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IEEE International Symposium on Information Theory (ISIT)

One-Shot Quantum State Redistribution and Quantum Markov Chains

Anurag Anshu, Shima Bab Hadiashar, Rahul Jain, Ashwin Nayak, Dave Touchette

Published Jul 12, 2021• 2 citations
Publisher

Abstract

We revisit the task of quantum state redistribution in the one-shot setting, and design a protocol for this task with communication cost in terms of a measure of distance from quantum Markov chains. More precisely, the distance is defined in terms of quantum max-relative entropy and quantum hypothesis testing entropy. Our result is the first to operationally connect one-shot quantum state redistribution and quantum Markov chains, and can be interpreted as an operational interpretation for a possible one-shot analogue of quantum conditional mutual information. The communication cost of our protocol is lower than all previously known ones and asymptotically achieves the well-known rate of quantum conditional mutual information. Thus, our work takes a step towards the important open question of near-optimal characterization of the one-shot quantum state redistribution. A full version of this paper is accessible at: https://arxiv.org/pdf/2104.08753.pdf

DOI
10.1109/isit45174.2021.9517813

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We revisit the task of quantum state redistribution in the one-shot setting, and design a protocol for this task with communication cost in terms of a measure of distance from quantum Markov chains. More precisely, the distance is defined in terms of quantum max-relative entropy and quantum hypothesis testing entropy. Our result is the first to operationally connect one-shot quantum state redistribution and quantum Markov chains, and can be interpreted as an operational interpretation for a possible one-shot analogue of quantum conditional mutual information. The communication cost of our protocol is lower than all previously known ones and asymptotically achieves the well-known rate of quantum conditional mutual information. Thus, our work takes a step towards the important open question of near-optimal characterization of the one-shot quantum state redistribution. A full version of this paper is accessible at: https://arxiv.org/pdf/2104.08753.pdf
We revisit the task of quantum state redistribution in the one-shot setting, and design a protocol for this task with communication cost in terms of a measure of distance from quantum Markov chains. More precisely, the distance is defined in terms of quantum max-relative entropy and quantum hypothesis testing entropy. Our result is the first to operationally connect quantum state redistribution and quantum Markov chains, and can be interpreted as an operational interpretation for a possible one-shot analogue of quantum conditional mutual information. The communication cost of our protocol is lower than all previously known ones and asymptotically achieves the well-known rate of quantum conditional mutual information. Thus, our work takes a step towards an optimal characterization of the resources required for one-shot quantum state redistribution, an important open problem in quantum Shannon theory.
This thesis consists of two parts: quantum compression and quantum learning theory. A common theme between these problems is that we study them through the lens of information theory. \n \nWe first study the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially higher communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed. We show that an ensemble given by Jain, Radhakrishnan, and Sen (ICALP'03) cannot be compressed by more than a constant number of qubits without shared entanglement, while in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost. \n \nNext, we study the task of quantum state redistribution, the most general version of compression of quantum states. We design a protocol for this task with communication cost in terms of a measure of distance from quantum Markov chains. More precisely, the distance is defined in terms of quantum max-relative entropy and quantum hypothesis testing entropy. Our result is the first to connect quantum state redistribution and Markov chains and gives an operational interpretation for a possible one-shot analogue of quantum conditional mutual information. The communication cost of our protocol is lower than all previously known ones and asymptotically achieves the well-known rate of quantum conditional mutual information. \n \nIn the last part, we focus on quantum algorithms for learning Boolean functions using quantum examples. We consider two commonly studied models of learning, namely, quantum PAC learning and quantum agnostic learning. We reproduce the optimal lower bounds by Arunachalam and de Wolf (JMLR’18) for the sample complexity of either of these models using information theory and spectral analysis. Our proofs are simpler than the previous ones and the techniques can be possibly extended to similar scenarios.
In quantum state redistribution as introduced by Luo and Devetak and Devetak and Yard, there are four systems of interest: the A system held by Alice; the B system held by Bob; the C system that is to be transmitted from Alice to Bob; and the R system that holds a purification of the state in the ABC registers. We give upper and lower bounds on the amount of quantum communication and entanglement required to perform the task of quantum state redistribution in a one-shot setting. Our bounds are in terms of the smooth conditional minand max-entropy, and the smooth max-information. The protocol for the upper bound has a clear structure, building on the work of Oppenheim: it decomposes the quantum state redistribution task into two simpler coherent state merging tasks by introducing a coherent relay. In the independent and identical (i.i.d.) asymptotic limit our bounds for the quantum communication cost converge to the quantum conditional mutual information I(C; R|B), and our bounds for the total cost converge to the conditional entropy H(C|B). This yields an alternative proof of optimality of these rates for quantum state redistribution in the i.i.d. asymptotic limit. In particular, we obtain a strong converse for quantum state redistribution, which even holds when allowing for feedback.
We study the problem of state redistribution both in the classical (shared randomness assisted) and quantum (entanglement assisted) one-shot settings and provide new upper bounds on the communication required. Our classical bounds are in terms of smooth-min and max relative entropies and the quantum bounds are in terms of max relative entropy and Renyi relative entropy of order 12\frac{1}{2}21​. We also consider a special case of this problem in the classical setting, previously studied by Braverman and Rao (2011). We show that their upper bound is optimal. In addition we provide an alternate protocol achieving a priory different looking upper bound. However, we show that our upper bound is essentially the same as their upper bound and hence also optimal. For the quantum case, we show that our achievability result is upper bounded by the achievability result obtained in Berta, Christandl, Touchette (2016).
Quantum information decoupling is a fundamental primitive in quantum information theory, underlying various applications in quantum physics. We prove a novel one-shot decoupling theorem formulated in terms of quantum relative entropy distance, with the decoupling error bounded by two sandwiched Rényi conditional entropies. In the asymptotic i.i.d. setting of standard information decoupling via partial trace, we show that this bound is ensemble-tight in quantum relative entropy distance and thereby yields a characterization of the associated decoupling error exponent in the low-cost-rate regime. Leveraging this framework, we derive several operational applications formulated in terms of purified distance: (i) single-letter expressions for the exact error exponents of quantum state merging in the low-entanglement-cost and high-entanglement-distillation-rate regimes, in terms of Petz-Rényi conditional entropies, and (ii) regularized expressions for achievable error exponents of entanglement distillation and quantum channel coding in terms of Petz-Rényi coherent informations. We further prove that these achievable bounds are tight for maximally correlated states and generalized dephasing channels, respectively, for the high distillation-rate/coding-rate regimes.