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IEEE International Symposium on Information Theory (ISIT)

Beating the Sum-Rate Capacity of the Binary Adder Channel with Non-Signaling Correlations

Omar Fawzi, Paul Fermé

Published Jun 26, 2022• 10 citations
Publisher

Abstract

We address the problem of coding for multiple-access channels (MACs) with the assistance of non-signaling correlations between parties. It is well-known that non-signaling assistance does not change the capacity of point-to-point channels. However, it was recently observed that one can construct MACs from two-player non-local games while relating the winning probability of the game to the capacity of the MAC. By considering games for which entanglement (a special kind of non-signaling correlation) increases the winning probability (e.g., the Magic Square game), this shows that for some specific kinds of channels, entanglement between the senders can increase the capacity.Here, we show that the increase in capacity from non-signaling assistance goes beyond such special channels and applies even to a simple deterministic MAC: the binary adder channel. In particular, we show that, with non-signaling assistance, a sum-rate of $\frac{{{{\log }_2}(72)}}{4} \simeq 1.5425$ can be reached with zero error, which beats the maximum classical sum-rate capacity of $\frac{3}{2}$.In order to achieve this, we show that efficient linear programs can be formulated to compute the success probability of the best non-signaling assisted code for a finite number of copies of a multiple-access channel. In particular, this can be used to give lower bounds on the zero-error non-signaling assisted capacity of multiple-access channels.

DOI
10.1109/isit50566.2022.9834699

Related papers

Multiple-Access Channel Coding With Non-Signaling Correlations

Omar Fawzi, Paul Fermé

We address the problem of coding for classical multiple-access channels (MACs) with the assistance of non-signaling correlations between parties. It is well-known that non-signaling assistance does not change the capacity of classical point-to-point channels. However, it was recently observed that one can construct MACs from two-player nonlocal games while relating the winning probability of the game to the capacity of the MAC. By considering games for which entanglement (a special kind of non-signaling correlation) increases the winning probability (e.g., the Magic Square game), this shows that for some specific kinds of channels, entanglement between the senders can increase the capacity. In this work, we make several contributions towards understanding the capacity region for MACs with the assistance of non-signaling correlations between the parties. We develop a linear program computing the optimal success probability for coding over$n$copies of a MAC$W$with size growing polynomially in$n$. Solving this linear program allows us to achieve inner bounds for MACs. Applying this method to the binary adder channel, we show that using non-signaling assistance, the sum-rate$\frac {\log _{2}(72)}{4} \simeq 1.5425$can be reached even with zero error, which beats the maximum sum-rate capacity of 1.5 in the unassisted case. For noisy channels, where the zero-error non-signaling assisted capacity region is trivial, we can use concatenated codes to obtain achievable points in the capacity region. Applied to a noisy version of the binary adder channel, we show that non-signaling assistance still improves the sum-rate capacity. Complementing these achievability results, we give an outer bound on the non-signaling assisted capacity region that has the same expression as the unassisted region except that the channel inputs are not required to be independent. Finally, we show that the capacity region with non-signaling assistance shared only between each sender and the receiver independently is the same as without assistance.

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On the Separation of Correlation-Assisted Sum Capacities of Multiple Access Channels

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The capacity of a channel characterizes the maximum rate at which information can be transmitted through the channel asymptotically faithfully. For a channel with multiple senders and a single receiver, computing its sum capacity is possible in theory, but challenging in practice because of the nonconvex optimization involved. To address this challenge, we investigate three topics in our study. In the first part, we study the sum capacity of a family of multiple access channels (MACs) obtained from nonlocal games. For any MAC in this family, we obtain an upper bound on the sum rate that depends only on the properties of the game when allowing assistance from an arbitrary set of correlations between the senders. This approach can be used to prove separations between sum capacities when the senders are allowed to share different sets of correlations, such as classical, quantum or no-signalling correlations. We also construct a specific nonlocal game to show that the approach of bounding the sum capacity by relaxing the nonconvex optimization can give arbitrarily loose bounds. Owing to this result, in the second part, we study algorithms for non-convex optimization of a class of functions we call Lipschitz-like functions. This class includes entropic quantities, and hence these results may be of independent interest in information theory. Subsequently, in the third part, we show that one can use these techniques to compute the sum capacity of an arbitrary two-sender MACs to a fixed additive precision in quasi-polynomial time. We showcase our method by efficiently computing the sum capacity of a family of two-sender MACs for which one of the input alphabets has size two. Furthermore, we demonstrate with an example that our algorithm may compute the sum capacity to a higher precision than using the convex relaxation.

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On the separation of correlation-assisted sum capacities of multiple access channels

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Computing the sum capacity of a multiple access channel (MAC) is a non-convex optimization problem. It is therefore common to compute an upper bound on the sum capacity using a convex relaxation. We investigate the performance of such a relaxation by considering a family of MACs obtained from nonlocal games. First, we derive an analytical upper bound on the sum capacity of such MACs, while allowing the senders to share any given set of correlations. Our upper bound depends only on the properties of the game available in practice, thereby providing a way to obtain separations between the sum capacity assisted by different sets of correlations. In particular, we obtain a bound on the sum capacity of the MAC obtained from the magic square game that is tighter than the previously known result. Next, we introduce a game for which the convex relaxation of the sum capacity can be arbitrarily loose, demonstrating the need to find other techniques to compute or bound the sum capacity. We subsequently propose an algorithm that can certifiably compute the sum capacity of any two-sender MAC to a given precision.

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The Multiple-Access Channel with Entangled Transmitters

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Communication over a classical multiple-access channel (MAC) with entanglement resources is considered, whereby two transmitters share entanglement resources a priori before communication begins. Leditzky et al. (2020) presented an example of a classical MAC, defined in terms of a pseudo telepathy game, such that the sum rate with entangled transmitters is strictly higher than the best achievable sum rate without such resources. Here, we establish inner and outer bounds on the capacity region for the general MAC with entangled transmitters, and show that the previous result can be obtained as a special case. It has long been known that the capacity region of the classical MAC under a message-average error criterion can be strictly larger than with a maximal error criterion (Dueck, 1978). We observe that given entanglement resources, the regions coincide. Furthermore, we address the combined setting of entanglement resources and conferencing, where the transmitters can also communicate with each other over rate-limited links. Using superdense coding, entanglement can double the conferencing rate.

arXiv:2303.10456•Mar 18, 2023•1 citation

The Capacity Region of the Multiple Access Channel with Non-Signaling Assistance

Yuhang Yao, Syed A. Jafar

The capacity region of the $K$-sender discrete memoryless multiple access channel (MAC) is fully characterized when non-signaling (NS) assistance is available to all $K$ transmitters and the receiver. It is shown to have the same form as the classical capacity region of the MAC, except that the input distribution is allowed to be arbitrarily dependent across the senders. In particular, the NS-assisted capacity region matches the natural generalization to $K$ senders of an outer bound that was previously established by Fawzi and Fermé for $K=2$ senders. Additionally, we provide examples of $K$-sender MACs where the multiplicative gain in capacity from NS-assistance is arbitrarily close to $K$. Combined with an upper bound from prior work, this establishes $K$ as the extremal value of the multiplicative gain from NS-assistance across all $K$-sender MAC settings.

cs.ITCross-listedarXiv:2608.13860•Aug 17, 2026•0 citations
We address the problem of coding for multiple-access channels (MACs) with the assistance of non-signaling correlations between parties. It is well-known that non-signaling assistance does not change the capacity of point-to-point channels. However, it was recently observed that one can construct MACs from two-player non-local games while relating the winning probability of the game to the capacity of the MAC. By considering games for which entanglement (a special kind of non-signaling correlation) increases the winning probability (e.g., the Magic Square game), this shows that for some specific kinds of channels, entanglement between the senders can increase the capacity.Here, we show that the increase in capacity from non-signaling assistance goes beyond such special channels and applies even to a simple deterministic MAC: the binary adder channel. In particular, we show that, with non-signaling assistance, a sum-rate of log⁡2(72)4≃1.5425\frac{{{{\log }_2}(72)}}{4} \simeq 1.54254log2​(72)​≃1.5425 can be reached with zero error, which beats the maximum classical sum-rate capacity of 32\frac{3}{2}23​.In order to achieve this, we show that efficient linear programs can be formulated to compute the success probability of the best non-signaling assisted code for a finite number of copies of a multiple-access channel. In particular, this can be used to give lower bounds on the zero-error non-signaling assisted capacity of multiple-access channels.
We address the problem of coding for classical multiple-access channels (MACs) with the assistance of non-signaling correlations between parties. It is well-known that non-signaling assistance does not change the capacity of classical point-to-point channels. However, it was recently observed that one can construct MACs from two-player nonlocal games while relating the winning probability of the game to the capacity of the MAC. By considering games for which entanglement (a special kind of non-signaling correlation) increases the winning probability (e.g., the Magic Square game), this shows that for some specific kinds of channels, entanglement between the senders can increase the capacity. In this work, we make several contributions towards understanding the capacity region for MACs with the assistance of non-signaling correlations between the parties. We develop a linear program computing the optimal success probability for coding overnnncopies of a MACWWWwith size growing polynomially innnn. Solving this linear program allows us to achieve inner bounds for MACs. Applying this method to the binary adder channel, we show that using non-signaling assistance, the sum-ratelog⁡2(72)4≃1.5425\frac {\log _{2}(72)}{4} \simeq 1.54254log2​(72)​≃1.5425can be reached even with zero error, which beats the maximum sum-rate capacity of 1.5 in the unassisted case. For noisy channels, where the zero-error non-signaling assisted capacity region is trivial, we can use concatenated codes to obtain achievable points in the capacity region. Applied to a noisy version of the binary adder channel, we show that non-signaling assistance still improves the sum-rate capacity. Complementing these achievability results, we give an outer bound on the non-signaling assisted capacity region that has the same expression as the unassisted region except that the channel inputs are not required to be independent. Finally, we show that the capacity region with non-signaling assistance shared only between each sender and the receiver independently is the same as without assistance.
The capacity of a channel characterizes the maximum rate at which information can be transmitted through the channel asymptotically faithfully. For a channel with multiple senders and a single receiver, computing its sum capacity is possible in theory, but challenging in practice because of the nonconvex optimization involved. To address this challenge, we investigate three topics in our study. In the first part, we study the sum capacity of a family of multiple access channels (MACs) obtained from nonlocal games. For any MAC in this family, we obtain an upper bound on the sum rate that depends only on the properties of the game when allowing assistance from an arbitrary set of correlations between the senders. This approach can be used to prove separations between sum capacities when the senders are allowed to share different sets of correlations, such as classical, quantum or no-signalling correlations. We also construct a specific nonlocal game to show that the approach of bounding the sum capacity by relaxing the nonconvex optimization can give arbitrarily loose bounds. Owing to this result, in the second part, we study algorithms for non-convex optimization of a class of functions we call Lipschitz-like functions. This class includes entropic quantities, and hence these results may be of independent interest in information theory. Subsequently, in the third part, we show that one can use these techniques to compute the sum capacity of an arbitrary two-sender MACs to a fixed additive precision in quasi-polynomial time. We showcase our method by efficiently computing the sum capacity of a family of two-sender MACs for which one of the input alphabets has size two. Furthermore, we demonstrate with an example that our algorithm may compute the sum capacity to a higher precision than using the convex relaxation.
Computing the sum capacity of a multiple access channel (MAC) is a non-convex optimization problem. It is therefore common to compute an upper bound on the sum capacity using a convex relaxation. We investigate the performance of such a relaxation by considering a family of MACs obtained from nonlocal games. First, we derive an analytical upper bound on the sum capacity of such MACs, while allowing the senders to share any given set of correlations. Our upper bound depends only on the properties of the game available in practice, thereby providing a way to obtain separations between the sum capacity assisted by different sets of correlations. In particular, we obtain a bound on the sum capacity of the MAC obtained from the magic square game that is tighter than the previously known result. Next, we introduce a game for which the convex relaxation of the sum capacity can be arbitrarily loose, demonstrating the need to find other techniques to compute or bound the sum capacity. We subsequently propose an algorithm that can certifiably compute the sum capacity of any two-sender MAC to a given precision.
In this work, we consider two-sender, one-receiver communication over a discrete memoryless multiple-access channel without feedback, where two senders may cooperate on channel coding by using preshared resources, such as shared randomness, quantum states and measurements, or nonlocal correlations. We present the capacity region when senders employ cooperative encoding with quantum and nonlocal resources, extending beyond shared randomness, and derive a sum rate that serves as a lower bound to the sum capacity; the lower bound is computable by exploiting specific strategies. We also compute the sum capacities for two instances. One is when senders apply local resources for cooperative encoding. The other is when senders exploit nonclassical resources for encoding against channels constructed by referring to nonlocal games; in this way, correlated noise other than independent errors occurs on code words. Comparing the exact sum capacities and lower bounds, we show that nonlocal and quantum resources for cooperative encoding enable higher sum capacities over local ones. The Clauser-Horne-Shimony-Holt and magic square games are considered for constructing multiple-access channels, and we demonstrate the usefulness of nonlocal and quantum resources to achieve higher-sum capacities.
Communication over a classical multiple-access channel (MAC) with entanglement resources is considered, whereby two transmitters share entanglement resources a priori before communication begins. Leditzky et al. (2020) presented an example of a classical MAC, defined in terms of a pseudo telepathy game, such that the sum rate with entangled transmitters is strictly higher than the best achievable sum rate without such resources. Here, we establish inner and outer bounds on the capacity region for the general MAC with entangled transmitters, and show that the previous result can be obtained as a special case. It has long been known that the capacity region of the classical MAC under a message-average error criterion can be strictly larger than with a maximal error criterion (Dueck, 1978). We observe that given entanglement resources, the regions coincide. Furthermore, we address the combined setting of entanglement resources and conferencing, where the transmitters can also communicate with each other over rate-limited links. Using superdense coding, entanglement can double the conferencing rate.
The capacity region of the KKK-sender discrete memoryless multiple access channel (MAC) is fully characterized when non-signaling (NS) assistance is available to all KKK transmitters and the receiver. It is shown to have the same form as the classical capacity region of the MAC, except that the input distribution is allowed to be arbitrarily dependent across the senders. In particular, the NS-assisted capacity region matches the natural generalization to KKK senders of an outer bound that was previously established by Fawzi and Fermé for K=2K=2K=2 senders. Additionally, we provide examples of KKK-sender MACs where the multiplicative gain in capacity from NS-assistance is arbitrarily close to KKK. Combined with an upper bound from prior work, this establishes KKK as the extremal value of the multiplicative gain from NS-assistance across all KKK-sender MAC settings.