Sequential Quantum Channel Discrimination Yonglong Li Christoph Hirche and Marco Tomamichel Abstract

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Sequential Quantum Channel Discrimination

Yonglong Li, Christoph Hirche, and Marco Tomamichel∗

Abstract

We consider the sequential quantum channel discrimination problem using adaptive and non-

adaptive strategies. In this setting the number of uses of the underlying quantum channel is

not ﬁxed but a random variable that is either bounded in expectation or with high probability.

We show that both types of error probabilities decrease to zero exponentially fast and, when

using adaptive strategies, the rates are characterized by the measured relative entropy between

two quantum channels, yielding a strictly larger region than that achievable by non-adaptive

strategies. Allowing for quantum memory, we see that the optimal rates are given by the

regularized channel relative entropy. Finally, we discuss achievable rates when allowing for

repeated measurements via quantum instruments and conjecture that the achievable rate region

is not larger than that achievable with POVMs by connecting the result to the strong converse

for the quantum channel Stein’s Lemma.

1 Introduction

Quantum hypothesis testing between diﬀerent quantum sources is of central importance to a variety

of quantum information processing tasks. In particular, discrimination of two quantum states has

long been an active research area in quantum information theory. Here, the goal is to ﬁnd tests

that give the optimal trade-oﬀ between two kinds of error probabilities, namely the probabilities of

false detection and false rejection. A typical setting is to consider an asymptotic scenario where an

inﬁnite number of copies of the state are available. For state discrimination this is a well explored

problem [1–9].

Sequential methods for the classical hypothesis testing problem were ﬁrst proposed in [10] and

have later been expanded into a subject called sequential analysis. The key merit of sequential

analysis is that the number of samples used in the statistical procedure is not ﬁxed in advance of

the statistical experiment and, given the tolerance error, the average number of samples needed is

much less than that in the ﬁxed-sample statistical experiment. In recent work, sequential hypothesis

testing was extended to the setting of quantum states [11,12].

Another problem with considerable recent progress is that of quantum channel discrimination,

which is a natural extension of the state version [13–16]. Most notably, these works determined

the optimal asymptotic rate in asymmetric quantum channel discrimination which is known as a

quantum Stein’s Lemma.

In this work, we combine these two ﬁelds and discuss sequential hypothesis testing between two

quantum channels, determining the optimal rate regions under certain expectation and probabilistic

constraints. We consider several diﬀerent strategies including non-adaptive and adaptive strategies,

and adaptive strategies with quantum memory. Ultimately showing that under the last set of

∗Y. Li and M. Tomamichel are with the Department of Electrical and Computer Engineering, National Univer-

sity of Singapore (NUS). C. Hirche is with Zentrum Mathematik, Technical University of Munich. C. Hirche and

M. Tomamichel are also with the Center for Quantum Technologies (CQT), NUS. (e-mails: {elelong,c.hirche,

marco.tomamichel}@nus.edu.sg).

arXiv:2210.11079v1 [cs.IT] 20 Oct 2022

strategies both errors decay exponentially at a rate given by the regularized channel relative entropy.

Finally, we give achievable and converse bounds in the most general setting of strategies using

quantum instruments potentially measuring the same states multiple times. The latter bounds

match conditional on a conjecture that is related to the strong converse of the quantum Stein’s

lemma for quantum channels.

Previous Results. Many of the early results on quantum state discrimination are reviewed in [1].

The generalization of Stein’s lemma [2,3] for quantum hypothesis testing establishes that the error

of the second kind decays exponentially with the Stein’s exponent given by the quantum relative

entropy when the ﬁrst kind of error is upper bounded by a given constant. On the other hand, if

both errors decrease exponentially, the optimal trade-oﬀ between the decay rates is governed by

the quantum Hoeﬀding bound [4–6]. In the Bayesian case, that is, the quantum state is prepared

according to some prior probability mass function, the total error probability decreases to zero

exponentially fast with exponent governed by the quantum Chernoﬀ exponent [7,8]. Beyond this,

second-order reﬁnements to the Stein’s exponent were derived in [17,18] and the moderate deviation

regime where one error probability decreases sub-exponentially has been analyzed in [19, 20].

In [21], Hayashi studied the classical channel discrimination problem and showed that adaptive

protocols do not improve the error exponents in the Stein, Chernoﬀ and Hoeﬀding regimes. In [22],

the authors studied the discrimination of an arbitrary quantum channel and a “replacer” channel

and showed that adaptive strategies provide no advantage over non-adaptive tensor-power strategies

asymptotically. In [13], the authors introduced the amortized quantum channel divergence between

quantum channels and showed that it is a general converse bound for the Stein’s exponent. For

several classes of channels [13] showed that the error exponents of adaptive protocols are the same

as those obtained by using non-adaptive protocols. In particular this applies to classical-quantum

channels, see also [14]. In [15] it was subsequently shown that the amortized channel relative entropy

is indeed also achievable using adaptive strategies. Finally, [16] showed that the amortized channel

relative entropy is equal to the regularized channel relative entropy, which in turn is achievable

by parallel strategies with quantum memory. This means that adaptive strategies are not more

powerful than parallel strategies in this setting. In [23], the authors introduces a new divergence

using the weighed geometric mean between two operators and derived the strong converse exponent

for quantum channel discrimination.

For classical hypothesis testing problems between two probability distributions P0and P1, when

the expected number of samples is bounded by n, it was shown in [24] that there exists a sequence

of tests—namely sequential probability ratio tests (SPRTs)—such that the exponents of the errors

of the ﬁrst and second kind simultaneously assume the extremal values D(P1kP0) and D(P0kP1).

This signiﬁcantly improves the classical Hoeﬀding bound of the error exponents [25,26] where if one

error exponent assumes its extremal value—the relative entropy—the other necessarily vanishes.

The sequential approach to quantum hypothesis testing was ﬁrst explored in [27]. In a recent

paper [11], it was shown that sequential quantum hypothesis test can reduce the number of samples

needed compared with the ﬁxed-length quantum hypothesis test. Also in [11], a converse was

shown, that is, the expectation of the number of quantum states in the sequential procedure was

lower bounded by a function of the tolerance error probabilities. In [12], the authors considered

the sequential hypothesis testing of two quantum states under a diﬀerent type of constraint on the

number of states used and proposed an adaptive strategy to achieve the lower bound given in [11].

Therefore, [12] characterized the regions of all achievable error exponents of the two kinds of error

probabilities.

mk+1

Oracle

σk+1

νk+1(mk+1, ρk+1 |ρk

1, mk

1, xk

∗

dk(mk

1, yk

1, ρk

Stop

0 or 1

ρk+1

Figure 1: The structure of a general adaptive sequential channel discrimination protocol without

quantum memory. At step k, the measurement outcome ykis considered together with all previous

measurement results yk−1

1and all previous choices of the agent, mk

1and ρk

1. The decision function

dkeither decides to stop (outputting the hypothesis 0 or 1) or continue. In the latter case a new

input state ρk+1 and a measurement mk+1 are sampled according to the distribution νk+1. Finally,

the channel oracle is called with input ρk+1, producing the output state σk+1 that will be measured

using mk+1.

Outline. The paper is structured as follows. In Section 2, we will introduce the notation used

throughout the paper and the mathematical formulation of the problem. In Section 3, the main

results and the corresponding proofs are presented.

2 Problem Formulation

2.1 Notation

In this work, we consider ﬁnite-dimensional quantum systems. Throughout the paper, A,B,C, etc,

denote quantum systems, but also the corresponding ﬁnite-dimensional Hilbert spaces. With |A|,

|B|,|C|, etc, we denote the dimensions of the corresponding quantum systems. Let L(A) be the

set of all linear operators from Ato A. A quantum channel NA→Bis a completely positive trace-

preserving linear map from L(A) to L(B) (for more details on quantum channels, see [28, Chapter

5]). Let DRA be the set of bipartite quantum states over the quantum system RA. Let Ybe some

ﬁnite alphabet. A set m={my:y∈ Y} of |A|×|A|positive-deﬁnite matrices is a positive operator

valued measure (POVM) if Py∈Y my=1A. A POVM m={my:y∈ Y} is called a projector

valued measure (PVM) if each myis a projector, that is, m2

y=my. Let MYbe the set of POVMs

with outcomes in Y.

2.2 The Sequential Quantum Channel Discrimination Problem

We consider the following binary channel discrimination problem:

H0:W=N0,A→BH1:W=N1,A→B,

where N0,A→Band N1,A→Bare two quantum channels from Ato B. Throughout the rest of the

paper, Nν,A→Bwill be denoted as Nνfor ν∈ {0,1}.

Let {Ri}n

i=1 be a sequence of ancilla systems, each of which is an identical copy of some ﬁnite

dimensional system R. Let Ybe a ﬁnite alphabet and let MYbe the set of POVMs whose

elements are of dimension |A||B|×|A||B|. Throughout the paper Mis used to denote a random

POVM and mis used to be a realization of M. For each k∈N, let dkbe a function from

MY× DRA × Ykto {0,1,∗}. Intuitively, dkis the decision function at time k: based on all

the input states, POVMs and outcomes before time k+ 1, dk6=∗means the testing procedure

stops before time k+ 1 and the experimenter makes the decision that Hdkis true; dk=∗means

the testing procedure continues. At time 1, the experimenter prepares the quantum state and the

POVM (ρR1A1

1, M1) randomly according to some probability measure ν1, and then passes the state

through the underlying quantum channel Wand obtains the output state σRB

1= idR1⊗ W(ρR1A

1).

Then the experimenter applies the POVM M1to σR1B1

1obtaining the outcome Y1. Then the

experimenter applies a decision function d1(ρR1A1

1, M1, Y1) to decide whether to accept H0or H1or

to continue the experiment. If d1=∗, then the experimenter chooses to continue the experiment

and prepares a new quantum state and a new POVM (ρR2A2

2, M2) according to some conditional

probability measure ν2(dρ, dm2|ρR1A1

1, M1, Y1) and obtains the output state σR2B2

2by passing ρR2A2

through the underlying quantum channel. Then the experimenter applies some random POVM

M2to σR2B2

2and obtains the outcome Y2. Based on (ρR1A1

1, ρR2A2

2, M2

1, Y 2

1), the experimenter

applies some decision function d2to decide whether to accept one of the hypothesis or continue

the experiment. This process continues until the experimenter accepts one of the hypothesis. Then

{(Mi, ρRiAi

i, Yi)}∞

i=1 is a sequence of random variables taking values in MY×DRA×Y. For notational

convenience, ρRA

kwill be abbreviated as ρk. The joint conditional density function of (Mk

1, ρk

1, Y k

µ(k)(mk

1, ρk

1, yk

1) =

j=1 nνj(dρ, dm|xj−1

1, mj−1

1)×Tr[W(ρj)mj(xj)]o,(1)

for any yk

1,ρk

1, and mk

1. The described strategies are inherently adaptive as the input state and

the measurement at each time depends on previous input states, measurements and outcomes of

the measurements. When {νj}∞

j=1 are probability measures on MY× DRA × Y, the strategies are

non-adaptive as the input state and the POVM chosen each time do not depend on the choices of

previous rounds. The ﬁrst time kwith dk6=∗is the number of uses of the underlying quantum

channel and is denoted by T. The stopping time Tis well deﬁned with respect to the ﬁltration

generated by {(Mj, ρj, Yj)}∞

j=1.

We call S= ({νk, dk}∞

k=1, T ) a sequential quantum channel discrimination strategy. In the

following we study sequences of sequential quantum channel discrimination strategies Sn, indexed

by n∈N. To simplify notation we use Pn,i to denote PSn,Nifor i∈ {0,1}. The notation En,i[·]

means that the expectation is taken with respect to the probability measure Pn,i. We consider two

types of constraints on the number of states Tnused during the test. The ﬁrst type of constraint

is the expectation constraint: maxi∈{0,1}En,i[Tn]≤n. In other words, the average number of copies

used in the testing procedure should be bounded by some number n. The second type of constraint

is the probabilistic constraint [29,30] maxi∈{0,1}Pn,i(Tn> n)< ε for some ﬁxed ε∈(0,1). In other

words, the number of copies of the state used during the testing procedure should be bounded by

some number nwith probability larger than 1 −ε.

We study the trade-oﬀ between the error probabilities (αn, βn) and either the expectation or the

probabilistic constraint on the number of copies of the state used during the test procedure. The

ﬁrst type of error is quantiﬁed by the probability that the experimenter declares that hypothesis

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SequentialQuantumChannelDiscriminationYonglongLi,ChristophHirche,andMarcoTomamichel*AbstractWeconsiderthesequentialquantumchanneldiscriminationproblemusingadaptiveandnon-adaptivestrategies.Inthissettingthenumberofusesoftheunderlyingquantumchannelisnotxedbutarandomvariablethatiseitherboundedinexpect...

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