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arXiv:0909.3839 · Physical Review A

Local content of bipartite qubit correlations

Cyril Branciard, Nicolas Gisin, Valerio Scarani

Published Feb 3, 2010• 11 citations
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Abstract

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.

DOI
10.1103/physreva.81.022103

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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 ∣\ensuremathψ(\ensuremathθ)\ensuremath⟩=cos\ensuremathθ∣00\ensuremath⟩+sin\ensuremathθ∣11\ensuremath⟩|\ensuremath{\psi}(\ensuremath{\theta})\ensuremath{\rangle}=\mathrm{cos}\ensuremath{\theta}|00\ensuremath{\rangle}+\mathrm{sin}\ensuremath{\theta}|11\ensuremath{\rangle}∣\ensuremathψ(\ensuremathθ)\ensuremath⟩=cos\ensuremathθ∣00\ensuremath⟩+sin\ensuremathθ∣11\ensuremath⟩, that their local content is pL‾(\ensuremathθ)=cos2\ensuremathθ\overline{{p}_{L}}(\ensuremath{\theta})=\mathrm{cos}2\ensuremath{\theta}pL​​(\ensuremathθ)=cos2\ensuremathθ. 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 $\ensuremath{\gamma}>2$.
We quantify the amount of non-locality contained in n noisy versions of the so-called Popescu–Rohrlich boxes ( PRBs ), i.e. bipartite systems violating the CHSH Bell inequality maximally. Following the approach by Elitzur, Popescu and Rohrlich, we measure the amount of non-locality of a system by representing it as a convex combination of a local behaviour, with maximal possible weight, and a non-signalling system. We show that the local part of n systems, each of which approximates a PRB with probability 1 − ε, is of order Θ(ε ⌈ n /2⌉ ) in the isotropic, and equal to (3ε) n in the maximally biased case.
A longstanding question in Bell nonlocality is whether finite-alphabet nonsignalling resources can reproduce the continuum of measurement correlations of partially entangled qubits. Here we resolve this problem by showing that every fixed pure two-qubit state admits an exact simulation of all local projective-measurement statistics using shared randomness and finitely many Popescu-Rohrlich (PR) boxes, without communication or postselection. The boxes operate in parallel, with a worst-case bound independent of the measurement settings: at most 128cot⁡4θ128\cot^4θ128cot4θ calls for the Schmidt state cos⁡θ∣00⟩+sin⁡θ∣11⟩\cosθ|00\rangle+\sinθ|11\ranglecosθ∣00⟩+sinθ∣11⟩, 0<θ<π/40<θ<π/40<θ<π/4. The key is an exact finite-selection method that reduces input-dependent continuous sampling to a choice from a fixed-length shared random list, implemented through a finite PR-box network. The construction extends to arbitrary mixed two-qubit states and local binary-outcome POVMs. This reveals a fundamental separation between the continuity of quantum measurement correlations and the alphabet size of the resources needed to simulate them, shedding light on the resource-theoretic structure of Bell nonlocality.
Entanglement between three or more parties exhibits a realm of properties unknown to two-party states. Bipartite states are easily classified using the Schmidt decomposition. The Schmidt coefficients of a bipartite pure state encompass all the nonlocal properties of the state and can be ``seen'' by looking at one party's density matrix only. Pure states of three and more parties, however, lack such a simple form. They have more invariants under local unitary transformations than any one party can ``see'' on their subsystem. These ``hidden nonlocalities'' will allow us to exhibit a class of multipartite states that cannot be distinguished from each other by any party. Generalizing a result of Bennett, Popescu, Rohrlich, Smolin, and Thapliyal, and using a recent result by Nielsen, we will show that these states cannot be transformed into each other by local actions and classical communication. Furthermore, we will use an orthogonal subset of such states to hint at applications to cryptography and illustrate an extension to quantum secret sharing [using recently suggested ((n,k))((n,k))((n,k))-threshold schemes].