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arXiv:2311.11820 · Quantum

Local Purity Distillation in Quantum Systems: Exploring the Complementarity Between Purity and Entanglement

Ray Ganardi, Piotr Masajada, Moein Naseri, Alexander Streltsov

Published Mar 20, 2025• 0 citations
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Abstract

Quantum thermodynamics and quantum entanglement represent two pivotal quantum resource theories with significant relevance in quantum information science. Despite their importance, the intricate relationship between these two theories is still not fully understood. Here, we investigate the interplay between entanglement and thermodynamics, particularly in the context of local cooling processes. We introduce and develop the framework of Gibbs-preserving local operations and classical communication. Within this framework, we explore strategies enabling remote parties to effectively cool their local systems to the ground state. Our analysis is centered on scenarios where only a single copy of a quantum state is accessible, with the ideal performance defined by the highest possible fidelity to the ground state achievable under these constraints. We focus on systems with fully degenerate local Hamiltonians, where local cooling aligns with the extraction of local purity. In this context, we establish a powerful link between the efficiency of local purity extraction and the degree of entanglement present in the system, a concept we define as purity-entanglement complementarity . Moreover, we demonstrate that in many pertinent scenarios, the optimal performance can be precisely determined through semidefinite programming techniques. Our findings open doors to various practical applications, including techniques for entanglement detection and estimation. We demonstrate this by evaluating the amount of entanglement for a class of bound entangled states.

DOI
10.22331/q-2025-03-20-1666

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Quantum thermodynamics and quantum entanglement represent two pivotal quantum resource theories with significant relevance in quantum information science. Despite their importance, the intricate relationship between these two theories is still not fully understood. Here, we investigate the interplay between entanglement and thermodynamics, particularly in the context of local cooling processes. We introduce and develop the framework of Gibbs-preserving local operations and classical communication. Within this framework, we explore strategies enabling remote parties to effectively cool their local systems to the ground state. Our analysis is centered on scenarios where only a single copy of a quantum state is accessible, with the ideal performance defined by the highest possible fidelity to the ground state achievable under these constraints. We focus on systems with fully degenerate local Hamiltonians, where local cooling aligns with the extraction of local purity. In this context, we establish a powerful link between the efficiency of local purity extraction and the degree of entanglement present in the system, a concept we define as purity-entanglement complementarity . Moreover, we demonstrate that in many pertinent scenarios, the optimal performance can be precisely determined through semidefinite programming techniques. Our findings open doors to various practical applications, including techniques for entanglement detection and estimation. We demonstrate this by evaluating the amount of entanglement for a class of bound entangled states.
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.
Entanglement is one of the most striking features of quantum mechanics, and yet it is not specifically quantum.More specific to quantum mechanics is the connection between entanglement and thermodynamics, which leads to an identification between entropies and measures of pure state entanglement.Here we search for the roots of this connection, investigating the relation between entanglement and thermodynamics in the framework of general probabilistic theories.We first address the question whether an entangled state can be transformed into another by means of local operations and classical communication.Under two operational requirements, we prove a general version of the Lo-Popescu theorem, which lies at the foundations of the theory of pure-state entanglement.We then consider a resource theory of purity where free operations are random reversible transformations, modelling the scenario where an agent has limited control over the dynamics of a closed system.Our key result is a duality between the resource theory of entanglement and the resource theory of purity, valid for every physical theory where all processes arise from pure states and reversible interactions at the fundamental level.As an application of the main result, we establish a one-to-one correspondence between entropies and measures of pure bipartite entanglement.The correspondence is then used to define entanglement measures in the general probabilistic framework.Finally, we show a duality between the task of information erasure and the task of entanglement generation, whereby the existence of entropy sinks (systems that can absorb arbitrary amounts of information) becomes equivalent to the existence of entanglement sources (correlated systems from which arbitrary amounts of entanglement can be extracted).
Entanglement distillation, the process of converting weakly entangled states into maximally entangled ones using Local Operations and Classical Communication (LOCC), is pivotal for robust entanglement-assisted quantum information processing in error-prone environments. A construction based on stabilizer codes offers an effective method for designing such protocols. By analytically investigating the effective action of stabilizer protocols for systems of prime dimension d , we establish a standard form for the output states of recurrent stabilizer-based distillation. This links the properties of input states, stabilizers, and encodings to the properties of the protocol. Based on those insights, we present a novel two-copy distillation protocol, applicable to all bipartite states in prime dimension, that maximizes the fidelity increase per iteration for Bell-diagonal states. The power of this framework and the protocol is demonstrated through numerical investigations, which provide evidence for superior performance in terms of efficiency and distillability of low-fidelity states compared to other well-established recurrence protocols. By elucidating the interplay between states, errors, and protocols, our contribution advances the systematic development of highly effective distillation protocols, enhancing our understanding of distillability.
Quantum communication relies on the existence of high quality quantum channels to exchange information. In practice, however, all communication links are affected by noise from the environment. Here we investigate the ability of quantum channels to perform quantum communication tasks by restricting the participants to use only local operations and one-way classical communication (one-way LOCC) along with the available quantum channel. In particular, a channel can be used to distill a highly entangled state between two parties, which further enables quantum or private communication. In this work, we invoke the framework of superchannels to study the distillation of a resourceful quantum state, such as a maximally entangled state or a private state, using multiple instances of a point-to-point quantum channel. We use the idea ofk-extendibility to obtain a semidefinite relaxation of the set of one-way LOCC superchannels and define a class of entanglement measures for quantum channels that decrease monotonically under such superchannels; therefore these measures, dubbed collectively the “unextendible entanglement of a channel”, yield upper bounds on several communication-theoretic quantities of interest in the regimes of resource distillation and zero error.We then generalize the formalism ofk-extendibility to bipartite superchannels, thus obtaining functions that are monotone under two-extendible superchannels. This allows us to analyze probabilistic distillation of ebits or secret key bits from a bipartite state when using a resourceful quantum channel. Moreover, we propose semidefinite programs to evaluate several of these quantities, providing a computationally feasible method of comparison between quantum channels for resource distillation.
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.
Abstract We study the task of entanglement distillation in the one-shot setting under different classes of quantum operations which extend the set of local operations and classical communication (LOCC). Establishing a general formalism which allows for a straightforward comparison of their exact achievable performance, we relate the fidelity of distillation under these classes of operations with a family of entanglement monotones and the rates of distillation with a class of smoothed entropic quantities based on the hypothesis testing relative entropy. We then characterise exactly the one-shot distillable entanglement of several classes of quantum states and reveal many simplifications in their manipulation. We show in particular that the ε-error one-shot distillable entanglement of any pure state is the same under all sets of operations ranging from one-way LOCC to separability-preserving operations or operations preserving the set of states with positive partial transpose, and can be computed exactly as a quadratically constrained linear program. We establish similar operational equivalences in the distillation of isotropic and maximally correlated states, reducing the computation of the relevant quantities to linear or semidefinite programs. We also show that all considered sets of operations achieve the same performance in environment-assisted entanglement distillation from any state.