# The Simultaneous Identification Capacity of the Classical--Quantum Multiple Access Channel

Here we discuss message identification, a problem formalized by Rudolf Ahlswede and Gunter Dueck, over a classical-quantum multiple access channel with two classical senders and one quantum receiver. We show that the simultaneous identification capacity, a capacity defined by Peter Löber, of this multiple access channel is equal to its message transmission capacity region.

## Authors

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## 1 Introduction

In message transmission, the receiver of the message is interested to know what exactly the message received reads. This differs from message identification such that the receiver of the message is only looking to answer the question, “Is this received message the one I am interested in?” The sender is free to choose the message they wish to send, and it is up to the receiver to determine this single bit of information.

Over classical channels it was shown by Rudolf Ahlswede, Gunter Dueck, et al that there exists identification codes that are doubly exponential in size with respect to the blocklength. Further, for classical-quantum channels, the result was extended by Peter Löber to show that there exists simultaneous identification codes which also are doubly exponential in size with respect to blocklength. Löber includes a restriction on his codes such that they must be simultaneous, which we provide the definition for in a latter section, but it was shown by Andreas Winter and Rudolf Ahlswede that this restriction can be dropped and so the single logarithmic scaled transmission capacity of a discrete, memoryless classical-quantum channel is equal to its doubly logarithmic scaled identification capacity.

Here we will firstly consider the classical-quantum multiple access channel with two classical senders and one quantum receiver (CCQ). We begin by introducing the framework for working with classical-quantum channels and review definitions for the message transmission setting.

Next we provide theory for the transmission capacity region of the CCQ channel using codes under a maximal error error criterion. In the case of the single sender-single receiver channel, there is no difference in capacity between using codes under a maximal error error criterion or an average error criterion, but it is known in the classical case that these two capacity regions are not equal [5]. Because use of maximal error codes are made in the proof of achievability, we need that the capacity region is not empty when the average error capacity region is non-empty, and this is therefore proved.

In the following section, we define the models for identification over a CCQ channel. We provide definitions for a restricted version of the problem, namely, simultaneous identification, a concept by Peter Löber [2]. Because the channel outputs a quantum state, multiple measurements for identification on the state cannot be performed. The simultaneous identification model overcomes this by performing all identification measurements at one time with a single measurement. With this, we can define a simultaneous identification capacity region for the CCQ channel and further determine a quantity for this region.

The achievability proof follows the technique of Ahlswede and Dueck to prove the achievability of the classical single sender-single receiver channel, that is, adding a small amount of randomness to a transmission code to transform it into a random identification (ID) code. We show that there exists realizations of the random ID code that achieve the desired capacity.

For the converse, we follow the technique of Peter Löber [2] and Yosef Steinberg [3] which uses resolvability theory, formalized by Te Sun Han and Sergio Verdú [6]. With this converse, we conclude that the simultaneous identification capacity of the CCQ multiple access channel is indeed equal to the transmission capacity region for memoryless CCQ channels.

## 2 Review of Message Transmission

In this section, we review the definitions for classical-quantum channels and their codes. By or we refer to finite alphabets. The -fold product set of an alphabet is denoted as . The space of quantum states with respect to a particular Hilbert space is denoted as and the set of linear operators on is denoted as

. Here we only consider finite dimensional, complex Hilbert spaces. The set of probability distributions on another set

is denoted as . Quantum channels, usually denoted as , in this report are always completely positive and trace preserving maps.

###### Definition 2.1 (CQ channel).

A classical-quantum (CQ) channel is a family of quantum channels

 {Wk:Xk→S(H⊗k)}k∈N.

We say that is a discrete memoryless CQ (DM-CQ) channel generated by if for all , for all ,

 Wk(xk)=k⨂i=1W(xi).
###### Definition 2.2 (CCQ channel).

A CCQ channel is a family of quantum channels

 {Wk:Xk×Yk→S(H⊗k)}k∈N.

We say that is a DM-CCQ channel generated by if for all and all , for all ,

 Wk(xk,yk)=k⨂i=1W(xi,yi).
###### Notation 2.3.

If is a discrete memoryless channel generated by a channel , we refer to it simply as , that is, without the bold face. We make this distinction since some lemmas or theorems will hold in general, but some are proved with the assumption of a discrete memoryless property.

###### Definition 2.4 (Channel state).

For CCQ channel , probability distributions and , and Hilbert spaces and with respective orthonormal bases and , the channel state is defined as

When , that is, and , we write for notational simplicity . We also define CQ channel states defined similarly, where we denote and for single sender channel states in a similar way, with a single distribution parameter over one orthonormal basis.

###### Definition 2.5 ((k,M,N)-code).

For a CCQ channel , a -code for classical message transmission is the family where , , and forms a POVM.

For a -code , the average error of transmission is defined as

 ¯¯¯e(C,Wk)\coloneqq1−1MNM,N∑m=1n=1\tr(DmnWk(xm,yn)),

and the maximal,

 e(C,Wk)\coloneqqmaxm,n1−\tr(DmnWk(xm,yn)).
###### Definition 2.6 (Achievable rate pair).

For a CCQ channel , we say , , is an achievable rate pair if for all , there exists a such that for all there exists a -code such that,

 1klogM≥R1−δ,1klogN≥R2−δ,¯¯¯e(C,Wk)≤ϵ. (1)

The capacity region for is defined as

 C(W)\coloneqq{(R1,R2)∣(R1,R2) is % an achievable rate pair}.

We say a -code achieves the rate pair if (1) holds for all .

###### Lemma 2.7.

For a CCQ channel , a -code with can be used as a or -code or with , where and are the CQ channels generated by taking the average output over one sender.

###### Proof.

Let be a -code with . Consider the channel

 WkM :Xk→S(H⊗k) :xk↦1NN∑n=1Wk(xk,yn)

Define and the code .

 ¯¯¯e (WkM,CM) =1−1MM∑m=1\tr(sm(Dmn)WkM(xm)) =1−1MNN∑n=1M∑m=1\tr((N∑n′=1Dmn′)Wk(xm,yn)) ≤1−1MNN∑n=1M∑m=1\tr(DmnWk(xm,yn)) ≤ϵ

Thus is a -code that has average error bounded by over the channel . Analogous arguments can be made to construct a -code with bounded average error over a channel

 WkN :Yk→S(H⊗k) :yk↦1MN∑n=1Wk(xkm,yk).

## 3 Maximal Message Transmission Error Capacity Region

In this section we give a brief analysis of the capacity region defined using the maximal message transmission error figure of merit, rather than the average message transmission error. We refer to this type of merit as the “maximal-error”.

###### Definition 3.1 (Maximal-error achievable rate pair).

For a CCQ channel , we say , , is a max-error achievable rate pair if for all , there exists a such that for all there exists a -code such that,

 1klogM≥R1−δ,1klogN≥R2−δ,e(C,Wk)≤ϵ. (2)

The max-error capacity region for is defined as

 Cmax(W)\coloneqq{(R1,R2)∣(R1,R2) is a max% -error achievable rate pair}.

We say a -code “max-error achieves” the rate pair if (2) holds for all .

###### Lemma 3.2 (Maximal error capacity region is closed).

For CCQ channel , is closed.

###### Proof.

Let be a sequence in that converge to some . By the convergence of , there exists an large enough such that for ,

 R1i0 ≥R1−δ/2 R2i0 ≥R2−δ/2.

Since is a max-error achievable rate pair, there exists a such that for all , there is a -code such that for all , and

 1klogM ≥R1i0−δ/2≥R1−δ 1klogN ≥R2i0−δ/2≥R2−δ.

Since was chosen arbitrarily, also achieves under maximum error criterion, hence and therefore is closed. ∎

###### Lemma 3.3 (Maximal error capacity region is convex).

For CCQ channel , is convex.

###### Proof.

Let and be two pairs in and , then there exists a such that for all , -code satisfies , , , and a such that for all , -code satisfies , , . We make the choice and such that the rate calculations below are satisfied. We show that for all that is a max-error achievable rate pair. The strategy is to construct a new code that sends first a message from code with uses of the channel and then a message from with uses of the channel. The maximum error of such a code is bounded by,

 \tr(Dmn⊗D′m′n′Wk(xm,yn)⊗Wk′(xm′,yn′))=\tr(DmnWk(xm,yn))\tr(D′m′n′Wk′(xm′,yn′))≤ϵ.

Let , then

 1k+k′ log(M⋅M′) ≥1k+k′log(2k(R1−δ1)2k′(R′1−δ2)) =kk+k′(R1−δ1)+k′k+k′(R′1−δ2) =αR1+(1−α)R′1−α(δ1+δ2)+ϵe =αR1+(1−α)R′1−δ,

where and can be chosen such that is arbitrarily close to with the inaccuracy. We can also choose and such that . We can make the same arguments to show that . Therefore is achievable and therefore is convex. ∎

###### Lemma 3.4 (Non-empty maximal error capacity region).

For CCQ channel , if is non-empty, then is also non-empty.

###### Proof.

Assume and and so and . From an average error code that achieves the rate pairs , we construct two codes that max-error achieve and respectively. Since is convex and closed, by a time sharing argument, the interior will thus be non-empty.

By definition, since , there exists a such that for all , the code satisfies, for all , , , and . Using this code, we construct a code in the following way. Since , we can write,

 ϵ ≥1−1MNM,N∑m=1n=1\tr(DmnWk(xm,yn)) =1NN∑n=1(1−1MM∑m=1\tr(DmnWk(xm,yn))).

Therefore, there exists at least one index such that

 1−1M∑m\tr(Dmn0Wk(xm,yn0))≤ϵ.

With this , we construct the code which achieves (under average error) . Note, the decoders no longer form a POVM, but can be modified to form a POVM via an expurgation as in [8, Ch. 16.5] with negligible effects for large . We transform this code to a max-error achieve . Assume without loss of generality that the codewords are ordered such that is non-decreasing. For , let . It holds,

 ϵ ≥1−1MM∑m=1\tr(Dmn0Wk(xm,yn0)) ≥1⌈2k(R1−δ)⌉M′−1∑m=11−\tr(Dmn0Wk(xm,yn0))≥0+1⌈2k(R1−δ)⌉⌈2k(R1−δ)⌉∑m=M′1−\tr(Dmn0Wk(xm,yn0))≥λ ≥1⌈2k(R1−δ)⌉(⌈2k/2(R1−δ)⌉)λ =λ/2.

Thus , as , and achieves maximal error rate . We can make analogous steps to find another code that achieves maximal error rate . By the convexity of , we have that, for all is an achievable maximal error rate and thus the interior of is not empty. ∎

## 4 Message Identification

In this section, we model message identification over a CCQ multiple access channel.

###### Definition 4.1 (Randomized (k,M)-ID-code).

For a CQ channel , a randomized -ID-code is a family , where are probability distributions, and for each , with .

For an -ID-code and CQ channel , we define two types of errors,

 e1(Cid,Wk) \coloneqqmaxm∈[M]1−∑xk∈XkPm(xk)\tr(ImWk(xk)), e2(Cid,Wk) \coloneqqmaxm,n∈[M]m≠n∑xk∈XkPm(xk)\tr(InWk(xk)).
###### Definition 4.2 ((k,M,N)-ID-code).

For a a CCQ channel , a -ID-code for classical message identification is the family where , , and such that for all and .

For a -ID-code , we define two types of errors,

 e1(Cid,Wk)\coloneqq maxm∈[M]n∈[N]1−∑xk∈Xkyk∈YkPm(xk)Qn(yk)\tr(ImnWk(xk,yk)), e2(Cid,Wk)\coloneqq maxm,m′∈[M],n,n′∈[N](m,n)≠(m′,n′)∑xk∈Xkyk∈YkPm(xk)Qn(yk)\tr(Im′n′Wk(xk,yk)).
###### Definition 4.3 (Simultaneous (k,M,N)-ID-code).

A -ID-code is called simultaneous if for there exists a POVM with subsets and such that for each and ,

 Imn=∑i∈Am∑j∈BnEij.
###### Definition 4.4 (Achievable ID-rate pair).

For a CCQ channel , we say , , is an achievable ID-rate pair if for all , there exists a such that for all , there is a -ID-code with

 1kloglogM≥R1−δ,1kloglogN≥R2−δ,e1(Cid,Wk)≤ϵ1,e2(Cid,Wk)≤ϵ2.

The ID capacity region of is defined as

 Cid (W)\coloneqq {(R1,R2)∣(R1,R2) is an achievable ID-rate % pair}.
###### Definition 4.5 (Achievable simultaneous ID-rate pair).

For a CCQ channel , we say , , we say is an achievable simultaneous ID-rate pair if for , there exists a such that for all there is a simultaneous -ID-code with

 1kloglogM≥R1−δ,1kloglogN≥R2−δ,e1(Csimid,Wk)≤ϵ1,e2(Csimid,Wk)≤ϵ2.

The simultaneous ID capacity region for a CCQ channel is defined as

 Csimid(W)\coloneqq {(R1,R2)∣(R1,R2) is an achievable sim. ID-% rate pair}.
###### Remark 4.6.

Since the simultaneous case is more restrictive, it is clear that

 Csimid(W)⊆Cid(W).
###### Theorem 4.7.

For a DM-CCQ channel generated by ,

 C(W)=Csimid(W)

## 5 Proof of Achievability

Here we give the proof of achievable for Theorem (4.7), and in the next section we give the converse proof. The proof here follows the approach taken by Ahlswede and Dueck in [7].

###### Theorem 5.1.

For a DM-CCQ channel generated by ,

 C(W)⊆Csimid(W).
###### Lemma 5.2 (Chernov-Höffding Bound).

Let

, and let a sequence of random variables

be such that for each , . Assume for each and the expectation value . Then it holds,

 Pr(M∑j=1ψj>λM)≤2−M⋅D(λ∥μ).

where is the relative entropy between the probability distributions and .

###### Lemma 5.3 (Transformator lemma for a DM-CCQ channel).

For the DM-CCQ channel generated by with an achievable rate , there is a simultaneous -ID-code that achieves the simultaneous ID-rate .

###### Proof.

Assume for DM-CCQ generated by that , otherwise the problem is reduced to the single sender case, a problem solved in [2]. For a rate pair , by definition, there exists a such that for all , there is a -code that achieves with , where as . Further, since int is non-empty, by Lemma 3.4, there is a non-trivial achievable maximal error rate pair and thus a such that for , there is a -code , and , with , with as , where the assumption that is made. We define and two families of maps

 A \coloneqq(Ai:[M′]→[M′′])Mi=1 B \coloneqq(Bj:[N′]→[N′′])Nj=1.

With these families of maps, we define an -ID-code where

 Pi(xm)\coloneqq{1M′if % ∃a∈[M′]:xm=u′a⋅u′′Ai(a)0otherwise,

and

 Qj(ym)\coloneqq{1N′if % ∃b∈[N′]:ym=v′b⋅v′′Bj(b)0otherwise.

The identifiers are defined as

 Iij\coloneqqM′∑a=1N′∑b=1D′ab⊗D′′Ai(a)Bj(b).

We show that with this structure, there exists a random construction of and such that there is a simultaneous ID-code which achieves the simultaneous ID-rate pair .

For and define the random variables such that

 Pr(Ua=u′a⋅u′′c)=1M′′

with , and such that

 Pr(Vb=v′b⋅v′′d)=1N′′

with . For , define

 ¯¯¯¯Ui \coloneqq{Ua}M′a=1 ¯¯¯¯Vj \coloneqq{Vb}N′b=1,

Let and

be the uniform distributions on

and respectively. Define the random identifier

 I(¯¯¯¯Ui,¯¯¯¯Vj)\coloneqqM′∑a=1N′∑b=1D(Ua,Vb)

where