# Universal random codes: Capacity regions of the compound quantum multiple-access channel with one classical and one quantum sender

We consider the compound memoryless quantum multiple-access channel (QMAC) with two sending terminals. In this model, the transmission is governed by the memoryless extensions of a completely positive and trace preserving map which can be any element of a prescribed set of possible maps. We study a communication scenario, where one of the senders shares classical message transmission goals with the receiver while the other sends quantum information. Combining powerful universal random coding results for classical and quantum information transmission over point-to-point channels, we establish universal codes for the mentioned two-sender task. Conversely, we prove that the two-dimensional rate region achievable with these codes is optimal. In consequence, we obtain a multi-letter characterization of the capacity region of each compound QMAC for the present transmission task.

## Authors

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11/01/2020

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## I Introduction

Real-world communication situations usually involve more than two communication parties. This rather general statement typically leads to highly nontrivial coding scenarios, when translated to information theory. A very basic situation in this field of topics is, when two or more sending parties are connected to a receiver via a multiple-access channel (MAC). The rate and performance one of the sending parties can achieve is in general strongly connected to the rates of the other parties, and finding code constructions which allow to determine the set of rate tuples which are asymptotically achievable is a task of some piquancy.
Early results which determined the average-error capacity region for classical message transmission over a memoryless classical MAC are due to Ahlswede ahlswede73 and Liao liao72 . Among others, these works stimulated a vital research in classical information theory (see csiszar11 , Chapter 14 for an overview). In case of the maximal error, the capacity region is still unknown while Dueck gave an example of a MAC where average and maximal error capacity regions are different dueck78 .
Regarding the setting, were the senders are connected to the receiver by a memoryless quantum multiple-access channel (QMAC), the first notable results include the paper winter99 . Therein the region of achievable rate pairs was determined for the case that all senders aim to convey classical messages. For other scenario where some of the senders aim to send quantum information, the achievable rates where characterized in yard08 , while the quantum capacity region (i.e. the set of rate achievable rate pairs if all senders send quantum information) was derived in horodecki07 ; yard08 .
However, all of the mentioned results were proven under the assumption, that the transmission c.p.t.p. map which governs the channel transmission is perfectly known to the senders as well as to the receiver. In this work, we try to impose more realistic assumptions on the channel model. We consider the compound memoryless QMAC, where the communication parties have no precise knowledge of the channel, but instead have only a priori knowledge of a set of channels, in which the generating map is contained. The consequence of their imprecise knowledge is, that they have to use “universal” codes, which perform well on all channels in the set.
The compound channel model which already has been studied in classical Shannon theory since the 1960s. In the domain of quantum Shannon theory, past research activities regarding compound quantum channels where mostly concentrated on point to point quantum channels, leading to determination of the classical capacity hayashi05 ; bjelakovic09 ; datta10 ; mosonyi15 , and the genuine quantum capacities bjelakovic08 ; bjelakovic09_a of a quantum channel.
Regarding compound quantum channels with having more than two users, the research was concentrated on private classical message transmission over wiretap channels, i.e. channels having one sender but two receiving parties.
Suitable codes for this situation were developed in boche14 generalizing techniques introduced for the case of classical compound wiretap channels in bjelakovic11 to the quantum setting.
The first codes for classical message transmission over compound classical-quantum MACs where provided in hirche16 .
In this article, we aim to extend the scope of multi-user quantum Shannon theory in the direction of models having more than one sender and involving channel uncertainty. We consider a QMAC with two sending parties and in the “hybrid” situation, where pursues the target of transmitting classical messages while sender aims for entanglement transmission.

### i.1 Outline

In Section III, we provide ourselves with precise definitions regarding the channel model and codes used in this work. Therein, we also state Theorem 6 which is the main result of this work which is a multi-letter characterization of the capacity region of the compound QMAC with a classical and a quantum sender. Section IV is of rather technical nature. We introduce random classical message transmission and entanglement transmission codes which are crucial ingredients for our reasoning. Section V contains the proofs of Theorem 6. In Section V.1 we construct suitable universal hybrid codes for the QMAC. These are obtained by combining ideas from yard08 with the universal random codes from the previous section. By providing the converse part of Theorem 6 in Section V.2, we complete our proof.

### i.2 Related work

The capacity regions of a perfectly known QMAC with one classical and one quantum sender (and moreover also the genuine quantum capacity regions of that channel model) where determined by Yard et al. in yard08 . The strategy used therein to derive codes being sufficient to prove the coding theorem is as follows. By combining known random coding results for classical message transmission from holevo98 , schumacher97 , and entanglement transmission devetak05 for single-user quantum channels in a sophisticated way, the authors constructed random codes for classical and quantum coding over the QMAC. Combining random codes to simultaneously achieve different transmission goals was long standard in classical multi-user Shannon theory, and can, in the quantum case traced back to winter01 , where the capacity regions of quantum multiple-access channels was determined in case that all senders wish to transmit classical messages. That a strategy in the mentioned manner is successful also in situations where classical and quantum transmission goals are to be accomplished simultaneously was first demonstrated in devetak05 . Therein, hybrid codes are constructed which allow to transmit classical and quantum information over a memoryless point-to-point quantum channel at the same time. The capacity region for simultaneous transmission of classical and quantum information was also shown to exceed the obvious region which to be achievable by time-sharing strategies for some quantum channels.
In both cases, suitable random codes already exist in the literature without having been exploited simultaneous transmission yet. In case of universal classical message transmission, independent first results can be found in bjelakovic09 , hayashi05 , and datta10 . In this work, we exploit the recent and very powerful techniques which where added to the aforementioned results in mosonyi15 . The random entanglement transmission codes we use in this work where developed in bjelakovic08 , bjelakovic09_a . The reader may note, that the approach pursued in devetak05 to derive good random quantum codes seems to be not suitable in case of compound quantum channels, as the discussion in Section VII of boche14 suggests. The random codes derived in bjelakovic08 stem from generalization of the codes in klesse07 which where derived in spirit of the so-called decoupling approach to the quantum capacity (see also hayden08 for a similar application of the decoupling idea.)

## Ii Notation and conventions

All Hilbert spaces which appear in this work are finite dimensional over the field of complex numbers equipped with the standard euclidean scalar product. For a Hilbert space , denotes the set of linear maps (or matrices), while denotes the set of density matrices, and the set of unitaries on . For an alphabet

(which we always assume to be of finite cardinality), we denote the simplex of probability distributions on

by . With a second Hilbert space , we denote by the set of completely positive (c.p.) trace non-increasing maps while is the notation for completely positive and trace preserving (c.p.t.p.) maps. For positive semi-definite matrices , we use the definition

 F(a,b) = ∥√a√b∥1

for the (quantum) fidelity. For a c.p.t.p. map and a density matrix , we use the entanglement fidelity defined by

where is any purification of . The von Neumann entropy of a state is defined by , and we will use in this work several entropic quantities which derive from it. For a bipartite state ,

 Ic(A⟩B,ρ):=S(ρB)−S(ρ), (1)

defines the coherent information of , while

 I(A;B,ρ):=S(ρA)+S(ρB)+S(ρ) (2)

is the quantum mutual information. We will employ the usual notation for systems, which have classical and quantum subsystems. E.g.

 ρXB:=∑x∈Xp(x)|x⟩⟨x|⊗ρx (3)

represents the preparation of a bipartite system, where one system is classical (with preparation being a probability distribution ) while is a density matrix for each outcome of

(the random variable with probability distribution p.) For a set

, we denote the closure of by . Moreover, we define for each the set by

 1l:={(1lx,1ly):(x,y)∈A}.

For each we moreover set

 Aδ:={x∈R+0×R+0: ∃y∈A:|x−y|≤δ}.

## Iii Basic definitions and main result

In this section we provide precise definitions of codes and capacity regions considered in this work. Let with three Hilbert spaces , , , under control of communication parties labelled by , , and . While , and act as senders for the channel, is designated as receiver. Let be a set of c.p.t.p. maps. If not otherwise specified, we do not assume further properties of the set (we especially do not demand to be a finite set.)
The compound memoryless quantum multiple access channel (QMAC) generated by (the compound QMAC for short) is given by the family of transmission maps. The above definition is interpreted as a channel model, where the transmission statistics for uses of the system is governed by , where can be any member of . We designate as sender transmitting classical messages. In this work, we consider two different coding scenarios which differ in the quantum transmission task performs. Adopting the terminology from yard08 consider Scenario I ( aims to perform entanglement generation), Scenario II ( aims to perform entanglement transmission). For the rest of the section we fix a set .

###### Definition 1 (Scenario-I code).

An -Scenario-I code for the compound QMAC is a family , where with an additional Hilbert spaces

• is a classical-quantum channel, i.e. for each .

• is a pure state.

• for each such that is a quantum channel.

In in the situation, where and the receiver perform entanglement transmission over the QMAC, we define

###### Definition 2 (Scenario-II code).

An -Scenario-II code for the compound QMAC is a family , where with an additional Hilbert spaces

• is a classical-quantum channel.

• for each , such that is trace preserving.

We next define the performance functions of the codes introduced above

 PI(C,M⊗n,m) := F(|m⟩⟨m|⊗Φ,idFB⊗D∘M⊗n(V(m)⊗Ψ)), and PII(C,M⊗n,m) := F(|m⟩⟨m|⊗Φ,idFB⊗D∘M⊗n(V(m)⊗Φ)).

and set for

 PII(C,M) := 1M1M1∑m=1 PII(C,M,m).
###### Definition 3 (Achievable rates).

Let . A pair of non-negative numbers is called an achievable Scenario-X rate for the compound QMAC , if for each exists a number , such that for each there is an -Scenario-X code for such that the conditions

1. for , and

are simultaneously fulfilled. We define the Scenario-X capacity region of the compound QMAC by

 CQX(M) := {(R1,R2)∈R+0×R+0: (R1,R2) achievable Scenario-X % rate for M}.

As an operational fact, following directly from the above definitions, we have the following.

###### Fact 4.

For each , and are compact convex subsets of .

Let . It holds .

###### Proof.

Let for an arbitrary but fixed blocklength , be an Scenario-II code. Let for fixed , , a spectral decomposition of given by

 idF⊗E(Φ) = N∑i=1λiΨi.

It holds for each

 F(|m⟩⟨m|⊗Φ,idF⊗D∘M⊗n∘id⊗nHA⊗E(V(m)⊗Φ)) =N∑i=1λi F(|m⟩⟨m|⊗Φ,idF⊗Dm∘M⊗n(Vm⊗Ψi)). (4)

If now is any index such that is maximal, the Scenario I code suffices by the inequality in (4). ∎

To enable us for concise statement of the main result, we introduce some more notation. Fix Hilbert spaces , and an alphabet . For given probability distribution , c.p.t.p. map , cq channel , pure state , we define an effective cqq state

 (5)

and a region

 ^C(1)(T,p,V,Ψ):={(R1,R2)∈R+0×R+0: R1≤I(X;C,ω) ∧ R2≤I(B⟩CX,ω)}, (6)

The following theorem is the main result this work.

###### Theorem 6.

Let . It holds

 CQI(M) = CQII(M) = cl(∞⋃l=1⋃p,V,Ψ⋂M∈M1l^C(1)(M⊗l,p,V,Ψ)) (7)
###### Remark 7.

The terms on the rightmost sides of the inclusion chains in the above theorem do not need convexification. The corresponding fact for the capacity region of the perfectly known QMAC was already proven in yard08 . For the reader’s convenience, we give an argument for the present case in Appendix B.

The proof of the equalities in Eq. (7) is split in several parts. Note, that the inequality is Fact 5. That the rightmost term in (7) is a subset of is the statement of Proposition 12. To complete the proof, we show, that is smaller than the rightmost term in Proposition 17.

## Iv Universal random codes for message and entanglement transmission

In this section, we state and discuss some universal random coding results for entanglement transmission and classical message transmission over single-sender channels. Most of the statements below, are already implicitly contained in the literature. However, the random nature of the codes where not explicitly stressed. Some additional properties which are connected to these random codes are revealed below, and may be useful in their own right.

### iv.1 Classical message transmission

For the reader’s convenience, we first introduce some terminology. A map with a (finite) alphabet and a Hilbert space is called a classical-quantum (cq) channel An classical message transmission code for is a family , where , and for each , with the additional property, that , if instances of the cq channel and the code are used for classical message transmission, we use the average transmission error

 ¯¯¯e(C,W⊗n):=1MM∑m=1 tr(1⊗nK−Dm)W⊗n(um)

as error criterion. The following proposition states existence of universal random message transmission codes for each given set of classical-quantum channels. Its proof can be extracted from mosonyi15 , where it was proven using the properties of quantum versions of the Renyi entropies together in combination with the Hayashi-Nagaoka random coding lemma hayashi03 .

###### Proposition 8 (Universal random cq codes without state knowledge mosonyi15 , Theorem 4.18).

Let be a set of classical-quantum channels, and . For each and large enough there exists an -random message transmission code which fulfills the following conditions.

1. is an independent family of random variables, each with distribution ,

2. , where , and

3. ,

where is a constant dependent on .

### iv.2 Entanglement transmission

In this paragraph we introduce universal coding results for the task of entanglement transmission, which were implicitly proven already in bjelakovic08 ,bjelakovic09_a . For a given quantum channel , an entanglement transmission code is a pair , where with a Hilbert space of dimension , , and are c.p.t.p. maps. The performance of the code is then measured by the entanglement fidelity , where is the maximally mixed state on . Their strategy to derive universal entanglement transmission codes for compound quantum channels was to generalize the decoupling lemma from klesse07 to achieve a one-shot bound for the performance in case of a finite set of channels for a fixed code subspace. A subsequent randomization over unitary transformations of that encoding led to random codes with achieving arbitrarily close to coherent information minimized over all possible channel states, given maximally mixed state on the input space. Further approximation using the so-called BSST lemma bennett02 approximating asymptotically each state by a sequence of maximally mixed state and a net approximation on the set of channels allowed to achieve the capacity of arbitrary compound quantum channels with these random codes. The the authors of bjelakovic08 , bjelakovic09_a applied a step of derandomization to end up with deterministic entanglement transmission codes sufficient to prove their coding theorem. We in turn, are explicitly interested in the random structure of the code. In the next proposition, we replicate the statement hidden in the proof of Lemma 9 in bjelakovic09_a

. Moreover, we notice, that the random code constructed in that proof have a very convenient property regarding the expected input state to the channel after encoding. It is a tensor product of the maximally mixed state appearing in the coherent information terms lower-bounding the rate.

###### Proposition 9 (cf. bjelakovic09_a , Lemma 9, boche17 ).

Let be a set of c.p.t.p. maps, a subspace of , . For each large enough exists an random entanglement transmission code , , such that

1. , where with being a purification of ,

2. , and

with a constant .

###### Remark 10.

In bjelakovic09 , actually a continuous random code distributed according to the Haar measure on the unitary group on the encoding subspace was constructed. To obtain the above finite version the Haar measure is replaced by a finitely supported measure which forms a so-called unitary design gross07 . This replacement is detailed in boche17 .

We notice, that in earlier work on perfectly known quantum channels (see e.g. devetak05 , devetak05b , hsieh10 ) usually a different type of random code was used. Instead of employing the random entanglement transmission codes from klesse07 or hayden08 based on the decoupling approach, the entanglement generation codes of devetak05 were used. These arise from a clever reformulation of private classical codes for a classical-quantum wiretap channel. In Appendix D in devetak05 , these codes where moreover further developed to approximately reproduce for a given by the random encoding.
However, we remark here, that establishing the results on this paper by generalizing the random codes in devetak05 is not very auspicious. As it was already noted in boche14 , that the method of generating entanglement generation codes from private classical codes employed in devetak05 seems not to carry over to the case of channel uncertainty.

###### Remark 11.

By applying the Proposition 9 to the special case , we obtain an alternative to the random codes from devetak05 in case of a perfectly known quantum channel. Alternatively one could also take the direct route to prove such a result and derive such codes directly from the original works klesse07 , hayden08 on the perfectly known channel.)

## V Proofs

### v.1 Inner bounds to the capacity regions

In this paragraph, we prove the achievability part of Theorem 6, i.e. the following statement.

###### Proposition 12 (Inner bound for the capacity region for uninformed users).

Let . It holds

 CQII(M) ⊃ cl(∞⋃l=1⋃p,V,Ψ⋂M∈M1l^C(1)(M⊗l,p,V,Ψ)) (8)

The main technical steps for proving the above assertion is done in the proof of the following proposition.

###### Proposition 13.

Let , a pure maximally entangled state, , a channel having only pure outputs. For each exists a number such that for each we find an Scenario-II code with

1. ,

2. , and

where is a strictly positive constant. The entropic quantities on the right hand sides of the first two inequalities are evaluated on the states

 ωs:=ω(Ts,p,V,Ψ)=∑x∈Xp(x)|x⟩⟨x|⊗idKB⊗Ts(V(x)⊗Φ) (s∈S)

Before we give a proof of the above proposition, we state a net approximation result which is used therein. We use the diamond norm defined on the set of maps from to by

 ∥N∥⋄:=maxa∈L(H⊗H)∥idH⊗N(a)∥1 (N∈L(H)). (9)

We will use

###### Lemma 14 (bjelakovic08 , Lemma 5.2).

Let . For each there is a set such that the following conditions are fulfilled

1. , and

2. for each exists such that .

###### Proof.

Fix , and set

We assume that and are both non-negative, otherwise the results follow either by trivial coding or reduction to a case of channel coding with a single sender. We define the state and a classical-quantum channel with outputs

 TA,s(x):=Ts(V(x)⊗πB) (x∈X) (10)

for each . If we fix the blocklength to be sufficiently large, we find by virtue of Proposition 8 a random message transmission code for the compound classical-quantum channel , where is an i.i.d. random family with generic distribution , rate and expected average message transmission error

 E ¯¯¯e(CA(U),T⊗nA,s)≤2−nc1 (11)

where is a strictly positive constant. Moreover we define for each a c.p.t.p. map by

 TB,s(τ):=∑x∈Xp(x)Ts(V(x)⊗τ)⊗|x⟩⟨x| (τ∈L(KB)).

Under the assumption of large enough blocklength , Proposition 9 assures us, that there exists a random entanglement transmission code for the compound quantum channel where is supported on a finite set , and which has rate

 1nlogM2 ≥ infs∈S I(B⟩CX,ωs)−δ = R2−δ,

such, that with a positive constant the expected entanglement fidelity can be bounded as

 EΛ [ infs∈S Fe(πF,˜DΛ∘T⊗nB,s∘EΛ)] ≥ 1−2−nc2. (12)

In addition, the expected density matrix resulting from the random encoding procedure is maximally mixed, i.e. For each pair of realizations of , we define an Scenario-II code using the decoding operations

 Dm,αu(x) := ˜Dα∘^Dm(u)(x) (x∈L(K⊗nC)).

with . Each of the codes defined above already is a Scenario-II code of suitable rates for classical message and entanglement transmission. To complete the proof of the proposition, we will lower-bound the the expected Scenario-II fidelity of the random code

. Fix, for the moment the channel state

. Let be the cq channel defined by the states

 T′A,sα(xn):=T⊗ns(V⊗n(xn)⊗Eα(πF)) (α∈A,xn∈Xn). (13)

Averaging over the random choice of the transmission statistics of is reproduced. Indeed, for each

 EΛT′A,sΛ(xn) = T⊗ns(V⊗n(xn)⊗EΛEα(πF)) = T⊗nA,s(xn). (14)

Since the average transmission error is an affine function of the cq channel, we have for each

 EUEΛ ¯¯¯e(C(u),T′A,sΛ) = EU¯¯¯e(C(u),T⊗nA,s)≤2−nc1. (15)

Define the cq channel defined by

 T′sα(xn):=id⊗nKB⊗T⊗ns(V⊗n(xn)⊗(id⊗nKB⊗Eα(Φ))) (xn∈Xn,α∈A) (16)

(note that the reduction of is, in fact, .) For each classical message , it holds

 EΛEU tr(id⊗nKB⊗^D(U)(T′sΛ(Um))) = EΛEU tr^Dm(U)T′A,sΛ(Um)≥1−2−nc1. (17)

The above inequality stems from the bound in (15) together with the observation, that by symmetry of the random selection procedure for the codewords, the expectation of the one-word message transmission error does not depend on the individual message . If we define

 γs(u,α) := 1−tr(1⊗nKB⊗^Dm(u))T′A,sα(um) (18)

we have by (18)

 EΛEU γs ≤2−nc1. (19)

Define for each realization of

 Γ(s)mm′(u,α) := id⊗nKB⊗^Dm(U)(T′A,sα(um)) (m,m′∈[M1]), (20) ^Γ(s)m(u,α) := M1∑m′=1Γ(s)mm′(u,α)⊗|um′⟩⟨um′|, and (21) Γ′(s)(xn,α) := id⊗nKB⊗T⊗ns(V⊗n(xn)⊗(Eα(Φ))⊗|xn⟩⟨xn| (xn∈Xn). (22)

Note, that if is an -valued random variable with distribution , that

 E~U Γ′(s)(~U,α) = (id⊗nKB⊗T⊗nB,s∘Eα)(Φ) (23)

holds by definition of . By the gentle measurement lemma (see Lemma 18 in Appendix A), we have

 ∥id⊗nKB⊗(id⊗nHC−^Dm(u))T′A,sα(um)∥1 ≤ 3√γs(u,α). (24)

Taking expectations on both sides of the inequality in (24), we arrive at

 EΛEU∥id⊗nKB⊗(id⊗nHC−^Dm(U))T′A,sα(Um)∥1 ≤ 3EΛEU√γs(u,α)≤3√EΛEUγs(u,α)≤3√2−nc1.

where the second inequality above is by Jensen’s inequality together with concavity of the square-root function. The last inequality is by the estimate in (

19). As a consequence of these bounds, we have

 EUEΛ ∥^Γ(s)m(U,Λ)−Γ′(s)(Um,Λ)∥1 ≤ 3⋅√2−nc1+2−nc1 ≤ 4⋅√2−nc1. (25)

for each message . Moreover, we can bound

 EΛEU F(Φ,id⊗nKB⊗˜DΛ(^Γ(s))) ≥ EΛEU (F(Φ,id⊗nKB⊗˜DΛ(Γ′(s)))− ∥id⊗nKB⊗˜D(Γ(s)(U,Λ)−Γ′(s)(Um,Λ))∥1) ≥EΛEU (Fe(Φ,~DΛ∘T⊗nB,s∘EΛ)− ∥Γ(s)(U,Λ)−Γ′(s)(Um,Λ)∥1) ≥EΛ Fe(Φ,~DΛ∘T⊗nB,s∘EΛ)−4⋅√2−nc1 ≥1−2−nc2−4⋅√2−nc1. (26)

The first line above is by application of Lemma 19 which can be found in Appendix A. The second is by using the equality in (23) together with monotonicity of the trace distance under taking partial traces. The third is by (25). The last estimate comes from (12). Putting all the estimates together, we can bound the expected Scenario-II fidelity. We have for each

 EUEΛ PII(C(U,Λ),T⊗ns,m) = F(|m⟩⟨m|⊗Φ,id⊗nKB⊗DU,Λ∘T⊗ns∘(id⊗nKA⊗EΛ)(V⊗n(Um)⊗Φ)) ≥1−∥|m⟩⟨m|−trF∘DUΛ∘T⊗ns(V⊗n(Um)⊗EΛ(π))∥1 −3⋅(1−F(Φ,id⊗nKB⊗trCM1∘DU,Λ∘T⊗ns∘(id⊗nKA⊗EΛ)(V⊗n(Um)⊗Φ)) ≥1−2−nc1+2⋅2−nc2+4√2−nc1+2−nc2. (27)

The first inequality is by Lemma 20 to be found in Appendix A. The second inequality is by … . The third inequality is by inserting the bounds from (15) and (26). Consequently, we have for each

 EPII(C(U,Λ),T⊗ns)≥1−2−