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arXiv:1706.08286 · IEEE Transactions on Information Theory

A Hypothesis Testing Approach for Communication Over Entanglement-Assisted Compound Quantum Channel

Anurag Anshu, Rahul Jain, Naqueeb Ahmad Warsi

Published Oct 16, 2018• 13 citations
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Abstract

We study the problem of communication over a compound quantum channel in the presence of entanglement. Classically, such a channel is modeled as a collection of conditional probability distributions wherein neither the sender nor the receiver is aware of the channel being used for transmission, except for the fact that it belongs to this collection. We provide near optimal achievability and converse bounds for this problem in the one-shot quantum setting in terms of the quantum hypothesis testing divergence. We also consider the case of informed sender, showing a one-shot achievability result that converges appropriately in the asymptotic and independent and identically distributed setting. Our achievability proof is similar in spirit to its classical counterpart. To arrive at our result, we use the technique of position-based decoding along with a new approach for constructing a union of two projectors, which might be of independent interest. We give another application of the union of projectors to the problem of testing composite quantum hypotheses.

DOI
10.1109/tit.2018.2876280

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We study the problem of communication over a compound quantum channel in the presence of entanglement. Classically, such a channel is modeled as a collection of conditional probability distributions wherein neither the sender nor the receiver is aware of the channel being used for transmission, except for the fact that it belongs to this collection. We provide near optimal achievability and converse bounds for this problem in the one-shot quantum setting in terms of the quantum hypothesis testing divergence. We also consider the case of informed sender, showing a one-shot achievability result that converges appropriately in the asymptotic and independent and identically distributed setting. Our achievability proof is similar in spirit to its classical counterpart. To arrive at our result, we use the technique of position-based decoding along with a new approach for constructing a union of two projectors, which might be of independent interest. We give another application of the union of projectors to the problem of testing composite quantum hypotheses.
We study the problem of communication over compound quantum channel in the presence of entanglement. Classically such channels are modeled as a collection of conditional probability distributions wherein neither the sender nor the receiver is aware of the channel being used for transmission, except for the fact that it belongs to this collection. We provide achievability and converse bounds for this problem in the one shot quantum setting in terms of quantum hypothesis testing relative-entropy. Our achievability proof is similar in spirit to its classical counterpart. To arrive at our result, we use the technique of position based decoding along with a new approach for constructing a union of two projectors, which can be of independent interest.
We study the problem of transmission of classical messages through a quantum channel in several network scenarios in the one-shot setting. We consider both the entanglement assisted and unassisted cases for the point to point quantum channel, the quantum multiple-access channel, the quantum channel with a state, and the quantum broadcast channel. We show that it is possible to near-optimally characterize the amount of communication that can be transmitted in these scenarios, using the position-based decoding strategy introduced in a prior study (A. Anshu, R. Jain, and N. Warsi, https://ieee.org/document/8399830). In the process, we provide a short and elementary proof of the converse for entanglement-assisted quantum channel coding in terms of the quantum hypothesis testing divergence [obtained earlier in the work of W. Matthews and S. Wehner, IEEE Trans. Inf. Theory 60, 7317–7329 (2014)]. Our proof has the additional utility that it naturally extends to various network scenarios mentioned above. Furthermore, none of our achievability results require a simultaneous decoding strategy, existence of which is an important open question in quantum Shannon theory.
Abstract Recently, a coding technique called position-based coding has been used to establish achievability statements for various kinds of classical communication protocols that use quantum channels. In the present paper, we apply this technique in the entanglement-assisted setting in order to establish lower bounds for error exponents, lower bounds on the second-order coding rate, and one-shot lower bounds. We also demonstrate that position-based coding can be a powerful tool for analyzing other communication settings. In particular, we reduce the quantum simultaneous decoding conjecture for entanglement-assisted or unassisted communication over a quantum multiple access channel to open questions in multiple quantum hypothesis testing. We then determine achievable rate regions for entanglement-assisted or unassisted classical communication over a quantum multiple-access channel, when using a particular quantum simultaneous decoder. The achievable rate regions given in this latter case are generally suboptimal, involving differences of Rényi-two entropies and conditional quantum entropies.
We exploit a coding technique called position-based coding in the entanglement-assisted setting to establish lower bounds for error exponents, lower bounds on the second-order coding rate, and one-shot lower bounds, and we also demonstrate that position-based coding can be a powerful tool for analyzing other communication settings. In particular, we reduce the quantum simultaneous decoding conjecture for entanglement-assisted or unassisted communication over a quantum multiple access channel to open questions in multiple quantum hypothesis testing. We then determine an achievable rate region for entanglement-assisted or unassisted classical communication over a quantum multiple-access channel, when using a particular quantum simultaneous decoder. The achievable rate regions given in this latter case are generally suboptimal, involving differences of Rényi-two entropies and conditional quantum entropies.
We study universal quantum codes for entanglement-assisted quantum communication over compound quantum channels. In this setting, sender and receiver do not know the specific channel that will be used for communication, but only know the set that the channel is selected from. We investigate different variations of the problem: uninformed users, informed receiver, informed sender, and feedback assistance. We derive single-letter formulas for all corresponding channel capacities. Our proofs are based on one-shot decoupling bounds and properties of smooth entropies.