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arXiv:quant-ph/0401048 · Physical Review Letters

MultipartiteSeparability Inequalities Exponentially Stronger than Local Reality Inequalities

Shasanka Mohan Roy

Published Jan 12, 2005• 42 citations
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Abstract

I show that separability of N-partite quantum states implies new inequalities on Bell correlations which are stronger than the corresponding Mermin-Roy-Singh local reality inequalities by a factor of 2((N-1)/2).
DOI
10.1103/physrevlett.94.010402

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We show that positivity of every partial transpose of NNN-partite quantum states implies inequalities on Bell correlations which are stronger than standard Bell inequalities by a factor of ${2}^{(N\ensuremath{-}1)∕2}$. A violation of the inequality implies that the system is in a bipartite distillable entangled state. It turns out that a family of NNN-qubit bound entangled states proposed by D\"ur [Phys. Rev. Lett. 87, 230402 (2001)] violates the inequality for $N\ensuremath{\geqslant}4$.
I derive separability inequalities for Bell correlations of observables in arbitrary pure or mixed NNN Qudit states in DND^NDN-dimensional state space. I find states (a continuum of states if D>3D>3D>3) including maximally entangled states which violate these inequalities by a factor 2N−12^{N-1}2N−1 ; local reality Bell inequalities are much weaker, their maximum violation being by a factor 2(N−1)/22^{(N-1)/2}2(N−1)/2. The separability inequalities allow tests of entanglement of unknown states using only the measured correlations .
A family of Bell-type inequalities is present, which are constructed directly from the "standard" Bell inequalities involving two dichotomic observables per site. It is shown that the inequalities are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits. Remarkably, our new inequalities can provide stronger non-locality tests in a sense that the local reality inequalities are exponentially stronger than the corresponding multipartite separability inequalities. This reveals that the exponential violation of local realism by separable states is an interesting consequence of quantum fluctuation of multipartite systems.
Chen (quant-ph/0611126) has recently claimed ``exponential violation of local realism by separable states", in the sense that multi-partite separable quantum states are supposed to give rise to correlations and fluctuations that violate a Bell-type inequality that Chen takes to be satisfied by local realism. However, this can not be true since all predictions (including all correlations and fluctuations) that separable quantum states give rise to have a local realistic description and thus satisfy all Bell-type inequalities, and this holds for all number of parties. Since Chen claims otherwise by presenting a new inequality, claimed to be a Bell-type one, which separable states supposedly can violate, there must be a flaw in the argumentation. I will expose this flaw, not merely for clarification of this issue, but perhaps even more importantly since it re-teaches us an old lesson John Bell taught us over 40 years ago. I will argue that this lesson provides us with a new morale especially relevant to modern research in Bell-type inequalities.
The correlations between local measurement results are characteristic for multi-partite quantum systems. Under the condition of local realism, which assumes that properties of physical systems exist independent of being measured, and that space-like separated actions on the system cannot influence each other, these correlations are constrained by Bell-inequalities. Quantum mechanics predicts correlations which are not explainable under these two assumptions. The resulting violation of Bell inequalities was originally studied for pairs of entangled particles (Einstein et al.,1935). Meanwhile these ideas were extended to three-and more partite quantum states in particular with respect to the different kinds of multi-particle entanglement and their implications on the obtainable correlations (Mermin, 1990). Here we present a method to construct characteristic Bell inequalities for particular four-qubit states. These inequalities are a versatile tool, not only to provide evidence for non-locality, but also for the characterization of the states, with respect to detection of genuine multipartite entanglement and to state discrimination. Exemplarily we apply our approach to the symmetric four qubit Dicke-state (Dicke, 1954) with two excitations. (A symmetric N-qubit Dicke state with M excitations is the equal superposition of N-qubit pure product states having M logical 1 's and (N-M) logical 0's.) Experimentally the state was observed with photons as carrier of the qubits, using polarization encoding.
The quantum Fisher information (QFI) of certain multipartite entangled quantum states is larger than what is reachable by separable states, providing a metrological advantage. Are these nonclassical correlations strong enough to potentially violate a Bell inequality? Here, we present evidence from two examples. First, we discuss a Bell inequality designed for spin-squeezed states which is violated only by quantum states with a large QFI. Second, we relax a well-known lower bound on the QFI to find the Mermin Bell inequality as a special case. However, a fully general link between QFI and Bell correlations is still open.