Classical codes for quantum broadcast channels

# Classical codes for quantum broadcast channels

Ivan Savov and Mark M. Wilde I.S. and M.M.W. were with the School of Computer Science, McGill University, Montréal, Québec, Canada when conducting parts of this research. I.S. is now with Minireference Publishing. M.M.W. is now with the Hearne Institute for Theoretical Physics, the Department of Physics and Astronomy, and the Center for Computation and Technology at Louisiana State University, Baton Rouge, LA 70803. This work was presented in part at the 2012 IEEE International Symposium on Information Theory.
###### Abstract

We present two approaches for transmitting classical information over quantum broadcast channels. The first technique is a quantum generalization of the superposition coding scheme for the classical broadcast channel. We use a quantum simultaneous nonunique decoder and obtain a proof of the rate region stated in [Yard et al., IEEE Trans. Inf. Theory 57 (10), 2011]. Our second result is a quantum generalization of the Marton coding scheme. The error analysis for the quantum Marton region makes use of ideas in our earlier work and an idea recently presented by Radhakrishnan et al. in arXiv:1410.3248. Both results exploit recent advances in quantum simultaneous decoding developed in the context of quantum interference channels.

## I Introduction

How can a broadcast station communicate separate messages to two receivers using a single antenna? Two well known strategies [1] for transmitting information over broadcast channels are superposition coding [2, 3] and Marton multicoding using correlated auxiliary random variables [4]. In this paper, we prove that these strategies can be adapted to the quantum setting by constructing random codebooks and matching decoding measurements that have asymptotically vanishing error in the limit of many uses of the channel.

Sending classical data over a quantum channel is one of the fundamental problems of quantum information theory [5]. Single-letter formulas are known for classical-quantum point-to-point channels [6, 7] and multiple access channels [8]. Classical-quantum channels are a useful abstraction for studying general quantum channels and correspond to the transmitters being restricted to classical encodings. Codes for classical-quantum channels (c-q channels), when augmented with an extra optimization over the possible input states, directly generalize to codes for quantum channels. Furthermore, it is known that classical encoding (coherent-state encoding using classical Gaussian codebooks) is sufficient to achieve the capacity of phase-insensitive quantum Gaussian channels, which is a realistic model for optical communication links [9, 10, 11].

Previous work on quantum broadcast channels includes [12, 13, 14]. Yard et al. consider both classical and quantum communication over quantum broadcast channels and state a superposition coding inner bound in their Theorem 1 similar to that stated in our Theorem 1 [12]. However, it is unclear to us whether the proof given for their Theorem 1 is complete (we elaborate on this point in what follows). Relying on Theorem 1 of [12], Ref. [13] discusses classical communication over a bosonic broadcast channel. Ref. [14] establishes a Marton rate region for quantum communication.

In this paper, we derive two achievable rate regions for classical-quantum broadcast channels by exploiting error analysis techniques developed in the context of quantum interference channels [15, 16]. In Section III, we prove achievability of the superposition coding inner bound (Theorem 1), by using a quantum simultaneous nonunique decoder at one of the receivers. In Section IV we prove that the quantum Marton rate region with no common message is achievable (Theorem 3). In the Marton coding scheme, the sub-channels to each receiver are essentially point-to-point, but it turns out that two techniques which we call the “projector trick” and “overcounting” [17] seem to be necessary in the proof. We discuss open problems and give an outlook for the future in Section V.

Note: The original justification for the quantum Marton region given in our earlier work [18] contained a gap, which was identified by Pranab Sen and relayed to us by Andreas Winter. This gap was addressed in the related paper [17], where an achievable region in the “one-shot” Marton coding setting was established. Here we show how to apply the overcounting method in order to close the aforementioned gap in our earlier work.

## Ii Preliminaries

### Ii-1 Notation

We denote classical random variables as , whose realizations are elements of the respective finite alphabets . Let , , denote their corresponding probability distributions. We denote quantum systems as , , and and their corresponding Hilbert spaces as , , and . We represent quantum states of a system  with a density operator , which is a positive semi-definite operator with unit trace. Let denote the von Neumann entropy of the state . A classical-quantum channel, , is represented by the set of possible output states , meaning that a classical input of leads to a quantum output . In a communication scenario, the decoding operations performed by the receivers correspond to quantum measurements on the outputs of the channel. A quantum measurement is a positive operator-valued measure (POVM) on the system , the output of which we denote . To be a valid POVM, the set of operators must all be positive semi-definite and sum to the identity: .

### Ii-2 Definitions and useful lemmas

We define a classical-quantum-quantum broadcast channel as the following map:

 x→ρB1B2x, (1)

where is a classical letter in an alphabet and is a density operator on the tensor product Hilbert space for systems and . The model is such that when the sender inputs a classical letter , Receiver 1 obtains system , and Receiver 2 obtains system . Since Receiver 1 does not have access to the part of the state , we model his state as , where denotes the partial trace over Receiver 2’s system.

###### Lemma 1 (Gentle Operator Lemma for Ensembles [19]).

Given an ensemble with expected density operator , suppose that an operator such that succeeds with high probability on the state :

 Tr{Λρ}≥1−ε. (2)

Then the subnormalized state is close in expected trace distance to the original state :

 EX{∥∥√ΛρX√Λ−ρX∥∥1}≤2√ε. (3)

The following lemma appears in [20, Lemma 2]. When using it for the square-root measurement in (8), we choose and .

###### Lemma 2 (Hayashi-Nagaoka).

The Hayashi-Nagaoka operator inequality applies to a positive operator and an operator where :

 I−(S+T)−12S(S+T)−12≤2(I−S)+4T. (4)

### Ii-3 Information processing task

The task of communication over a broadcast channel is to use independent instances of the channel in order to communicate with Receiver 1 at a rate and to Receiver 2 at a rate . More specifically, the sender chooses a pair of messages from message sets , where , and encodes these messages into an -symbol codeword suitable as input for the channel uses.

The output of the channel is a quantum state of the form:

 N⊗n(xn(m1,m2)) ≡ρBn1Bn2xn(m1,m2). (5)

where To decode the message intended for him, Receiver 1 performs a POVM on the system , the output of which we denote . Receiver 2 similarly performs a POVM on the system , and the random variable associated with the outcome is denoted .

An error occurs whenever either of the receivers decodes the message incorrectly. The probability of error for a particular message pair is

 pe(m1,m2)≡Tr{(I−Λm1⊗Γm2)ρBn1Bn2xn(m1,m2)}, (6)

where the operator represents the complement of the correct decoding outcome.

###### Definition 1.

An broadcast channel code consists of a codebook and two decoding POVMs and such that the average probability of error is bounded from above as

 ¯¯¯pe ≡1|M1||M2|∑m1,m2pe(m1,m2)≤ε. (7)

A rate pair is achievable if there exists an quantum broadcast channel code for all and sufficiently large .

When devising coding strategies for c-q channels, the main obstacle to overcome is the construction of a decoding POVM that correctly decodes the messages. Given a set of positive operators which are suitable for detecting each message, we can construct a POVM by normalizing them using the square-root measurement [6, 7]:

 Λm ≡(∑kΠ′k)−12Π′m(∑kΠ′k)−12. (8)

Thus, the search for a decoding POVM is reduced to the problem of finding positive operators apt at detecting and distinguishing the output states produced by each of the possible input messages ( and for some small ).

## Iii Superposition coding inner bound

One possible strategy for the broadcast channel is to send a message at a rate that is low enough so that both receivers are able to decode. Furthermore, if we assume that Receiver 1 has a better reception signal, then the sender can encode a further message superimposed on top of the common message that Receiver 1 will be able to decode given the common message. The sender encodes the common message at rate using a codebook generated from a probability distribution , and the additional message for Receiver 1 at rate using a conditional codebook with distribution .

###### Theorem 1 (Superposition coding inner bound).

A rate pair is achievable for the quantum broadcast channel in (1) if it satisfies the following inequalities:

 R1 ≤I(X;B1|W)θ, (9) R2 ≤I(W;B2)θ, (10) R1+R2 ≤I(X;B1)θ, (11)

where the above information quantities are with respect to a state of the form

 ∑w,xpW(w)pX|W(x|w) |w⟩⟨w|W⊗|x⟩⟨x|X⊗ρB1B2x. (12)

It suffices to take the cardinality of the alphabet for to be no larger than , where is the input alphabet of the channel.

###### Proof:

The idea of the proof given below is to exploit superposition encoding and a quantum simultaneous nonunique decoder for the decoding of the first receiver [2, 3]. We use a standard HSW decoder for the second receiver [6, 7]. The cardinality bound follows directly from Appendix A of [12].

Codebook generation. The sender randomly and independently generates sequences according to the product distribution

 pWn(wn)≡n∏i=1pW(wi). (13)

For each sequence , the sender then randomly and conditionally independently generates sequences according to the product distribution:

 pXn|Wn(xn|wn(m2))≡n∏i=1pX|W(xi|wi(m2)). (14)

The sender then transmits the codeword if she wishes to send .

POVM Construction. We now describe the POVMs that the receivers employ in order to decode the transmitted messages. First consider the state we obtain from (12) by tracing over the system:

 ρWXB1=∑w,xpW(w) pX|W(x|w) |w⟩⟨w|W⊗|x⟩⟨x|X⊗ρB1x. (15)

Further tracing over the system gives

 ρWB1 =∑wpW(w) |w⟩⟨w|W⊗σB1w, (16)

where . For the first receiver, we exploit a square-root decoding POVM as in (8) based on the following positive operators:

 Π′m1,m2≡Π ΠWn(m2) ΠXn(m1,m2) ΠWn(m2) Π, (17)

where we have made the abbreviations

 Π≡ΠBn1ρ,δ,ΠWn(m2)≡ΠBn1σWn(m2),δ,ΠXn(m1,m2)≡ΠBn1ρXn(m1,m2),δ. (18)

The above projectors are weakly typical projectors [5, Section 14.2.1] defined with respect to the states , , and .

Consider now the state in (12) as it looks from the point of view of Receiver 2. If we trace over the and systems, we obtain the following state:

 ρWB2=∑wpW(w) |w⟩⟨w|W⊗σB2w, (19)

where . For the second receiver, we exploit a standard HSW decoding POVM that is with respect to the above state—it is a square-root measurement as in (8), based on the following positive operators:

 Π′Bn2m2=ΠBn2ρ,δ ΠB2σWn(m2),δ ΠBn2ρ,δ, (20)

where the above projectors are weakly typical projectors defined with respect to and .

Error analysis. We now analyze the expectation of the average error probability for the first receiver with the POVM defined by (8) and (17):

 (21)

Due to the above exchange between the expectation and the average and the symmetry of the code construction (each codeword is selected randomly and independently), it suffices to analyze the expectation of the average error probability for the first message pair , i.e., the last line above is equal to

 EXn,Wn{Tr{(I−ΓBn11,1)ρB1Xn(1,1)}}. (22)

Using the Hayashi-Nagaoka operator inequality (Lemma 2 in the appendix), we obtain the following upper bound on this term:

 2EXn,Wn{Tr{(I−Π′1,1)ρB1Xn(1,1)}}+4∑(m1,m2)≠(1,1)EXn,Wn{Tr{Π′m1,m2ρB1Xn(1,1)}}. (23)

We begin by bounding the first term above. Consider the following chain of inequalities:

 EXn,Wn{% Tr{Π′1,1ρB1Xn(1,1)}} =EXn,Wn{%Tr{Π ΠWn(1)ΠXn(1,1) ΠWn(1) Π ρB1Xn(1,1)}} (24) ≥EXn,Wn{Tr{ΠXn(1,1)ρB1Xn(1,1)}}−EXn,Wn{∥∥ρB1Xn(1,1)−Π ρB1Xn(1,1) Π∥∥1} −EXn,Wn{∥∥ρB1Xn(1,1)−ΠWn(1) ρB1Xn(1,1) ΠWn(1)∥∥1} (25) ≥1−ε−4√ε, (26)

where the first inequality follows from the inequality

 Tr{Λρ}≤Tr{Λσ}+∥ρ−σ∥1, (27)

which holds for all subnormalized states and , and such that . The second inequality follows from the Gentle Operator Lemma for ensembles (see Lemma 1 in the appendix) and the properties of typical projectors for sufficiently large .

We now focus on bounding the second term in (23). We can expand this term as follows:

 ∑m1≠1EXn,Wn{Tr{Π′m1,1 ρB1Xn(1,1)}}+∑m1,m2≠1EXn,Wn{Tr{Π′m1,m2 ρB1Xn(1,1)}}. (28)

Consider the first term in (28):

 ∑m1≠1EXn,Wn{Tr{Π′m1,1 ρB1Xn(1,1)}} (29) =∑m1≠1EXn,WnTr{Π ΠWn(1) ΠXn(m1,1) ΠWn(1) Π ρB1Xn(1,1)} (30) ≤2n[H(B1|WX)+δ]∑m1≠1EXn,Wn{Tr[Π ΠWn(1)ρXn(m1,1)ΠWn(1)Π ρB1Xn(1,1)]} (31) =2n[H(B1|WX)+δ]∑m1≠1EWn{Tr[ΠWn(1)EXn|Wn{ρXn(m1,1)} ΠWn(1) Π EXn|Wn{ρB1Xn(1,1)}Π ]} (32) =2n[H(B1|WX)+δ]∑m1≠1EWn{Tr{Π ΠWn(1)σWn(1) ΠWn(1) Π σWn(1)}} (33) ≤2n[H(B1|WX)+δ]2−n[H(B1|W)−δ]∑m1≠1EWn{Tr{Π ΠWn(1) Π σWn(1)}} (34) ≤2n[H(B1|WX)+δ]2−n[H(B1|W)−δ] ∑m1≠1EWn{Tr{σWn(1)}} (35) ≤2−n[I(X;B1|W)−2δ] M1. (36)

The first inequality is due to the projector trick inequality [21, 16, 15] which states that

 ΠXn(m1,1) ≤2n[H(B1|WX)+δ]ρB1Xn(m1,1). (37)

Note that this inequality is a straightforward consequence of the following standard typicality operator inequality and the fact that and commute:

 2−n[H(B1|WX)+δ]ΠXn(m1,1)≤ΠXn(m1,1)ρB1Xn(m1,1)ΠXn(m1,1). (38)

The second inequality follows from the properties of typical projectors:

 ΠWn(1)σWn(1) ΠWn(1)≤2−n[H(B1|W)−δ]ΠWn(1). (39)

Now consider the second term in (28):

 ∑m1,m2≠1EXn,Wn{Tr{Π′m1,m2 ρB1Xn(1,1)}} (40) =∑m1,m2≠1EXn,Wn{Tr[ΠΠWn(m2)ΠXn(m1,m2)ΠWn(m2)Π ρB1Xn(1,1)]} (41) =∑m1,m2≠1Tr[EXn,Wn{ΠWn(m2) ΠXn(m1,m2) ΠWn(m2)}ΠEXn,Wn{ρB1Xn(1,1)}Π ] (42) =∑m1,m2≠1Tr{EXn,Wn{ΠWn(m2)ΠXn(m1,m2)ΠWn(m2)} Πρ⊗nΠ} (43) ≤2−n[H(B1)−δ]∑m1,m2≠1Tr[EXn,Wn{ΠWn(m2)ΠXn(m1,m2)ΠWn(m2)}Π] (44) =2−n[H(B1)−δ]∑m1,m2≠1EXn,WnTr[ΠXn(m1,m2)ΠWn(m2)ΠΠWn(m2)] (45) ≤2−n[H(B1)−δ]∑m2≠1, m1EXn,Wn{Tr{ΠXn(m1,m2)}} (46) ≤2−n[H(B1)−δ] 2n[H(B1|WX)+δ] M1 M2 (47) =2−n[I(WX;B1)−2δ] M1 M2 (48) =2−n[I(X;B1)−2δ] M1 M2. (49)

The equality follows from the way the codebook is constructed (i.e., the Markov chain ), as discussed also in [16]. This completes the error analysis for the first receiver.

For the second receiver, the decoding error analysis follows from the HSW coding theorem. We now present this for completeness and tie the coding theorem together so that the sender and two receivers can agree on a strategy that has asymptotically vanishing error probability in the large limit. The following bound holds for the expectation of the average error probability for the second receiver if is sufficiently large:

 EXn,Wn{1M2∑m2Tr{(I−ΛBn2m2)ρBn2Xn(m1,m2)}} (50) =EWn{1M2∑m2Tr{(I−ΛBn2m2)EXn|Wn{ρBn2Xn(m1,m2)}}} (51) =EWn{1M2∑m2Tr{(I−ΛBn2m2)σBn2Wn(m2)}} (52) ≤2(ε+2√ε)+4 [2−n[I(W;B2)−2δ] M2], (53)

where the last line follows from an analysis similar to that given above.

Putting everything together, the joint POVM performed by both receivers is of the form:

 ΓBn1m1,m2⊗ΛBn2m′2, (54)

and the expectation of the average error probability for both receivers is bounded from above as

 EXn,Wn1M1M2∑m1,m2Tr{(I−ΓBn1m1,m2⊗ΛBn2m2)ρBn1Bn2Xn(m1,m2)} (55) (56) +EXn,Wn{1M1M2∑m1,m2Tr{(I−ΛBn2m2)ρBn2Xn(m1,m2)}} (57) (58) +4 [2−n[I(X;B1|W)−2δ] M1+2−n[I(X;B1)−2δ] M1 M2], (59)

where the first inequality follows from the following “union bound” operator inequality:

 IBn1Bn2−ΓBn1m1,m2⊗ΛBn2m2≤(IBn1Bn2−ΓBn1m1,m2⊗IBn2)+(IBn1Bn2−IBn1⊗ΛBn2m2), (60)

and the second inequality follows from our previous estimates. Thus, as long as the sender chooses the message sizes and such that , , and , then there exists a particular code with asymptotically vanishing average error probability in the large limit. ∎

###### Remark 2.

It is unclear to us whether the proof of [12, Theorem 1] is complete. These authors begin their proof by claiming that the region in Theorem 1 is equivalent to the following region:

 R1 ≤I(X;B1|W)θ, (61) R2 ≤I(W;B2)θ, (62) R2 ≤I(W;B1)θ. (63)

The regions certainly intersect at the corner point associated with their successive decoding strategy, but the full regions for a fixed distribution do not coincide in general. The proof of [12, Theorem 1] demonstrates achievability of all rates in the rectangular part of Receiver 1’s region given in our Theorem 1. With our simultaneous decoding non-unique decoding strategy, we can achieve any rate in the triangular part of this region as well, which could be useful if the first constraint above on Receiver 2 is looser than the second constraint above on Receiver 2. In such a case, the successive decoding strategy from [12, Theorem 1] would not be able to achieve the rate if , but the simultaneous decoding strategy can. It might be the case that the proof of [12, Theorem 1] could be completed by choosing particular coding distributions and taking unions over the resulting regions, but this is not discussed there.

## Iv Marton coding scheme

We now prove that the Marton inner bound is achievable for quantum broadcast channels. The Marton scheme depends on auxiliary random variables and , multicoding, and the properties of strongly typical sequences and projectors. The proof depends on some ideas originally presented in [18] and critically on the “overcounting” technique recently presented in [17].

###### Theorem 3 (Marton inner bound).

Let be a classical-quantum broadcast channel and be a deterministic function. The following rate region is achievable:

 R1 ≤ I(U1;B1)θ, (64) R2 ≤ I(U2;B2)θ, (65) R1+R2 ≤ I(U1;B1)θ+I(U2;B2)θ−I(U1;U2)θ, (66)

where the information quantities are with respect to the state:

 θU1U2B1B2=∑u1,u2p(u1,u2)|u1⟩⟨u1|U1⊗|u2⟩⟨u2|U2⊗ρB1B2f(u1,u2). (67)

It suffices to take the cardinalities and of and to be no larger than the cardinality of the channel’s input alphabet : i.e., .

Define the following states:

 ρB1f(u1,u2) ≡TrB2{ρB1B2f(u1,u2)}, (68) ωB1u1 ≡∑u2pU2|U1(u2|u1)ρB1f(u1,u2), (69) ¯¯¯ρB1 ≡∑u1pU1(u1)ωB1u1. (70)

Codebook construction. Define two auxiliary indices and , and let and . For each , generate a sequence  independently and randomly according to the product distribution

 pUn1(un1)≡n∏i=1pU1(u1,i). (71)

Similarly, for each , generate a sequence  independently and randomly according to the product distribution

 pUn2(un2)≡n∏i=1pU2(u2,i). (72)

Partition the sequences into different bins, each of which we label as . Partition the sequences into different bins, each of which we label as . For each message pair, the sender selects a sequence pair , where is the strongly typical set for . The scheme is such that each sequence is taken from the appropriate bin and the sender demands that they are strongly jointly-typical (otherwise admitting failure by just sending the first sequence pair in the bin). The codebook is deterministically constructed from , by applying the function .

Transmission. Let and