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IEEE International Symposium on Information Theory (ISIT)

Generalized resource theory of purity: one-shot purity distillation with local noisy operations and one way classical communication

Sayantan Chakraborty, Aditya Nema, Francesco Buscemi

Published Jun 25, 2023• 2 citations
Publisher

Abstract

We investigate the problem of producing local pure states by performing local noisy operations assisted by one-way classical communication on a given bipartite mixed state in the one-shot setting. We consider the following two scenarios:1)Scenario I: A party, say Alice, is provided with a single copy of some quantum state ρAon system A. The task for Alice is to extract pure qubit states using only noisy operations on A. We call this task purity concentration.2)Scenario II: Two parties, Alice and Bob possess the A and B sub-systems, respectively, of a given bipartite quantum state ρAB. They are allowed to perform any local noisy operations and communicate via a one-way dephasing (i.e., classical) channel. The task for them is to design a protocol using these resources such that together they can extract pure local qubit states from the shared state ρAB. We call this task local purity distillation.

DOI
10.1109/isit54713.2023.10206692

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We investigate the problem of producing local pure states by performing local noisy operations assisted by one-way classical communication on a given bipartite mixed state in the one-shot setting. We consider the following two scenarios:1)Scenario I: A party, say Alice, is provided with a single copy of some quantum state ρAon system A. The task for Alice is to extract pure qubit states using only noisy operations on A. We call this task purity concentration.2)Scenario II: Two parties, Alice and Bob possess the A and B sub-systems, respectively, of a given bipartite quantum state ρAB. They are allowed to perform any local noisy operations and communicate via a one-way dephasing (i.e., classical) channel. The task for them is to design a protocol using these resources such that together they can extract pure local qubit states from the shared state ρAB. We call this task local purity distillation.
In this paper we revisit the problem of distributed purity distillation in the one-shot setting. The problem considers a bipartite quantum state ρABshared between two parties Alice and Bob, who are allowed local unitary operations and one-way classical communication. The goal is to extract pure qubit states using these given resources, in their respective laboratories. A caveat of this problem is that all known protocols which recover the optimal number of pure qubit states require some pure qubits at the beginning at the protocol, to be used as ancilla. In this paper, we present a one-shot purity distillation protocol which requires a supply of ancilla qubits at 0 rate in the asymptotic iid setting, and little to no ancilla in the one-shot setting. We show that our protocol outperforms the best known purity distillation algorithms in terms of requirement of additional ancilla, while maintaining the same rate of pure states distilled.
We consider the task of distilling local purity from a noisy quantum state ρABC, wherein we provide a protocol for three parties, Alice, Bob and Charlie, to distill local purity (at a rate P) from many independent copies of a given quantum state ρABC. The three parties have access to their respective subsystems of ρABC, and are only allowed to use local unitary operations. In addition, Alice and Bob can communicate with Charlie using a one-way multiple-access dephasing channel of link rates R1and R2, respectively. The objective of the protocol is to minimize the usage of the dephasing channel (in terms of rates R1and R2) while maximizing the asymptotic purity that can be jointly distilled from ρABC. To achieve this, we employ ideas from distributed measurement compression protocols, and in turn, characterize a set of sufficient conditions on (P, R1,R2) in terms of quantum information theoretic quantities such that P amount of purity can be distilled using rates R1and R2.
Local pure states represent a fundamental resource in quantum information theory. In this work we obtain one-shot achievable bounds on the rates for local purity distillation, in the single-party and in the two-party cases. In both situations, local noisy operations are freely available, while in the two-party case also one-way classical communication can be used. In addition, in both situations local pure ancillas can be borrowed, as long as they are discounted from the final net rate of distillation. The one-shot rates that we obtain, written in terms of mutual information-like quantities, are shown to recover in the limit the asymptotic i.i.d. rates of Devetak [PRA, 2005], up to first order analysis.
Pure states are an important resource in many quantum information processing protocols. However, even making a fixed pure state, say ∣0⟩|0\rangle∣0⟩, in the laboratory requires a considerable amount of effort. Often one ends up with a mixed state ρρρ whose classical description is nevertheless known. Hence it is important to develop protocols that extract a fixed pure state from a known mixed state. In this work, we study the problem of extracting a fixed pure state ∣0⟩A′∣0⟩B′|0\rangle^{A'} |0\rangle^{B'}∣0⟩A′∣0⟩B′ from a known pure state ρABρ^{AB}ρAB distributed between two parties AAA and BBB. Here, A′A'A′, B′B'B′ are subspaces of AAA, BBB and the total amount of purity extracted is log⁡∣A′∣+log⁡∣B′∣\log |A'| + \log |B'|log∣A′∣+log∣B′∣. The parties can borrow local pure ancilla, apply local unitary operations and send a message from AAA to BBB through a dephasing channel. If local pure ancilla is borrowed, it must be subtracted in order to properly account for the purity extracted. We obtain the most efficient achievable bounds on one shot distributed purity extraction, in terms of the rate of local ancilla borrowed by the protocol, while distilling pure qubits at the best known rate. Our protocols borrow little to no local pure ancilla. Our bounds improve upon the existing bounds for this problem in both one shot as well as asymptotic iid settings. In particular they subsume all the asymptotic iid results of Devetak and Krovi-Devetak. In addition, we derive upper bounds for the rate of distillation in the one shot setting, which nearly match our achievable bounds.
We study the process of assisted work distillation. This scenario arises when two parties share a bipartite quantum state ρ_{AB} and their task is to locally distill the optimal amount of work when one party is restricted to thermal operations, whereas the other can perform general quantum operations and they are allowed to communicate classically. We demonstrate that this question is intimately related to the distillation of classical and quantum correlations. In particular, we show that the advantage of one party performing global measurements over many copies of ρ_{AB} is related to the nonadditivity of the entanglement of formation. We also show that there may exist work bound in the quantum correlations of the state that is only extractable under the wider class of local Gibbs-preserving operations.
Secret-key distillation from quantum states and channels is a central task of interest in quantum information theory, as it facilitates private communication over a quantum network. Here, we study the task of secret-key distillation from bipartite states and point-to-point quantum channels using local operations and one-way classical communication (one-way LOCC). We employ the resource theory of unextendible entanglement to study the transformation of a bipartite state under one-way LOCC, and we obtain several efficiently computable upper bounds on the number of secret bits that can be distilled from a bipartite state using one-way LOCC channels; these findings apply not only in the one-shot setting but also in some restricted asymptotic settings. We extend our formalism to private communication over a quantum channel assisted by forward classical communication. We obtain efficiently computable upper bounds on the one-shot forward-assisted private capacity of a channel, thus addressing a question in the theory of quantum-secured communication that has been open for some time now. Our formalism also provides upper bounds on the rate of private communication when using a large number of channels in such a way that the error in the transmitted private data decreases exponentially with the number of channel uses. Moreover, our bounds can be computed using semidefinite programs, thus providing a computationally feasible method to understand the limits of private communication over a quantum network.