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arXiv:1012.6044 · Communications in Mathematical Physics

One-Shot Decoupling

Frédéric Dupuis, Mario Berta, Jürg Wullschleger, Renato Renner

Published Mar 20, 2014• 136 citations
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Abstract

If a quantum system A , which is initially correlated to another system, E , undergoes an evolution separated from E , then the correlation to E generally decreases. Here, we study the conditions under which the correlation disappears (almost) completely, resulting in a decoupling of A from E . We give a criterion for decoupling in terms of two smooth entropies, one quantifying the amount of initial correlation between A and E , and the other characterizing the mapping that describes the evolution of A . The criterion applies to arbitrary such mappings in the general one-shot setting. Furthermore, the criterion is tight for mappings that satisfy certain natural conditions. One-shot decoupling has a number of applications both in physics and information theory, e.g., as a building block for quantum information processing protocols. As an example, we give a one-shot state merging protocol and show that it is essentially optimal in terms of its entanglement consumption/production.

DOI
10.1007/s00220-014-1990-4

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IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2019•3 citations
If a quantum system A , which is initially correlated to another system, E , undergoes an evolution separated from E , then the correlation to E generally decreases. Here, we study the conditions under which the correlation disappears (almost) completely, resulting in a decoupling of A from E . We give a criterion for decoupling in terms of two smooth entropies, one quantifying the amount of initial correlation between A and E , and the other characterizing the mapping that describes the evolution of A . The criterion applies to arbitrary such mappings in the general one-shot setting. Furthermore, the criterion is tight for mappings that satisfy certain natural conditions. One-shot decoupling has a number of applications both in physics and information theory, e.g., as a building block for quantum information processing protocols. As an example, we give a one-shot state merging protocol and show that it is essentially optimal in terms of its entanglement consumption/production.
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.
If a quantum system A, which is initially correlated to another system, E, undergoes an evolution separated from E, then the correlation to E generally decreases. Here, we study the conditions under which the correlation disappears (almost) completely, resulting in a decoupling of A from E. We give a criterion for decoupling in terms of two smooth entropies, one quantifying the amount of initial correlation between A and E, and the other characterizing the mapping that describes the evolution of A. The criterion applies to arbitrary such mappings in the general one-shot setting. Furthermore, the criterion is tight for mappings that satisfy certain natural conditions. Decoupling has a number of applications both in physics and information theory, e.g., as a building block for quantum information processing protocols. As an example, we give a one-shot state merging protocol and show that it is essentially optimal in terms of its entanglement consumption/production.
The decoupling technique was originally developed for information-theoretical purposes. It describes the conditions under which the correlations in a bipartite state disappear if one part undergoes an evolution separated from the other. In the past years there has been enormous progress in understanding the foundations of statistical mechanics from first principles of quantum mechanics. By use of the decoupling technique we are able to reproduce and generalize major results of this development and to approach open problems. As a first application of the decoupling technique we generalize the result of [Popescu et al., Nat. Phys. 2, 754-758 (2006)] about the apparent validity of the postulate of equal a priori probabilities to states which may be correlated to a reference. We express it in a form which allows to apply recent results about random two-qubit interactions. We give a criterion for the apparent validity of the postulate which is tight up to differences between different entropy measures. Similarly, we generalize the result of [Linden et al., Phys. Rev. E 79, 061103 (2009)] about the independence of the temporal average of a quantum mechanical system of its initial state to initial states which may be correlated to a reference. We develop a criterion for whether a quantum-mechanical system has “forgotten” about its initial state which is tight up to differences between different entropy measures. We find that comparing two local entropies of just one particular state tells us whether generic initial states of the system have already evolved to the same state or not. After developing new bounds on the times which are necessary for entropy changes, we are able to provide lower bounds on the times which are necessary for a system to become independent of its initial state. We discover an intimate connection between a system becoming independent of its initial state and a loss of an observer’s knowledge about the state of the system. As a further application of the decoupling technique we find sufficient conditions under which a system stays close to its initial state for all times, thereby extending the result of [Gogolin et al., Phys. Rev. Lett. 106, 040401 (2011)].
Abstract We introduce a task that we call partial decoupling, in which a bipartite quantum state is transformed by a unitary operation on one of the two subsystems and then is subject to the action of a quantum channel. We assume that the subsystem is decomposed into a direct-sum-product form, which often appears in the context of quantum information theory. The unitary is chosen at random from the set of unitaries having a simple form under the decomposition. The goal of the task is to make the final state, for typical choices of the unitary, close to the averaged final state over the unitaries. We consider a one-shot scenario, and derive upper and lower bounds on the average distance between the two states. The bounds are represented simply in terms of smooth conditional entropies of quantum states involving the initial state, the channel and the decomposition. Thereby we provide generalizations of the one-shot decoupling theorem. The obtained result would lead to further development of the decoupling approaches in quantum information theory and fundamental physics.
Quantum information decoupling is a pillar of quantum information theory, underlying quantum communication, error correction, and information recovery. However, its existing formulations rely on random unitary operations tied to finite-dimensional assumptions on quantum systems. Here we establish a decoupling framework for arbitrary separable, possibly infinite-dimensional systems. For finite-dimensional inputs and arbitrary separable reference/output systems, we derive error bounds on one-shot decoupling for completely positive maps in terms of sandwiched Rényi conditional entropies. For partial-trace decoupling with finite-dimensional references, the error exponent is optimal up to the known critical rate. To handle infinite-dimensional inputs, we assume finite entropy of the manipulated system. We construct finite-rank projections on independent and identically distributed (IID) states to restrict randomization to a projected finite-dimensional subspace, while ensuring high success probability by including auxiliary components in the discarded subsystem at asymptotically negligible cost. This leads to an infinite-dimensional IID partial-trace decoupling protocol achieving optimal asymptotic dimension rates. As an application, we construct an infinite-dimensional quantum-state-merging protocol, a mother protocol of quantum information theory. Under finite entropy of Alice's marginal, it achieves the same quantum-communication and total-cost rates as in finite dimensions, governed by mutual information and conditional entropy. These results show that the operational interpretation of entropic quantities through these achievable rates constitutes a universal principle beyond finite dimensions. More broadly, our framework provides foundational tools for exploring quantum information regardless of whether we model the physical world using finite- or infinite-dimensional spaces.
We introduce a task that we call partial decoupling, in which a bipartite quantum state is transformed by a unitary operation on one of the two subsystems and then is subject to the action of a quantum channel. We assume that the subsystem is decomposed into a direct-sum-product form, which often appears in the context of quantum information theory. The unitary is chosen at random from the set of unitaries having a simple form under the decomposition. The goal of the task is to make the final state, for typical choices of the unitary, close to the averaged final state over the unitaries. We consider a one-shot scenario, and derive upper and lower bounds on the average distance between the two states. The bounds are represented simply in terms of smooth conditional entropies of quantum states involving the initial state, the channel and the decomposition. Thereby we provide generalizations of the one-shot decoupling theorem. The obtained result would lead to further development of the decoupling approaches in quantum information theory and fundamental physics.