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arXiv:1204.2982 · Physical Review A

Local content of all pure two-qubit states

Samuel Portmann, Cyril Branciard, Nicolas Gisin

Published Jul 11, 2012• 37 citations
arXivHTMLPDF

Abstract

The local (nonlocal) content in the sense of Elitzur, Popescu, and Rohrlich [Phys. Lett. A 162, 25 (1992)] is a natural measure for the locality (nonlocality) of quantum states. Its computation is in general difficult, even in low dimensions, and is one of the few open questions about pure two-qubit states. We present a complete solution to this long-lasting problem.

DOI
10.1103/physreva.86.012104

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The local (nonlocal) content in the sense of Elitzur, Popescu, and Rohrlich [Phys. Lett. A 162, 25 (1992)] is a natural measure for the locality (nonlocality) of quantum states. Its computation is in general difficult, even in low dimensions, and is one of the few open questions about pure two-qubit states. We present a complete solution to this long-lasting problem.
One of the last open problems concerning two qubits in a pure state is to find the exact local content of their correlation, in the sense of Elitzur, Popescu, and Rohrlich (EPR2) [A. C. Elitzur, S. Popescu, and D. Rohrlich, Phys. Lett. A162, 25 (1992)]. We propose an EPR2 decomposition that allows us to prove, for a wide range of states $|\ensuremath{\psi}(\ensuremath{\theta})\ensuremath{\rangle}=\mathrm{cos}\ensuremath{\theta}|00\ensuremath{\rangle}+\mathrm{sin}\ensuremath{\theta}|11\ensuremath{\rangle}$, that their local content is $\overline{{p}_{L}}(\ensuremath{\theta})=\mathrm{cos}2\ensuremath{\theta}$. We also share reflections on how to possibly extend this result to all two-qubit pure states.
The local and nonlocal contents of nonlocal probability distributions are studied using the approach of Elitzur, Popescu, and Rohrlich [A. Elitzur, S. Popescu, and D. Rohrlich, Phys. Lett. A 162, 25 (1992)]. This work focuses on distributions that can be obtained by single-copy von Neumann measurements on bipartite quantum systems. For pure two-qubit states $|\ensuremath{\Psi}(\ensuremath{\theta})⟩=\text{cos}\text{ }\ensuremath{\theta}|00⟩+\text{sin}\text{ }\ensuremath{\theta}|11⟩$ (with $\text{cos}\text{ }\ensuremath{\theta}\ensuremath{\ge}\text{sin}\text{ }\ensuremath{\theta}$), the local content of the corresponding probability distribution is found to lie between $1\ensuremath{-}\text{sin}\text{ }2\ensuremath{\theta}$ and $\text{cos}\text{ }2\ensuremath{\theta}$. For the family $|\ensuremath{\Psi}(\ensuremath{\gamma})⟩=(|00⟩+|11⟩+\ensuremath{\gamma}|22⟩)/\sqrt{2+{\ensuremath{\gamma}}^{2}}$ of two-qutrit states, nonzero local content is found for $\ensuremath{\gamma}>2$.
One of the most fascinating consequences of quantum theory is non‐locality, i.e., the fact that the measurement outcomes of (spatially separated) parts of a system can have a correlation unexplainable by shared classical information. Note that at the same time, these correlations are non‐signaling and do not allow for message transmission. Popescu and Rohrlich have defined a non‐local box as a “basic building block of non‐locality” and initiated a systematic study of non‐local correlations and their applications. However, they left open the question of whether any non‐signaling correlation can be simulated by such non‐local boxes. We show that the answer is yes with respect to arbitrarily accurate approximations.
A set of nonlocal correlations that have come to be known as a Popescu-Rohrlich (PR) box suggest themselves as a natural unit of nonlocality, much as a singlet is a natural unit of entanglement. We present two results relevant to this idea. One is that a wide class of multipartite correlations can be simulated using local operations on PR boxes only. We show this with an explicit scheme, which has the interesting feature that the number of PR boxes required is related to the computational resources necessary to represent a function defining the multipartite box. The second result is that there are quantum multipartite correlations, arising from measurements on a cluster state, that cannot be simulated with n PR boxes, for any n.