Computing Sum of Sources over a Classical-Quantum MAC

We consider the problem of communicating a general bivariate function of two classical sources observed at the encoders of a classical-quantum multiple access channel. Building on the techniques developed for the case of a classical channel, we propose and analyze a coding scheme based on coset codes. The proposed technique enables the decoder recover the desired function without recovering the sources themselves. We derive a new set of sufficient conditions that are weaker than the current known for identified examples. This work is based on a new ensemble of coset codes that are proven to achieve the capacity of a classical-quantum point-to-point channel.

Authors

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02/21/2022

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I Introduction

Early research in quantum state discrimination led to the investigation of the information carrying capacity of quantum states. Suppose Alice - a sender - can prepare any one of the states in the collection and Bob - the receiver - has to rely on a measurement to infer the label of the state, then what is the largest sub-collection of states that Bob can distinguish perfectly? Studying this question in a Shannon-theoretic sense, Schumacher, Westmoreland [199707PhyRev_SchWes] and Holevo [199801TIT_Hol] characterized the exponential growth of this sub-collection, thereby characterizing the capacity of a classical-quantum (CQ) point-to-point (PTP) channel. In the following years, generalizations of this question with multiple senders and/or receivers have been studied with an aim of characterizing the corresponding information carrying capacity of quantum states in network scenarios [winter2001capacity].

In this work, we consider the problem of computing functions of information sources over a CQ multiple access channel (MAC). Let model a CQ-MAC. Sender - the party having access to the choice of label - observes a classical information stream . The pairs

are independent and identically distributed (IID) with a single-letter joint distribution

. The receiver, who is provided with the prepared quantum state, intends to reconstruct a specific function of the information observed by the senders. The question of interest is under what conditions, specified in terms of the CQ-MAC, and , can the receiver reconstruct the desired function losslessly?

The conventional approach to characterizing sufficient conditions for this problem relies on enabling the receiver reconstruct the pair of classical source sequences. Since the receiver is only interested in recovering the bivariate function , and not the pair, this approach can be strictly sub-optimal. Can we exploit this and design a more efficient communication strategy, thereby weakening the set of sufficient conditions? In this work, we present one such communication strategy for a general CQ-MAC that is more efficient than the conventional approach. This strategy is based on asymptotically good random nested coset codes. We analyze its performance and derive new sufficient conditions for a general problem instance and identify examples for which the derived conditions are strictly weaker.

Our findings here are built on the ideas developed in the classical setting. Focusing on a source coding formulation, i.e. a noiseless MAC, Körner and Marton [197903TIT_KorMar] devised an ingenious coding technique that enabled the receiver recover the sum of the sources without recovering either source. In [200710TIT_NazGas], the linearity of the Körner-Marton (KM) source coding map was further exploited to enable the receiver recover the sum of the sources using only the sum of the KM indices, not even requiring the pair. Leveraging this observation and focusing on the subclass of additive MACs, specific MAC channel coding techniques are devised in [200710TIT_NazGas] that enabled the receiver recover the sum of two channel coding message indices.

The techniques of [197903TIT_KorMar], [200710TIT_NazGas] are instances of a broader framework of coding strategies. Decoding functions of sources or channel inputs efficiently require codes endowed with algebraic closure properties. To emphasize, the conventional approach of deriving inner bounds/achievable rate region by analyzing expected performance of IID random codes is incapable of yielding performance limits - capacity or rate-distortion regions as the case may be- in network communication scenarios. To improve upon this, it is necessary to analyze the expected performance of random codes endowed with algebraic closure properties. In a series of works [2020Bk_PraPadShi], an information theoretic study of the latter codes has been carried out yielding new inner bounds for multiple network communication scenarios.

In this work, we embark on developing these ideas in the CQ setup. After having provided the problem statement in Sec. II, we focus on a simplified CQ MAC and illustrate the core idea of our coding scheme. The latter relies on developing a nested coset code (NCC) based communication scheme for a CQ PTP channel and analyzing its performance (Sec. IV). Leveraging this building block, we design and analyze the performance of an NCC-based coding scheme for computing sum over a general CQ-MAC (Sec. VI). Going further we generalize this idea for computing arbitrary functions over a general CQ-MAC.

Ii Preliminaries and Problem Statement

We supplement the notation in [2013Bk_Wil] with the following. For positive integer , . For a Hilbert space , and denote the collection of linear, positive and density operators acting on , respectively. The von Neumann entropy of a density operator is denoted by . Given any ensemble , the Holevo information [holevo] is denoted as . A POVM acting on is a collection of positive operators that form a resolution of the identity: , where is a finite set. We employ an underline notation to aggregate objects of similar type. For example, denotes , denotes , denotes the Cartesian product .

Consider a (generic) CQ-MAC specified through (i) finite sets , (ii) Hilbert space , and (iii) a collection of density operators. This CQ-MAC is employed to enable the receiver reconstruct a bivariate function of the classical information streams observed by the senders. Let be finite sets and distributed with PMF models the pair of information sources observed at the encoders. Specifically, sender observes the sequence and the sequence are IID with single-letter PMF . The receiver aims to recover the sequence losslessly, where is a specified function.

A CQ-MAC code of block-length for recovering consists of two encoders maps , and a POVM

. The average error probability of the CQ-MAC code

is

 ¯¯¯ξ(cf) = 1−∑s–n:f(s–n)=rnWnS1S2(sn1,sn2)\tr(λrnρ⊗nc,s–n)

where , where for .

A function of the sources is said to be reconstructible over a CQ-MAC if for , a sequence such that .

In this article, we are concerned with the problem of characterizing sufficient conditions under which a function of the sources is reconstructible over a generic MAC . One of our findings - Proposition 2 - provides a characterization of sufficient conditions in terms of a computable function of the associated objects- density operators that characterize the CQ-MAC, function and the source distribution .

As we shall see, the specific problem of computing sum of sources will play an important role in our work. In this case, is a finite field with elements and the receiver aims to reconstruct where denotes addition in . A CQ-MAC code of block-length for recovering the sum consists of two encoders maps , and a POVM .

Restricting to a sum, we say the sum of sources over field is reconstructible over a CQ-MAC if and the function is reconstructible over the CQ-MAC. The problem of characterizing sufficient conditions under which a sum of sources is reconstructible over a CQ-MAC plays an important role in this work. One of our findings - Theorem 2 - provides a computable characterization of a set of sufficient conditions under which a sum of sources is reconstructible over a CQ-MAC. As the reader will note, this encapsulates the central element of our characterization in Proposition 2.

We also formalize the notions of a CQ-PTP and CQ-MAC codes for communicating uniform messages. A CQ-MAC code for a CQ-MAC consists of (i) index sets , (ii) encoder maps and a decoding POVM . For , we let where for .

A CQ-PTP code for a CQ-PTP consists of (i) an index set , (ii) and encoder map and a decoding POVM . For , we let where .

Iii The Central Idea

Let us consider the specific problem of reconstructing the sum of sources each taking values in . We begin by reviewing the KM coding scheme for the case of a noiseless classical MAC. It was shown in [197903TIT_KorMar] the existence of linear code with a parity matrix and decoder map such that , for any , and sufficiently large , so long as . This implies that a receiver equipped with the decoding map can recover the sum if it possesses the sum of the Körner-Marton indices .

We are therefore led to building an efficient CQ-MAC coding scheme that enables the receiver only reconstruct the sum of the two message indices. Indeed, if the two senders send the KM indices to such a CQ-MAC channel code and the receiver employs the above source decoder on the decoded sum of the KM indices, it can recover the sum of sources. To illustrate the design of the desired CQ-MAC channel code, let us consider a CQ-MAC wherein and the collection satisfies whenever . Consider a CQ-PTP where for any satisfying . Suppose we are able to communicate over this CQ-PTP via a linear CQ-PTP code . Specifically, suppose there exists a generator matrix and a POVM so that . for any and sufficiently large , where where . We can then use this linear CQ-PTP code as our desired CQ-MAC channel code. Indeed, observe that, suppose both senders employ this same linear CQ-PTP code, then sender maps its KM index to the channel codeword . Observe that the structure of the CQ-MAC implies . If the receiver employs the POVM designed for the CQ-PTP, it ends up decoding the sum of the KM indices , and consequently, recover the sum of the sources.

A careful analysis of the above idea reveals that two MAC channel codes employed by the encoders do not ‘blow up’ when added, is crucial to the efficiency of the above scheme. A linear code being algebraically closed enables this. However, the codewords of a random linear code are uniformly distributed and cannot achieve the capacity of an arbitrary classical PTP channel, let alone a CQ-PTP channel. We are therefore forced to enlarge a linear code to identify sufficiently many codewords of the desired empirical distribution. We are thus led to a

nested coset code (NCC)[201301arXivComputation_PadPra]. A NCC comprises of cosets of a coarse linear code within a fine code. Within each coset, we can identify a codeword of the desired empirical distribution. We choose as many cosets as the number of messages. Analogous to our illustration above where we chose a linear code that achieves the capacity of the CQ-PTP , our first step (Sec. IV) is to design a NCC with its POVM that can achieve capacity of an arbitrary CQ PTP. Our second step is to endow both senders with this same NCC and analyze decoding the sum of the messages. This gets us to our next challenge - How do we analyze decoding their message sum, for a general CQ-MAC for which does not necessarily imply . In Sec. V, we address this challenge, leverage our findings in Sec. IV and generalize the idea for a general CQ-MAC.

Iv Nested Coset Codes Achieve Capacity of CQ-PTP

We begin by formalizing the structure of an NCC.

Definition 1.

An NCC built over a finite field comprises of (i) generator matrices ,

(ii) a bias vector

, an encoder map . We let denote elements in the range space of the generator matrix .

Definition 2.

A CQ-PTP code is an NCC CQ-PTP if there exists an NCC such that for all .

Theorem 1.

Given a CQ-PTP and a PMF on , there exists a CQ-PTP code such that (i) , (ii) is a NCC CQ-PTP, (iii) and for all sufficiently large.

Proof.

In order to achieve a rate , the standard approach is to pick codewords uniformly and independently from . However, the resulting code is not algebraically closed. On the other hand, if we pick a random generator matrix , with , whose entries from are IID uniform, then its range space - the resulting collection of codewords - are uniformly distributed and pairwise independent but not typical.

To satisfy the dual requirements of algebraically closure and typicality, we observe the following. If a collection of codewords are uniformly distributed in and pairwise independent, as we found the range space of to be, then the expected number of codewords that are typical is . This indicates that if we pick a generator matrix with entries uniformly distributed and IID, such that , then its range space will contain codewords that are -typical. The latter codewords can be used for communication.

Each coset of where will play an analogous role as a single codeword in a conventional IID random code. Just as we pick of the latter, we consider cosets of within a larger linear code with generator matrix with . The messages index the cosets of . A predetermined element in each coset that is typical is the assigned codeword for the message and chosen for communication.111The reader is encouraged to relate to the bounds stated in theorem statement and induced bounds on the rate of communication . A formal proof we provide below has two parts - error probability analysis for a generic fixed code followed by an upper bound on the latter via code randomization.

Upper bound on Error Prob. for a generic fixed code : Consider a generic NCC with its range space . We shall use this and define a CQ-PTP code that is an NCC CQ-PTP. Towards that end, let and

 s(m)\ensurestackMath\stackon[1pt]=Δ{{a∈VK:vn(a,m)∈Tnδ(pV)}if θ(m)≥1{0k}if θ(m)=0,

for each . For , a predetermined element is chosen. On receiving message , the encoder prepares the quantum state and is communicated. The encoding map is therefore determined via the collection .

Towards specifying the decoding POVM let be a spectral decomposition for . We let . For any , let be the conditional typical projector as in [2013Bk_Wil, Defn. 15.2.4] with respect to the ensemble and distribution . Similarly, let be the (unconditional) typical projector of the state as defined in [2013Bk_Wil, Defn. 15.1.3]. For , we let . We let , where

 λa,m\ensurestackMath\stackon[1pt]=Δ(∑^a∈Vk∑^m∈Vlγ^a,^m)−1/2γa,m(∑~a∈Vk∑~m∈Vlγ~a,~m)−1/2, (1)

and . Since , we have . The latter lower bound implies . The same lower bound coupled with the definition of the generalized inverse implies . We thus have . It can now be verified that is a POVM. In essence, the elements of this POVM is identical to the standard POVMs except the POVM elements corresponding to a coset have been added together. Indeed, since each coset corresponds to one message, there is no need to disambiguate within the coset.

We have thus associated an NCC and a collection with a CQ-PTP code. The error probability of this code is

 q−l∑m∈Itr((I−∑a∈Vkλa,m)ρ⊗nm)≤q−l∑m∈Itr((I−λam,m)ρ⊗nm). (2)

Denoting event , its complement and the associated indicator functions respectively, a generic term in the RHS of the above sum satisfies

 tr((I−λam,m)ρ⊗nm)1Ec+tr((I−λam,m)ρ⊗nm)1E≤1Ec+3∑i=1T2i,

where

 T21=2\tr((I−γam,m)ρ⊗nm)1E,T22=4∑^a≠am\tr(γ^a,mρ⊗nm)1E,andT23=4∑^m≠m∑~a\tr(γ~a,^mρ⊗nm)1E,

where we have used Hayashi-Nagaoka inequality [hayashi2003general].

Distribution of the Random Code : The objects and the collection specify an NCC CQ-PTP code unambiguously. A distribution for a random code is therefore specified through a distribution of these objects. We let upper case letters denote the associated random objects, and obtain

 P(GI=gI,GO/I=gO/IBn=bn,Am=am:m∈S(m))=q−(k+l+1)n∏m∈Vl1Θ(m),

and analyze the expectation of and the terms in regards to the above random code. We begin by For this, we provide the following proposition.

Proposition 1.

There exist such that for all sufficiently small and sufficiently large , we have , if , where as .

Proof.

The proof follows from Appendix B of [MACwithStates_ArunPad_SandeepPra] with the identification of . ∎

We now consider . Deriving an upper bound on is by deriving a lower bound . This follows by an argument that is colloquially referred to as ‘pinching’. Lemma 2 in Appendix A proves the existence of such that for sufficiently large . We now analyze . Denoting the event

 J\ensurestackMath\stackon[1pt]=Δ{Θ(m)≥1,Vn(^a,^m)=^xnAm=d,Vn(d,m)=xn}⊆K\ensurestackMath\stackon[1pt]=Δ{Vn(^a,^m)=^xnVn(d,m)=xn} (3)

we perform the following steps.

 EP[T22]=∑^a∈VkEP[tr(Γ^a,mρ⊗nm)1{θ(m)≥1}1{^a≠Am}] =∑d∈Vk∑^a∈Vk∑xn∈Tnδ(pV)∑^xn∈VnE[tr(Γ^a,mρ⊗nm)1^a≠d1J] =∑d∈Vk∑^a≠d∑xn∈Tnδ(pV)∑^xn∈VnE[% tr(Γ^a,mρ⊗nm)1J]

where the restriction of the summation to is valid since forces the choice such that . Going further, we have

 EP[T22] =∑d,^a∈Vk^a≠d∑xn∈Tnδ(pV)∑^xn∈Tnδ(pV)E[tr(πρπ^xnπρρ⊗nxn)1J] =∑d,^a:^a≠d∑xn∈Tnδ(pV)∑^xn∈Tnδ(pV)tr(πρπ^xnπρρ⊗nxn)P(J) (a)≤∑d,^a:^a≠d∑^xn∈Tnδ(pV)tr(π^xnπρ)P(J)2−n[S(ρ)−H(pV)+ϵV] (b)≤∑d,^a:^a≠d∑^xn∈Tnδ(pV)tr(π^xnπρ)P(K)2−n[S(ρ)−H(pV)+ϵV] (c)=∑d,^a:^a≠d∑^xn∈Tnδ(pV)tr(π^xnπρ)1q2n2−n[S(ρ)−H(pV)+ϵV] (d)≤2−n[χ({pV;ρv})+ϵV−2H(pV)−2knlogq+2logq], (4)

where the restriction of the summation to follows from the fact that is the zero projector if , (a) follows from the operator inequality found in [2017BkOnline_Wil, Eqn. 20.34, 15.20], (b) follows from Def. 3, (c) follows from pairwise independence of the distinct codewords, and (d) follows from and [2017BkOnline_Wil, Eqn. 15.77] and as . We now derive an upper bound on . We have

 EP[T23] =∑d,^a∈Vk∑^m≠m∑xn,^xn∈Tnδ(pV)E[tr(πρΠ^a,^mπρρ⊗nAm,m)1J] =∑d,^a∈Vk∑^m≠m∑xn,^xn∈Tnδ(pV)tr(π^xnπρρ⊗nxnπρ)P(J) ≤∑d,^a∈Vk∑^m≠m∑^xn∈Tnδ(pV)tr(π^xnπρ)P(J)2−n[S(ρ)−H(pV)+ϵV] ≤∑d,^a∈Vk∑