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The Decoupling Theorem

Frédéric Dupuis, Mario Berta, Jürg Wullschleger, Renato Renner

Published Jan 1, 2011• 12 citations
OpenAlex

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. 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.

Related papers

One-Shot Decoupling

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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.

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Tight any-shot quantum decoupling

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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.

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Understanding Equipartition and Thermalization from Decoupling

Adrian Hutter

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)].

•Jan 1, 2011•6 citations

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Consider a bipartite system, of which one subsystem, A , undergoes a physical evolution separated from the other subsystem, R . One may ask under which conditions this evolution destroys all initial correlations between the subsystems A and R , i.e. decouples the subsystems. A quantitative answer to this question is provided by decoupling theorems , which have been developed recently in the area of quantum information theory. This paper builds on preceding work, which shows that decoupling is achieved if the evolution on A consists of a typical unitary, chosen with respect to the Haar measure, followed by a process that adds sufficient decoherence. Here, we prove a generalized decoupling theorem for the case where the unitary is chosen from an approximate two-design. A main implication of this result is that decoupling is physical, in the sense that it occurs already for short sequences of random two-body interactions, which can be modeled as efficient circuits. Our decoupling result is independent of the dimension of the R system, which shows that approximate two-designs are appropriate for decoupling even if the dimension of this system is large.

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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.

arXiv:1903.05796•Communications in Mathematical Physics•Jul 16, 2021•12 citations

One-Shot Randomized and Nonrandomized Partial Decoupling

Eyuri Wakakuwa, Yoshifumi Nakata

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.

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. 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.
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.
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)].
Consider a bipartite system, of which one subsystem, A , undergoes a physical evolution separated from the other subsystem, R . One may ask under which conditions this evolution destroys all initial correlations between the subsystems A and R , i.e. decouples the subsystems. A quantitative answer to this question is provided by decoupling theorems , which have been developed recently in the area of quantum information theory. This paper builds on preceding work, which shows that decoupling is achieved if the evolution on A consists of a typical unitary, chosen with respect to the Haar measure, followed by a process that adds sufficient decoherence. Here, we prove a generalized decoupling theorem for the case where the unitary is chosen from an approximate two-design. A main implication of this result is that decoupling is physical, in the sense that it occurs already for short sequences of random two-body interactions, which can be modeled as efficient circuits. Our decoupling result is independent of the dimension of the R system, which shows that approximate two-designs are appropriate for decoupling even if the dimension of this system is large.
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.
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.