Witnessing non-Markovianity by Quantum Quasi-Probability Distributions Moritz F. Richter1 Raphael Wiedenmann1and Heinz-Peter

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Witnessing non-Markovianity by Quantum

Quasi-Probability Distributions

Moritz F. Richter1, Raphael Wiedenmann1and Heinz-Peter

Breuer12

1Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, D-79104

Freiburg, Germany

2EUCOR Centre for Quantum Science and Quantum Computing, University of

Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany

October 2022

Abstract. We employ frames consisting of rank-one projectors (i.e. pure quantum

states) and their induced informationally complete quantum measurements (IC-

POVMs) to represent generally mixed quantum states by quasi-probability

distributions. In the case of discrete frames on ﬁnite dimensional systems this results in

a vector like representation by quasi-probability vectors, while for the continuous frame

of coherent states in continuous variable (CV) systems the approach directly leads

to the celebrated representation by Glauber-Sudarshan P-functions and Husimi Q-

functions. We explain that the Kolmogorov distances between these quasi-probability

distributions lead to upper and lower bounds of the trace distance which measures

the distinguishability of quantum states. We apply these results to the dynamics

of open quantum systems and construct a non-Markovianity witness based on the

Kolmogorov distance of the P- and Q-functions. By means of several examples we

discuss the performance of this witness and demonstrate that it is useful in the regime

of high entropy states for which a direct evaluation of the trace distance is typically

very demanding. For Gaussian dynamics in CV systems we even ﬁnd a suitable non-

Markovianity measure based on the Kolmogorov distance between the P-functions

which can alternatively be used as a witness for non-Gaussianity.

Keywords: Open quantum systems, non-Markovian dynamics, quantum measurement,

IC-POVM, frames, quasi-probability distributions, Kolmogorov distance

arXiv:2210.06058v1 [quant-ph] 12 Oct 2022

Witnessing non-Markovianity by Quantum Quasi-Probability Distributions 2

1. Introduction

A topical issue in the quantum theory of open systems [1] is the deﬁnition, detection

and quantiﬁcation of non-Markovian dynamics, i.e. dynamical processes determined

by memory eﬀects [2, 3, 4]. One approach to the deﬁnition of non-Markovianity in the

quantum regime is based on the concept of an information ﬂow between the open system

and its environment [5]. According to this concept Markovian dynamics is characterized

by a continuous ﬂow of information from the open system to its environment, while non-

Markovian behavior features a ﬂow of information from the environment back to the

open system. To formulate these ideas in mathematical terms one has to introduce a

suitable measure for the information inside the open system. A natural such measure is

given by the trace distance [6] between pairs of open system states, since this distance

has a direct interpretation in terms of the distinguishability of the quantum states [7].

Thus, a quantum process is said to be Markovian if the trace distance and, hence, the

distinguishablility decreases monotonically in time and, vice versa, it is non-Markovian

if the trace distance shows a non-monotonic behavior, implying a temporary increase of

the distinguishability which characterizes the information backﬂow. We note that there

are also entropic measures for this information with similar properties [8, 9]. As a further

concept one introduces the dynamical maps Λt,t≥0, which describe the evolution in

time of the density matrix ρof the open system as ρ(t) = Λt[ρ(0)]. This map is known

to be completely positive and trace preserving (CPTP) and has the important property

to be a contraction for the trace distance [10, 6]. This leads to the following measure

for the degree of memory eﬀects of a certain pair of initial states ρ1and ρ2of the open

system [5, 11],

N(ρ1,ρ2,{Λt}) := Z

σ≥0

dt σ(ρ1,ρ2, t)(1)

with σ(ρ1,ρ2, t) := d

dtdtr(Λt[ρ1],Λt[ρ2]),(2)

quantifying the total backﬂow of information form the environment into the system.

Obviously, this quantity depends on the chosen pair of initial states. To make this

quantity independent of the initial state pair and to obtain a function which only depends

on the family of dynamical maps, one can take the maximum over all initial state pair

leading to the non-Markovianity measure

N({Λt}) := max

ρ1,ρ2N(ρ1,ρ2,{Λt}).(3)

This information ﬂow approach to quantum non-Markovianity has been applied

to many theoretical models and experimental systems (see, e.g., the reviews [2, 3]

and references therein). However, most applications deal with relatively simple open

systems, describing for example a qubit or a small number of interacting qubits (e.g. a

spin-chain) linearly coupled to bosonic or fermionic reservoirs with a structured spectral

density. For more complicated systems not only the determination of the dynamical

evolution is much more involved, even if it can be represented in terms of an exact or

Witnessing non-Markovianity by Quantum Quasi-Probability Distributions 3

approximate master equation, but also the calculation of the trace distance between

the quantum states can become highly demanding, especially for inﬁnite dimensional

continuous variable (CV) systems. In this paper we want to tackle these diﬃculties by

investigating non-Markovianity measures or at least non-Markovianity witnesses which

are more suitable for CV systems, employing representations of quantum states by means

of quasi-probability distributions.

To this end, we start in section 2 by introducing a vector-like representation of

quantum states based on generalized bases on bounded operators consisting of a set of

ﬁxed but not mutually orthogonal pure states which we will address as quantum frames.

We show that for any quantum state its decomposition into such a quantum frame can

be understood as a quasi-probabilistic mixture, and we will recapitulate how one can

connect these quantum frames to so called informationally complete positive operator

valued measure or IC-POVM [12, 13], a class of quantum measurements especially useful

in quantum tomography and even experientially realizable [14, 15]. The frame based

and the IC-POVM based representation of a quantum state can then be used in section

3 to deﬁne distance measures for quantum states by applying the Kolmogorov distance

for probability distribution. We then will show how to use these measures in order to

approximate the trace distance.

While section 2 considers ﬁnite dimensional quantum systems only, we will

reformulate in section 4 the idea for the continuous quantum frame of coherent states

in CV systems and its connection to the Glauber-Sudarshan P-function and the Husimi

Q-function [16]. Again we apply the Kolmogorov distance to these quasi-probability

distributions in order to approximate the trace distance between given quantum states.

We also discuss the performance of this approximation with the help of a class of

randomly generated Gaussian states [17]. On the basis of this approximation we will

construct in section 5 a suitable witness for the non-Markovianity in CV systems using

P- and Q-functions (subsection 5.1) and illustrate this by means of the example of a

non-Markovian quantum oscillator (subsection 5.2). In subsection 5.3 we focus on a

special class of dynamical maps which preserve the Gaussianity of an initial quantum

state and discuss the time evolution of the Kolmogorov distance between P-functions

under those maps. This will lead us to a novel measure for non-Markovianity for the

case of Gaussian dynamics, since the Kolmogorov distance turns out to be contracting

under Gaussian CPTP maps. Finally we draw our conclusions in section 6 and give an

outlook on possible future studies, where one could apply the witnesses and measures

of non-Markovianity developed here.

2. Quantum Frames and IC-POVM

We begin with a brief introduction to frames in general Hilbert spaces in order to clarify

the terminology and to sketch basic concepts. For more details, the reader is referred

to Ref. [18]. A frame F={|fii} ⊂ H in a Hilbert space His a set of vectors satisfying

span {F} =H. The immediate question now is: What is the diﬀerence to a basis in H?

Witnessing non-Markovianity by Quantum Quasi-Probability Distributions 4

The answer is the following:

•Firstly, a basis has to be a minimal set spanning Hwhile a frame might be

overcomplete. In this sense a frame is just a generalized notion of a basis.

•Secondly, when speaking of a basis we usually mean an orthonormal basis (ONB)

for which hfi|fji=δij . Although mathematically a basis is neither required to be

orthogonal nor normalized, we will use the term frame to stress that orthonormality

of its elements is not assumed.

The possible non-orthogonality causes the major technical diﬀerences between ONBs

and frames. An important object in frame theory is the frame operator S:H → H

deﬁned by

S|vi=X

ihfi|vi|fii.(4)

In case of an ONB the frame operator is simply the identity operator while for general

frames it is not. Later on, this frame operator will provide the link between a frame

induced decomposition and a frame induced measurement of a state. Furthermore, we

can deﬁne a canonical dual frame via the frame operator as

F:= {|˜

fii=S−1|fii} with |vi=X

ih˜

fi|vi|fii.(5)

Again, in the case of an ONB we ﬁnd ˜

F=Fand, hence, an ONB can be addressed as a

minimal and self-dual frame. However, although orthogonality simpliﬁes decomposition

and spanning signiﬁcantly, it might be of advantage to use non-orthogonal bases with

certain other useful properties instead as we will do in this paper (see below). Even

further, it might be desirable to have a spanning set, i.e. a frame, which covers some

part of the vector space at hand more densely such that vectors one wants to decompose

are already closer (in a certain sense) to some frame elements and have decompositions

more pronounced on a single such element.

Let us now consider a quantum system with ﬁnite-dimensional Hilbert space H.

We denote the space of bounded operators on Hby B(H), which becomes a Hilbert

spaces on its own via the Hilbert-Schmidt scalar product

hA, BiHS := tr hA†Bi.(6)

The physical states of the quantum system are represented by density matrices ρwhich

are Hermitian and positive operators with trace one [6]. Pure states of the system are

represented by rank-one projections ρ=|ϕihϕ|, where |ϕi∈His a normalized state

vector. Any density operator has a spectral decomposition of the form ρ=Pipi|iihi|

with ~p = (..., pi, ...)Ta probability distribution and {|ii} an ONB in H. Note that two

diﬀerent mixed quantum states not only diﬀer in their probability distribution ~p but

typically have diﬀerent ONBs in their spectral decomposition, too. By contrast, the

concept of frames allows the following deﬁnition which will give a decomposition into a

ﬁxed set of pure states.

Witnessing non-Markovianity by Quantum Quasi-Probability Distributions 5

Deﬁnition 1. A frame F:= {|ψiihψi|} ⊂ B(H)consisting of pure quantum states is

called a quantum frame.

In the following we will assume Fto be minimal (i.e. the |ψiihψi|form a basis in

B(H)which, however, cannot be orthogonal). Accordingly, any density matrix can be

decomposed as

ρ=X

fi|ψiihψi|,(7)

where we have

ρ†=ρ⇔fi∈R∀i(8)

tr [ρ]=1 ⇔X

fi= 1 (9)

hϕ|ρ|ϕi ≥ 0⇔X

fi|hψi|ϕi|2≥0.(10)

Note that the last line has to be fulﬁlled for all |ϕi ∈ H in order to ensure the positivity

of ρ. The coeﬃcients fiare referred to as frame decomposition coeﬃcients (FDCs).

We see that using quantum frames we can decompose any mixed quantum state quasi-

probabilistically (i.e. all fiare real and sum up to one but might be negative) into a

ﬁxed set of pure states. Furthermore, ~

f= (...,fi, . . .)Tis a vector like representation

of the mixed quantum state at hand which we will call the frame vector and indicate

by means of

f'ρ(11)

the one-to-one correspondence between the frame vector and the density matrix. The

frame operator is represented by

Sij = tr [|ψiihψi||ψjihψj|] = |hψi|ψji|2,(12)

where S~

fis the frame vector of S(ρ).

A given quantum frame F={|ψiihψi|} can also be associated to a rank-one

informationally complete positive operator valued measure or IC-POVM given by {Ei:=

ci|ψiihψi|} such that PiEi=1[13, 12]. Such an IC-POVM is a mathematical expression

of a tomographic quantum measurement mapping a quantum state ρto a probability

distribution ~p with [6]

pi= tr [Eiρ] = X

tr [|ψiihψi||ψjihψj|] = X

Sijfj=: X

Mijfj,(13)

where we have introduced the matrix Mwith elements Mij =1

ciSij. This means that Ei

is the eﬀect operator of outcome iwhile piis the probability to measure this outcome if

the system at hand is prepared in state ρ. The term informationally complete refers to

the fact that due to the underlying frame structure diﬀerent quantum states always have

diﬀerent IC-POVM probability distributions as well, i.e. the vector ~p encodes complete

information about its quantum state ρand hence is a vector like representation of it,

too,

~p 'ρ,(14)

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Witnessingnon-MarkovianitybyQuantumQuasi-ProbabilityDistributionsMoritzF.Richter1,RaphaelWiedenmann1andHeinz-PeterBreuer121PhysikalischesInstitut,UniversitätFreiburg,Hermann-Herder-Straÿe3,D-79104Freiburg,Germany2EUCORCentreforQuantumScienceandQuantumComputing,UniversityofFreiburg,Hermann-Herder-Str...

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Witnessing non-Markovianity by Quantum Quasi-Probability Distributions Moritz F. Richter1 Raphael Wiedenmann1and Heinz-Peter

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