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Accessibility between Entangled States

Fumiaki Morikoshi

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Entanglement, a weird quantum correlation between separate physical systems, is an indispensable resource for quantum information processing. As with other physical resources, the key to understanding its full potential is to quantify it. In this paper, I review our recent insights into the quantification of entanglement based on the operational notion of accessibility between physical states. In particular, I show that the partially ordered structure of mixed entangled states underlies the non-uniqueness of their entanglement measure.

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Entanglement, a weird quantum correlation between separate physical systems, is an indispensable resource for quantum information processing. As with other physical resources, the key to understanding its full potential is to quantify it. In this paper, I review our recent insights into the quantification of entanglement based on the operational notion of accessibility between physical states. In particular, I show that the partially ordered structure of mixed entangled states underlies the non-uniqueness of their entanglement measure.
Ordering physical states is the key to quantifying some physical property of the states uniquely. Bipartite pure entangled states are totally ordered under local operations and classical communication (LOCC) in the asymptotic limit and uniquely quantified by the well-known entropy of entanglement. However, we show that mixed entangled states are partially ordered under LOCC even in the asymptotic limit. Therefore, non-uniqueness of entanglement measure is understood on the basis of an operational notion of asymptotic convertibility.
Quantum entanglement is a key physical resource in quantum information processing that allows for performing basic quantum tasks such as teleportation and quantum key distribution, which are impossible in the classical world. Ever since the rise of quantum information theory, it has been an open problem to quantify entanglement in an information-theoretically meaningful way. In particular, every previously defined entanglement measure bearing a precise information-theoretic meaning is not known to be efficiently computable, or if it is efficiently computable, then it is not known to have a precise information-theoretic meaning. In this Letter, we meet this challenge by introducing an entanglement measure that has a precise information-theoretic meaning as the exact cost required to prepare an entangled state when two distant parties are allowed to perform quantum operations that completely preserve the positivity of the partial transpose. Additionally, this entanglement measure is efficiently computable by means of a semidefinite program, and it bears a number of useful properties such as additivity and faithfulness. Our results bring key insights into the fundamental entanglement structure of arbitrary quantum states, and they can be used directly to assess and quantify the entanglement produced in quantum-physical experiments.
Entanglement is one of most peculiar characteristic of Quantum Mechanics (probably the other most mentioned is the particle tunneling through a barrier) because it entails correlations that supersedes classical correlations. It is conceived as the genuine quantum characteristic by many researchers and quite recently it has emerged as a physical resource to produce non-classical task, like quantum teleportation, quantum cryptography and quantum information. Entanglement could be distributed (efficient quantum communication is equivalent to efficient entanglement distribution (Plenio & Virmani, 2007)), concentrated (given an amount of entanglement, equal to the full content of entanglement in n pairs of identical entangled pure state systems, it is possible, using local operations on each system, to concentrate the total amount of entanglement into a smaller number m, m < n, of maximally pure entangled state system (Bennett, et al.; 1996)) and used to perform many quantum information tasks useful to overcome technical restrictions present on classical communication. In this chapter, we review the entanglement of quantum mechanical systems. First, in Section 2, we give a characterization of entanglement in terms of its special features as resource and its mathematical structure. Then, in section 3, we review some of the Bell inequalities. In Sections 4 and 5, we review some of the most important entanglement measures published for two and three entangled systems. After that, in Section 6, we give a characterization of quantum gates and operators by its entanglement power, i. e. the amount of entanglement that they can produce when acting on an arbitrary state. Finally, in Section 7, we briefly review the experimental detection of entanglement.
Quantum entanglement, a fundamental concept in quantum physics, has been elucidated through the development of Bell’s inequalities. Recent advancements have enabled controlled manipulation and measurement of entangled quantum states. This chapter provides a concise overview of entanglement’s mathematical underpinnings, its manipulation, and quantification. Special attention is given to quantification methods and their implications in quantum information. Key principles and measures are presented to facilitate a foundational understanding. Readers are encouraged to supplement this overview with comprehensive review articles and primary literature for further insights.
Quantum entanglement is the defining feature of composite quantum systems, underpinning correlations beyond classical randomness and enabling quantum communication, cryptography, and error correction. Surveys treat entanglement, Bell nonlocality, contextuality, and decoherence in isolation or narrow subcommunities. This review advances a unified perspective analysing these phenomena as distinct but interrelated manifestations of nonclassical correlations with an operational structure. We organise bipartite entanglement through separability criteria, Schmidt decomposition, and partial-transpose techniques, and develop tools for nonlocal correlations based on Bell inequalities, Tsirelson’s bound, nonlocal games, and self-testing, emphasising device-independent protocols. We then examine contextuality and magic states as resources for quantum computational advantage beyond entanglement, clarifying their relationship to nonlocality. Finally, we characterise decoherence as environment-induced entanglement selecting classical records, and use this perspective to specify which aspects of the measurement problem it addresses and which require interpretational or dynamical extensions. Integrating these dimensions, the review clarifies how notions of nonclassicality relate, diverge, and underpin quantum technologies.