On the near-optimality of one-shot classical communication over quantum channels

# On the near-optimality of one-shot classical communication over quantum channels

Anurag Anshu111Center for Quantum Technologies, National University of Singapore, Singapore. a0109169@u.nus.edu   Rahul Jain222Center for Quantum Technologies, National University of Singapore and MajuLab, UMI 3654, Singapore. rahul@comp.nus.edu.sg   Naqueeb Ahmad Warsi333Center for Quantum Technologies, National University of Singapore and School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore and IIITD, Delhi. warsi.naqueeb@gmail.com
###### Abstract

We study the problem of transmission of classical messages through a quantum channel in several network scenarios in the one-shot setting. We consider both the entanglement assisted and unassisted cases for the point to point quantum channel, quantum multiple-access channel, quantum channel with state and the quantum broadcast channel. We show that it is possible to near-optimally characterize the amount of communication that can be transmitted in these scenarios, using the position-based decoding strategy introduced in a prior work [1]. In the process, we provide a short and elementary proof of the converse for entanglement-assisted quantum channel coding in terms of the quantum hypothesis testing divergence (obtained earlier in [2]). Our proof has the additional utility that it naturally extends to various network scenarios mentioned above. Furthermore, none of our achievability results require a simultaneous decoding strategy, existence of which is an important open question in quantum Shannon theory.

## 1 Introduction

Understanding the limits of communication through various models of channels is a central aspect of classical information theory. Some landmark results in this direction are the models of point to point channel [3], multiple access channel [4, 5], channel with a state [6] and broadcast channel [7]. The diversity of scenarios in which information theory can be applied has led to various settings in which the problem of channel coding is studied. Below we discuss two settings relevant to this work.

• Asymptotic and i.i.d. setting: Here, the senders are allowed to use the channel multiple times in a memoryless fashion and the goal is to obtain bounds on the rate of transmission for an arbitrarily large number of channel uses, as the error is made to go to zero. It is highly desirable that the resulting bounds are single letter, that is, they do not require unbounded optimization in their computation. Without this restriction, it would be possible to obtain tight characterization of the capacity of all of the aforementioned channel settings [8, Section 4.3].

• One-shot setting: Here, the senders are allowed to use the channel only once, which can arise in many practical scenarios. It is desirable to obtain bounds on the amount of communication which are near optimal. That is, a communication cost (or cost region for multiple messages) may be obtained which is a converse cost for any protocol that makes an error (in terms of the probability of incorrectly decoding the messages) and there exists a protocol that achieves the cost (where is of the order of ) up to some additive factors.

Quantum information theory generalizes the models of classical channels in various ways, by introducing channels that can take quantum inputs and produce quantum outputs or by allowing new resources such as quantum entanglement. Several works have studied the problem of transmission of quantum information through a point to point quantum channel in the asymptotic and i.i.d. setting, both in the entanglement assisted case [9] and the entanglement unassisted cases ([10, 11] for the transmission of classical information and [12, 13, 14] for the transmission of quantum information). In the entanglement assisted case, the transmission of classical information is equivalent to the transmission of quantum information up to a factor of , due to the duality between quantum teleportation [15] and super-dense coding [16]. In the entanglement unassisted case, the duality is lost and we have two different aforementioned scenarios for the transmission of classical information and quantum information. In this work, we shall focus on the transmission of classical information for both of the entanglement assisted and unassisted cases.

Several quantum network scenarios have also been studied in the asymptotic and i.i.d. setting, such as the quantum multiple access channel [17, 18, 19, 20, 21], the quantum broadcast channel [22, 23, 24, 25] and quantum channel with state [26, 27]. In most of these cases (both the entanglement assisted and unassisted), a single letter characterization is not known. Some exceptions, where a single letter characterization is known, are the entanglement assisted point to point quantum channel [9], the classical-quantum multiple access channel [17] and the entanglement assisted quantum channel with state [26, 27].

Classical communication over the point to point channel has been studied in several works in the one-shot setting [27, 28, 29, 30, 31, 1]. These results have been extended to the quantum network scenarios in the works [26, 23, 27, 1]. However, in all the cases except for the point to point channel (both entanglement assisted [1] and entanglement unassisted [29]), a near-optimal one-shot characterization is not known. An interesting variant, where the communicating parties are equipped with arbitrary non-local correlations, has also been considered in the works [32, 33], providing improvements to the entanglement assisted case (which is a weaker non-local resource).

In this work, we provide a near-optimal one-shot characterization for many quantum network scenarios, using the position-based decoding strategy introduced in [1]. In contrast with [1], we do not require the convex-split technique [34] for our achievability results. Our results, as obtained in Sections 3 and 4, are summarized below.

• Point to point quantum channel: A converse bound for the entanglement assisted case has been given in [2] and a nearly matching achievability result has been obtained recently in [1]. We provide a short proof of the converse in [2]. This also considerable simplifies an alternative proof given in [1, arXiv version 2], which was inspired by the analogous asymptotic and i.i.d. result [35, Section 21.5] and used a one-shot analogue of the chain rule for the conditional quantum mutual information. We are able to avoid the use of any such chain rule in our converse proof, by considering the quantum hypothesis testing divergence between appropriate quantum states. Our proof technique has the utility that it easily extends to various network scenarios. As an application, we recover the one-way case of [36, Theorem 1.1].

For the entanglement unassisted case, we provide a similar characterization to that given in [29]. Our characterization has the property that it is of similar form for other entanglement unassisted network scenarios.

• Quantum channel with state: We provide a near optimal characterization for this channel in the one-shot setting, with a tight dependence on the error of decoding. The optimization involved in our bound is comparable to the optimization involved in earlier known results [26, 27, 1]. It is not clear if our bound attains a single letter expression in the asymptotic and i.i.d. setting, in contrast with the asymptotic and i.i.d. form of the bounds given in [26, 27, 1]. On the other hand, we show as a corollary that the achievability bound given in [1] for the quantum channel with state is near optimal in the one-shot setting. Same feature is not known for the one-shot achievability bounds in [26, 27].

We also provide near-optimal bounds for this channel for the entanglement unassisted case, with the property that the registers involved in our bounds have dimension comparable to that of the input and output registers of the channel.

• Quantum broadcast channel: In a similar fashion to the quantum channel with state, we provide a near optimal characterization for this channel in the one-shot setting (discussing the case of one sender and two receivers, as the results similarly extend to more than two receivers), with a tight dependence on the error of decoding. The optimization involved in our bound is comparable to the optimization involved in earlier known results [23, 27, 1]. We note that the asymptotic and i.i.d. analogue of our converse result is implicit in [23, Theorem 3]. It is not clear if our bound attains a single letter expression in the asymptotic and i.i.d. setting, which is also the case for the asymptotic and i.i.d. form of the bounds given in [23, 27, 1]. On the other hand, we show as a corollary that the achievability bound given in [1] for the quantum channel with state is near optimal in the one-shot setting. Same feature is not known for the one-shot achievability bound in [23, 27].

We also provide near-optimal bounds for this channel for the entanglement unassisted case, with the property that the registers involved in our bounds have dimension comparable to that of the input and output registers of the channel.

• Quantum multiple access channel: We provide a new converse bound for the multiple access channel with two senders and one receiver (which can easily be extended to the case of more than two senders). We show how to achieve this bound in two different ways (both of which can easily be extended to the case of more than two senders). The first way uses the pretty good measurement technique of Hayashi and Nagaoka [37] and has a tight dependence on the error of decoding one of the messages (at the cost of quadratic loss on the error of decoding the other message). The second way uses the sequential decoding strategy of Sen [38] (with the quantitatively improved version of [39]; see also the related works [40, 41] and the recent improvement [42]) and has a tight dependence on the error of decoding both messages up to multiplicative constants. As far as we know, this is a first instance where the sequentially decoding strategy gives a better dependence on the overall error of decoding the messages in comparison to the pretty good measurement. Furthermore, our achievability results do not need a simultaneous decoding strategy [20, 43, 44]. It is not clear if this bound leads to a single letter characterization in the asymptotic and i.i.d. setting, a situation similar to the other known bound for the entanglement assisted quantum multiple access channel in the asymptotic and i.i.d. setting [18].

We also provide near-optimal bounds for the entanglement unassisted case, with the property that the registers involved in our bounds have dimension comparable to that of the input and output registers of the channel.

## 2 Preliminaries

In this section we set our notations, make the definitions and state the facts that we will need later for our proofs.

Consider a finite dimensional Hilbert space endowed with an inner product . The norm of an operator on is and norm is . For hermitian operators , the notation implies that is a positive semi-definite operator. A quantum state (or a density matrix or a state) is a positive semi-definite matrix on with trace equal to . It is called pure if and only if its rank is . A sub-normalized state is a positive semi-definite matrix on with trace less than or equal to . Let be a unit vector on , that is . With some abuse of notation, we use to represent the state and also the density matrix , associated with . Given a quantum state on , the support of , called is the subspace of spanned by all eigenvectors of with non-zero eigenvalues. For quantum states on , the notation means that the support of is contained in the support of .

A quantum register is associated with some Hilbert space . Define . Let represent the set of all linear operators acting on the set of quantum states acting on the Hilbert space . We denote by , the set of quantum states on the Hilbert space . State with subscript indicates . If two registers are associated with the same Hilbert space, we shall represent the relation by . Composition of two registers and , denoted , is associated with Hilbert space . For two quantum states and , represents the tensor product (Kronecker product) of and . The identity operator on (and associated register ) is denoted .

Let . We define

 ρB:=TrA(ρAB):=∑i(⟨i|⊗IB)ρAB(|i⟩⊗IB),

where is an orthonormal basis for the Hilbert space . The state is referred to as the marginal state of . Unless otherwise stated, a missing register from subscript in a state will represent partial trace over that register. Given a , a purification of is a pure state such that . Purification of a quantum state is not unique.

A quantum map is a completely positive and trace preserving (CPTP) linear map (mapping states in to states in ). A unitary operator is such that . An isometry is such that and , where is a projection on . The set of all unitary operations on register is denoted by .

We shall consider the following information theoretic quantities. Let .

 F(ρA,σA)\lx@stackreldef=∥√ρA√σA∥1.
2. Purified distance ([47]). For ,

 P(ρA,σA)=√1−F2(ρA,σA).
3. -ball. For ,

 Bε(ρA)\lx@stackreldef={ρ′A∈D(A)| P(ρA,ρ′A)≤ε}.
4. Relative entropy ([48]). For such that ,

 D(ρA∥σA)\lx@stackreldef=Tr(ρAlogρA)−Tr(ρAlogσA).
5. Smooth quantum hypothesis testing divergence ([28], see also [37]). For and ,

 DεH(ρA∥σA)\lx@stackreldef=sup0⪯Λ⪯IA,Tr(ΛρA)≥1−εlog(1Tr(ΛσA)).
6. Max-relative entropy ([49]). For such that ,

 Dmax(ρA∥σA)\lx@stackreldef=inf{λ∈R:ρA⪯2λσA}.

We will use the following facts.

###### Fact 1 (Triangle inequality for purified distance, [47, 50]).

For states ,

 P(ρA,σA)≤P(ρA,τA)+P(τA,σA).
###### Fact 2 (Monotonicity under quantum operations, [51],[52]).

For quantum states , , and quantum operation , it holds that

 F(E(ρ),E(σ))≥F(ρ,σ)andDεH(ρ∥σ)≥DεH(E(ρ)∥E(σ)).
###### Fact 3 ([1]).

Let be quantum states and be an operator. Then .

###### Fact 4 (Gentle measurement lemma,[53, 54]).

Let be a quantum state and be an operator. Then

 F(ρ,AρATr(A2ρ))≥√Tr(A2ρ).

Following fact is analogous to the gentle measurement lemma (Fact 4).

###### Fact 5.

Consider a quantum state and a measurement such that . Let . Then .

###### Fact 6 (Hayashi-Nagaoka inequality, [37]).

Let be positive semi-definite operators and . Then

 I−(S+T)−12S(S+T)−12⪯(1+c)(I−S)+(2+c+1c)T.
###### Fact 7 ([29]).

Let and . It holds that

 DεH(ρA∥σA)≤D(ρA∥σA)1−ε.
###### Fact 8 (Sequential measurement, [38, 39]).

Let be a quantum state and be projectors. Let . Then

 Tr(Π′kΠ′k−1…Π′1ρΠ′1Π′2…Π′k)≥1−4∑iTr(Πiρ).
###### Fact 9.

Let . Let be a quantum state such that and . Then for any quantum state ,

 DεH(ρMM′∥ρM⊗σM′)≥log|M|.
###### Proof.

Setting , consider

 Tr(ΛMM′ρM⊗σM′)=∑m⟨m|MρM|m⟩M⋅⟨m|M′σM′|m⟩M′=1|M|∑m⟨m|M′σM′|m⟩M′=1|M|.

Further, . The bound now follows from the definition of . ∎

###### Fact 10 (Neumark’s theorem, [55]).

For any POVM acting on a system there exists a unitary and an orthonormal basis such that for all quantum states , we have

 Tr[U†SP(IS⊗|i⟩⟨i|P)USP(ρS⊗|0⟩⟨0|P)]=Tr[MiρS].
###### Fact 11.

Let and be quantum states. Then

 DεH(|ρ⟩⟨ρ|A∥σA)=sup0⪯Λ⪯IA:rk(Λ)=1,Tr(Λ|ρ⟩⟨ρ|A)≥1−εlog(1Tr(ΛσA)),

where is the rank of the operator .

###### Proof.

We apply Neumark’s theorem (Fact 10) to rewrite the smooth quantum hypothesis divergence as

 supΠ:Π2=Π,Tr(Π|ρ⟩⟨ρ|A⊗|0⟩⟨0|P)≥1−εlog(1Tr(ΠσA⊗|0⟩⟨0|P)). (1)

Fix a such that , define . Then

 Tr(|ϕ⟩⟨ϕ|⋅|ρ⟩⟨ρ|A⊗|0⟩⟨0|P)=⟨ρ|A⟨0|PΠ|ρ⟩A|0⟩P⋅⟨ρ|A⟨0|PΠ|0⟩P|ρ⟩A⟨ρ|A⟨0|PΠ|0⟩P|ρ⟩A=⟨ρ|A⟨0|PΠ|0⟩P|ρ⟩A≥1−ε.

Further, since , we have . Thus, . Thus, the projector achieving the supremum in Equation 1 has rank . The proof now follows by defining , which satisfies . ∎

## 3 Entanglement assisted quantum coding

### 3.1 Point to point quantum channel

Alice wants to communicate a classical message chosen from to Bob over a quantum channel such that Bob is able to decode the correct message with probability at least , for all message . To accomplish this task Alice and Bob also share entanglement between them. Let the input to Alice be given in a register . We now make the following definition.

###### Definition 1.

Let be the shared entanglement between Alice () and Bob (). An -entanglement assisted code for the quantum channel consists of

• An encoding map for Alice.

• A decoding operation for Bob, with being the output register such that for all ,

 Pr(M′≠m|M=m)≤ε.

The following converse was shown in [2]. We provide a simpler proof with the utility that it can be easily extended to complex network scenarios.

###### Theorem 1.

Fix a quantum channel and . For any -entanglement assisted code for this quantum channel, it holds that

 R≤max|ψ⟩AB′minσBDεH(NA→B(ψAB′)∥σB⊗ψB′).
###### Proof.

We will prove the upper bound for uniform distribution over the messages. Fix a quantum state . Let be the quantum state after Alice’s encoding. There exists a register that purifies into the pure state . Let be the quantum state after the action of the channel and . From Facts 9 and 2, we have

 R ≤ DεH(ϕMM′∥ϕM⊗D(σB⊗ρEB))≤DεH(ρMBEB∥ρM⊗σB⊗ρEB) = DεH(ρMBEB∥ρMEB⊗σB)≤DεH(ρMBEBF∥ρMEBF⊗σB),

where we have used the facts that and . Since register is obtained by an action of the channel , we have

 R≤DεH(NA→B(ψMAEBF)∥ψMEBF⊗σB).

Setting and optimizing over all , we conclude the converse. ∎

Following achievability was shown in [1], which is near optimal.

###### Theorem 2 ([1]).

Fix a quantum channel and . There exists an -entanglement assisted code for this channel if

 R≤maxψAB′Dε+δH(NA→B(ψAB′)∥NA→B(ψA)⊗ψB′)−log1δ.

The error of can be improved to , by tuning the parameter in Hayashi-Nagaoka inequality (Fact 6), as noted in [44].

Success probability for entanglement assisted communication over noiseless channel. Now, we recover the result in [36, Theorem 1.1] for one way protocols, as an application of Theorem 1.

###### Corollary 1.

For any - entanglement assisted code for the identity channel , it holds that

 1−ε≤|A|22R.
###### Proof.

We apply Theorem 1 with to obtain

 R≤max|ψ⟩AB′DεH(|ψ⟩⟨ψ|AB′∥IA|A|⊗ψB′).

Let be the Schmidt decomposition of such that and let . It holds that . From Fact 11, let (with some abuse of notation) be the rank one operator achieving the optimum for . We recall that need not be normalized. Since and

 ⟨Π|Π′IA|A|⊗ψB′Π′|Π⟩=⟨Π|IA|A|⊗ψB′Π′|Π⟩≤⟨Π|IA|A|⊗ψB′|Π⟩,

we have that . Thus, we expand such that . The condition translates to . Further,

 ⟨Π|IA|A|⊗ψB′|Π⟩=1|A|∑i|aiλi|2.

By Cauchy-Schwartz inequality,

 1−ε≤|∑iaiλi|2≤|A|∑i|aiλi|2⟹1−ε|A|≤∑i|aiλi|2.

Hence, it holds that

 ⟨Π|IA|A|⊗ψB′|Π⟩≥1−ε|A|2,

for any feasible choice of . The inequality is achieved when and , which also satisfies the constraints . Hence, we conclude that

 R≤max|ψ⟩AB′DεH(|ψ⟩⟨ψ|AB′∥IA|A|⊗ψB′)≤log|A|21−ε.

### 3.2 Quantum channel with state

Alice wants to communicate a classical message chosen from to Bob over a quantum channel such that Bob is able to decode the correct message with probability at least . Alice shares entanglement with the channel as well. This model in the classical setting is known as the Gel’fand-Pinsker channel.

###### Definition 2.

Let be the shared entanglement between Alice and Bob and let be the state shared between Alice and Channel. An -entanglement assisted code for the quantum channel consists of

• An encoding operation for Alice.

• A decoding operation for Bob, with being the output register such that for all ,

 Pr(M′≠m|M=m)≤ε.

We have the following converse.

###### Theorem 3.

Fix a quantum channel with state and an . For every -entanglement assisted code for this channel, it holds that

 R≤maxψASB′:ψSB′=τS⊗ψB′minσBDεH(NAS→B(ψASB′)∥ψB′⊗σB).
###### Proof.

We will prove the upper bound for uniform distribution over the messages. Fix a quantum state . Let be the quantum state after Alice’s encoding and be the quantum state after the action of the channel. Let . From Facts 9 and 2,

 R ≤ DεH(ϕMM′∥ϕM⊗D(σB⊗ρEB))≤DεH(ρMBEB∥ρM⊗σB⊗ρEB) = DεH(ρMBEB∥ρMEB⊗σB),

where we have used the facts that and . Now, observe that and . Setting , we conclude that

 R≤DεH(NAS→B(ψASB′)∥ψB′⊗σB),

where . ∎

As a corollary of above converse, we obtain the following converse statement, which matches (up to some constants) the achievability result given in [1, Theorem 5].

###### Corollary 2.

Fix a quantum channel with state and an . For every -entanglement assisted code for this channel, it holds that

 R≤maxψASB′:ψS=τS(minσBDεH(NAS→B(ψASB′)∥ψB′⊗σB)−Dmax(ψSB′∥ψS⊗ψB′)).
###### Proof.

If , then . Thus, the optimization in above statement is over a larger set, as compared to that given in Theorem 3. ∎

The utility of [1, Theorem 5] is that it yields a single letter expression in the asymptotic and i.i.d. setting, as shown in [27] using different techniques. It is also possible to directly achieve the bound given in Theorem 3, as we show below. The utility of this bound is that it is of the form similar to that for multiple access channel and broadcast channel, both of which are one-shot optimal.

###### Theorem 4.

Fix a quantum channel with state and . There exists an -entanglement assisted code for this channel, if

 R≤maxψASB′:ψSB′=τS⊗ψB′DεH(NAS→B(ψASB′)∥NAS→B(ψAS)⊗ψB′)−log4εδ2.
###### Proof.

Fix a quantum state achieving the optimum above such that . Let be a purification of . Alice and Bob share copies of in registers . Let be a purification of . Let be an isometry such that .

Encoding: To send the message , Alice prepares the pure state by applying the isometry on the registers and sends register through the channel.

Decoding and error analysis: Bob performs the position-based decoding strategy across the registers. Let be the operator achieving the optimum in the definition of . Define

 Λ(m)\lx@stackreldef=IB′1⊗IB′2⊗⋯ΠBB′m⊗⋯⊗IB′2R,

and

 Ω(m)\lx@stackreldef=⎛⎜⎝∑m′∈[1:2R]Λ(m′)⎞⎟⎠−12Λ(m)⎛⎜⎝∑m′∈[1:2R1]Λ(m′)⎞⎟⎠−12.

Bob applies the measurement to decode .

Error analysis: Employing Hayashi-Nagaoka inequality (Fact 6), we have

 Pr{M′≠m∣M=m} ≤(1+c)Tr((I−ΠBB′)NAS→B(ψASB′))+(2+c+1c)⋅2R1Tr(ΠBB′NAS→B(ψAS)⊗ψB′) ≤(1+c)ε+4c2R1−DεH(NAS→B(ψASB′)∥NAS→B(ψAS)⊗ψB′) ≤ε+2δ,

where we choose .

This completes the proof. ∎

Alice wishes to communicate message pair simultaneously to Bob and Charlie over a quantum broadcast channel, where is intended for Bob and is intended for Charlie, such that both Bob and Charlie output the correct message with probability at least .

###### Definition 3.

Let and be the shared entanglement between Alice and Bob and Alice and Charlie respectively. An entanglement assisted code for the quantum broadcast channel consists of

• An encoding operation for Alice.

• A pair of decoding operations , and , with being the output registers, such that for all

 Pr(M′1≠m1|M1=m1)≤ε1,Pr(M′2≠m2|M2=m2)≤ε2.

We have the following converse.

###### Theorem 5.

Fix a quantum channel and . For any - entanglement assisted code for this channel, there exist registers and a quantum state satisfying such that

 R1≤minσBDε1H(TrCNA→BC(ψAB′)∥σB⊗ψB′),
 R2≤minτCDε2H(TrBNA→BC(ψAC′)∥τC⊗ψC′).
###### Proof.

We will prove the upper bound for uniform distribution over the messages. Fix quantum states . Let be the quantum state after Alice’s encoding and be the quantum state after the action of the channel. Let and . From Facts 9 and 2,

 R1≤Dε1H(ϕBM1M′1∥ϕBM1⊗D(σB⊗ρEB))≤Dε1H(ρM1BEB∥ρM1⊗σB⊗ρEB)=Dε1H(ρM1BEB∥ρM1EB⊗σB),

where we have used the facts that and