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arXiv:1109.4348 · New Journal of Physics

Decoupling with unitary approximate two-designs

Oleg Szehr, Frédéric Dupuis, Marco Tomamichel, Renato Renner

Published May 15, 2013• 65 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/15/5/053022

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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.
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 prove a new concentration result for non-catalytic decoupling by showing that, for suitably large ttt, applying a unitary chosen uniformly at random from an approximate ttt-design on a quantum system followed by a fixed quantum operation almost decouples, with high probability, the given system from another reference system to which it may initially have been correlated. Earlier works either did not obtain high decoupling probability, or used provably inefficient unitaries, or required catalytic entanglement for decoupling. In contrast, our approximate unitary designs always guarantee decoupling with exponentially high probability and, under certain conditions, lead to computationally efficient unitaries. As a result we conclude that, under suitable conditions, efficiently implementable approximate unitary designs achieve relative thermalisation in quantum thermodynamics with exponentially high probability. We also show the scrambling property of black hole, when the black hole evolution is according to pseudorandom approximate unitary ttt-design, as opposed to the Haar random evolution considered earlier by Hayden-Preskill.
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 prove a new concentration result for non-catalytic decoupling by showing that, for suitably largettt, applying a unitary chosen uniformly at random from an approximatettt-design on a quantum system followed by a fixed quantum operation almost decouples, with high probability, the given system from another reference system to which it may initially have been correlated. Earlier works either did not obtain high decoupling probability, or used provably inefficient unitaries, or required catalytic entanglement for decoupling. In contrast, our approximate unitary designs always guarantee decoupling with exponentially high probability and, under certain conditions, lead to computationally efficient unitaries. As a result we conclude that, under suitable conditions, efficiently implementable approximate unitary designs achieve relative thermalisation in quantum thermodynamics with exponentially high probability. We also show the scrambling property of black hole, when the black hole evolution is according to pseudorandom approximate unitaryttt-design, as opposed to the Haar random evolution considered earlier by Hayden-Preskill.
Decoupling theorems have proven useful in various applications in the area of quantum information theory. This thesis builds upon preceding work by Frédéric Dupuis [arXiv:1012.6044v1], where a general decoupling theorem is obtained and its implications for quantum coding theory are studied. At first we generalize this theorem to the case where the average is taken over an approximate unitary 2-design. The second part of this project tackles the question whether or not it is possible to decorrelate CQ-states with classical operations. We obtain results similar to the pivotal Leftover Hash Lemma. Finally we analyze the decoupling power of permutation operators in a fully quantum context and show a general procedure that yields decoupling theorems with permutations.
We investigate decoupling, one of the most important primitives in quantum Shannon theory, by replacing the uniformly distributed random unitaries commonly used to achieve the protocol, with repeated applications of random unitaries diagonal in the Pauli- Z and - X bases. This strategy was recently shown to achieve an approximate unitary 2 -design after a number of repetitions of the process, which implies that the strategy gradually achieves decoupling. Here, we prove that even fewer repetitions of the process achieve decoupling at the same rate as that with the uniform ones, showing that rather imprecise approximations of unitary 2 -designs are sufficient for decoupling. We also briefly discuss efficient implementations of them and implications of our decoupling theorem to coherent state merging and relative thermalisation.