Научная статья на тему 'Evaluation of non-unital qubit channel capacities'

Evaluation of non-unital qubit channel capacities Текст научной статьи по специальности «Физика»

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Ключевые слова
QUBIT CHANNEL / NON-UNITAL CHANNEL / HOLEVO CAPACITY / КУБИТНЫЙ КАНАЛ / НЕУНИТАЛЬНЫЙ КАНАЛ / ПРОПУСКНАЯ СПОСОБНОСТЬ ХОЛЕВО

Аннотация научной статьи по физике, автор научной работы — Filippov Sergey Nikolaevich

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We have applied quantum Sinkhorn's theorem to non-unital qubit channels and derived lower and upper bounds on the classical capacity of such channels.

Текст научной работы на тему «Evaluation of non-unital qubit channel capacities»

УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА.

_ СЕРИЯ ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ

2018, Т. 160, кн. 2 С. 258-265

ISSN 2541-7746 (Print) ISSN 2500-2198 (Online)

UDK 530.145

EVALUATION OF NON-UNITAL QUBIT CHANNEL CAPACITIES

S.N. Filippov

Institute of Physics and Technology, Russian Academy of Sciences, Moscow, 117218 Russia Moscow Institute of Physics and Technology, Dolgoprudny, 141701 Russia

Abstract

We have applied quantum Sinkhorn's theorem to non-unital qubit channels and derived lower and upper bounds on the classical capacity of such channels. Keywords: qubit channel, non-unital channel, Holevo capacity

Introduction

Transmission of classical information through quantum channels has been covered in a number of papers [1-5] and reviews [6, 7]. In brief, if R G [0,1] is an achievable rate of information transmission, then n qubits effectively allow to transmit 2nR classical messages.

The encoder assigns an n-qubit density operator g(n) to each message i. The n-qu-bit density operator is a positive semidefinite operator with unit trace, which acts on 2n dimensional Hilbert space H.2n . In the process of information transmission, each qubit is transmitted through a quantum channel $, which is a completely positive and trace preserving map. Therefore, the output state of n qubits reads $®n[g(n)]. The decoder is a measurement device described by a positive operator-valued measure, which assigns a positive-semidefinite operator Mjn (acting on 2n-dimensional Hilbert space) to each observed outcome j G {1,..., N}. Let p(j\i) be the probability of observing outcome j G {0,1,..., N} if the original message is i, then by the quantum-mechanical rule

p(n)(j\i) = tr[^(n)Mjn)].

N N

Condition M(n) = I guarantees p(n)(j\i) = 1. The maximum confusion proba-j=i j=i bility reads

Perr (n, N)= max (1 - p(n)(j \j)) .

j = 1,...,N \ J

R G [0,1] is called an achievable rate of information transmission if

lim perr(n, 2nR) =0.

n—

By the classical capacity C($) of quantum channel $ we understand the supremum of achievable rates:

C($) = sup { R : lim perr(n, 2nR) = 0) .

I n—J

The celebrated result in quantum information theory is that

C ($)

lim 1 Cx ($®n),

(1)

where the quantity Cx(^) is expressed through all possible ensembles of density operators {pk, Pk} and the von Neumann entropy S(p) = —tr(plog2p) by formula

Cx(^)

sup

{pk ,pk }

$ £pk] "XPkSt^Pk])

We will refer to Cx as the Holevo capacity of quantum channel ^.

Calculation of classical capacity C($) is complicated in general. In this paper, we find lower and upper bounds on the capacity of general qubit channels.

1. Relation between unital and non-unital qubit channels

Let A and B be two operators acting on H2. By $a we denote a completely positive map $a[X ] = AX At, i.e., a map with a single Kraus operator A. Analogously, $B [X] = BXBt. Hereafter, j denotes the Hermitian conjugation.

Suppose that $ is a qubit map, which belongs to the interior of the cone of positivity preserving maps. Then, [8] states that there exist positive definite operators A and B acting on H2 , such that the map

Y = $A o $ o $B (2)

is unital, i.e., Y(I) = I, the identity operator. This result was also anticipated earlier as a quantum Sinkhorn's theorem [9]. In addition, if $ is completely positive and trace preserving, then Y is completely positive and trace preserving too. For the given non-unital qubit channel $ , the particular form of operators A and B is derived in [10, 11]. Since A and B are nondegenerate, formula (2) implies that

$ = $A-1 O Y O $b_i ,

i.e., all non-boundary non-unital qubit channels $ can be decomposed into a concatenation of three completely positive maps $b-i , Y, $a-i , with Y being unital.

On the other hand, for any unital qubit channel Y there exist unitary operators V and W, such that [12]

Y = $W o A O $v ,

where the quantum channel A has a so-called diagonal form in the basis of conventional Pauli operators 1,01,02,03:

1 1 3

A[X] = 2tr[X]I + 2 X) Aitr[oiX]oi. (3)

i=l

Parameters Ai, X2, A3 in (3) are real and satisfy the constraint 1 ± A3 ^ |Ai ± A2I as A is completely positive [12].

Clearly, the classical capacities of channels Y and A coincide. Moreover, since the additivity hypothesis holds true for unital qubit channels [13], the classical capacity equals the Holevo capacity and reads

C(Y) = C(A) = Cx(A) = 1 - ^1 (l - ma^ I^ , (4)

where h(x) = —xlog2X — (1 — x)log2 (1 — x).

In what follows, we relate the classical capacity of non-unital qubit channel $ with the classical capacity of unital qubit channel Y, which is given by formula (4).

2. Bounds on the classical capacity of non-unital qubit channels

Proposition 1. Let us suppose that $ is a non-unital qubit channel, such that the qubit map Y = $A o $ o $B is unital. Then, C($) > C(Y) - 2log2(||A||||B||).

Proof. Let {g(n), M(n)}N=1 be the optimal code of size N = 2"Rt for the composite channel Y®n such that lim perr Y(n, 2"Rt ) = 0.

n—

Consider a set of modified input states

^n) = B®n0(n)(Bt)®n

6%

tr[B®ng(n)(Bt)®n]

and a modified positive operator-valued measure {j ^ M(n)}N= 0 with elements

N ( At)®nM(n) A®n

M(n) = I - = (A ' Ark A ■ j = 1.-.n,

j=1 11 11

where || = ||TO = max (^\XtX\$) is the operator norm. It is not hard to see

that M0n) is positive semidefinite.

Using the modified code, let each qubit be transmitted through the channel $. Then, the probability to observe outcome j = 0 provided input message i equals

tr s A®n$®n

pjn)(jli)=tr \g(n)M(jn}1

(At)®nM((n)}

tr[ß®nejn)(ßt)®n]yAy2n Since o $ o $B = T, we get

Vn)(J|i) = L ' = p KJV''

tr[B®ng(n)(B t)®n ]|A|2n trlB®n^(n)(Bt)®n]||A||2n,

where p(n)(j\i) is the probability to get outcome j G {1,..., N} for the input message i G {1,..., N} in the original optimal protocol for channel Y®n.

Observation of the outcome j =0 in the modified protocol would be treated as an unsuccessful event, whereas observation of the outcome j G {1,...,N} leads to a successful identification of the message because p(n)(j\i) ^ Sj if n ^ ^. The probability to observe nonzero outcome j equals

N 1 1

p(n) = E p(n)(j\i) =-^-^ 1

j=i

tr[B®ngjn)(Bt)®n]^A\\2^ (\\A\\ \\B\

\2n •

Utilizing the modified protocol, one can transmit information only in the case of successful events J =0, so the average number of successfully transmitted messages N equals

NV = pjn)N = Pjn)2nRY £ 2njRY-2i°g2jIHI H^HiO Therefore, the considered protocol enables one to achieve the rate

R £ Rr - 21og2(\\A\\\\B\\) (5)

by utilizing the channel $.

If Ry ^ C(Y) and one observes the successful event (j = 0), then the maximum error probability in the modified protocol

Perr(n, N) = . max^ (l - = . maxN (l - p(n) (j\j)) - 0 if n - m.

Taking the supremum on both sides of eq. (5) with requirement perr(n, N) =

0, we get

C($) > C(Y) - 2log2M||B||).

In the proof of proposition 1, we have used only the relation Y = ◦ $ ◦ $S . Instead, if we use the relation $ = $a-i ◦ Y o $s-i, we immediately get C(Y) ^ C($) — 2log2(||A-1||||B-1||). Therefore, we immediately obtain the upper bound on capacity C($).

Proposition 2. Let us suppose that $ is non-unital qubit channel and its decomposition through the unital qubit channel Y reads $ = $a-i ◦ Y o $s-i . Then, C($) < C(Y) + 2log2(||A-1||||B-1||).

Combining propositions 1 and 2, we get the following result.

Corollary 1. Let $ be a unital qubit channel belonging to the interior of positive qubit maps, then there exist positive definite operators A and B acting on H2, such that the map Y = o $ o $s is unital and

C(Y) — 2log2(||A||||B||) < C($) < C(Y) + 2log2(||A-1||||B-1||).

Proof. The statement straightforwardly follows from the decomposition existence [8] and propositions 1 and 2. □

3. Four-parameter non-unital qubit channels

Consider a non-unital qubit channel of the form

$[X] = 2 ^tr[X](J + tsas) + J2 Xij >

where ts and A1,A2,As are real parameters, which in addition to the condition of complete positivity also satisfy the inequality that \ts\ + \As\ < 1. It guarantees that $ is an interior point of the cone of positive qubit maps. In [11], the explicit form of decomposition $ = $^-i ◦ Y o $B-i is provided:

A = 2 ( V(1 + \ts\)2 — AS 0

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V(1+ ts)2 — A2 + ^(1 — ts)2 — AS V 0 ^(1 — \ts\)2 — A2

B

1 , 1 0

V1 + tsxs + \Asxs\

0 1

x3 = —t3

y/1 + tsxs — \Asxs\ J

1 — t2 + A2 + VTT1 + ts)2 — A3][(1 — ts )2 — A2]

1 — t2 — A2 + ^[(1 + ts)2 — A2][(1 — ts)2 — A2],

and the unital qubit map Y = A has the form (3) with parameters

Ai = , —Xl, =, (6)

+ A3)2 -t§ + V(i - A3)2 -t3

A2 = , =, (7)

V(1 + A3)2 -12 + V(i - A3)2 -13

A3 = --^--2 • (8)

(7(i + A3)2 -12 W(i - A3)2 -13)

We explicitly find the operator norms

2

PH =-, 2 2 , (9)

1 + /(1 -M)2 - A3 + V(1 + It3|)2 - A3

»■ (io)

» » Vi + ¿3x3 - IA3X3I'

\\b-i\\ = v/i+t3xmA3X3T. (12)

By substituting these norms in corollary 1, we find the lower and upper bounds on capacity C($).

Proposition 3. The classical capacity of non-unital qubit channel

1 1 3

] = -tr[X](I + ¿303) + Ajtrfog]aj

2 ^ ^ 2 ■. 1

with \t3\ + |As| < 1 satisfies

' 1

C($) > 1 - ^ 1(l - .max l^lf)- 2log2(||A||||B||),

C($) < 1 - ^2(1 - ^O) +2loga(yA-1yyS

where Aj, i = 1, 2, 3 are given by formulas (6)-(8) and ||A||, ||A 1|, ||B||, HB 1| are given by formulas (9)-(12).

Conclusions

We have obtained new lower and upper bounds on the classical capacities of non-unital qubit channels. We must emphasize that the obtained result holds true for the regularized version of the Holevo capacity, formula (1). Our proofs are based on the seminal relation between unital and non-unital qubit channels, which was developed in [8] and [11]. This relation can be productive in other research areas as well, for instance, in the study of divisibility of qubit dynamical maps [14-18] and their tensor

products [19], in the study of entanglement annihilation [20-24] and absolutely separating quantum channels [25], in the study of quantum capacities and other types of capacities [6, 7]. For practical applications, we have derived lower and upper bounds for a four-parameter family of non-unital qubit channels. For instance, this family covers generalized amplitude damping channels, which originate from the processes of emission and absorption due to interaction of the two-level system (qubit) with a reservoir of finite temperature [26].

Acknowledgements. The study was supported by the Russian Foundation for Basic Research (project no. 16-37-60070 mol-a-dk).

References

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23. Filippov S.N., Melnikov A.A., Ziman M. Dissociation and annihilation of multipartite entanglement structure in dissipative quantum dynamics. Phys. Rev. A, 2013, vol. 88, no. 6, art. 062328, pp. 1-11. doi: 10.1103/PhysRevA.88.062328.

24. Filippov S.N. PPT-inducing, distillation-prohibiting, and entanglement-binding quantum channels. J. Russ. Laser Res., 2014, vol. 35, no. 5, pp. 484-491. doi: 10.1007/s10946-014-9451-2.

25. Filippov S.N., Magadov K.Y., Jivulescu M.A. Absolutely separating quantum maps and channels. New J. Phys., 2017, vol. 19, art. 083010, pp. 1-19. doi: 10.1088/1367-2630/aa7e06.

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Recieved October 23, 2017

Filippov Sergey Nikolaevich, Candidate of Physical and Mathematical Sciences, Senior Researcher; Associate Professor, Department of Theoretical Physics, Institute of Physics and Technology of Russian Academy of Sciences

Nahimovskii pr. 34, Moscow, 117218 Russia Moscow Institute of Physics and Technology

Institutskiy per. 9, Dolgoprudny, Moscow Region, 141701 Russia E-mail: sergey.filippov@phystech.edu

УДК 530.145

Анализ пропускной способности неунитальных кубитных каналов

С.Н. Филиппов

Физико-технологический институт Российской академии наук, г. Москва, 117218, Россия Московский физико-технический институт, г. Долгопрудным, 141701, Россия

Аннотация

Квантовая теорема Синкхорна применяется к неунитальных кубитным каналам. Находятся верхняя и нижняя границы для классической пропускной способности таких каналов.

Ключевые слова: кубитный канал, неунитальный канал, пропускная способность Холево

Поступила в редакцию 23.10.17

Филиппов Сергей Николаевич, кандидат физико-математических наук, старший научный сотрудник; доцент кафедры теоретической физики

Физико-технологический институт Российской академии наук

Нахимовский пр., д. 36 корп. 1, г. Москва, 117218, Россия Московский физико-технический институт

Институтский пер., д. 9, г. Долгопрудный, Московская обл., 141701, Россия E-mail: sergey.filippov@phystech.edu

I For citation: Filippov S.N. Evaluation of non-unital qubit channel capacities. Uchenye ( Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, \ no. 2, pp. 258-265.

Для цитирования: Filippov S.N. Evaluation of non-unital qubit channel capacities // Учен. зап. Казан. ун-та. Сер. Физ.-матем. науки. - 2018. - Т. 160, кн. 2. - С. 258-265.

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