overline{f(s)}\mathrm{d}A_{u}(s)=a(\mathds{1}_{[0,t[}fu),\quad\int_{0}^{t}f(s)\mathrm{d}A_{u}^{*}(s)=a^{*}(\mathds{1}_{[0,t[}fu),\quad\mbox{and}\quad\int_{0}^{t}f(s)\mathrm{d}\Lambda_{r}(s)=\mathrm{d}\Gamma(\operatorname{mult}(\mathds{1}_{[0,t[}f\
r)).$

Let $\mathbf{e}$ be the canonical basis of $\mathbb{C}^{k}$, let
$H\in\mathcal{B}(\mathcal{H})$ be such that $H=H^{*}$, and let
$\mathbf{L}\in\mathcal{B}(\mathcal{H})\otimes\mathbb{C}^{k}$. Then we define
$(s,t)\mapsto U_{t,s}$ to be the two parameter unitary group solution, for
$t\geq s$, of

$\mathrm{d}U_{t,s}=\left\\{\left(-iH-\frac{1}{2}\mathbf{L}^{*}.\mathbf{L}\right)\mathrm{d}t+\mathbf{L}.\mathrm{d}\mathbf{A}_{\mathbf{e}}^{*}(t)-\mathrm{d}\mathbf{A}_{\mathbf{e}}(t).\mathbf{L}^{*}\right\\}U_{t,s},\quad
U_{s,s}={\rm{id}}.$ (2.5)

Corollary 26.4 in [56] ensures that a solution exists and is a unitary group.
For $s=0$, we denote $U_{t,0}=U_{t}$. Note that For each $t\geq s>0$,
$U_{t,s}$ is a unitary operator from $\mathcal{H}\otimes\Gamma$ to itself.

Physically, this group models the interaction between the atom described by
$\mathcal{H}$ and the field described by the quantum noises. The law
transformation we are concerned with is the one resulting from the unitary
transformation of the initial state by the unitary $U_{T}$. Hence $U_{T}$ is
the non-commutative version of Girsanov’s transform.

Let us now introduce the filtered measurable space we work on:

######  Definition 1.

Let $\mathrm{D}$ be the set of càdlàg functions
$\mathbb{R}_{+}\to\mathbb{R}^{k}$ equipped with the Skorhokhod topology. Let
$\mathcal{F}$ be its Borel $\sigma$-algebra and
$(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}}$ its usual filtration. Then
$(\mathrm{D},\mathcal{F},(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}})$ is a
filtered measurable space.

As the definition of a probability measure relies on the spectral theorem for
commutative von Neumann algebras, we need to map our unbounded noise operators
to bounded ones. For that purpose, for any function $f\in\Omega$, let

$\displaystyle\iota(f):\mathbb{R}_{+}$ $\displaystyle\to\mathbb{C}^{k}$
$\displaystyle t$
$\displaystyle\mapsto((i+f_{1}(t))^{-1},\dotsc,(i+f_{k}(t))^{-1}).$

Note that $\iota$ is injective from $\mathrm{D}$ to the set of
$\mathbb{C}^{k}$ valued functions of $\mathbb{R}_{+}$. It is moreover bounded
with respect to the supremum norm. We can now define probability measure of
our main concern:

######  Definition 2.

Fix $q\in\\{1,\dotsc,k\\}$ and let $\mathbf{u}$ be an orthonormal basis of
$\mathbb{C}^{k}$. Let $\rho\in\mathcal{D}(\mathcal{H})$. For any
$T\in\mathbb{R}_{+}$, let $\mathbb{P}_{T}$ be the probability measure on
$(\mathrm{D},\mathcal{F}_{T})$ pushed forward by $\iota^{-1}$ of the spectral
measure of the smallest commutative von Neumann algebra containing

$(i+B_{u_{i}}(t))^{-1}\quad\mbox{and}\quad(i+\Lambda_{u_{j}}(t))^{-1},\qquad
t\in[0,T[,\ i=1,\dotsc,q,\ j=q+1,\dotsc,k,$

with respect to
$\Psi_{T}=U_{T}(\rho\otimes|\Omega\rangle\langle\Omega|)U_{T}^{*}$. Let
$\mathbb{P}\equiv\mathbb{P}_{\rho\otimes\Omega}$ be the unique extension of
$(\mathbb{P}_{t})_{t\in\mathbb{R}_{+}}$ to $(\mathrm{D},\mathcal{F})$.

The fact that $(\mathbb{P}_{t})_{t\in\mathbb{R}_{+}}$ is a consistent family
and thus can be extended to $\mathbb{P}$ by the Kolmogorov extension theorem
is a standard result in quantum filtering; see [13].

Note that Corollary 26.4 in [56] holds for more general quantum stochastic
differential equations than Eq. (2.5). In particular, it can involve gauge
processes $\Lambda_{r}$. However, we can restrict ourselves to Eq. (2.5)
without loss of generality: According to the corollary, fixing
$\mathbf{S}\in{\cal B}(\mathcal{H})\otimes M_{k}(\mathbb{C})$ as a unitary
operator, we could have used the unitary group $(s,t)\mapsto V_{t,s}$, which
is the solution of

$\mathrm{d}V_{t,s}=\left\\{(-iH-\frac{1}{2}\mathbf{L}^{*}.\mathbf{L})\mathrm{d}t+\mathbf{L}.\mathrm{d}\mathbf{A}_{\mathbf{e}}^{*}(t)-\mathrm{d}\mathbf{A}_{\mathbf{e}}(t).\mathbf{L}^{*}\mathbf{S}+\mathop{\rm
Tr}\nolimits_{\mathbb{C}^{k}}[(\mathbf{S}-{\rm{id}})\mathrm{d}\mathbf{\Lambda}_{\mathbf{ee}^{*}}(t)]\right\\}V_{t,s},\quad
V_{s,s}={\rm{id}},$ (2.6)

to define a reference state $\Phi_{T}$ that depends on the extra parameter
${\bf S}\in\mathcal{B}(\mathcal{H})\otimes M_{k}(\mathbb{C})$. However, using
$\Omega\equiv\Omega_{\Gamma(L^{2}([0,t[))}\otimes\Omega_{\Gamma(L^{2}([t,+\infty[))}$,
$A_{u}(t+\Delta t)-A_{u}(t)\equiv a(\mathds{1}_{[t,t+\Delta t[})$,
$\Lambda_{r}(t+\Delta
t)-\Lambda_{r}(t)\equiv\mathrm{d}\Gamma(r\mathds{1}_{[t,t+\Delta t[})$ and
$a(\mathds{1}_{[t,t+\Delta
t[})\Omega_{\Gamma(L^{2}([t,+\infty[))}=\mathrm{d}\Gamma(r\mathds{1}_{[t,t+\Delta
t[})\Omega_{\Gamma(L^{2}([t,+\infty[))}=0$ for any $\Delta t>0$ and $t\geq 0$,
we obtain that $\Phi_{T}$ and $\Psi_{T}$ are both solutions to the same
differential equation. Moreover, since this differential equation has a unique
solution, we get

$\Phi_{T}=\Psi_{T},\quad\forall T\geq 0.$

Therefore, the extra parameter $\bf S$ does not change the reference state
$\Psi_{T}$. Hence, the measure $\mathbb{P}$ is independent of $\bf S$ and we
set it to ${\rm{id}}$ in the remainder of the article.

The proof of the next proposition is standard in quantum filtering. It can be
found in [13] for example.

######  Proposition 1.

Fix $q\in\\{1,\dotsc,k\\}$ and let $\mathbf{u}$ be an orthonormal basis of
$\mathbb{C}^{k}$. Let $\mathbf{X}$ and $\mathbf{Y}$ be the operator valued
functions defined by

$X_{i}(t)=U_{t}^{*}B_{u_{i}}(t)U_{t}\quad\mbox{and}\quad
Y_{j}(t)=U_{t}^{*}\Lambda_{u_{j}}(t)U_{t},\qquad t\in[0,T[,\ i=1,\dotsc,q,\
j=q+1,\dotsc,k,$

respectively. Then, $\mathbf{X}$ and $\mathbf{Y}$ are solutions of

$\mathrm{d}X_{i}(t)=U_{t}^{*}\big{(}L_{u_{i}}+L_{u_{i}}^{*}\big{)}U_{t}\mathrm{d}t+\mathrm{d}B_{u_{i}}(t),\quad
X_{i}(0)=0,$

and

$\mathrm{d}Y_{j}(t)=\mathrm{d}\Lambda_{u_{j}}(t)+U_{t}^{*}L_{u_{j}}U_{t}\mathrm{d}A_{u_{j}}^{*}(t)+U_{t}^{*}L_{u_{j}}^{*}U_{t}\mathrm{d}A_{u_{j}}(t)+U_{t}^{*}L_{u_{j}}^{*}L_{u_{j}}U_{t}\mathrm{d}t,\quad
Y_{j}(0)=0,$

respectively, with $L_{u_{i}}\in\mathcal{B}(\mathcal{H})$ defined by

$\displaystyle L_{u_{i}}=({\rm{id}}_{\mathcal{B}(\mathcal{H})}\otimes\langle
u_{i}|)\mathbf{L}\,.$ (2.7)

Furthermore, the measure $\mathbb{P}$ is the push forward by $\iota^{-1}$ of
the spectral measure, with respect to
$\rho\otimes|\Omega\rangle\langle\Omega|$, of the smallest commutative von
Neumann algebra containing

$(i+X_{i}(t))^{-1}\quad\mbox{and}\quad(i+Y_{j}(t))^{-1},\qquad
t\in\mathbb{R}_{+},\ i=1,\dotsc,q,\ j=q+1,\dotsc,k.$

In the present article we study the concentration properties of the measure
$\mathbb{P}$, or, equivalently, the processes $\bf X$ and $\bf Y$. Note that
for a fixed ${\bf L}\in\mathcal{B}(\mathcal{H})\otimes\mathbb{C}^{k}$ and
initial state $\rho\in\mathcal{D}(\mathcal{H})$, these two processes and their
joint law $\mathbb{P}$ depend only on the choice of the unitary basis $\bf u$
and the index $q\in\\{1,\dotsc,k\\}$.

The fact that the set of operators
$\\{X_{i}(t),Y_{j}(t):i=1,\dotsc,q;j=q+1,\dotsc,k;t\in\mathbb{R}_{+}\\}$
defines a commutative family is not obvious, but follows from the fact that
$\bf u$ is an orthogonal basis and
$U_{t+s}^{*}B_{u_{i}}(t)U_{t+s}=U_{t}^{*}B_{u_{i}}(t)U_{t}$ and
$U_{t+s}^{*}\Lambda_{u_{j}}(t)U_{t+s}=U_{t}^{*}\Lambda_{u_{j}}(t)U_{t}$ for
any $i,j$ and $t,s\in\mathbb{R}_{+}$ – see [13] for example.

Whenever $\dim\mathcal{H}=1$, the operators $L_{u_{j}}$ are complex numbers,
and from Theorem 1 we obtain that $\bf X$ is a $q$-dimensional Brownian motion
plus a deterministic drift linear in time, whereas $\bf Y$ is an independent
$(k-q)$-tuple of independent Poisson processes with fixed intensity.

###  2.2 Quantum trajectories and classical Girsanov’s transform

The measure $\mathbb{P}$ of Definition 2 can also be defined using commutative
stochastic calculus and the usual Girsanov theorem. Let $q\in\\{1,\dotsc,k\\}$
and the orthonormal basis $\mathbf{u}$ of $\mathbb{C}^{k}$ be the defining
parameters of $\mathbb{P}$. Let $(\sigma_{t})_{t\in\mathbb{R}_{+}}$ be the
solution to the stochastic differential equation (SDE)
$\sigma_{0}=\rho\in\mathcal{D}$ and

$\begin{split}\mathrm{d}\sigma_{t}=&\mathcal{L}^{*}(\sigma_{t-})\mathrm{d}t\\\
&+\sum_{i=1}^{q}(L_{u_{i}}\sigma_{t-}+\sigma_{t-}L_{u_{i}})\mathrm{d}W_{i}(t)\\\
&+\sum_{j=q+1}^{k}\left(L_{u_{j}}\sigma_{t-}L_{u_{j}}^{*}-\sigma_{t-}\right)[\mathrm{d}N_{j}(t)-\mathrm{d}t],\end{split}$
(2.8)

with

$\mathcal{L}^{*}:X\mapsto-i[H,X]+\mathop{\rm
Tr}\nolimits_{\mathbb{C}^{k}}[\mathbf{L}X\mathbf{L}^{*}]-\frac{1}{2}(\mathbf{L}^{*}.\mathbf{L}X+X\mathbf{L}^{*}.\mathbf{L}),$

so that $\mathop{\rm Tr}\nolimits\circ\mathcal{L}^{*}=0$. The processes
$W_{1},\dotsc,W_{q}$ are independent Brownian motions and
$N_{q+1},\dotsc,N_{k}$ are independent Poisson process of unit intensities,
independent of the Brownian motions $W_{1},\dotsc,W_{q}$. Denote $\mathbb{Q}$
the probability measure on $(\mathrm{D},\mathcal{F})$ of
$t\mapsto(W_{1}(t),\dotsc,W_{q}(t),N_{k+1}(t),\dotsc,N_{k}(t))$ and
$\mathbb{Q}_{t}$ its restriction to $\mathcal{F}_{t}$. The process
$t\mapsto\sigma_{t}$ is positive semi-definite valued (see [13, 4]) and allows
us to recover $\mathbb{P}$ as a Girsanov transform of $\mathbb{Q}$.

######  Proposition 2.

Fix $q\in\\{1,\dotsc,k\\}$ and let $\mathbf{u}$ be an orthonormal basis of
$\mathbb{C}^{k}$. Let $\rho\in\mathcal{D}$ and $\sigma$ be the solution of
(2.8). Let $Z:t\mapsto\mathop{\rm Tr}\nolimits[\sigma_{t}]$. Then $Z$ is a
positive Doléans-Dade exponential martingale and for any $t\in\mathbb{R}_{+}$,

$\mathrm{d}\mathbb{P}_{t}=Z_{t}\mathrm{d}\mathbb{Q}_{t}.$

###### Proof.

See [13, 4]. ∎

The process $\varrho:t\mapsto\sigma_{t}/\mathop{\rm Tr}\nolimits[\sigma_{t}]$
lives in the set of density matrices $\mathcal{D}$. It can be shown (see [13,
4]) that with respect to $\mathbb{P}$, $\varrho$ is a quantum trajectory in
the sense that it is the unique solution to

$\begin{split}\mathrm{d}\varrho_{t}=&\mathcal{L}^{*}(\varrho_{t-})\mathrm{d}t\\\
&+\sum_{i=1}^{q}\left(L_{u_{i}}\varrho_{t-}+\varrho_{t-}L_{u_{i}}-\mathop{\rm
Tr}\nolimits[(L_{u_{i}}+L_{u_{i}}^{*})\rho_{t-}]\varrho_{t-}\right)\mathrm{d}B_{u_{i}}(t)\\\
&+\sum_{j=q+1}^{k}\left(\frac{L_{u_{j}}\varrho_{t-}L_{u_{j}}^{*}}{\mathop{\rm
Tr}\nolimits[L_{u_{j}}^{*}L_{u_{j}}\varrho_{t-}]}-\varrho_{t-}\right)[\mathrm{d}N_{u_{j}}(t)-\mathop{\rm
Tr}\nolimits[L_{u_{j}}^{*}L_{u_{j}}\varrho_{t-}]\mathrm{d}t].\end{split}$
(2.9)

Here $B_{u_{1}},\dotsc,B_{u_{q}}$ are $q$ independent Brownian motions and
$N_{q+1},\dots,N_{k}$ are $k-q$ inhomogeneous Poissons processes with
stochastic intensities $\mathop{\rm
Tr}\nolimits[L_{u_{q+1}}^{*}L_{u_{q+1}}\rho_{t-}]\mathrm{d}t,\dotsc,\mathop{\rm
Tr}\nolimits[L_{u_{k}}^{*}L_{u_{k}}\rho_{t-}]\mathrm{d}t$ respectively, such
that for each $j\in\\{q+1,\dotsc,k\\}$, $t\mapsto
N_{u_{j}}(t)-\int_{0}^{t}\mathop{\rm
Tr}\nolimits[L_{u_{j}}^{*}L_{u_{j}}\rho_{s-}]\mathrm{d}s$ is a martingale.
Moreover, following [13], we deduce that with respect to $\mathbb{P}$, for any
$i\in\\{1,\dotsc q\\}$

$X_{i}\sim t\mapsto B_{u_{i}}(t)-\int_{0}^{t}\mathop{\rm
Tr}\nolimits[(L_{u_{i}}+L_{u_{i}}^{*})\varrho_{s-}]\mathrm{d}s$

and for any $j\in\\{q+1,\dotsc,k\\}$,

$Y_{j}\sim N_{j}.$

This shows the relationship between the measurement signals $\mathbf{X}$ and
$\mathbf{Y}$ and the underlying quantum system state $\varrho$. The state
$\varrho$ determines the drift of the measurement signal $\mathbf{X}$ and the
jump intensities of the measurement signal $\mathbf{Y}$. In particular, since
$\mathbb{E}[\varrho_{t}]=\mathrm{e}^{t\mathcal{L}^{*}}\rho$,

$\mathbb{E}[X_{i}]=\mathop{\rm
Tr}\nolimits\left[(L_{u_{i}}+L_{u_{i}}^{*})\int_{0}^{t}\mathrm{e}^{s\mathcal{L}^{*}}\rho\,\mathrm{d}s\right]\quad\quad{and}\quad\mathbb{E}[Y_{j}]=\mathop{\rm
Tr}\nolimits\left[L_{u_{j}}^{*}L_{u_{j}}\int_{0}^{t}\mathrm{e}^{s\mathcal{L}^{*}}\rho\,\mathrm{d}s\right].$
(2.10)

The operator $\mathcal{L}^{*}$ in particular is the generator of a semigroup
of trace preserving completely positive maps. In the next section we detail
the properties of such semigroups and their duals.

###  2.3 Quantum Markov semigroups

In this section we give the definition of a quantum Markov semi-group, discuss
its relationship to non-commutative Girsanov transforms and quantum
trajectories and present some properties of symmetric quantum Markov semi-
groups.

####  2.3.1 Quantum Markov semi-groups

An open quantum system is said to undergo Markovian dynamics if its
interaction with its environment is memoryless. In this case its dynamics is
modeled by a quantum Markov semi-group (or quantum dynamical semi-group).

######  Definition 3 (QMS).

A quantum Markov semi-group (QMS) $t\mapsto e^{t\mathcal{L}}$ is a uniformly
continuous semi-group of completely positive111A map $\Phi:\mathcal{B}({\cal
H})\to\mathcal{B}({\cal H})$ is completely positive if and only if
$\Phi\otimes{\rm{id}}_{M_{n}(\mathbb{C})}$ is a positive map for any natural
number $n$. maps from $\mathcal{B}({\cal H})$ to itself that preserves the
identity: $e^{t\mathcal{L}}({\rm{id}}_{{\cal H}})={\rm{id}}_{{\cal H}}$.

The generator $\mathcal{L}$ of a QMS can always be written in Lindblad form
([49, 35]).

######  Theorem 2.

If $\mathcal{L}$ is the generator of a $\operatorname{QMS}$, then there exist
$H\in\mathcal{B}_{\operatorname{sa}}({\cal H})$, $k\in\mathbb{N}$ and
$\mathbf{L}:{\cal H}\to{\cal H}\otimes\mathbb{C}^{k}$, such that

$\mathcal{L}:X\mapsto
i[H,X]+\mathbf{L}^{*}(X\otimes{\rm{id}}_{\mathbb{C}^{k}})\mathbf{L}-\frac{1}{2}(\mathbf{L}^{*}.\mathbf{L}\,X+X\,\mathbf{L}^{*}.\mathbf{L}).$
(2.11)

Conversely, if $\mathcal{L}$ is as above, then it generates a
$\operatorname{QMS}$.

The generator $\mathcal{L}$ is called a Lindbladian. The operator
$\mathcal{L}_{D}:X\mapsto\mathcal{L}(X)-i[H,X]$ is sometimes called the
dissipator. The dual $\mathcal{L}^{*}$ of $\mathcal{L}$ with respect to the
Hilbert-Schmidt inner product is the generator of a semi-group of completely
positive maps preserving the trace. In particular,
$e^{t\mathcal{L}^{*}}\mathcal{D}({\cal H})\subset\mathcal{D}({\cal H})$. By
abuse of notation we also call $t\mapsto e^{t\mathcal{L}^{*}}$ a QMS.

Note that Theorem 2 shows that the operators $\mathcal{L}^{*}$ in Eqs. (2.8)
and (2.9) are generators of a QMS. Conversely, for any QMS, we can define a
process $\varrho$, which is a solution of a SDE like (2.9), such that
$\mathbb{E}[\varrho_{t}]=\mathrm{e}^{t\mathcal{L}^{*}}\rho$. Such a process
$\varrho$ is then called an unraveling of the QMS. Unravelings are used as
numerical tools to study QMS ([18, 34]).

More importantly for us, any QMS can be unitarily dilated ([41, Theorem 7.3]):

######  Theorem 3 (Dilation).

Let $(s,t)\mapsto U_{t,s}$ be the two parameter unitary group solution of Eq.
(2.5) with the same $H$ and $\mathbf{L}$ as in the definition of the generator
$\mathcal{L}$. Then for any $X\in\mathcal{B}(\mathcal{H})$,

$\mathrm{e}^{t\mathcal{L}}(X)=\mathop{\rm
Tr}\nolimits_{\Gamma}[U_{t,0}^{*}(X\otimes{\rm{id}}_{\Gamma})U_{t,0}({\rm{id}}_{{\cal
H}}\otimes\Omega)].$

The unitary group $(s,t)\mapsto U_{t,s}$ is called a dilation of the QMS.

Note that dilations and unravelings are not unique. The next proposition,
translated from [62, Proposition 7.4], characterizes those $H$ and
$\mathbf{L}$ which define the same $\operatorname{QMS}$.

######  Proposition 3.

The operator couples $(H,\mathbf{L})\in\mathcal{B}_{\operatorname{sa}}({\cal
H})\times\mathcal{B}({\cal H};{\cal H}\otimes\mathbb{C}^{k})$ and
$(H^{\prime},\mathbf{L}^{\prime})\in\mathcal{B}_{\operatorname{sa}}({\cal
H})\times\mathcal{B}({\cal H};{\cal H}\otimes\mathbb{C}^{k})$ define the same
$\operatorname{QMS}$ if and only if there exists a $k\times k$ unitary matrix
$U$, a vector $\mathbf{c}\in\mathbb{C}^{k}$ and a real constant
$E\in\mathbb{R}$ such that

$\mathbf{L}^{\prime}=({\rm{id}}_{\cal H}\otimes U)\mathbf{L}+{\rm{id}}_{\cal
H}\otimes\mathbf{c}$

and

$H^{\prime}=H-\operatorname{Im}(\mathbf{L}^{*}({\rm{id}}_{\cal
H}\otimes\mathbf{c}))+E.$

####  2.3.2 Dirichlet form

A QMS generically has a unique stationary state [38], i.e., there exists a
unique $\sigma\in\mathcal{D}({\cal H})$ such that $\mathcal{L}^{*}(\sigma)=0$.
If this stationary state $\sigma$ is in addition of full rank, the
corresponding QMS is called _primitive_. In that case, one can show [14] that
every initial state $\rho$ converges to $\sigma$ when evolved with respect to
the QMS. From now on we will assume $t\mapsto e^{t\mathcal{L}}$ to be
primitive and denote by $\sigma$ its invariant state with full-rank.

Let us first equip $\mathcal{B}({\cal H})$ with a Hilbert space structure.

######  Definition 4 (KMS inner product).

The KMS inner product on $\mathcal{B}({\cal H})$ is defined for
$X,Y\in\mathcal{B}({\cal H})$ by

$\langle X,Y\rangle_{\operatorname{KMS}}=\mathop{\rm
Tr}\nolimits[\sigma^{\frac{1}{2}}X^{*}\sigma^{\frac{1}{2}}Y].$

Equipped with this inner product, $\mathcal{B}({\cal H})$ is a Hilbert space.

As we most often use the KMS inner product, when no confusion is possible, we
may sometimes write
$\langle.,.\rangle_{\sigma}\equiv\langle.,.\rangle_{\operatorname{KMS}}$ to
emphasize the state $\sigma$ with respect to which the inner product is
defined. In what follows, we also refer to the norm associated to
$\langle.,.\rangle_{\operatorname{KMS}}$ as $\|.\|_{L^{2}(\sigma)}$ and refer
to the corresponding non-commutative weighted $L^{2}$ space as
$L^{2}(\sigma)$. It will also be convenient to define the following operator:

$\displaystyle\Gamma_{\sigma}:{\cal B}({\cal H})\to{\cal B}({\cal H})\,,\qquad
X\mapsto\sigma^{\frac{1}{2}}X\sigma^{\frac{1}{2}}\,.$ (2.12)

For a map $\Phi:\mathcal{B}({\cal H})\to\mathcal{B}({\cal H})$ we denote by
$\Phi^{\operatorname{KMS}}$ its dual with respect to the KMS inner product.

While we primarily work with KMS inner products, we will sometimes also use
the GNS inner product.

######  Definition 5 (GNS inner product).

The GNS inner product on $\mathcal{B}({\cal H})$ is defined for
$X,Y\in\mathcal{B}({\cal H})$ by

$\langle X,Y\rangle_{\operatorname{GNS}}=\mathop{\rm Tr}\nolimits[\sigma
X^{*}Y].$

Equipped with this inner product, $\mathcal{B}({\cal H})$ is a Hilbert space.

A last possible inner product we would like to mention here is the BKM inner
product. While we do not use it, it is relevant to the study of semi-groups
that are gradient flows of the entropy ([16]).

######  Definition 6 (BKM inner product).

The BKM inner product on $\mathcal{B}({\cal H})$ is defined for
$X,Y\in\mathcal{B}({\cal H})$ by

$\langle X,Y\rangle_{\operatorname{BKM}}=\int_{0}^{1}\mathop{\rm
Tr}\nolimits[\sigma^{1-t}X^{*}\sigma^{t}Y]\,\mathrm{d}t.$

Equipped with this inner product, $\mathcal{B}({\cal H})$ is a Hilbert space.

We can now define the Dirichlet form associated to the generator
$\mathcal{L}$.

######  Definition 7 (Dirichlet form).

The (symmetrized) Dirichlet form of the QMS generator $\mathcal{L}$ is defined
for any $X\in\mathcal{B}({\cal H})$ by

$\mathcal{E}_{\mathcal{L}}(X)=-\frac{1}{2}\big{(}\langle
X,\mathcal{L}(X)\rangle_{\operatorname{KMS}}+\langle\mathcal{L}(X),X\rangle_{\operatorname{KMS}}\big{)}.$

Note that if $\mathcal{L}$ is symmetric with respect to the KMS inner product,
we recover the usual (non-symmetrized) Dirichlet form. In the next section we
further investigate these symmetric generators of QMS.

In the rest of the article, to any map
$T:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$,
$T^{\operatorname{KMS}}$ denotes its dual with respect to the KMS inner
product.

####  2.3.3 Quantum detailed balance

For a QMS, the notion of symmetry is most often referred to through its
physical interpretation of detailed balance. There exist different definitions
of quantum detailed balance (QDB) depending on the inner product used on
$\mathcal{B}({\cal H})$. We focus on the GNS, BKM and KMS notions of QDB.

######  Definition 8.

The QMS, or equivalently its generator, is said to verify KMS, GNS or BKM QDB
if it is symmetric with respect to the respective inner product.

The GNS notion of QDB is the most restrictive in the sense that

$\mbox{GNS QDB}\implies\mbox{KMS and BKM QDB}.$

A proof of this implication can be found in [16, Theorem 2.9]. Counterexamples
to KMS QDB implying GNS QDB can be found in [16, Appendix B]. Similarly,
counterexamples to BKM QDB implying GNS QDB can be constructed. In [10], an
example of a BKM and KMS symmetric map that is not GNS symmetric is provided.
One can also prove that the BKM and KMS QDB notions are not comparable,
meaning that there exist maps that are one and not the other in both cases
(see Appendix B).

The notions of detailed balance can be extended by requiring symmetry only up
to a unitary or anti-unitary mapping [10]. Using this generalization it was
proven in the same reference that KMS QDB is equivalent to the vanishing of
some notion of entropy production. However, we do not consider this
generalization here as it would dramatically obscure our discussion.

We mentioned in Proposition 3 that a generator $\mathcal{L}$ can be defined
using different operators $H$ and $\mathbf{L}$. Following [29] and [1, Theorem
4.4], KMS QDB singles out a subclass of these operators. From now on, we
define the modular operator

$\displaystyle\Delta\equiv\Delta_{\sigma}:{\cal B}({\cal H})\to{\cal B}({\cal
H})\,,\qquad X\mapsto\sigma X\sigma^{-1}\,.$ (2.13)

######  Theorem 4.

The generator $\mathcal{L}$ verifies $\operatorname{KMS}$ $\operatorname{QDB}$
if and only if there exists $H\in\mathcal{B}_{\operatorname{sa}}({\cal H})$
and $\mathbf{L}:{\cal H}\to{\cal H}\otimes\mathbb{C}^{k}$ such that Eq. (2.11)
holds and

$(\Delta^{\frac{1}{2}}\circ\operatorname{adj}\otimes{\rm{id}}_{\mathbb{C}^{k}})\mathbf{L}=\mathbf{L},$

where $\operatorname{adj}:\mathcal{B}({\cal H})\to\mathcal{B}({\cal H});\
X\mapsto X^{*}$ and

$H=\frac{i}{2}\int_{0}^{\infty}e^{-t\sigma^{\frac{1}{2}}}[\mathbf{L}^{*}.\mathbf{L},\sigma^{\frac{1}{2}}]e^{-t\sigma^{\frac{1}{2}}}\mathrm{d}t=\frac{i}{2}\tanh\circ\log(\Delta^{\frac{1}{4}})(\mathbf{L}^{*}.\mathbf{L}).$

Assuming GNS QDB imposes a finer structure [28].

######  Theorem 5.

Let $d=\dim{\cal H}$. The generator $\mathcal{L}$ verifies
$\operatorname{GNS}$ $\operatorname{QDB}$ if and only if there exists
$\mathbf{L}:{\cal H}\to{\cal H}\otimes\mathbb{C}^{d^{2}}$ such that Eq. (2.11)
holds with $H=0$ and

$(\Delta\otimes{\rm{id}}_{\mathbb{C}^{d^{2}}})\mathbf{L}=({\rm{id}}_{\cal
H}\otimes D)\mathbf{L}$

for some diagonal $d^{2}\times d^{2}$ matrix $D$ unitarily equivalent to the
modular operator $\Delta$.

Note that the operators $\mathbf{L}$ of Theorems 4 and 5 are generally
distinct. In the next section we will use the Dirichlet form to derive
concentration inequalities for the probability measures of Definition 2.
Assuming KMS QDB, we then show that these inequalities are optimal.

##  3 Bounding the tail probability of indirect measurements

Consider a system $S$ with associated Hilbert space ${\cal H}_{S}$, which is
coupled to an environment $E$ described by the Fock space $\Gamma$ such that
the evolution of the combined system $S\vee E$ is described by the unitary
evolution $U_{t}$ defined through the quantum stochastic differential equation
(cf. Eq. (2.5))

$\mathrm{d}U_{t}U_{t}^{*}=\left(-iH-\frac{1}{2}\mathbf{L}^{*}.\mathbf{L}\right)\mathrm{d}t+\mathbf{L}.\mathrm{d}\mathbf{A}_{\mathbf{e}}^{*}(t)-\mathrm{d}\mathbf{A}_{\mathbf{e}}(t).\mathbf{L}^{*},\quad
U_{0}={\rm{id}}\ .$ (3.1)

This combined evolution corresponds to – given that the environment is in the
Fock space vacuum state $\Omega$ – an evolution of the reduced system $S$
described by a quantum Markov semigroup $t\mapsto\mathrm{e}^{t\mathcal{L}}$
(cf. Section 2.3.1). We will assume here and in the following that
$t\mapsto\mathrm{e}^{t{\cal L}}$ is primitive, i.e., has a unique full-rank
stationary state denoted by $\sigma$.

The goal is to indirectly measure observables of the system $S$. In the
following, we will consider observables of the form

$\displaystyle
O^{B}(u):=\sum_{j=1}^{k}u_{j}\,(L_{j}+L_{j}^{*})=L_{u}+L_{u}^{*}\qquad\text{
and }\qquad O^{P}(u):=L_{u}^{*}L_{u}$ (3.2)

for some $u\in\mathbb{R}^{k}$ with $\|u\|=1$, where $L_{j}$ denotes the $j$-th
component of $\mathbf{L}$, $j\in\\{1,\dots,k\\}$, and $L_{u}$ is defined as in
Equation 2.7. For the indirect measurement of the observables, here the system
which is physically measured is a subsystem (i.e., a subset of the modes) of
the environment $E$. Alternatively, one could introduce a separate measurement
system $M$ associated to additional bosonic modes. To perform an indirect
measurement of observables on $S$, the modes of $M$ would then have to be
coupled to the system $S\vee E$ such that the resulting combined evolution of
the system $S\vee E\vee M$ is of the form (3.1), but with the vectors
$\mathbf{A}_{\mathbf{e}}(t)$, $\mathbf{A}_{\mathbf{e}}^{*}(t)$ enlarged to
also incorporate the modes of the measurement system, which are then coupled
to the observables of interest.

Let $1\leq q\leq\ell$ be natural numbers, and let
$\mathbf{u}:=\\{u_{1},\cdots,u_{\ell}\\}$ be a set of orthogonal normalized
vectors in $\mathbb{R}^{k}$. We consider the following vector of observables
to be (indirectly) measured:

$\mathbf{O}(\mathbf{u})=\left(O^{B}(u_{1}),\dots,O^{B}(u_{q}),O^{P}(u_{q+1}),\dots,O^{P}(u_{\ell})\right)\,.$

We recall that the corresponding processes can be simultaneously measured (cf.
Section 2.1.1). In order to distinguish the Brownian and Poisson parts, we
introduce the notations $\mathbf{u}^{B}:=(u_{1},\dots,u_{q})$ and
$\mathbf{u}^{P}:=(u_{q+1},\dots,u_{\ell})$. The corresponding estimator
$\mathbf{E}$ is then defined as

$\displaystyle\mathbf{E}_{t,\mathbf{u}}=\frac{1}{t}\,U_{t}^{*}(\mathbf{A}_{\mathbf{u}^{B}}(t)+\mathbf{A}_{\mathbf{u}^{B}}^{*}(t),{\Lambda}_{\mathbf{u}^{P}}(t))\,U_{t}\equiv\frac{1}{t}\,(\mathbf{X}(t),\mathbf{Y}(t))\,,$
(3.3)

where $\mathbf{X}$ and $\mathbf{Y}$ are the operator valued functions defined
in Proposition 1. We denote by
${m}_{\mathbf{u}}=(m_{\mathbf{u}^{B}}^{B},m_{\mathbf{u}^{P}}^{P})\in\mathbb{R}^{\ell}$
the vector with entries

$\displaystyle\left({m}_{\mathbf{u}}\right)_{j}\equiv{m}_{u_{j}}\coloneqq\left\\{\begin{aligned}
&\mathop{\rm Tr}\nolimits(\sigma\,O^{B}(u_{j}))\ ,&\quad
j\in\\{1,\dots,q\\}\;\;\\\ &\mathop{\rm Tr}\nolimits(\sigma\,O^{P}(u_{j}))\
,&\quad j\in\\{q+1,\dots,\ell\\}\,.\end{aligned}\right.$

###  3.1 The main upper bound

Given $r\in\mathbb{R}^{\ell}_{+}$, we are interested in upper bounding the
probability

$\displaystyle\mathbb{P}(\cap_{i}\\{E_{t,u_{i}}-m_{u_{i}}\geq r_{i}\\})\,.$
(3.4)

To this end, let us define $\Phi_{u_{i}}(X):=L_{u_{i}}^{*}X+XL_{u_{i}}$,
$i\leq q$, and $\Psi_{u_{j}}(X):=L_{u_{j}}^{*}XL_{u_{j}}$, $j\geq q+1$.
Furthermore, let

$\displaystyle f_{u_{i}}^{B}(X):=\frac{1}{2}\,\langle
X,\,(\Phi_{u_{i}}+\Phi_{u_{i}}^{\operatorname{KMS}})(X)\rangle_{\sigma}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ i\leq q$
$\displaystyle f_{u_{j}}^{P}(X):=\frac{1}{2}\,\langle
X,\,(\Psi_{u_{j}}+\Psi_{u_{j}}^{\operatorname{KMS}})(X)\rangle_{\sigma}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ j\geq q+1\,,$

and $({f}_{\mathbf{u}^{B}}^{B})_{i}(X):=f_{u_{i}}^{B}(X)$,
$({f}_{\mathbf{u}^{P}}^{P})_{j}(X):=f_{u_{j}}^{P}(X)$.

We recall that the relative entropy between (possibly non-normalized) mass
functions $\mathbf{p}$ and $\mathbf{q}$ is defined as

$\displaystyle
D(\mathbf{p}\|\mathbf{q}):=\sum_{l}\,p_{l}\,\ln\Big{(}\frac{p_{l}}{q_{l}}\Big{)}-p_{l}+q_{l}\geq
0\,.$

Note that $D(\mathbf{p}\|\mathbf{q})=0$ if and only if
$\mathbf{p}=\mathbf{q}$, and that it may be infinite if $q_{l}=0$ and
$p_{l}>0$ for some $l$.

The next theorem constitutes the main result of this section. Its proof is
postponed to Section 4.

######  Theorem 6.

Let $t\mapsto\mathrm{e}^{t{\cal L}}$ be a primitive quantum Markov semigroup
on the finite dimensional matrix algebra $M_{n}(\mathbb{C})$ with invariant
state $\sigma$. Then, for all $t\geq 0$, all initial states $\rho\in{\cal
D}(\mathbb{C}^{n})$, any ${r}\in\mathbb{R}_{+}^{\ell}$ and any family
$\mathbf{u}=\\{u_{1},\dots,u_{\ell}\\}$ of orthogonal, normalized vectors in
$\mathbb{C}^{k}$:

$\displaystyle\frac{\mathbb{P}(\cap_{i}\\{E_{t,u_{i}}-m_{u_{i}}\geq
r_{i}\\})}{\|\Gamma_{\sigma}^{-1}(\rho)\|_{L^{2}(\sigma)}}$ (3.5)
$\displaystyle\qquad\leq\exp\Big{(}-t\underset{\|X\|_{L^{2}(\sigma)}=1}{\inf}\,\big{\\{}{\cal
E}(X)+\frac{1}{2}\,\|{r}^{B}+{m}_{\mathbf{u}^{B}}^{B}-{f}_{\mathbf{u}^{B}}^{B}(X)\|^{2}+D({r}^{P}+{m}_{\mathbf{u}^{P}}^{P}\|{f}_{\mathbf{u}^{P}}^{P}(X))\big{\\}}\Big{)}\,.$

In Section 3.2, we show that this concentration bound is optimal in the sense
that for KMS symmetric semigroups and appropriately chosen Kraus operators,
for large times we obtain a large deviation principle for $\mathbb{P}$ with
rate function given by

$\displaystyle
I_{\mathbf{u}}({s}):=\underset{\|X\|_{L^{2}(\sigma)}=1}{\inf}\,\big{\\{}{\cal
E}(X)+\tfrac{1}{2}\,\|{s}^{B}-{f}^{B}_{\mathbf{u}^{B}}(X)\|^{2}+D({s}^{P}\|{f}^{P}_{\mathbf{u}^{P}}(X))\big{\\}}.$
(3.6)

###  3.2 Reversible semigroups: optimal concentration bound and large
deviation principle

Following the proofs of [42], for any ${\lambda}\in\mathbb{R}^{\ell}$,

$e({\lambda}):=\lim_{t\to\infty}\frac{1}{t}\ln\mathbb{E}(\exp(t\,{\lambda}.\mathbf{E}_{t,\mathbf{u}}))$

exists and ${\lambda}\mapsto e({\lambda})$ is differentiable everywhere. It
follows then from the Gärtner-Ellis theorem that the estimator $\mathbf{E}$
verifies a large deviation principle in the sense that for any Borel set
$B\subset\mathbb{R}^{\ell}$,

$-\inf_{{s}\in\operatorname{int}(B)}J({s})\leq\liminf_{t\to\infty}\frac{1}{t}\ln\mathbb{P}(\mathbf{E}_{t,\mathbf{u}}\in
B)\leq\limsup_{t\to\infty}\frac{1}{t}\ln\mathbb{P}(\mathbf{E}_{t,\mathbf{u}}\in
B)\leq-\inf_{{s}\in\operatorname{cl}(B)}J({s})$

with $\operatorname{int}(B)$ the interior of $B$, $\operatorname{cl}(B)$ the
closure of $B$ and

$\displaystyle
J:{s}\mapsto\sup_{{\lambda}\in\mathbb{R}^{\ell}}{\lambda}.{s}-e({\lambda})$
(3.7)

taking values in $[0,+\infty]$ and being lower semi-continuous and convex as
the supremum of affine continuous functions. The next theorem provides
sufficient conditions ensuring $J=I$.

######  Theorem 7.

Assume $\mathcal{L}$ verifies $\operatorname{KMS}$ $\operatorname{QDB}$. Let
$H\in\mathcal{B}({\cal H})$ and $\mathbf{L}:{\cal H}\to{\cal
H}\otimes\mathbb{C}^{k}$ be as in Theorem 4 and let $(s,t)\mapsto U_{t,s}$ be
the two parameter unitary group solution of Eq. (2.5) with these two
operators. Then for $U_{T}=U_{T,0}$, any orthonormal family
$\mathbf{u}=\\{u_{1},\dots,u_{\ell}\\}$ of $\mathbb{R}^{k}$ and any
$q\in\\{1,\dotsc,\ell\\}$, the measure $\mathbb{P}$ of Definition 2 is such
that for any ${s}\in\mathbb{R}^{\ell}$

$J(s)=I(s)$

with $J:\mathbb{R}^{k}\to[0,+\infty]$ the large deviation principle rate
function defined by Eq. (3.7) and $I:\mathbb{R}^{k}\to[0,+\infty]$ the
concentration bound defined by Eq. (3.6).

###### Proof.

Since
$(\Delta^{\frac{1}{2}}\circ\operatorname{adj}\otimes{\rm{id}}_{\mathbb{C}^{k}})\mathbf{L}=\mathbf{L}$
and for any $i\in\\{1,\dotsc,\ell\\}$ $u_{i}$ is a real unit vector,
$\sigma^{\frac{1}{2}}L_{u_{i}}^{*}\sigma^{-\frac{1}{2}}=L_{u_{i}}$. It then
follows that $\Phi_{i}:X\mapsto L_{u_{i}}^{*}X+XL_{u_{i}}$ and
$\Psi_{i}:X\mapsto L_{u_{i}}^{*}XL_{u_{i}}$ are both KMS symmetric. Therefore,
for any $\lambda\in\mathbb{R}^{\ell}$, the perturbed generator
$\mathcal{L}_{\lambda,\mathbf{u}}$ defined in Eq. (4.2) in the proof of
Theorem 6 is KMS symmetric. Hence the spectrum of
$\mathcal{L}_{\lambda,\mathbf{u}}$ is real and

$\max\\{{\operatorname{Re}}(x):x\in{\rm
sp}\mathcal{L}_{\lambda,\mathbf{u}}\\}=\max\\{x:x\in{\rm
sp}\mathcal{L}_{\lambda,\mathbf{u}}\\}=\sup_{\|X\|_{L^{2}(\sigma)}=1}\langle
X,\mathcal{L}_{\lambda,\mathbf{u}}(X)\rangle_{\operatorname{KMS}}=-\inf_{\|X\|_{L^{2}(\sigma)}=1}\mathcal{E}_{\lambda}(X).$

From a direct extension of [42],
$e(\lambda)=\max\\{{\operatorname{Re}}(x):x\in{\rm
sp}\mathcal{L}_{\lambda,\mathbf{u}}\\}$. Hence,

$e(\lambda)=-\inf_{\|X\|_{L^{2}(\sigma)}=1}\mathcal{E}_{\lambda}(X).$

Since $J$ is defined as the Legendre transform of the left hand side and,
following the proof of Theorem 6, particularly Eq. (4.15), $I$ is the Legendre
transform of the right hand side, we deduce $J=I$. ∎

The last theorem proves that the bound in Theorem 6 is optimal. Indeed, if
${r}\in\operatorname{int}(\\{x\in\mathbb{R}_{+}^{\ell}:I({m}_{\mathbf{u}}+x)<\infty\\})$
the convexity of $I$ implies it is continuous in a neighborhood of
${m}_{\mathbf{u}}+{r}$, and therefore minimal in ${m}_{\mathbf{u}}+{r}$ on
both the interior and closure of
$\\{{s}\in\mathbb{R}^{\ell}:s_{i}-m_{u_{i}}\geq r_{i},\,\forall
i\in\\{1,\dotsc,\ell\\}\\}$. It follows that the inequalities in the large
deviation principle are saturated and

$\lim_{t\to\infty}\frac{1}{t}\ln\mathbb{P}(\cap_{i}\\{E_{t,u_{i}}-m_{i}\geq
r_{i}\\})=-I({m}_{\mathbf{u}}+r).$

##  4 Proof of Theorem 6

This section is dedicated to the proof of Theorem 6. In Section 4.1, we show
that there exists an upper bound on the tail probability (3.4) that depends on
a certain perturbed semigroup. As pointed out in the subsequent Section 4.2,
we can in particular bound the tail probaility in terms of the
$L^{2}(\sigma)\to L^{2}(\sigma)$ contraction of this perturbed semigroup.
Moreover, we show how to further bound the latter contraction in terms of the
Dirichlet form of the generator ${\cal L}$ corresponding to the original
semigroup.

###  4.1 The quantum perturbed semigroup

In this section, we show an upper bound on the tail probability in terms of a
perturbed semigroup:

######  Proposition 4.

Let $t\mapsto\mathrm{e}^{t{\cal L}}$ be a primitive quantum Markov semigroup
on the finite dimensional matrix algebra $M_{n}(\mathbb{C})$ with invariant
state $\sigma$. Furthermore, let $1\leq q\leq\ell$,
$\lambda\equiv(\lambda^{B},\lambda^{P})\in\mathbb{R}_{+}^{\ell}$, with
$\lambda^{B}=(\lambda_{1},...,\lambda_{q})$ and
$\lambda^{P}=(\lambda_{q+1},...,\lambda_{\ell})$, and let
$\mathbf{u}\equiv(\mathbf{u}^{B},\mathbf{u}^{P})$, with
$\mathbf{u}^{B}=\\{u_{1},\dots,u_{q}\\}$ and
$\mathbf{u}^{P}=\\{u_{q+1},\dots,u_{\ell}\\}$, be a family of orthogonal,
normalized vectors in $\mathbb{C}^{k}$. Then, for all $t\geq 0$, all initial
states $\rho\in{\cal D}(\mathbb{C}^{n})$, any
$r\in\mathbb{R}_{+}^{\ell},\lambda$, and $\mathbf{u}$, we have

$\displaystyle\mathbb{P}(\cap_{i}\\{E_{t,u_{i}}-m_{u_{i}}\geq
r_{i}\\})\leq\mathop{\rm Tr}\nolimits(\rho\,\mathrm{e}^{t{\cal
L}_{\lambda,\mathbf{u}}}(\operatorname{id}))\,\mathrm{e}^{-t\,\lambda.(m_{\mathbf{u}}+r)}\,,$
(4.1)

where

$\displaystyle\mathcal{L}_{{\lambda},\mathbf{u}}(X)\coloneqq\mathcal{L}(X)+{\lambda}^{B}.\left(\mathbf{L}_{\mathbf{u}^{B}}^{*}X+X\mathbf{L}_{\mathbf{u}^{B}}+\frac{{\lambda}^{B}}{2}X\right)+\sum_{j=q+1}^{\ell}\,(\mathrm{e}^{\lambda_{j}}-1)L_{u_{j}}^{*}XL_{u_{j}}\,.$
(4.2)

Before we prove Proposition 4 in its full generality in Section 4.1.3, we
first consider two different simplified setups in Sections 4.1.1 and 4.1.2.
These setups treat the special cases of the estimator
$\mathbf{E}_{t,\mathbf{u}}$ corresponding to a single-mode Brownian motion
($\mathbf{E}_{t,\mathbf{u}}\equiv E_{t,u_{1}^{B}}$), and a single-mode Poisson
process ($\mathbf{E}_{t,\mathbf{u}}\equiv E_{t,u_{1}^{P}}$) respectively.
These simplified examples will be instructive for the more general
multivariate case (with both kinds of processes combined).

####  4.1.1 Brownian motion

In this section we consider the special case of the estimator

$\displaystyle\mathbf{E}_{t,\mathbf{u}}\equiv
E_{t,u_{1}^{B}}=\frac{1}{t}U_{t}^{*}(A_{u_{1}^{B}}(t)+A_{u_{1}^{B}}^{*}(t))U_{t}=\frac{1}{t}X_{1}(t)\,,$

which corresponds to a single-mode Brownian motion (cf. Theorem 1). We will
denote $u:=u_{1}^{B}$ and $X(t):=X_{1}(t)$ for simplicity throughout this
section.

######  Proposition 5.

Let $t\mapsto\mathrm{e}^{t{\cal L}}$ be a primitive quantum Markov semigroup
on $M_{n}(\mathbb{C})$ with invariant state $\sigma$. Then, for all $t\geq 0$,
all initial states $\rho\in{\cal D}(\mathbb{C}^{n})$, any
$r,\lambda\in\mathbb{R}_{+}$ and $u\in S^{k-1}(\mathbb{R})$, we have

$\displaystyle\mathbb{P}\left(X(t)/t-m_{u}^{B}>r\right)\leq\mathop{\rm
Tr}\nolimits\big{(}\rho\,\mathrm{e}^{t{\cal
L}_{\lambda,u}^{B}}({\rm{id}})\big{)}\,\mathrm{e}^{-\lambda
t(m_{u}^{B}+r)}\,,$

where

$\displaystyle{\cal
L}_{\lambda,u}^{B}(X):=\mathcal{L}(X)+\lambda(L_{u}^{*}X+XL_{u})+\frac{\lambda^{2}}{2}X\,.$

###### Proof.

Let $\phi_{t,u}^{B}(\lambda):=\mathbb{E}[\exp(i\lambda X(t))]$ be the
characteristic function of $X(t)$. By Proposition 1 we may write

$\displaystyle\phi_{t,u}^{B}(\lambda)=\mathop{\rm
Tr}\nolimits\left[\rho\otimes\Omega\ \,\mathrm{e}^{i\lambda X(t)}\right]\,,$

where the expression on the right side of the above equation is well-defined
since $\mathrm{e}^{i\lambda X(t)}$ is unitary. Furthermore, by the definition
of $X(t)$, we have that

$\displaystyle\phi_{t,u}^{B}(\lambda)$ $\displaystyle=\mathop{\rm
Tr}\nolimits\left[\rho\otimes\Omega\
\mathrm{e}^{i\frac{\lambda}{2}U_{t}^{*}(A_{u}(t)+A_{u}^{*}(t))U_{t}}\,{\rm{id}}\,\mathrm{e}^{i\frac{\lambda}{2}U_{t}^{*}(A_{u}(t)+A_{u}^{*}(t))U_{t}}\right]$
$\displaystyle=\mathop{\rm Tr}\nolimits\left[\rho\otimes\Omega\
U_{t}^{*}\mathrm{e}^{i\frac{\lambda}{2}(A_{u}(t)+A_{u}^{*}(t))}\,{\rm{id}}\,\mathrm{e}^{i\frac{\lambda}{2}(A_{u}(t)+A_{u}^{*}(t))}U_{t}\right]$
$\displaystyle=\mathop{\rm
Tr}\nolimits\left[\rho\,{\Phi_{t,u}^{B}}^{(i\lambda)}({\rm{id}})\right]\,,$

where the family of operators $t\mapsto{\Phi_{t,u}^{B}}^{(i\lambda)}$ on
$M_{n}(\mathbb{C})$ is given by

$\displaystyle{\Phi_{t,u}^{B}}^{(i\lambda)}(X):=\mathop{\rm
Tr}\nolimits_{\Gamma}\left[\Omega\
{V_{t,u}^{B}}^{*}(-\lambda)XV_{t,u}^{B}(\lambda)\right]\,,$ (4.3)

with
$V_{t,u}^{B}(\lambda):=\mathrm{e}^{i\frac{\lambda}{2}(A_{u}(t)+A_{u}^{*}(t))}U_{t}$
unitary. This is a strongly continuous semigroup with respect to $t$, with
generator (cf. Appendix A.1)

$\displaystyle\mathcal{L}_{i\lambda,u}^{B}(X)\coloneqq\mathcal{L}(X)+i\lambda\,(L_{u}^{*}X+XL_{u})-\frac{\lambda^{2}}{2}X\,.$
(4.4)

Moreover, the expression of its generator ensures that for any $t$,
${\Phi_{t,u}^{B}}^{(i\lambda)}$ is an entire analytic function in $\lambda$.
This in particular also implies that the characteristic function
$\phi_{t,u}^{B}(\lambda)$ is entire analytic in $\lambda$, whence it can be
written as a Fourier integral on the whole complex plane (cf. Theorem 7.1.1 of
[50]), and therefore the Laplace transform of $X(t)$,
$\varphi_{t,u}^{B}(\lambda):=\mathbb{E}[\exp(\lambda X(t))]$, is well-defined.
In particular, the latter can be expressed in terms of the analytic
continuation ${\Phi_{t,u}^{B}}^{(\lambda)}$ of
${\Phi_{t,u}^{B}}^{(i\lambda)}$:

$\displaystyle\varphi_{t,u}^{B}(\lambda)=\phi_{t,u}^{B}(-i\lambda)=\mathop{\rm
Tr}\nolimits\left[\rho\,{\Phi_{t,u}^{B}}^{(\lambda)}({\rm{id}})\right]\,.$

Hence, using that for any $\lambda>0$, $x\in\mathbb{R}\mapsto\exp(\lambda
x)\in\mathbb{R}_{+}$ is strictly increasing, together with Markov’s
inequality, we get for any $r\geq 0$ and $\lambda>0$

$\displaystyle\mathbb{P}\left(X(t)/t-m_{u}^{B}>r\right)$
$\displaystyle=\mathbb{P}\left(\mathrm{e}^{\lambda X(t)}>\mathrm{e}^{\lambda
t\left(m_{u}^{B}+r\right)}\right)$
$\displaystyle\leq\varphi_{t,u}^{B}(\lambda)\,\mathrm{e}^{-\lambda
t\left(m_{u}^{B}+r\right)}$ $\displaystyle=\mathop{\rm
Tr}\nolimits\left[\rho\,{\Phi_{t,u}^{B}}^{(\lambda)}({\rm{id}})\right]\,\mathrm{e}^{-\lambda
t\left(m_{u}^{B}+r\right)}$

as claimed. ∎

######  Remark 1.

  1. (a)

Note that we have only established a bound on the right tail of $X(t)/t$.
However, a derivation analogous to the one in the above proof also yields a
bound for the left tail: For any $r>0$ and $\lambda>0$,

$\mathbb{P}(X(t)/t-m_{u}^{B}<-r)\leq\varphi_{t,u}^{B}(-\lambda)\,\mathrm{e}^{\lambda
t(m_{u}^{B}-r)}.$

Using this, together with Proposition 5, a double-sided bound can be
established.

  2. (b)

Note that we may write

$\displaystyle{\Phi_{t,u}^{B}}^{(i\lambda)}(X)$ $\displaystyle=\mathop{\rm
Tr}\nolimits_{\Gamma}\left[\Omega\ U_{t}^{*}XU_{t}\mathrm{e}^{i\lambda
U_{t}^{*}(A_{u}(t)+A_{u}^{*}(t))U_{t}}\right]\,,$

where we used that the operators $A_{u}(t)+A_{u}^{*}(t)$ act on $\Gamma$ only,
as well as the unitarity of $U_{t}$. Thus the semigroup
$t\mapsto{\Phi_{t,u}^{B}}^{(i\lambda)}$ emerges from the quantum Markov
semigroup $t\mapsto\mathrm{e}^{t{\cal L}}$ via the introduction of the
perturbation (by Proposition 1 and linearity in $u$)

$\displaystyle\mathrm{e}^{i\lambda
X(t)}=\exp\left(i\lambda\left(\int_{0}^{t}U_{s}(O^{B}(u))U_{s}^{*}\,\mathrm{d}s+A_{u}(t)+A_{u}^{*}(t)\right)\right)\,,$

i.e., up to Brownian motion, a perturbation corresponding to the time-averaged
evolution of $O^{B}(u)\equiv L_{u}+L_{u}^{*}$.

####  4.1.2 Poisson process

As in the previous section, here we consider a special case of the estimator
$\mathbf{E}_{t,\mathbf{u}}$:

$\displaystyle\mathbf{E}_{t,\mathbf{u}}\equiv
E_{t,u_{1}^{P}}=\frac{1}{t}U_{t}^{*}\Lambda_{u_{1}^{P}}(t)U_{t}=\frac{1}{t}Y_{1}(t)\,.$

We will denote $u:=u_{1}^{P}$ and $Y(t):=Y_{1}(t)$ for simplicity throughout
this section.

######  Proposition 6.

Let $t\mapsto\mathrm{e}^{t{\cal L}}$ be a primitive quantum Markov semigroup
on $M_{n}(\mathbb{C})$ with invariant state $\sigma$. Then, for all $t\geq 0$,
all initial states $\rho\in{\cal D}(\mathbb{C}^{n})$, any
$r,\lambda\in\mathbb{R}_{+}$ and $u\in S^{k-1}(\mathbb{R})$, we have

$\displaystyle\mathbb{P}\left(Y(t)/t-m_{u}^{P}>r\right)\leq\mathop{\rm
Tr}\nolimits\big{[}\rho\,\mathrm{e}^{t{\cal
L}_{\lambda,u}^{P}}({\rm{id}})\big{]}\,\mathrm{e}^{-\lambda
t(m_{u}^{P}+r)}\,,$

where

$\displaystyle{\cal
L}_{\lambda,u}^{P}(X):=\mathcal{L}(X)+(\mathrm{e}^{\lambda}-1)L_{u}^{*}XL_{u}\,.$
(4.5)

###### Proof.

The proof of Proposition 6 follows the same lines as the proof of Proposition
5. First of all, we relate the characteristic function
$\phi_{t,u}^{P}(\lambda):=\mathbb{E}[\exp(i\lambda Y(t))]$ of $Y(t)$ to the
strongly continuous semigroup $t\mapsto{\Phi_{t,u}^{P}}^{(i\lambda)}$ on
$M_{n}(\mathbb{C})$,

$\displaystyle{\Phi_{t,u}^{P}}^{(i\lambda)}(X):=\mathop{\rm
Tr}\nolimits_{\Gamma}\left[\Omega\
{V_{t,u}^{P}}^{*}(-\lambda)XV_{t,u}^{P}(\lambda)\right]$ (4.6)

with
$V_{t,u}^{P}(\lambda):=\mathrm{e}^{i\frac{\lambda}{2}\Lambda_{u}(t)}U_{t}$
unitary and generator (cf. Appendix A.2)

$\displaystyle\mathcal{L}_{i\lambda,u}^{P}(X)\coloneqq\mathcal{L}(X)+(\mathrm{e}^{i\lambda}-1)L_{u}^{*}XL_{u}\,,$

using similar arguments as in the case of Brownian motion (cf. Section 4.1.1).
From this, again following the same argumentation as in the previous section
treating single-mode Brownian motion, we get the well-defined Laplace
transform of $Y(t)$ of the form

$\displaystyle\varphi_{t,u}^{P}(\lambda):=\mathbb{E}(\exp(\lambda
Y(t)))=\phi_{t,u}^{P}(-i\lambda)=\mathop{\rm
Tr}\nolimits\left[\rho\,{\Phi_{t,u}^{P}}^{(\lambda)}({\rm{id}})\right]\,,$

and the properties of the exponential function, together with Markov’s
inequality, subsequently yield the claimed bound on the tail probability. ∎

######  Remark 2.

  1. (a)

Note that, as in the case of Brownian motion, the bound on the left tail is
established analogously, yielding for any $r>0$ and $\lambda>0$:

$\mathbb{P}(Y(t)/t-m_{u}^{P}<-r)\leq\varphi_{t,u}^{P}(-\lambda)\,\mathrm{e}^{\lambda
t(m_{u}^{P}-r)}.$

Using this, together with Proposition 6, a double-sided bound can be
established.

  2. (b)

Note that, similarly to the Brownian motion setting, the semigroup
$t\mapsto{\Phi_{t,u}^{P}}^{(i\lambda)}$ can be seen as a perturbed version of
the quantum Markov semigroup $t\mapsto\mathrm{e}^{t{\cal L}}$. More precisely,
we may write

$\displaystyle{\Phi_{t,u}^{P}}^{(i\lambda)}(X)$ $\displaystyle=\mathop{\rm
Tr}\nolimits_{\Gamma}\left[\Omega\ U_{t}^{*}XU_{t}\mathrm{e}^{i\lambda
U_{t}^{*}\Lambda_{u}(t)U_{t}}\right]\,,$

where we used that the operators $\Lambda_{u}(t)$ act on $\Gamma$ only, as
well as the unitarity of $U_{t}$. Here the perturbation corresponds to the
time-averaged evolution of the observable $O^{P}(u)\equiv L_{u}^{*}L_{u}$ (up
to an additive stochastic part, cf. Proposition 1, and noting that $Y(t)$ is
linear in $uu^{*}$).

####  4.1.3 Proof of Proposition 4

Let us now consider the more general setup of Proposition 4 and the general
estimator $\mathbf{E}_{t,\mathbf{u}}\equiv(\mathbf{X}(t),\mathbf{Y}(t))/t$.
Recall that $1\leq q\leq\ell$,
$\lambda\equiv(\lambda^{B},\lambda^{P})\in\mathbb{R}_{+}^{\ell}$, with
$\lambda^{B}=(\lambda_{1},...,\lambda_{q})$ and
$\lambda^{P}=(\lambda_{q+1},...,\lambda_{\ell})$, and
$\mathbf{u}\equiv(\mathbf{u}^{B},\mathbf{u}^{P})$, with
$\mathbf{u}^{B}=\\{u_{1},\dots,u_{q}\\}$ and
$\mathbf{u}^{P}=\\{u_{q+1},\dots,u_{\ell}\\}$, is a family of orthogonal,
normalized vectors in $\mathbb{C}^{k}$.

To prove Proposition 4, we proceed as in both the special cases of Sections
4.1.1 and 4.1.2: We first relate the characteristic function
$\phi_{t,\mathbf{u}}(\lambda):=\mathbb{E}[\exp(i\lambda.(\mathbf{X}(t),\mathbf{Y}(t)))]$
of the vector $(\mathbf{X}(t),\mathbf{Y}(t))$ to a perturbed semigroup
$t\mapsto{\Phi_{t,\mathbf{u}}}^{(i\lambda)}(X)$, which then yields a well-
defined expression of the Laplace transform of the same vector, whence the
properties of the exponential function, together with Markov’s inequality,
yield the claimed bound.

In particular, the perturbed semigroup
$t\mapsto{\Phi_{t,\mathbf{u}}}^{(i\lambda)}(X)$ on $M_{n}(\mathbb{C})$ is
given by

$\displaystyle{\Phi_{t,\mathbf{u}}}^{(i\lambda)}(X):=\mathop{\rm
Tr}\nolimits_{\Gamma}\left[\Omega\
{V_{t,\mathbf{u}}}^{*}(-\lambda)XV_{t,\mathbf{u}}(\lambda)\right]\,,$ (4.7)

with
$V_{t,\mathbf{u}}(\lambda):=\mathrm{e}^{\frac{i}{2}\lambda.(\mathbf{A}_{\mathbf{u}^{B}}(t)+\mathbf{A}_{\mathbf{u}^{B}}^{*}(t),\Lambda_{\mathbf{u}^{P}}(t))}U_{t}$
unitary. The generator of this strongly continuous semigroup with respect to
$t$ is (cf. Appendix A.3)

$\displaystyle\mathcal{L}_{i{\lambda},\mathbf{u}}(X)\coloneqq$
$\displaystyle\mathcal{L}(X)+i{\lambda}^{B}.\left(\mathbf{L}_{\mathbf{u}^{B}}^{*}X+X\mathbf{L}_{\mathbf{u}^{B}}+\frac{i{\lambda}^{B}}{2}X\right)+\sum_{j=q+1}^{\ell}\,(\mathrm{e}^{i\lambda_{j}}-1)L_{u_{j}}^{*}XL_{u_{j}}\,.$
(4.8)

The identity

$\displaystyle\phi_{t,\mathbf{u}}(\lambda)=\mathop{\rm
Tr}\nolimits\left[\rho\,{\Phi_{t,\mathbf{u}}}^{(i\lambda)}({\rm{id}})\right]\,,$

the resulting form of the Laplace transform of
$(\mathbf{X}(t),\mathbf{Y}(t))$,

$\displaystyle\varphi_{t,\mathbf{u}}(\lambda):=\mathbb{E}[\exp(\lambda.(\mathbf{X}(t),\mathbf{Y}(t)))]=\phi_{t,\mathbf{u}}(-i\lambda)=\mathop{\rm
Tr}\nolimits\left[\rho\,{\Phi_{t,\mathbf{u}}}^{(\lambda)}({\rm{id}})\right]\,,$

and also the final resulting bound on the tail probability are then
established analogously to the case of single-mode Brownian motion (cf.
Section 4.1.1) and single-mode Poisson process (cf. Section 4.1.2).

######  Remark 3.

Note that, as in the special cases of Sections 4.1.1 and 4.1.2, we can also
get a double-sided bound using the same techniques as in the proof of
Proposition 4.

###  4.2 The upper bound

Here we upper bound the probability that the estimator
$\mathbf{E}_{t,\mathbf{u}}$ defined in Equation 3.3 is away from the mean
vector ${m}_{\mathbf{u}}$ in terms of the Dirichlet form of ${\cal L}$. First,
we rewrite more explicitly the perturbed generator:

$\displaystyle\mathcal{L}_{{\lambda},\mathbf{u}}(X)$
$\displaystyle=\mathcal{L}(X)+\sum_{j=1}^{q}\,\lambda_{j}\,(L_{u_{j}}^{*}X+XL_{u_{j}}+\frac{1}{2}\lambda_{j}X)+\sum_{j=q+1}^{\ell}\,\big{(}\mathrm{e}^{\lambda_{j}}-1\big{)}\,L_{u_{j}}^{*}XL_{u_{j}}$
(4.9)
$\displaystyle\quad=\sum_{i=1}^{k}L_{i}^{*}XL_{i}-\frac{1}{2}\\{L_{i}^{*}L_{i},X\\}+\sum_{j=1}^{q}\,\lambda_{j}\,(L_{u_{j}}^{*}X+XL_{u_{j}}+\frac{1}{2}\lambda_{j}X)+\sum_{j=q+1}^{\ell}\big{(}\mathrm{e}^{\lambda_{j}}-1\big{)}\,L_{u_{j}}^{*}XL_{u_{j}}\,.$

Next, we let $\lambda^{B}(\mathbf{u}^{B})\in\mathbb{R}^{k}$ be the vector of
components
$\lambda^{B}(\mathbf{u}^{B})_{i}=\sum_{j=1}^{q}\lambda_{j}(u_{j})_{i}$. Since
for each $j$, $u_{j}$ is assumed to be normalized, we further have that
$\|\lambda^{B}(\mathbf{u}^{B})\|_{2}=\|\lambda^{B}\|_{2}$. Then,

$\displaystyle\sum_{i=1}^{k}$ $\displaystyle
L_{i}^{*}XL_{i}+\sum_{j=1}^{q}\,\lambda_{j}\,(L_{u_{j}}^{*}X+XL_{u_{j}}+\frac{1}{2}\lambda_{j}X)=\sum_{i=1}^{k}(L_{i}+\lambda^{B}(\mathbf{u}^{B})_{i})^{*}X(L_{i}+\lambda^{B}(\mathbf{u}^{B})_{i})-\frac{1}{2}\|\lambda^{B}\|_{2}^{2}\,X\,.$
(4.10)

Therefore

$\displaystyle{\cal
L}_{\lambda,\mathbf{u}}(X)=\Psi_{{\lambda},\mathbf{u}}(X)-\frac{1}{2}\|\lambda^{B}\|_{2}^{2}X-\frac{1}{2}\sum_{i=1}^{k}\\{L_{i}^{*}L_{i},X\\}\,,$
(4.11)

for some completely positive map $\Psi_{{\lambda},\mathbf{u}}$ defined as

$\displaystyle\Psi_{{\lambda},\mathbf{u}}(X):=\sum_{j=q+1}^{\ell}(e^{\lambda_{j}}-1)L_{u_{j}}^{*}XL_{u_{j}}+\sum_{i=1}^{k}\,(L_{i}+\lambda^{B}(\mathbf{u}^{B})_{i})^{*}X(L_{i}+\lambda^{B}(\mathbf{u}^{B})_{i})\,.$
(4.12)

Then, for
$\rho:=\Gamma_{\sigma}(X)\equiv\sigma^{\frac{1}{2}}X\sigma^{\frac{1}{2}}$, we
have

$\displaystyle\varphi_{t,\mathbf{u}}({\lambda})=\mathop{\rm
Tr}\nolimits\big{[}\rho\,\Phi^{{\lambda}}_{t,\mathbf{u}}({\rm{id}})\big{]}\leq\|X\|_{L^{2}(\sigma)}\,\|{\Phi}_{t,\mathbf{u}}^{{\lambda}}:\,L^{2}(\sigma)\to
L^{2}(\sigma)\|\,.$ (4.13)

By the Lumer-Philips theorem we can upper bound the last operator norm as
follows:

$\displaystyle\|{\Phi}_{t,\mathbf{u}}^{{\lambda}}:\,L^{2}(\sigma)\to
L^{2}(\sigma)\|\leq\exp\big{(}{-t\underset{\|X\|_{L^{2}(\sigma)}=1}{\inf}{\cal
E}_{{\lambda}}(X)}\big{)}\,,$ (4.14)

where ${\cal E}_{{\lambda},\mathbf{u}}(X):=-\frac{1}{2}(\langle X,{\cal
L}_{{\lambda},\mathbf{u}}(X)\rangle_{\sigma}+\langle{\cal
L}_{{\lambda},\mathbf{u}}(X),X\rangle_{\sigma})$ denotes the (symmetrized)
Dirichlet form of ${\cal L}_{{\lambda},\mathbf{u}}$. Optimizing over
${\lambda}$, we end up with

$\displaystyle\mathbb{P}(\cap_{i}\\{E_{t,u_{i}}-m_{u_{i}}\geq
r_{i}\\})\leq\,\|\Gamma_{\sigma}^{-1}(\rho)\|_{L^{2}(\sigma)}\,\exp\Big{(}-t\,\underset{{\lambda}}{\sup}\,\,\,\underset{\|X\|_{L^{2}(\sigma)}=1}{\inf}\,\big{\\{}{\cal
E}_{{\lambda},\mathbf{u}}(X)+{\lambda}.({r}+{m}_{\mathbf{u}})\big{\\}}\Big{)}\,.$
(4.15)

Following Equation 4.11, ${\cal L}_{{\lambda},\mathbf{u}}+{\cal
L}_{{\lambda},\mathbf{u}}^{\rm KMS}$ is the generator of a completely positive
semi-group. Hence, by the Perron–Frobenius Theorem, the infinimum is a minimum
that is reached for $X$ positive semi-definite. The infinimum over $X$ can
thus be restricted to $X$ positive semi-definite. Inspired by this
observation, we define the function
$g:\mathbb{R}_{+}^{\ell_{B}+\ell_{P}}\times{\cal D}({\cal H})\to\mathbb{R}$ as

$\displaystyle g({\lambda},\gamma):={\cal
E}_{{\lambda},\mathbf{u}}(\Gamma_{\sigma}^{-\frac{1}{2}}(\gamma^{\frac{1}{2}}))+{\lambda}.({r}+{m}_{\mathbf{u}})\,\,,$

so that the optimization in the exponential on the right-hand side of (4.15)
turns out to be equivalent to $\sup_{{\lambda}}\inf_{\gamma\in{\cal D}({\cal
H})}g({\lambda},\gamma)$.

######  Lemma 1.

The function $g$ is convex in the state $\gamma$ and concave in the parameter
${\lambda}$.

###### Proof.

The concavity in ${\lambda}$ can be directly verified from (4.9). To prove the
convexity in $\gamma$, we consider the Dirichlet form

$\displaystyle{\cal E}_{{\lambda},\mathbf{u}}(X)$
$\displaystyle=-\frac{1}{2}\langle X,\,({\cal L}_{{\lambda},\mathbf{u}}+{\cal
L}_{{\lambda},\mathbf{u}}^{\operatorname{KMS}})(X)\rangle_{\sigma}$
$\displaystyle=-\frac{1}{2}\langle
X,\,(\Psi_{{\lambda},\mathbf{u}}+\Psi_{{\lambda},\mathbf{u}}^{\operatorname{KMS}})(X)\rangle_{\sigma}+\frac{\|\lambda^{B}\|^{2}}{2}\,\|X\|_{L^{2}(\sigma)}^{2}-\frac{1}{2}\langle
X,\,LX+XL^{\prime}\rangle_{\sigma}\,,$

for some operators $L,L^{\prime}$ depending on $\mathbf{u}$ and $\lambda$.
Since the KMS-dual of a completely positive map is completely positive, the
map
$\Psi_{{\lambda},\mathbf{u}}+\Psi_{{\lambda},\mathbf{u}}^{\operatorname{KMS}}$
has a Kraus decomposition which we denote by
$(\Psi_{{\lambda},\mathbf{u}}+\Psi_{{\lambda},\mathbf{u}}^{\operatorname{KMS}})(X):=\sum_{l}K_{l}^{*}XK_{l}$.
Now, for $X=\Gamma_{\sigma}^{-\frac{1}{2}}(\sqrt{\gamma})$,
$\|X\|_{L^{2}(\sigma)}=1$ and denoting
$\widetilde{K}_{l}:=\sigma^{-\frac{1}{4}}K_{l}\sigma^{\frac{1}{4}}$, we have

$\displaystyle{\cal
E}_{{\lambda},\mathbf{u}}(\Gamma_{\sigma}^{-\frac{1}{2}}(\sqrt{\gamma}))=-\frac{1}{2}\sum_{l}\,\mathop{\rm
Tr}\nolimits\Big{[}\sqrt{\gamma}\,\widetilde{K}_{l}^{*}\sqrt{\gamma}\widetilde{K}_{l}\Big{]}+\frac{1}{2}\|{\lambda}^{B}\|^{2}-\frac{1}{2}\,\mathop{\rm
Tr}\nolimits\big{[}\Gamma_{\sigma}^{-\frac{1}{2}}(\sqrt{\gamma})\sigma^{\frac{1}{2}}\big{[}L\Gamma_{\sigma}^{-\frac{1}{2}}(\sqrt{\gamma})+\Gamma_{\sigma}^{-\frac{1}{2}}(\sqrt{\gamma})L^{\prime}\big{]}\sigma^{\frac{1}{2}}\big{]}\,.$

By the Ando-Lieb concavity theorem (see Theorem 5.15 of [62]), the first sum
over $l$ is convex in $\gamma$. Moreover,

$\displaystyle\mathop{\rm
Tr}\nolimits\big{[}\Gamma_{\sigma}^{-\frac{1}{2}}(\sqrt{\gamma})\sigma^{\frac{1}{2}}\big{[}L\Gamma_{\sigma}^{-\frac{1}{2}}(\sqrt{\gamma})+\Gamma_{\sigma}^{-\frac{1}{2}}(\sqrt{\gamma})L^{\prime}\big{]}\sigma^{\frac{1}{2}}\big{]}=\mathop{\rm
Tr}\nolimits\big{[}\gamma(\sigma^{\frac{1}{4}}L\sigma^{-\frac{1}{4}}+\sigma^{-\frac{1}{4}}L^{\prime}\sigma^{\frac{1}{4}})\big{]}$

is linear in $\gamma$. This ends the proof. ∎

We are now ready to prove Theorem 6:

###### Proof of Theorem 6.

Thanks to the previous Lemma, we can use Sion’s minimax theorem in order to
swap the minimization in $X$ and the suppremum in $\lambda$ in (4.15), so that

$\displaystyle\mathbb{P}(\cap_{i}\\{E_{t,u_{i}}-m_{u_{i}}\geq
r_{i}\\})\leq\,\|\Gamma_{\sigma}^{-1}(\rho)\|_{L^{2}(\sigma)}\,\exp\Big{(}-t\,\underset{\|X\|_{L^{2}(\sigma)}=1}{\inf}\,\underset{{\lambda}}{\sup}\big{\\{}{\cal
E}_{{\lambda},\mathbf{u}}(X)+{\lambda}.({r}+{m}_{\mathbf{u}})\big{\\}}\Big{)}\,.$

A simple optimization over $\lambda$ then yields:

$\displaystyle\sup_{{\lambda}}\\{{\cal
E}_{{\lambda},\mathbf{u}}(X)+{\lambda}.({r}+{m}_{\mathbf{u}})\\}={\cal
E}(X)+\frac{1}{2}\,\|{r}^{B}+{m}_{\mathbf{u}^{B}}^{B}-{f}_{\mathbf{u}^{B}}^{B}(X)\|^{2}+D({r}^{P}+{m}_{\mathbf{u}^{P}}^{P}\|{f}_{\mathbf{u}^{P}}^{P}(X))\,,$

and the result follows. ∎

##  5 Concentration via transportation and functional inequalities

The goal of this section is to prove concentration for the tail probability in
Theorem 6 by means of non-commutative functional and transportation cost
inequalities. The main tool that we use is a lower bound on the Dirichlet form
in terms of a quantum generalization of the Wasserstein distance. In Section
5.1, we introduce a definition for the quantum Wasserstein distance which
generalizes various quantities recently introduced in the community of quantum
information theorists. Section 5.2 consists of a short review on functional
and transportation cost inequalities, which we then use in Section 5.3 to
derive our concentration bounds. In this section, we exclusively assume that
our primitive QMS $t\mapsto\mathrm{e}^{t\mathcal{L}}$ over a finite
dimensional Hilbert space is GNS symmetric.

###  5.1 Quantum transportation cost distances

Given a complete separable metric space $({\cal X},d)$, let $c:{\cal
X}\times{\cal X}\to[0,\infty]$ be a lower semicontinuous function such that
$c(x,x)=0$ for all $x\in{\cal X}$. The function $c$ is referred to as the cost
function. Given such a cost function, the transportation cost $T_{c}$ is
defined on the space of probability measures over ${\cal X}$ by

$\displaystyle T_{c}(\mu,\nu):=\inf_{\pi\in\Omega(\mu,\nu)}\,\iint_{{\cal
X}^{2}}\,c(x,y)\,\pi(\mathrm{d}x,\mathrm{d}y)\,,$ (5.1)

where the infimum is taken over the set of couplings $\Omega(\mu,\nu)$ of
$\mu$ and $\nu$. Whenever the cost function $c$ is equal to a power $d^{p}$ of
the metric, $p\geq 1$, the transportation cost is usually denoted by $W_{p}$
and called the $L^{p}$ Wasserstein distance. Transportation cost distances
admit a dual representation, also known as the Kantorovich duality theorem
[60]:

$\displaystyle
T_{c}(\mu,\nu):=\sup_{(u,v)\in\Phi_{c}}\,\int\,u\mathrm{d}\nu-\int\,v\,\mathrm{d}\mu\,,$
(5.2)

where $\Phi_{c}:=\\{(u,v)\in{\cal B}({\cal X}):\,u(x)-v(y)\leq
c(x,y)\,,\forall(x,y)\in{\cal X}^{2}\\}$ and ${\cal B}({\cal X})$ denotes the
space of Borel-measurable, real bounded functions over ${\cal X}$. This
functional characterization is the one which Bobkov and Goetze used in their
pioneering work [12] on concentration inequalities. The dual expression was
later extended as follows [36, 37]:

$\displaystyle{\cal T}_{\Phi}(\nu,\mu)=\sup_{(u,v)\in\Phi}\int
u\,\mathrm{d}\nu-\int v\,\mathrm{d}\mu\,,$ (5.3)

where $\Phi\subset{\cal B}({\cal X})^{2}$ is a non-empty set such that (i)
$u\leq v$ for all $(u,v)\in\Phi$; and (ii) for all probability measures
$\nu_{1},\nu_{2}$, there exists $(u,v)\in\Phi$ such that $\int
u\,\mathrm{d}\nu_{1}-\int v\,\mathrm{d}\nu_{2}\geq 0$. In the quantum setting,
various extensions of transportation costs have been recently proposed [43,
57, 17, 23, 21]. As in the classical setting, these Wasserstein distances have
in common that they can be written in terms of a supremum over test
observables satisfying some linear constraints. Here, we adopt the approach of
[36, 37] and propose a unifying definition. For the sake of simplicity, we
will restrict ourselves to finite dimensional systems:

######  Definition 9.

Let ${\cal H}$ be a finite dimensional Hilbert space. Then, given a subset
$\Phi$ of ${\cal B}_{\operatorname{sa}}({\cal H})^{2}$ such that

  * (i)

$X\leq Y$ for all $(X,Y)\in\Phi$ ;

  * (ii)

for all $\omega_{1},\omega_{2}\in{\cal D}({\cal H})$, there exists
$(X,Y)\in\Phi$ such that $\mathop{\rm Tr}\nolimits[\omega_{1}X]-\mathop{\rm
Tr}\nolimits[\omega_{2}Y]\geq 0$ ;

the quantum transportation cost distance ${\cal T}_{\Phi}:{\cal D}({\cal
H})\times{\cal D}({\cal H})\to[0,\infty]$ is defined as

$\displaystyle{\cal
T}_{\Phi}(\omega_{1},\omega_{2}):=\sup_{(X,Y)\in\Phi}\,\mathop{\rm
Tr}\nolimits[\omega_{1}X]-\mathop{\rm Tr}\nolimits[\omega_{2}Y]\,.$

######  Example 1 (Trace distance).

When $\Phi:=\\{(T,T)|\,0\leq T\leq{\rm{id}}\\}$ is the set of quantum effects,
${\cal T}_{\Phi}$ is the trace distance.

######  Example 2 (Wasserstein distance from Lindblad evolutions).

Here we fix a set
$\\{\partial_{j}\equiv[L_{j},.]\\}_{j\in\mathcal{J}}=\\{[L_{j}^{*},.]\\}_{j\in\mathcal{J}}$
of derivations compatible with a full-rank state $\sigma$, i.e., for which
there exists $\\{\omega_{j}\\}_{j\in\mathcal{J}}$ such that for all
$j\in\mathcal{J}$

$\displaystyle\sigma\,L_{j}=\mathrm{e}^{-\omega_{j}}L_{j}\sigma\,,\quad
L_{j}^{*}\sigma=\mathrm{e}^{\omega_{j}}\sigma L_{j}^{*}\,.$

Then the Wasserstein distance of order $1$ between two states
$\rho,\omega\in\mathcal{D}(\mathcal{H})$ was defined in [57] as

$\displaystyle W_{1,{\cal
L}}(\omega_{1},\omega_{2}):=\sup_{\|X\|_{\operatorname{Lip}}\leq
1}\,|\mathop{\rm Tr}\nolimits[\omega_{1}X]-\mathop{\rm
Tr}\nolimits[\omega_{2}X]|\,,$ (5.4)

where

$\|X\|_{\operatorname{Lip}}:=\Big{(}\sum_{j\in\mathcal{J}}(\mathrm{e}^{-\omega_{j}/2}+\mathrm{e}^{\omega_{j}/2})\,\|\partial_{j}X\|_{\infty}^{2}\Big{)}^{1/2}\,.$

Here ${\cal L}$ stands for the generator of the GNS-symmetric quantum Markov
semigroup obtained from taking the operators $L_{j}$ to be its Lindblad
operators. This is nothing but the transportation cost ${\cal T}_{\Phi}$ for
$\Phi=\\{(X,X)$, $\|X\|_{\operatorname{Lip}}\leq 1\\}$. A variant of this
distance for symmetric generators, where the Lipschitz constant is based on
the non-commutative gradient associated with the generator considered, can
also be found in [31].

######  Example 3 (Quantum Ornstein distance).

More recently, a new quantum Wasserstein distance on the $n$-fold tensor
product ${\cal H}^{\otimes n}$ was proposed in [21] (see also [22]). It can be
expressed in its Kantorovich dual form as

$\displaystyle
W_{1}^{\operatorname{Orn}}(\rho,\sigma):=\sup_{\|X\|_{\operatorname{Lip}}^{\operatorname{Orn}}\leq
1}\mathop{\rm Tr}\nolimits[X(\rho-\sigma)]\,,$

where

$\displaystyle\|X\|_{\operatorname{Lip}}^{\operatorname{Orn}}:=2\max_{i\in[n]}\min_{H^{(i)}\in{\cal
B}_{\operatorname{sa}}({\cal H}_{i^{c}})}\big{\|}H-{\rm{id}}_{i}\otimes
H^{(i)}\big{\|}_{\infty}\,.$

###  5.2 Functional and transportation cost inequalities

Here we fix a primitive, GNS symmetric QMS $t\mapsto e^{t{\cal L}}$ with
invariant state $\sigma$. Following the standard classical notations of [37],
we denote the Dirichlet form of a normalized, positive semi-definite operator
$X\in L^{2}(\sigma)$ as

$\displaystyle I_{{\cal L}}(\rho):={\cal E}(X)\,,$

where
$\rho=(\sigma^{\frac{1}{4}}X\sigma^{\frac{1}{4}})^{2}\equiv(\Gamma_{\sigma}^{\frac{1}{2}}(X))^{2}$.
This extends the definition of the Fisher-Donsker-Varadhan information
$I(\nu|\mu):={\cal E}_{L}(\sqrt{f})$ with $\nu=f\mu$ as defined in Equation
1.5. It is also the quantity that arises for example on the right-hand side of
Equation 3.5. This observation is at the core of the main result of the
section, namely the derivation of concentration bounds for quantum
trajectories based on the transportation cost-information inequalities which
we define now (see [37] for classical analogues).

######  Definition 10.

Let $t\mapsto\mathrm{e}^{t{\cal L}}$ be a primitive quantum Markov semigroup
with invariant state $\sigma$, ${\cal T}_{\Phi}$ a quantum transportation cost
distance and $\alpha:[0,\infty)\to[0,\infty]$ a function that is left-
continuous, increasing and such that $\alpha(0)=0$. Then the triple $({\cal
L},{\cal T}_{\Phi},\alpha)$ is said to satisfy a quantum transportation cost-
information inequality if the following holds: for any $\rho\in{\cal D}({\cal
H})$,

$\displaystyle\alpha({\cal T}_{\Phi}(\rho,\sigma))\leq I_{{\cal L}}(\rho)\,.$
($\alpha({\operatorname{{\cal T}_{\Phi})I}}$)

Next, we provide examples of triples $({\cal L},{\cal T}_{\Phi},\alpha)$
satisfying a quantum transportation cost-information inequality. We do so by
relating the latter to previously studied quantum functional and
transportation cost inequalities. We recall that the quantum entropy
functional for any $X\geq 0$ is defined as

$\displaystyle\operatorname{Ent}_{2,\sigma}(X):=\mathop{\rm
Tr}\nolimits\Big{[}\Gamma_{\sigma}^{\frac{1}{2}}(X)^{2}\,(\ln\Gamma_{\sigma}^{\frac{1}{2}}(X)^{2}-\ln\sigma)\Big{]}-\|X\|_{L^{2}(\sigma)}^{2}\,\ln\|X\|_{L^{2}(\sigma)}^{2}\,.$

Whenever $\|X\|_{L^{2}(\sigma)}=1$, the quantum entropy functional is equal to
the relative entropy

$\displaystyle D(\rho\|\sigma):=\mathop{\rm
Tr}\nolimits\big{[}\rho\,(\ln\rho-\ln\sigma)\big{]}\,,$

for $\rho:=\Gamma_{\sigma}^{\frac{1}{2}}(X)^{2}$. Next, the entropy production
of the semigroup is defined for any $\rho\in{\cal D}({\cal H})$ as

$\displaystyle\operatorname{EP}_{\mathcal{L}}(\rho):=-\mathop{\rm
Tr}\nolimits\big{[}\mathcal{L}^{*}(\rho)(\ln\rho-\ln\sigma)\big{]}\,\,.$

Finally, we recall that the variance of $X$ is defined as

$\displaystyle\operatorname{Var}_{\sigma}(X):=\|X-\mathop{\rm
Tr}\nolimits[\sigma X]\|_{L^{2}(\sigma)}^{2}\,.$

Then the QMS is said to satisfy:

  * (i)

A logarithmic Sobolev inequality if there exists a constant $\alpha_{2}>0$
such that, for all $X\geq 0$,

$\displaystyle\alpha_{2}\,\operatorname{Ent}_{2,\sigma}(X)\leq\mathcal{E}(X)\,.$
(LSI($\alpha_{2}$))

We denote the best constant achieving this bound by $\alpha_{2}({\cal L})$.

  * (ii)

A modified logarithmic Sobolev inequality if there exists a constant
$\alpha_{1}>0$ such that, for any $\rho\in{\cal D}({\cal H})$:

$\displaystyle
4\alpha_{1}\,D(\rho\|\sigma)\leq\operatorname{EP}_{\mathcal{L}}(\rho)\,.$
(MLSI($\alpha_{1}$))

We denote the best constant achieving this bound by $\alpha_{1}({\cal L})$.

  * (iii)

A transportation cost inequality if there exists $c>0$ such that, for all
$\rho\in{\cal D}({\cal H})$:

$\displaystyle W_{1,{\cal L}}(\rho,\sigma)\leq\sqrt{2c\,D(\rho\|\sigma)}\,.$
(TC($c$))

  * (iv)

A transportation cost-information inequality if there exists $C>0$ such that,
for all $\rho\in{\cal D}({\cal H})$:

$\displaystyle W_{1,{\cal
L}}(\rho,\sigma)\leq\sqrt{2C\,\operatorname{I}_{\mathcal{L}}(\rho)}\,;$
(TI(C))

in other words, ($\alpha({\operatorname{{\cal T}_{\Phi})I}}$) holds for
$\alpha(r)=\frac{r^{2}}{2C}$, $E_{*}(\rho)=\sigma$ for all $\rho$ and ${\cal
T}_{\Phi}:=W_{1,{\cal L}}$.

  * (v)

A Poincaré inequality if there exists $\lambda>0$ such that, for all
$X\in{\cal B}({\cal H})$:

$\displaystyle\lambda\,\operatorname{Var}_{\sigma}(X)\leq\mathcal{E}(X)\,.$
(PI($\lambda$))

The best constant achieving this inequality is the spectral gap of ${\cal L}$,
which we denote by $\lambda({\cal L})$.

The following proposition regroups known results connecting the inequalities
introduced above, and relating them to the notion of a quantum transportation
cost-information inequality:

######  Proposition 7.

Let $t\mapsto\mathrm{e}^{t{\cal L}}$ be a finite dimensional,
$\operatorname{GNS}$ symmetric quantum Markov semigroup. Then the following
implications hold:

$\displaystyle\operatorname{LSI}(\alpha_{1})\quad$
$\displaystyle\Rightarrow\quad\operatorname{MLSI}(2\alpha_{1})$ (i)
$\displaystyle\operatorname{MLSI}(\alpha_{1})\quad$
$\displaystyle\Rightarrow\quad\operatorname{TC}((4\alpha_{1})^{-1})$ (ii)
$\displaystyle\operatorname{TC}(c)+\operatorname{LSI}(\alpha_{2})\quad$
$\displaystyle\Rightarrow\quad\operatorname{TI}\Big{(}\frac{c}{\alpha_{2}}\Big{)}$
(iii) $\displaystyle\operatorname{MLSI}(\alpha_{1})\quad$
$\displaystyle\Rightarrow\quad\operatorname{PI}(2\alpha_{1})\,.$ (iv)

###### Proof.

The proof of (i) and (iv) can be found in [45, 15]. (ii) was derived in [57].
Finally (iii) and (iv) are direct consequences of the definitions of LSI and
MLSI. ∎

######  Corollary 1.

Let $t\mapsto\mathrm{e}^{t{\cal L}}$ be a finite dimensional,
$\operatorname{GNS}$ symmetric quantum Markov semigroup, and assume that
$\operatorname{LSI}(\alpha_{2})$ holds. Then
$\operatorname{TI}(8^{-1}\alpha_{2}^{-2})$ holds.

######  Remark 4.

In a paper to appear [32], one of the authors proves the that the
transportation-information inequality is satisfied under a certain condition
of positivity of a non-commutative version of Ollivier’s coarse Ricci
curvature lower bound [54], hence generalizing a result by Fathi and Shu [30].
The latter is satisfied e.g., for a family of quantum Gibbs samplers.

Next, we also prove the following transportation cost-information inequality
based on the Poincaré inequality (see Theorem 3.1 in [37]):

######  Proposition 8.

The Poincaré inequality $\operatorname{PI(\lambda)}$ implies the following
transportation cost-information inequality: for any $\rho\in{\cal D}({\cal
H})$,

$\displaystyle\|\rho-\sigma\|_{1}^{2}\,\leq\,\frac{4}{\lambda}\,I_{{\cal
L}}(\rho)\,.$ (5.5)

###### Proof.

We first prove the following rudimentary inequality: for all $X\geq 0$,

$\displaystyle\|\Gamma_{\sigma}(X)-\sigma\|_{1}^{2}\leq
4\operatorname{Var}_{\sigma}(I_{2,1}(X))\,,$ (5.6)

where
$I_{2,1}(X):=\Gamma_{\sigma}^{-\frac{1}{2}}\big{[}(\Gamma_{\sigma}(X))^{\frac{1}{2}}\big{]}$.
Indeed,

$\displaystyle\|\rho-\sigma\|_{1}$
$\displaystyle=\frac{1}{2}\big{\|}\big{\\{}\sqrt{\rho}-\sqrt{\sigma},\sqrt{\rho}+\sqrt{\sigma}\big{\\}}\big{\|}_{1}=\frac{1}{2}\,\|\\{\sigma^{\frac{1}{4}}(\sigma^{-\frac{1}{4}}\sqrt{\rho}\,\sigma^{-\frac{1}{4}}-{\rm{id}})\,\sigma^{\frac{1}{4}},\,\sqrt{\rho}+\sqrt{\sigma}\\}\|_{1}$
$\displaystyle\leq\frac{1}{2}\,\big{(}\|\sigma^{\frac{1}{4}}(\sigma^{-\frac{1}{4}}\sqrt{\rho}\,\sigma^{-\frac{1}{4}}-{\rm{id}})\,\sigma^{\frac{1}{4}}\,(\sqrt{\rho}+\sqrt{\sigma})\|_{1}+\|(\sqrt{\rho}+\sqrt{\sigma})\,\sigma^{\frac{1}{4}}(\sigma^{-\frac{1}{4}}\,\sqrt{\rho}\,\sigma^{-\frac{1}{4}}-{\rm{id}})\,\sigma^{\frac{1}{4}}\|_{1}\big{)}$
$\displaystyle\leq\|\sigma^{\frac{1}{4}}(\sigma^{-\frac{1}{4}}\,\sqrt{\rho}\,\sigma^{-\frac{1}{4}}-{\rm{id}})\,\sigma^{\frac{1}{4}}\|_{2}\,\|\sqrt{\rho}+\sqrt{\sigma}\|_{2}$
$\displaystyle\leq
2\,\|\sigma^{-\frac{1}{4}}\,\sqrt{\rho}\,\sigma^{-\frac{1}{4}}-{\rm{id}}\|_{L^{2}(\sigma)}\,,$

and Equation 5.6 follows after taking the square and choosing
$\rho=\Gamma_{\sigma}(X)$. Equation (5.5) follows from a direct application of
Poincaré’s inequality and the definition of $I_{\cal L}(\rho)$. ∎

###  5.3 Concentration of trajectories

Concentration of measure is the phenomenon according to which almost all the
points of a set are close to a subset of positive measure. More precisely, let
$({\cal X},d)$ be a metric space, and $\mu$ a probability measure on the Borel
sets ${\cal B}({\cal X})$. Then, given a set $A\in{\cal B}({\cal X})$ such
that $\mu(A)\geq 1/2$, the complement $(A^{r})^{c}$ of its r-enlargements
$A^{r}=\\{x\in{\cal X};d(x,A)\leq r\\}$ should rapidly decay with $r$. This is
typically the case when a transportation cost inequality of the following form
is satisfied for the measure $\mu$: there exists a positive function $\alpha$
such that for any other measure $\nu\ll\mu$,

$\displaystyle\alpha(W_{1}(\nu,\mu))\leq D(\nu\|\mu)\,.$

This connection was first proved by Marton in [51] by a beautiful geometric
argument. A more analytical argument based on bounds on the Laplace transform
of $\mu$ was established later by Bobkov and Götze who further proved the
equivalence between Gaussian concentration of $\mu$ and the corresponding
transportation cost inequality for the Wasserstein distance and
$\alpha(r)=r^{2}$ [12]. In [57], the authors extended the approach of Bobkov
and Götze to the quantum setting and proved that the transportation cost
inequality for the Wasserstein distance defined in (5.4) and $\alpha(r)=r^{2}$
implies Gaussian concentration of Lipschitz quantum observables. Similar
proofs can also be found in [31, 21, 22].

In [37], the functional analytical approach of Bobkov and Götze was extended
to Markov processes. There, concentration for observables evolving along the
stochastic process was proven to be equivalent to the existence of a
transporation-information inequality. The main theorem of this section is
inspired by Theorem 2.2 of [37]:

######  Theorem 8.

Let $t\mapsto\mathrm{e}^{t{\cal L}}$ be a primitive,
$\operatorname{GNS}$-symmetric finite dimensional quantum Markov semigroup on
${\cal B}({\cal H})$ with invariant state $\sigma$, and let $\Phi\subset{\cal
B}_{\operatorname{sa}}({\cal H})^{2}$ be as in Definition 9. Assume further
that $\alpha({\operatorname{{\cal T}_{\Phi}})\operatorname{I}}$ holds with
$\alpha(r):=\frac{r^{2}}{2C}$. Then, for any initial state $\rho\in{\cal
D}({\cal H})$, $t>0$, $u\in\mathbb{C}^{k}$, and any indirectly measured
observable $O^{B}(u):=L_{u}+L^{*}_{u}$ such that the observable
$\widetilde{O}^{B}(u):=\Delta_{\sigma}^{\frac{1}{4}}(L^{*}_{u})+\Delta_{\sigma}^{-\frac{1}{4}}(L_{u})$
satisfies
$(\sqrt{C}\widetilde{O}^{B}(u),\sqrt{C}\widetilde{O}^{B}(u))\in\Phi$,

$\displaystyle\mathbb{P}\Big{(}\frac{1}{t}\int_{0}^{t}\mathop{\rm
Tr}\nolimits[O^{B}(u)\rho_{s}]\,ds+\|u\|\,\frac{W_{t}}{t}>\mathop{\rm
Tr}\nolimits[\sigma
O^{B}(u)]+r\Big{)}\leq\|\Gamma_{\sigma}^{-1}(\rho)\|_{\mathbb{L}_{2}(\sigma)}\,\exp\Big{(}-\frac{t\,r^{2}}{4}\Big{)}\,.$
(5.7)

###### Proof.

Without loss of generality, we assume that $\|u\|=1$ and $\mathop{\rm
Tr}\nolimits[\sigma O^{B}(u)]=0$. Then, by Theorem 6, we have

$\displaystyle\frac{\mathbb{P}\Big{(}\frac{1}{t}\int_{0}^{t}\mathop{\rm
Tr}\nolimits[O^{B}(u)\rho_{s}]\,ds+\frac{W_{t}}{t}>r\Big{)}}{\|\Gamma_{\sigma}^{-1}(\rho)\|_{L^{2}(\sigma)}}\,\leq\,\exp\Big{(}-t\,\underset{\rho}{\inf}\,\big{\\{}I_{\cal
L}(\rho)+\frac{1}{2}\,\big{(}r-\mathop{\rm
Tr}\nolimits[\rho\,\widetilde{O}^{B}(u)]\big{)}^{2}\big{\\}}\Big{)}\,,$ (5.8)

where
$\widetilde{O}^{B}(u):=\Delta_{\sigma}^{\frac{1}{4}}(L^{*}_{u})+\Delta_{\sigma}^{-\frac{1}{4}}(L_{u})$.
Next, by $\alpha({\cal T}_{\Phi})\operatorname{I}$, we can further bound
$I_{\cal L}(\rho)$ as follows:

$\displaystyle I_{\cal L}(\rho)\geq\frac{{\cal
T}_{\Phi}(\rho,\sigma)^{2}}{2C}\geq\frac{\mathop{\rm
Tr}\nolimits[\rho\,\widetilde{O}^{B}(u)]^{2}}{2}\,,$ (5.9)

where we used that
$(\sqrt{C}\widetilde{O}^{B}(u),\sqrt{C}\widetilde{O}^{B}(u))\in\Phi$ and that
$\mathop{\rm Tr}\nolimits[\sigma\,\widetilde{O}^{B}(u)]=\mathop{\rm
Tr}\nolimits[\sigma\,O^{B}(u)]=0$. Therefore,

$\displaystyle\frac{\mathbb{P}\Big{(}\frac{1}{t}\int_{0}^{t}\mathop{\rm
Tr}\nolimits[O^{B}(u)\,\rho_{s}]\,ds+\frac{W_{t}}{t}>r\Big{)}}{\|\Gamma_{\sigma}^{-1}(\rho)\|_{L^{2}(\sigma)}}\,\leq\exp\Big{(}-t\,\inf_{\rho}\,\frac{\mathop{\rm
Tr}\nolimits[\rho\,\widetilde{O}^{B}(u)]^{2}}{2}+\frac{\big{(}r-\mathop{\rm
Tr}\nolimits[\rho\,\widetilde{O}^{B}(u)]\big{)}^{2}}{2}\Big{)}\leq\exp\Big{(}-\frac{t\,r^{2}}{4}\Big{)}\,,$

where the last bound follows from the two-points inequality
$2(a^{2}+b^{2})\geq(a-b)^{2}$. The result follows. ∎

######  Remark 5.

Observe that we do not assume $u\in S^{k-1}(\mathbb{C})$ above, but instead
simply $u\in\mathbb{C}^{k}$. This is done up to a scaling of the Brownian
motion $W_{t}\to\|u\|\,W_{t}$, where $\|\cdot\|$ is the canonical $2$-norm.

######  Corollary 2.

With the notations of Theorem 8, we have that, under $\operatorname{TI}(C)$,
for any $u\in\mathbb{C}^{k}$ and any indirectly measured observable
$O^{B}(u):=L_{u}+L^{*}_{u}$ and all $t,r>0$

$\displaystyle\mathbb{P}\Big{(}\frac{1}{t}\int_{0}^{t}\mathop{\rm
Tr}\nolimits[O^{B}(u)\,\rho_{s}]\,ds+\|u\|\,\frac{W_{t}}{t}>\mathop{\rm
Tr}\nolimits[\sigma\,O^{B}(u)]+r\Big{)}\leq\|\Gamma_{\sigma}^{-1}(\rho)\|_{\mathbb{L}_{2}(\sigma)}\,\exp\left(-\frac{t\,r^{2}}{2\big{(}1+C\|\widetilde{O}^{B}(u)\|_{\operatorname{Lip}}^{2}\big{)}}\right)\,,$
(5.10)

where we recall that
$\widetilde{O}^{B}(u)=\Delta_{\sigma}^{\frac{1}{4}}(L^{*}_{u})+\Delta_{\sigma}^{-\frac{1}{4}}(L_{u})$.

###### Proof.

This directly follows from a simple modification of the proof of Theorem 8:
first, (5.9) is replaced by the bound

$\displaystyle I_{\cal L}(\rho)\geq\frac{W_{1,{\cal
L}}(\rho,\sigma)^{2}}{2C}\geq\frac{\mathop{\rm
Tr}\nolimits[\rho\,\widetilde{O}^{B}(u)]^{2}}{2C\,\|\widetilde{O}^{B}(u)\|_{\operatorname{Lip}}^{2}}\,.$

The result follows after replacing the use of the inequality
$2(a^{2}+b^{2})\geq(a-b)^{2}$ by
$\big{[}\big{(}\frac{a}{b}\big{)}^{2}+(r-a)^{2}\big{]}\geq\frac{r^{2}}{1+b^{2}}$,
with $a=\mathop{\rm Tr}\nolimits[\rho\widetilde{O}^{B}(u)]$ and
$b=\sqrt{C}\|\widetilde{O}^{B}(u)\|_{\operatorname{Lip}}$. ∎

##### Concentration from Poincaré inequality

We now prove the following weaker concentration bound depending on the gap of
${\cal L}$ (see also Theorem 3.1 in [37]).

######  Theorem 9.

Let $t\mapsto\mathrm{e}^{t{\cal L}}$ be a finite dimensional, primitive,
$\operatorname{KMS}$ symmetric quantum Markov semigroup. Then, for any initial
state $\rho\in{\cal D}({\cal H})$, $t>0$, $u\in\mathbb{C}^{k}$ and any
indirectly measured observable $O^{B}(u):=L_{u}+L^{*}_{u}$,

$\displaystyle\mathbb{P}_{\rho}\Big{(}\,\frac{1}{t}\,\int_{0}^{t}\,\mathop{\rm
Tr}\nolimits[O^{B}(u)\,\rho_{s}]\,ds+\|u\|\,\frac{W_{t}}{t}\geq\mathop{\rm
Tr}\nolimits[\sigma\,O^{B}(u)]+r\Big{)}\leq\|\Gamma_{\sigma}^{-1}(\rho)\|_{L^{2}(\sigma)}\,\exp\left(-\frac{\lambda({\cal
L})\,t\,r^{2}}{2\big{(}\lambda({\cal
L})+2\|\widetilde{O}^{B}(u)\|_{\infty}^{2}\big{)}}\right)\,,$

where we recall that
$\widetilde{O}^{B}(u)=\Delta_{\sigma}^{\frac{1}{4}}(L^{*}_{u})+\Delta_{\sigma}^{-\frac{1}{4}}(L_{u})$.
and where $\lambda({\cal L})$ is the gap of ${\cal L}$.

###### Proof.

The proof follows the same lines as those of the proof of Corollary 2, and is
a direct consequence of Theorem 6, Proposition 8 and the dual formulation of
the trace norm. ∎

##  6 Examples

###  6.1 Depolarizing channel

We first consider the simplest QMS, namely the depolarizing semigroup on
${\cal B}({\cal H})$, which is defined for any full-rank state $\sigma$ by

$\displaystyle\mathrm{e}^{t{\cal
L}_{\sigma}}(X):=(1-\mathrm{e}^{-t}){\rm{id}}\mathop{\rm Tr}\nolimits[\sigma
X]+\mathrm{e}^{-t}X\,.$ (6.1)

One can readily check that the semigroup is primitive with unique invariant
state $\sigma$, and that it is GNS symmetric with respect to $\sigma$. The LSI
and MLSI constants of the semi-group $t\mapsto e^{t{\cal L}_{\sigma}}$ were
computed in [52, 9]. In particular, we have

$\displaystyle\alpha_{2}({\cal
L}_{\sigma})=\frac{1-2{s}_{\min}(\sigma)}{\ln\big{(}s_{\min}(\sigma)^{-1}-1\big{)}}\,,\qquad\qquad\alpha_{2}({\cal
L}_{d^{-1}{\rm{id}}})=\frac{d-2}{d\ln(d-1)}\,.$ (6.2)

where $s_{\min}(\sigma)$ denotes the minimal eigenvalue of $\sigma$. For sake
of simplicity, we restrict ourselves to the case when $\sigma$ is the
maximally mixed state on ${\cal H}$: $\sigma=d^{-1}{\rm{id}}$, where $d$
stands for the dimension of ${\cal H}$. In that case, the Lindblad operators
of ${\cal L}_{\sigma}$ can be chosen as
$L_{xy}:=d^{-\frac{1}{2}}|x\rangle\langle y|$ for an arbitrary orthonormal
basis $\\{|x\rangle\\}_{x}$ of ${\cal H}$. In this case, any self-adjoint
operator $O$ can be interpreted as an indirectly measured observable

$\displaystyle O\equiv
O^{B}(u)=\sum_{x,y=1}^{d}\,\frac{u_{x,y}}{\sqrt{d}}\,\Big{(}|x\rangle\langle
y|+|y\rangle\langle x|\Big{)}\,.$ (6.3)

Without loss of generality, the basis $\\{|x\rangle\\}_{x}$ can be chosen as
to diagonilize the observable $O$, so that the eigenvalues $\\{O_{x}\\}_{x}$
satisfy $O_{x}=\delta_{x,y}\,\frac{u_{x,y}}{\sqrt{d}}$. Therefore

$\displaystyle\|u\|^{2}=\sum_{x,y}|u_{x,y}|^{2}=d\,\sum_{x}\,O_{x}^{2}=d\|O\|_{2}^{2}\,.$

By Corollary 1, we then have that the depolarizing semi-group satisfies the
transport-information inequality with constant $C=8^{-1}\alpha_{2}({\cal
L}_{\sigma})^{-2}$. Therefore, from Corollary 2, we derive the following
concentration for the quantum trajectories:

######  Corollary 3.

In the notations of Corollary 2, for any observable $O$ with eigenvalues
$\\{O_{x}\\}_{x}$ and all $t,r>0$,

$\displaystyle\mathbb{P}\Big{(}\frac{1}{t}\int_{0}^{t}\mathop{\rm
Tr}\nolimits[O\,\rho_{s}]\,ds+\sqrt{d}\|O\|_{2}\,\frac{W_{t}}{t}>\frac{1}{d}\mathop{\rm
Tr}\nolimits[\,O]+r\Big{)}\leq
d\,\exp\left(-\frac{2(d-2)^{2}\,t\,r^{2}}{4(d-2)^{2}+\sum_{x,y}(O_{x}-O_{y})^{2}\,{d^{2}\ln(d-1)^{2}}}\right)\,.$

###### Proof.

In light of the paragraph before Corollary 3, it remains to express the
Lipschitz constant of $O$. Since $\sigma=d^{-1}{\rm{id}}$, the constants
$\\{\omega_{x,y}\\}_{x,y}$ in the definition of the Lipschitz constant are all
equal to $0$. Without loss of generality, we can chose the basis
$\\{|x\rangle\\}_{x}$ in which $O$ is diagonal. Then

$\displaystyle\|O\|_{\operatorname{Lip}}^{2}=\sum_{x,y}\,2\,\big{\|}\big{[}|x\rangle\langle
y|,O\big{]}\|_{\infty}^{2}=2\,\sum_{x,y}\,(O_{x}-O_{y})^{2}\,.$

The result follows. ∎

###  6.2 Tensorization

In the classical framework, when two semigroups each satisfy a (modified)
logarithmic Sobolev inequality with constants $\alpha({\cal L}_{1})$ and
$\alpha({\cal L}_{2})$ respectively, then their tensor product will also
satisfy the same inequality with a constant $\alpha=\min(\alpha({\cal
L}_{1}),\alpha({\cal L}_{2}))$. This property is known as dimension-free
tensorization. In the case of a transportation-cost or transportation-
information inequality, the constant will rather scale as the sum of the
constants of the local systems. Since logarithmic Sobolev inequalities imply
transportation cost ones, the former provide a much stronger notion of
tensorization when satisfied.

Tensorization is a much more subtle property to prove in the quantum realm. In
general, neither the tensorization of the logarithmic Sobolev constant, nor
that of the modified logarithmic Sobolev constant is known to hold for general
classes of semigroups. Some exceptions for the former include the class of
primitive qubit unital semigroups [46] or the qubit depolarizing semigroup
$t\mapsto\mathrm{e}^{t{\cal L}_{\sigma}}$ [9]. Similarly, the modified
logarithmic Sobolev constant is only known to tensorize for a few cases
including e.g., the quantum Ornstein Uhlenbeck semigroup [16, 55].

Thankfully, new techniques have been introduced in order to deal with the
current lack of a proof of tensorization of the aforementioned quantum
functional inequalities. In the case of the logarithmic Sobolev constant, the
following was proved in [58, Theorem 9]:

######  Lemma 2.

Let $N\in\mathbb{N}$ and for any $k\in\\{1,...,N\\}$ let
$t\mapsto\mathrm{e}^{t{\cal L}_{k}}$ be a primitive $\operatorname{QMS}$
acting on ${\cal B}({\cal H})$ verifying $\operatorname{KMS}$
$\operatorname{QDB}$ with respective invariant state $\sigma_{k}$ and spectral
gap $\lambda_{k}$. Then, the logarithmic Sobolev constant $\alpha_{2}({\cal
L}^{(N)})$ of the product $\operatorname{QMS}$
$t\mapsto\mathrm{e}^{t\sum_{k}{\cal L}_{k}\otimes{\rm{id}}_{k^{c}}}$ satisfies

$\displaystyle\frac{\min_{k}\\{\lambda_{k}\\}}{\ln\big{(}d^{4}\max_{k}\\{\|\sigma_{k}^{-1}\|_{\infty}\\}\big{)}+11}\leq\alpha_{2}({\cal
L}^{(N)})\leq\frac{\min_{k}\\{\lambda_{k}\\}}{2}\,,$ (6.4)

where $d:=\operatorname{dim}({\cal H})$. In particular, the lower bound in
(6.4) is independent of the number $N$ of subsystems.

We also mention in passing the recent advances in proving tensorization of the
modified logarithmic Sobolev constant beyond the primitive case in [33]. Lemma
2 can be used in combination with Corollary 2 to provide a tensorization
result for the concentration bounds derived in Section 5:

######  Corollary 4.

Assume that the $\operatorname{QMS}$ $t\mapsto\mathrm{e}^{t\sum_{k=1}^{n}{\cal
L}_{k}\otimes{\rm{id}}_{k^{c}}}$ has $\operatorname{LSI}$ constant
$\alpha_{2}$. Furthermore, we denote by $\\{\omega_{k,j}\\}_{j\in\mathcal{J}}$
the Bohr frequencies of ${\cal L}_{k}$, and by
$\\{L_{k,j}\\}_{j\in\mathcal{J}}$ its corresponding Lindblad operators. Then,
for any observable $O_{{u}}=\big{(}L_{{u}}+L_{{u}}^{*}\big{)}$ with
$u=(u_{k,j})_{k,j}\in\mathbb{C}^{n|{\cal J}|}$, we have

$\displaystyle\mathbb{P}\Big{(}\frac{1}{t}\int_{0}^{t}\mathop{\rm
Tr}\nolimits[O_{{u}}\rho_{s}]\,ds+\|u\|\,\frac{\,W_{t}}{t}>\mathop{\rm
Tr}\nolimits[\otimes_{k}\sigma_{k}O_{{u}}]+r\Big{)}\leq\|\Gamma_{\otimes_{k}\sigma_{k}}^{-1}(\rho)\|_{L^{2}(\otimes_{k}\sigma_{k})}\,\exp\left(-\frac{4\alpha_{2}^{2}\,t\,r^{2}}{8\alpha_{2}^{2}+n\alpha(u)}\right)\,,$
(6.5)

where
$\alpha({u}):=2\,|\mathcal{J}|\max_{k,j}\mathrm{e}^{\omega_{k,j}/2}\|\sum_{i}u_{k,i}\big{[}L_{k,j},\Delta_{\sigma_{k}}^{\frac{1}{4}}(L_{k,i}^{*})+\Delta_{\sigma_{k}}^{-\frac{1}{4}}(L_{k,i})\big{]}\big{\|}_{\infty}^{2}$.

###### Proof.

We denote $\sigma:=\otimes_{k}\sigma_{k}$. By Corollary 2, we have that if the
tensor product semigroup satisfies a transportation-information inequality
then (6.5) follows as long as the Lipschitz constant on the right-hand side of
(5.10) is upper bounded by $\alpha({u})$. This follows from:

$\displaystyle\|\widetilde{O}^{B}(u)\|_{\operatorname{Lip}}^{2}$
$\displaystyle=\sum_{k=1}^{n}\sum_{j\in\mathcal{J}}\,\big{(}\mathrm{e}^{-\omega_{k,j}/2}+\mathrm{e}^{\omega_{k,j}/2}\big{)}\,\big{\|}\big{[}L_{k,j},\Delta_{\sigma}^{\frac{1}{4}}(L_{{u}}^{*})+\Delta_{\sigma}^{-\frac{1}{4}}(L_{{u}})\big{]}\big{\|}_{\infty}^{2}$
(6.6)
$\displaystyle=\sum_{k=1}^{n}\sum_{j\in\mathcal{J}}\,\big{(}\mathrm{e}^{-\omega_{k,j}/2}+\mathrm{e}^{\omega_{k,j}/2}\big{)}\,\big{\|}\sum_{i}u_{k,i}\big{[}L_{k,j},\Delta_{\sigma_{k}}^{\frac{1}{4}}(L_{k,i}^{*})+\Delta_{\sigma_{k}}^{-\frac{1}{4}}(L_{k,i})\big{]}\big{\|}_{\infty}^{2}$
(6.7) $\displaystyle\leq
n2\,|\mathcal{J}|\max_{k,j}\mathrm{e}^{\omega_{k,j}/2}\|\sum_{i}u_{k,i}\big{[}L_{k,j},\Delta_{\sigma_{k}}^{\frac{1}{4}}(L_{k,i}^{*})+\Delta_{\sigma_{k}}^{-\frac{1}{4}}(L_{k,i})\big{]}\big{\|}_{\infty}^{2}\,.$
(6.8)

The result then follows from Corollary 1 which establishes that transport-
information is implied by LSI. ∎

###  6.3 Gibbs samplers

Tensorization can be thought of as a property of non-interacting systems or of
systems at infinite temperature, for which the evolution can be written as a
tensor power of local channels. Proofs of LSI/MLSI for quantum interacting
spin systems (a.k.a. Gibbs samplers) have recently attracted the attention of
the community [6, 7, 2]. More recently, it was shown that Gibbs states over
arbitrary graphs satisfy a transportation cost inequality at large enough
temperature [22]. Building on the techniques of [22], one of the authors
proves in an article to appear that a certain class of Gibbs samplers
satisfies the transportation cost-information inequality at high enough
temperature.

More precisely, let $G=(V,E)$ be a graph with $n=|V|$, and let ${\cal
H}_{V}:=\bigotimes_{v\in V}{\cal H}_{v}$ be the Hilbert space of a local
quantum system, with ${\cal H}_{v}:=\mathbb{C}^{d}$ for all $v\in V$. The
interactions are modeled through the Hamiltonian
$H:=\sum_{A\subset\Lambda}h_{A}\otimes 1_{A^{c}}$, where each local self-
adjoint operator $h_{A}$ satisfies $\|h_{A}\|\leq 1$ and is supported on the
region $A\subset V$. Here, we also assume that the Hamiltonian is of finite-
range, which means that the size and diameter of any region $A$ appearing in
the decomposition of $H$ is uniformly bounded by a constant $r>0$ independent
of $n$. We also assume the interaction to be commuting, i.e.,
$[h_{A},h_{A^{\prime}}]=0$ for all $A,A^{\prime}$. Next, we define the Gibbs
state $\omega$ associated to $H$ at inverse temperature $\beta>0$ as

$\displaystyle\omega:=\frac{e^{-\beta H}}{\mathop{\rm
Tr}\nolimits\big{[}e^{-\beta H}\big{]}}\,.$ (6.9)

A Gibbs sampler is a locally defined quantum channel which prepares an
approximation of the Gibbs state $\omega$ starting from any initial state on
${\cal H}_{V}$. The efficiency of the Gibbs sampler depends on the time it
takes to reach the approximating state. Here, we consider the heat-bath
generator which is defined as follows: for a given site $v\in V$, we denote
the composition of the partial trace $\mathop{\rm Tr}\nolimits_{v}$ on $v$
with the Petz recovery map of $v$ as

$\displaystyle\Psi_{v}^{*}(\rho)=\Phi_{v}^{*}\circ\mathop{\rm
Tr}\nolimits_{v}(\rho)=\omega^{\frac{1}{2}}\omega_{v^{c}}^{-\frac{1}{2}}\,(\rho_{v^{c}}\otimes
I_{v})\,\omega_{v^{c}}^{-\frac{1}{2}}\omega^{\frac{1}{2}}\,,$ (6.10)

where $\omega$ is the Gibbs state of the Hamiltonian $H$, and where we denoted
by $\omega_{A}$ the reduced state on the subregion $A\subseteq V$. Clearly,
when $H$ is made of commuting terms, the map $\Psi_{v}$ acts non-trivially on
the neighborhood of $v$, which is defined as

$N_{v}:=\bigcup\left\\{A\in E:v\in A\right\\}\,.$ (6.11)

Next, we introduce the generator of the heat-bath dynamics

$\displaystyle{\cal L}_{V}:=\sum_{v\in V}{\cal L}_{v}\,,$ (6.12)

where ${\cal L}_{v}:=\Psi_{v}-{\rm{id}}$. The quantum Markov semigroup
$t\mapsto e^{t{\cal L}_{V}^{*}}$ generated by ${\cal L}_{V}^{*}$ converges to
$\omega$ as $t\to\infty$. The following result is a direct consequence of [32,
Proposition 4.13 and Proposition 6.1].

######  Lemma 3.

There exists an inverse temperature $\beta_{c}>0$ such that, for any
$\beta<\beta_{c}$, the $\operatorname{QMS}$ $t\mapsto e^{t{\cal L}_{V}}$
satisfies the transportation cost-information inequality with respect to the
quantum Ornstein distance with constant $C=\mathcal{O}(n)$: for any quantum
state $\rho$,

$\displaystyle W_{1}^{\operatorname{Orn}}(\rho,\omega)\leq\sqrt{2C\,I_{{\cal
L}_{V}}(\rho)}\,.$

In what follows, we denote again by $k$ the number of Linblad operators
contained in $\mathcal{L}_{V}$. Clearly, $k=\mathcal{O}(|V|)$. Combining the
previous Lemma with the main concentration bound (5.7), we find the following
corollary:

######  Corollary 5.

For any initial state $\rho\in{\cal D}({\cal H})$, $t>0$,
$u\in\mathbb{C}^{k}$, and any indirectly measured observable
$O^{B}(u):=L_{u}+L^{*}_{u}$,

$\displaystyle\mathbb{P}\Big{(}\frac{1}{t}\int_{0}^{t}\mathop{\rm
Tr}\nolimits[O^{B}(u)\rho_{s}]\,ds+\|u\|\,\frac{W_{t}}{t}>\mathop{\rm
Tr}\nolimits[\omega
O^{B}(u)]+r\Big{)}\leq\,\exp\left(\frac{\beta\|H\|_{\infty}}{2}-\frac{t\,r^{2}}{2\big{(}1+C(\|\widetilde{O}^{B}(u)\|_{\operatorname{Lip}}^{\operatorname{Orn}})^{2}\big{)}}\right)\,,$
(6.13)

where $C=\mathcal{O}(n)$ was introduced in Lemma 3, and where
$\widetilde{O}^{B}(u):=\Delta_{\sigma}^{\frac{1}{4}}(L^{*}_{u})+\Delta_{\sigma}^{-\frac{1}{4}}(L_{u})$.

###### Proof.

The result follows directly from Corollary 2 up to the replacement of
$\|.\|_{\operatorname{Lip}}$ to
$\|.\|_{\operatorname{Lip}}^{\operatorname{Orn}}$ and the estimate:

$\displaystyle\|\Gamma_{\sigma}^{-1}(\rho)\|_{L^{2}(\sigma)}=\big{\|}e^{\frac{\beta
H}{4}}\rho e^{\frac{\beta H}{4}}\big{\|}_{2}\leq
e^{\frac{\beta\|H\|_{\infty}}{2}}\,.$

∎

Another Gibbs sampler which models the thermalization of a quantum system
weakly interacting with a large reservoir is the so-called Davies dynamics
[20]. In [44], it is proved that Davies dynamics are gapped at any inverse
temperature $\beta>0$ in 1D and on regular lattices below a threshold inverse
temperature $\beta_{c}>0$. This result was extended to the MLSI in the 1D case
in a paper to appear [5]. However, it is still open whether these dynamics
also satisfy a transportation cost-information inequality under reasonable
assumptions.

##### Acknowledgement

The research of T.B. has been supported by ANR project ESQUISSE
(ANR-20-CE47-0014-01) of the French National Research Agency (ANR). The
research of T.B. and C.R. has been supported by ANR project QTraj
(ANR-20-CE40-0024-01) of the French National Research Agency (ANR). C.R.
acknowledges the support of the Munich Center for Quantum Sciences and
Technology, as well as the Humboldt Foundation. The research of L.H. was
partially supported by the Swiss National Science Foundation (Grant No.
P2EZP2-188093).

##  Appendix A The quantum perturbed generators

Here we present the computational steps for the derivation of the generators
$\mathcal{L}_{i\lambda,u}^{B}$, $\mathcal{L}_{i\lambda,u}^{P}$, and
$\mathcal{L}_{i\lambda,\mathbf{u}}$ in Section 4.1 for single-mode Brownian
motion (cf. Section 4.1.1), single-mode Poisson processes (cf. Section 4.1.2),
and the general multi-mode case with both kinds of processes included (cf.
Section 4.1.3), respectively.

As it will be used in the derivations below, we recall the following product
rule [56]: Let $\mathbf{E}_{t},\mathbf{F}_{t},\mathbf{G}_{t},\mathbf{H}_{t}$
and
$\mathbf{E}^{\prime}_{t},\mathbf{F}^{\prime}_{t},\mathbf{G}^{\prime}_{t},\mathbf{H}^{\prime}_{t}$
be appropriate families of operators on ${\cal H}_{S}\otimes\Gamma$, and let
$X_{t},X^{\prime}_{t}$ be defined by the differential equations

$\displaystyle\operatorname{d}X_{t}$
$\displaystyle=\mathbf{E}_{t}.\operatorname{d}\mathbf{A}_{\mathbf{u}}(t)+\mathbf{F}_{t}.\mathbf{A}_{\mathbf{u}}^{*}(t)+\mathbf{G}_{t}.\operatorname{d}\mathbf{\Lambda}_{\mathbf{u}\mathbf{u}^{*}}(t)+\mathbf{H}_{t}\operatorname{d}t\,,$
$\displaystyle\operatorname{d}X^{\prime}_{t}$
$\displaystyle=\mathbf{E}^{\prime}_{t}.\operatorname{d}\mathbf{A}_{\mathbf{u}}(t)+\mathbf{F}^{\prime}_{t}.\mathbf{A}_{\mathbf{u}}^{*}(t)+\mathbf{G}^{\prime}_{t}.\operatorname{d}\mathbf{\Lambda}_{\mathbf{u}\mathbf{u}^{*}}(t)+\mathbf{H}^{\prime}_{t}\operatorname{d}t\,.$

Then

$\displaystyle\operatorname{d}(X_{t}X^{\prime}_{t})=(\operatorname{d}X_{t})X^{\prime}_{t}+X_{t}(\operatorname{d}X^{\prime}_{t})+(\operatorname{d}X_{t})(\operatorname{d}X^{\prime}_{t})\,,$
(A.1)

where

$\displaystyle(\operatorname{d}X_{t})X^{\prime}_{t}=\mathbf{E}_{t}X^{\prime}_{t}.\operatorname{d}\mathbf{A}_{\mathbf{u}}(t)+\mathbf{F}_{t}X^{\prime}_{t}.\operatorname{d}\mathbf{A}_{\mathbf{u}}^{*}(t)+\mathbf{G}_{t}X^{\prime}_{t}.\operatorname{d}\mathbf{\Lambda}_{\mathbf{u}\mathbf{u}^{*}}(t)+\mathbf{H}_{t}X^{\prime}_{t}\operatorname{d}t\,,$

and $(\operatorname{d}X_{t}).(\operatorname{d}X^{\prime}_{t})$ is evaluated
according to the quantum Itô rules (2.4). Furthermore, the following