Zero-error Communication Research Articles

Covariant Quantum Combinatorics with Applications to Zero-Error Communication

We develop the theory of quantum (a.k.a. noncommutative) relations and quantum (a.k.a. noncommutative) graphs in the finite-dimensional covariant setting, where all systems (finite-dimensional C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras) carry an action of a compact quantum group G, and all channels (completely positive maps preserving a certain G-invariant functional) are covariant with respect to the G-actions. We motivate our definitions by applications to zero-error quantum communication theory with a symmetry constraint. Some key results are the following: (1) We give a necessary and sufficient condition for a covariant quantum relation to be the underlying relation of a covariant channel. (2) We show that every quantum confusability graph with a G-action (which we call a quantum G-graph) arises as the confusability graph of a covariant channel. (3) We show that a covariant channel is reversible precisely when its confusability G-graph is discrete. (4) When G is quasitriangular (this includes all compact groups), we show that covariant zero-error source-channel coding schemes are classified by covariant homomorphisms between confusability G-graphs.

Zero-Error Communication Over Adversarial MACs

We consider zero-error communication over a two-transmitter deterministic adversarial multiple access channel (MAC) governed by an adversary who has access to the transmissions of both senders (hence called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">omniscient</i> ) and aims to maliciously corrupt the communication. None of the encoders, jammer and decoder is allowed to randomize using private or public randomness. This enforces a combinatorial nature of the problem. Our model covers a large family of channels studied in the literature, including all deterministic discrete memoryless noisy or noiseless MACs. In this work, given an arbitrary two-transmitter deterministic omniscient adversarial MAC, we characterize when the capacity region 1) has nonempty interior (in particular, is two-dimensional); 2) consists of two line segments (in particular, has empty interior); 3) consists of one line segment (in particular, is one-dimensional); 4) or only contains (0, 0) (in particular, is zero-dimensional). This extends a recent result by Wang, Budkuley, Bogdanov and Jaggi (2019) from the point-to-point setting to the multiple access setting. Indeed, our converse arguments build upon their generalized Plotkin bound and involve delicate case analysis. One of the technical challenges is to take care of both “joint confusability” and “marginal confusability”. In particular, the treatment of marginal confusability does <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">not</i> follow from the point-to-point results by Wang et al. Our achievability results follow from random coding with expurgation.

Optimal measurement structures for contextuality applications

The Kochen-Specker (KS) theorem is a cornerstone result in the foundations of quantum mechanics describing the fundamental difference between quantum theory and classical non-contextual theories. Recently specific substructures termed 01-gadgets were shown to exist within KS proofs that capture the essential contradiction of the theorem. Here, we show these gadgets and their generalizations provide an optimal toolbox for contextuality applications including (i) constructing classical channels exhibiting entanglement-assisted advantage in zero-error communication, (ii) identifying large separations between quantum theory and binary generalized probabilistic theories, and (iii) finding optimal tests for contextuality-based semi-device-independent randomness generation. Furthermore, we introduce and study a generalization to definite prediction sets for more general logical propositions, that we term higher-order gadgets. We pinpoint the role these higher-order gadgets play in KS proofs by identifying these as induced subgraphs within KS graphs and showing how to construct proofs of state-independent contextuality using higher-order gadgets as building blocks. The constructions developed here may help in solving some of the remaining open problems regarding minimal proofs of the Kochen-Specker theorem.

Quantum Graphs as Quantum Relations

The “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We use this perspective to interpret the Knill–Laflamme error-correction conditions (Knill and Laflamme in Theory of quantum error-correcting codes. Phys Rev A 55:900-911, 1997) in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke’s noncommutative graph homomorphisms (Stahlke in Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans Inf Theory 62:554–577, 2016) and Duan, Severini, and Winter’s noncommutative bipartite graphs (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013), and to realize the noncommutative confusability graph associated to a quantum channel (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013) as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the “mixed” setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.

Zero-Error Coding via Classical and Quantum Channels in Sensor Networks.

Today’s sensor networks need robustness, security and efficiency with a high level of assurance. Error correction is an effective communicational technique that plays a critical role in maintaining robustness in informational transmission. The general way to tackle this problem is by using forward error correction (FEC) between two communication parties. However, by applying zero-error coding one can assure information fidelity while signals are transmitted in sensor networks. In this study, we investigate zero-error coding via both classical and quantum channels, which consist of n obfuscated symbols such as Shannon’s zero-error communication. As a contrast to the standard classical zero-error coding, which has a computational complexity of , a general approach is proposed herein to find zero-error codewords in the case of quantum channel. This method is based on a n-symbol obfuscation model and the matrix’s linear transformation, whose complexity dramatically decreases to . According to a comparison with classical zero-error coding, the quantum zero-error capacity of the proposed method has obvious advantages over its classical counterpart, as the zero-error capacity equals the rank of the quantum coefficient matrix. In particular, the channel capacity can reach n when the rank of coefficient matrix is full in the n-symbol multilateral obfuscation quantum channel, which cannot be reached in the classical case. Considering previous methods such as low density parity check code (LDPC), our work can provide a means of error-free communication through some typical channels. Especially in the quantum case, zero-error coding can reach both a high coding efficiency and large channel capacity, which can improve the robustness of communication in sensor networks.

Reliable and Secure Multishot Network Coding Using Linearized Reed-Solomon Codes

Multishot network coding is considered in a worst-case adversarial setting in which an omniscient adversary with unbounded computational resources may inject erroneous packets in up to $t$ links, erase up to $\rho$ packets, and wire-tap up to $\mu$ links, all throughout $\ell$ shots of a linearly-coded network. Assuming no knowledge of the underlying linear network code (in particular, the network topology and underlying linear code may be random and change with time), a coding scheme achieving zero-error communication and perfect secrecy is obtained based on linearized Reed-Solomon codes. The scheme achieves the maximum possible secret message size of $ \ell n^\prime - 2t - \rho - \mu $ packets for coherent communication, where $ n^\prime $ is the number of outgoing links at the source, for any packet length $ m \geq n^\prime $ (largest possible range). By lifting this construction, coding schemes for non-coherent communication are obtained with information rates close to optimal for practical instances. The required field size is $ q^m $, where $ q > \ell $, thus $ q^m \approx \ell^{n^\prime} $, which is always smaller than that of a Gabidulin code tailored for $ \ell $ shots, which would be at least $ 2^{\ell n^\prime} $. A Welch-Berlekamp sum-rank decoding algorithm for linearized Reed-Solomon codes is provided, having quadratic complexity in the total length $n = \ell n^\prime $, and which can be adapted to handle not only errors, but also erasures, wire-tap observations and non-coherent communication. Combined with the obtained field size, the given decoding complexity is of $ \mathcal{O}(n^{\prime 4} \ell^2 \log(\ell)^2) $ operations in $ \mathbb{F}_2 $.

A compositional approach to quantum functions

We introduce a notion of quantum function and develop a compositional framework for finite quantum set theory based on a 2-category of quantum sets and quantum functions. We use this framework to formulate a 2-categorical theory of quantum graphs, which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero-error communication and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups. We show that the 2-categories of quantum sets and quantum graphs are semisimple. We analyze dualisable and invertible 1-morphisms in these 2-categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs.

Two-Party Zero-Error Function Computation with Asymmetric Priors

We consider a two party network where each party wishes to compute a function of two correlated sources. Each source is observed by one of the parties. The true joint distribution of the sources is known to one party. The other party, on the other hand, assumes a distribution for which the set of source pairs that have a positive probability is only a subset of those that may appear in the true distribution. In that sense, this party has only partial information about the true distribution from which the sources are generated. We study the impact of this asymmetry on the worst-case message length for zero-error function computation, by identifying the conditions under which reconciling the missing information prior to communication is better than not reconciling it but instead using an interactive protocol that ensures zero-error communication without reconciliation. Accordingly, we provide upper and lower bounds on the minimum worst-case message length for the communication strategies with and without reconciliation. Through specializing the proposed model to certain distribution classes, we show that partially reconciling the true distribution by allowing a certain degree of ambiguity can perform better than the strategies with perfect reconciliation as well as strategies that do not start with an explicit reconciliation step. As such, our results demonstrate a tradeoff between the reconciliation and communication rates, and that the worst-case message length is a result of the interplay between the two factors.

On Zero-Error Communication via Quantum Channels in the Presence of Noiseless Feedback

We initiate the study of zero-error communication via quantum channels when the receiver and sender have at their disposal a noiseless feedback channel of unlimited quantum capacity, generalizing Shannon's zero-error communication theory with instantaneous feedback. We first show that this capacity is a function only of the linear span of Choi-Kraus operators of the channel, which generalizes the bipartite equivocation graph of a classical channel, and which we dub "non-commutative bipartite graph". Then we go on to show that the feedback-assisted capacity is non-zero (with constant activating noiseless communication) if and only if the non-commutative bipartite graph is non-trivial, and give a number of equivalent characterizations. This result involves a far-reaching extension of the "conclusive exclusion" of quantum states [Pusey/Barrett/Rudolph, Nature Phys. 8:475-478]. We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede. We demonstrate this bound to have many good properties, including being additive and given by a minimax formula. We also prove that this quantity is the entanglement-assisted capacity against an adversarially chosen channel from the set of all channels with the same Choi-Kraus span, which can also be interpreted as the feedback-assisted unambiguous capacity. The proof relies on a generalization of the "Postselection Lemma" [Christandl/Koenig/Renner, PRL 102:020504] that allows to reflect additional constraints, and which we believe to be of independent interest. We illustrate our ideas with a number of examples, including classical-quantum channels and Weyl diagonal channels, and close with an extensive discussion of open questions.

Certifying Equality With Limited Interaction

The equality problem is usually one's first encounter with communication complexity and is one of the most fundamental problems in the field. Although its deterministic and randomized communication complexity were settled decades ago, we find several new things to say about the problem by focusing on three subtle aspects. The first is to consider the expected communication cost (at a worst-case input) for a protocol that uses limited interaction--i.e., a bounded number of rounds of communication--and whose error probability is zero or close to it. The second is to treat the false negative error rate separately from the false positive error rate. The third is to consider the information cost of such protocols. We obtain asymptotically optimal rounds-versus-cost tradeoffs for equality: both expected communication complexity and information complexity scale as $$\Theta ({{\mathrm{ilog}}}^{r-1} n)$$ź(ilogr-1n), where r is the number of rounds and $${{\mathrm{ilog}}}^k n = \log \log \cdots \log n$$ilogkn=loglogźlogn, with k logs. These bounds hold even when the false negative rate approaches 1. For the case of zero-error communication cost, we obtain essentially matching bounds, up to a tiny additive constant. We also provide some applications. As an application of our information cost bounds, we obtain new bounded-round randomized lower bounds for the Intersection problem, in which there are two players who hold subsets $$S,T \subseteq [n]$$S,T⊆[n]. In many realistic scenarios, the sizes of S and T are significantly smaller than n, so we impose the constraint that $$|S|, |T| \le k$$|S|,|T|≤k. We study the minimum number of bits the parties need to communicate in order to compute the entire intersection set $$S \cap T$$SźT, using r rounds. We show that any r-round protocol has information cost (and thus communication cost) $$\Omega (k {{\mathrm{ilog}}}^r k)$$Ω(kilogrk) bits. We also give an O(r)-round protocol achieving $$O(k{{\mathrm{ilog}}}^r k)$$O(kilogrk) bits, which for $$r = \log ^* k$$r=logźk gives a protocol with O(k) bits of communication. This is in contrast to other basic problems such as computing the union or symmetric difference, for which $$\Omega (k\log (n/k))$$Ω(klog(n/k)) bits of communication is required for any number of rounds.

Direct Sum Fails for Zero-Error Average Communication

We show that in the model of zero-error communication complexity, direct sum fails for average communication complexity as well as for external information complexity. Our example also refutes a version of a conjecture by Braverman et al. that in the zero-error case amortized communication complexity equals external information complexity. In our examples the underlying distributions do not have full support. One interpretation of a distribution of non full support is as a promise given to the players (the players have a guarantee on their inputs). This brings up the issue of promise versus non-promise problems in this context.

Characterizing the Performance of xor Games and the Shannon Capacity of Graphs

In this Letter we give a set of necessary and sufficient conditions such that quantum players of a two-party XOR game cannot perform any better than classical players. With any such game, we associate a graph and examine its zero-error communication capacity. This allows us to specify a broad new class of graphs for which the Shannon capacity can be calculated. The conditions also enable the parametrization of new families of games that have no quantum advantage for arbitrary input probability distributions, up to certain symmetries. In the future, these might be used in information-theoretic studies on reproducing the set of quantum nonlocal correlations.

Zero-Error Capacity of a Class of Timing Channels

We analyze the problem of zero-error communication through timing channels that can be interpreted as discrete-time queues with bounded waiting times. The channel model includes the following assumptions: 1) Time is slotted, 2) at most $ N $ "particles" are sent in each time slot, 3) every particle is delayed in the channel for a number of slots chosen randomly from the set $ \{0, 1, \ldots, K\} $, and 4) the particles are identical. It is shown that the zero-error capacity of this channel is $ \log r $, where $ r $ is the unique positive real root of the polynomial $ x^{K+1} - x^{K} - N $. Capacity-achieving codes are explicitly constructed, and a linear-time decoding algorithm for these codes devised. In the particular case $ N = 1 $, $ K = 1 $, the capacity is equal to $ \log \phi $, where $ \phi = (1 + \sqrt{5}) / 2 $ is the golden ratio, and the constructed codes give another interpretation of the Fibonacci sequence.

Multiparty Zero-Error Classical Channel Coding With Entanglement

We study the effects of quantum entanglement on the performance of two classical zero-error communication tasks among multiple parties. Both tasks are generalizations of the two-party zero-error channel-coding problem, where a sender and a receiver want to perfectly communicate messages through a one-way classical noisy channel. If the two parties are allowed to share entanglement, there are several positive results that show the existence of channels for which they can communicate strictly more than what they could do with classical resources. In the first task, one sender wants to communicate a common message to multiple receivers. We show that if the number of receivers is greater than a certain threshold then entanglement does not allow for an improvement in the communication for any finite number of uses of the channel. On the other hand, when the number of receivers is fixed, we exhibit a class of channels for which entanglement gives an advantage. The second problem we consider features multiple collaborating senders and one receiver. Classically, cooperation among the senders might allow them to communicate on average more messages than the sum of their individual possibilities. We show that whenever a channel allows single-sender entanglement-assisted advantage, then the gain extends also to the multi-sender case. Furthermore, we show that entanglement allows for a peculiar amplification of information which cannot happen classically, for a fixed number of uses of a channel with multiple senders.

Entanglement can Increase Asymptotic Rates of Zero-Error Classical Communication over Classical Channels

It is known that the number of different classical messages which can be communicated with a single use of a classical channel with zero probability of decoding error can sometimes be increased by using entanglement shared between sender and receiver. It has been an open question to determine whether entanglement can ever increase the zero-error communication rates achievable in the limit of many channel uses. In this paper we show, by explicit examples, that entanglement can indeed increase asymptotic zero-error capacity, even to the extent that it is equal to the normal capacity of the channel.

Zero-Error Channel Capacity and Simulation Assisted by Non-Local Correlations

Shannon's theory of zero-error communication is re-examined in the broader setting of using one classical channel to simulate another exactly, and in the presence of various resources that are all classes of non-signalling correlations: Shared randomness, shared entanglement and arbitrary non-signalling correlations. Specifically, when the channel being simulated is noiseless, this reduces to the zero-error capacity of the channel, assisted by the various classes of non-signalling correlations. When the resource channel is noiseless, it results in the "reverse" problem of simulating a noisy channel exactly by a noiseless one, assisted by correlations. In both cases, 'one-shot' separations between the power of the different assisting correlations are exhibited. The most striking result of this kind is that entanglement can assist in zero-error communication, in stark contrast to the standard setting of communicaton with asymptotically vanishing error in which entanglement does not help at all. In the asymptotic case, shared randomness is shown to be just as powerful as arbitrary non-signalling correlations for noisy channel simulation, which is not true for the asymptotic zero-error capacities. For assistance by arbitrary non-signalling correlations, linear programming formulas for capacity and simulation are derived, the former being equal (for channels with non-zero unassisted capacity) to the feedback-assisted zero-error capacity originally derived by Shannon to upper bound the unassisted zero-error capacity. Finally, a kind of reversibility between non-signalling-assisted capacity and simulation is observed, mirroring the famous "reverse Shannon theorem".

Entanglement-assisted zero-error capacity is upper-bounded by the Lovászϑfunction

The zero-error capacity of a classical channel is expressed in terms of the independence number of some graph and its tensor powers. This quantity is hard to compute even for small graphs such as the cycle of length seven, so upper bounds such as the Lov\'asz theta function play an important role in zero-error communication. In this paper, we show that the Lov\'asz theta function is an upper bound on the zero-error capacity even in the presence of entanglement between the sender and receiver.

Universal Secure Network Coding via Rank-Metric Codes

The problem of securing a network coding communication system against an eavesdropper adversary is considered. The network implements linear network coding to deliver n packets from source to each receiver, and the adversary can eavesdrop on \mu arbitrarily chosen links. The objective is to provide reliable communication to all receivers, while guaranteeing that the source information remains information-theoretically secure from the adversary. A coding scheme is proposed that can achieve the maximum possible rate of n-\mu packets. The scheme, which is based on rank-metric codes, has the distinctive property of being universal: it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying network code. The only requirement of the scheme is that the packet length be at least n, which is shown to be strictly necessary for universal communication at the maximum rate. A further scenario is considered where the adversary is allowed not only to eavesdrop but also to inject up to t erroneous packets into the network, and the network may suffer from a rank deficiency of at most \rho. In this case, the proposed scheme can be extended to achieve the rate of n-\rho-2t-\mu packets. This rate is shown to be optimal under the assumption of zero-error communication.

Detection experiment of decision errors caused by quantum noise in an optical signal system

Decision errors caused by quantum noise in an optical signal system have been experimentally detected. Such an experiment is necessary in order to facilitate the development of a future advanced zero error optical communication system. The apparatus consisted of two low dark-count photoelectron multiplying tube detectors which were used to detect two separate polarization components (X, Y) of manually controlled static binary polarization shift keying (PoISK) signals. With this system, events of the output state (0, 0) (the error events caused by quantum noise) were observed. Undesired error events for zero-error communication were also observed. Such events are caused by external noise such as polarization beam splitter leaks or dark-count (current) noise. However, the number of these error events was reduced by the suppression of polarization beam splitter leak noise.

Repeated communication and Ramsey graphs

Studies the savings afforded by repeated use in two zero-error communication problems. 1. Channel coding: proving a correspondence between Ramsey numbers and independence numbers of normal graph powers, the authors show that some channels can communicate exponentially more bits in two uses than they can in one use, and that this is essentially the largest possible increase. Using probabilistic constructions of self-complementary Ramsey graphs, the authors show that similar results hold even when the number of transmissible bits is large relative to the channel's size. 2. Dual-source coding: using probabilistic colorings of directed line graphs, the authors show that there are dual sources where communicating one instance requires arbitrarily many bits, yet communicating many instances requires at most two bits per instance. For dual sources where the number of bits required for a single instance is comparable to the source's size, they employ probabilistic constructions of self-complementary Ramsey graphs that are also Cayley graphs to show that conveying two instances may require only a logarithmic number of additional bits over that needed to convey one instance. >