Smooth Entropy Bounds on One-Shot Quantum State Redistribution

Berta, Mario; Christandl, Matthias; Touchette, Dave

doi:10.1109/tit.2016.2516006

Cited by 36 publications

(92 citation statements)

References 30 publications

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“…We use a new achievability bound for a communication cost in terms of one-shot entropies. The proof of this result appears in a joint work with Berta and Christandl [11]. Independently of our work, similar upper bounds on one-shot state redistribution have been obtained by Datta, Hsieh and Oppenheim [21].…”

Section: Protocol Compression and Direct Sumsupporting

confidence: 82%

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Quantum Information Complexity

Touchette

2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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We define a new notion of information cost for quantum protocols, and a corresponding notion of quantum information complexity for bipartite quantum tasks. These are the fully quantum generalizations of the analogous quantities for bipartite classical tasks that have found many applications recently, in particular for proving communication complexity lower bounds and direct sum theorems. Finding such a quantum generalization of information complexity was one of the open problems recently raised by Braverman (STOC'12).Previous attempts have been made to define such a quantity for quantum protocols, with particular applications in mind; our notion differs from these in many respects. First, it directly provides a lower bound on the quantum communication cost, independent of the number of rounds of the underlying protocol. Secondly, we provide an operational interpretation for quantum information complexity: we show that it is exactly equal to the amortized quantum communication complexity of a bipartite task on a given input. This generalizes a result of Braverman and Rao (FOCS'11) to quantum protocols. Along the way to prove this result, we even strengthens the classical result in a bounded round scenario, and also prove important structural properties of quantum information cost and complexity.We prove that using this definition leads to the first general direct sum theorem for bounded round quantum communication complexity. Previous direct sum results in quantum communication complexity either held for some particular classes of functions, or were general but only held for single-round protocols. We also discuss potential applications of the new quantities to obtain lower bounds on quantum communication complexity.

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Section: Protocol Compression and Direct Sumsupporting

confidence: 82%

“…Entanglement is deemed free for the compression. We make use of a one-shot state redistribution achievability bound that we derived in joint work with Berta and Christandl [11]. The bound in that work is stated in terms of one-shot entropies.…”

Section: Protocol Compression At Information Costmentioning

confidence: 99%

Quantum Information Complexity

Touchette

2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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“…Regarding the existing algorithms, the achievability bounds of E(ψ) of the corresponding entanglementassisted LOCC algorithms can be calculated from the analyses in Refs. [38], [40]- [44], [47]. Given > 0, these achievability bounds are in the form E(ψ) = · · · + O log 1 as → 0, which diverges to infinity as higher fidelity is pursued.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Quantum State Merging for Arbitrarily Small-Dimensional Systems

Yamasaki

Murao

2019

IEEE Trans. Inform. Theory

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Recent advances in quantum technology facilitate the realization of information processing using quantum computers at least on the small and intermediate scales of up to several dozens of qubits. We investigate entanglement cost required for one-shot quantum state merging, aiming at quantum state transformation on these scales. In contrast to existing coding algorithms achieving nearly optimal approximate quantum state merging on a large scale, we construct algorithms for exact quantum state merging so that the algorithms are applicable to any given state of an arbitrarily-small-dimensional system. In the algorithms, entanglement cost can be reduced depending on a structure of the given state derived from the Koashi-Imoto decomposition. We also provide improved converse bounds for exact quantum state merging achievable for qubits but not necessarily achievable in general. As for approximate quantum state merging, we obtain algorithms and improved converse bounds by applying smoothing to those for exact state merging. Our results are applicable to distributed quantum information processing and multipartite entanglement transformation on small and intermediate scales.Index Terms-Quantum state merging, multipartite entanglement transformation, small and intermediate scale. I. INTRODUCTIONT HE ERA of small-and intermediate-scale quantum computers of up to several dozens of qubits is approaching due to advances in quantum technology. There exists, however, technical difficulty in increasing the number of lownoise qubits built in one quantum device [1]. For further scaling up, distributed quantum information processing using multiple quantum devices connected by a network for quantum communication is considered to be promising [2], [3]. Aimed at efficient quantum information processing, coding algorithms for quantum communication tasks in such a distributed setting should be designed to be suitable for transferring quantum states on these small and intermediate scales.Quantum state merging [4], [5] is a task playing crucial roles in distributed quantum information processing [6]-[8] and multipartite entanglement transformations [9]-[13]. Originally, state merging, or state redistribution [14], [15] as a generalized task including state merging, was introduced in the context of quantum Shannon theory, and it has applied to the analyses of

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“…In this section, we consider state redistribution with feedback [8], where the state redistribution protocol consists of M rounds of forward and backward quantum communication between Alice and Bob. The initial state of the protocol is again the pure state ψ ABCR ⊗ Φ k T A T B , where systems A and C are with Alice, B is with Bob, R is an inaccessible reference system, and Φ k…”

Section: State Redistribution With Feedbackmentioning

confidence: 99%

Strong converse theorems using Rényi entropies

Leditzky

Wilde

Datta

2016

Journal of Mathematical Physics

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We use a Rényi entropy method to prove strong converse theorems for certain information-theoretic tasks which involve local operations and quantum or classical communication between two parties. These include state redistribution, coherent state merging, quantum state splitting, measurement compression with quantum side information, randomness extraction against quantum side information, and data compression with quantum side information. The method we employ in proving these results extends ideas developed by Sharma [41], which he used to give a new proof of the strong converse theorem for state merging. For state redistribution, we prove the strong converse property for the boundary of the entire achievable rate region in the (e, q)-plane, where e and q denote the entanglement cost and quantum communication cost, respectively. In the case of measurement compression with quantum side information, we prove a strong converse theorem for the classical communication cost, which is a new result extending the previously known weak converse. For the remaining tasks, we provide new proofs for strong converse theorems previously established using smooth entropies. For each task, we obtain the strong converse theorem from explicit bounds on the figure of merit of the task in terms of a Rényi generalization of the optimal rate. Hence, we identify candidates for the strong converse exponents for each task discussed in this paper. To prove our results, we establish various new entropic inequalities, which might be of independent interest. These involve conditional entropies and mutual information derived from the sandwiched Rényi divergence. In particular, we obtain novel bounds relating these quantities, as well as the Rényi conditional mutual information, to the fidelity of two quantum states.

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Smooth Entropy Bounds on One-Shot Quantum State Redistribution

Cited by 36 publications

References 30 publications

Quantum Information Complexity

Quantum Information Complexity

Quantum State Merging for Arbitrarily Small-Dimensional Systems

Strong converse theorems using Rényi entropies

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