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arXiv:quant-ph/0606238

Local extraction of EPR entanglement from classical systems

Dagomir Kaszlikowski, Vlatko Vedral

Published Jun 28, 2006• 1 citation
arXivHTMLPDF

Abstract

Coherent states with large amplitudes are traditionally thought of as the best quantum mechanical approximation of classical behavior. Here we argue that, far from being classical, coherent state are in fact highly entangled. We demonstrate this by showing that a general system of indistinguishable bosons in a coherent state can be used to entangle, by local interactions, two spatially separated and distinguishable non-interacting quantum systems. Entanglement can also be extracted in the same way from number states or any other nontrivial superpositions of them.

DOI
10.48550/arxiv.quant-ph/0606238

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Coherent states with large amplitudes are traditionally thought of as the best quantum mechanical approximation of classical behavior. Here we argue that, far from being classical, coherent state are in fact highly entangled. We demonstrate this by showing that a general system of indistinguishable bosons in a coherent state can be used to entangle, by local interactions, two spatially separated and distinguishable non-interacting quantum systems. Entanglement can also be extracted in the same way from number states or any other nontrivial superpositions of them.
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In early days of quantum theory it was believed that the results of measurements performed on two distant physical systems should be uncorrelated thus their quantum state should be separable it means described by a simple tensor product of the individual local state vectors or a tensor product of individual local density operators. It was shown many years ago by EPR that two systems which interacted in the past and separated afterwards had to be described in most cases by particular non-separable states which are called entangled. It was noticed by Zanardi et al. that a Hilbert space of possible state vectors of compound physical system could be partitioned in different way by introducing various tensor product structures induced by the experimentally accessible observables (interactions and measurements). Therefore a separable state in one partition could become entangled in different partition. In this sense the entanglement was relative to a particular set of experimental capabilities. Continuing this line of thought Torre et al. claimed to prove that for any separable state there existed strongly correlated physical observables therefore all quantum states were entangled. The same claims one may find in several recent papers. In this paper we will discuss in operational way the differences existing between separable, non-separable and entangled states and we will show that the conclusions of Torre et al. were unjustified. A sufficient condition for entanglement is the violation of BI-CHSH and/or steering inequalities. Since there exist experiments outside the quantum physics violating these inequalities therefore in the operational approach one cannot say that the entanglement is an exclusive quantum phenomenon.
We study the problem of extracting EPR pairs from a general source of entanglement. Suppose Alice and Bob share a bipartite state R which is reasonably to perfect EPR pairs. The only information Alice and Bob possess is a lower bound on the fidelity of R and a maximally entangled state. They wish to purify R using local operations and classical communication and output a state that is arbitrarily close to EPR pairs. We prove that on average, Alice and Bob cannot increase the fidelity of the input state significantly. On the other hand, there exist that may fail with a small probability, and otherwise will output states arbitrarily close to EPR pairs with very high probability. These come from the purity-testing protocols of Barnum et al.
Locally indistinguishable states are useful to distribute information among spatially separated parties such that the information is locked. This implies that the parties are not able to extract the information completely via local operations and classical communication (LOCC), while it might be possible via LOCC when the parties share entanglement. In this work, we consider an information distribution protocol using orthogonal states for $m\ensuremath{\ge}3$ spatially separated parties such that even if any $k\ensuremath{\le}(m\ensuremath{-}1)$ parties collaborate still the information cannot be revealed completely. Such a protocol is useful to understand up to what extent the encoded information remains locked. However, if required, the parties can share entanglement and extract the information completely by LOCC. To make the process resource efficient, it should consume less number of entangled states. We show that though the set of states, which are locally indistinguishable across every bipartition, are sufficient for the above protocol, they may consume a higher number of entangled states when aiming for complete information extraction. We establish this by constructing a class of locally indistinguishable sets of orthogonal states which can be employed to accomplish the above protocol and these sets consume less number of entangled states, compared with the former sets, for complete information extraction. In fact, this difference in the number of required entangled states for complete information extraction grows linearly with the number of parties. This study sheds light on suitable use of local indistinguishability property of quantum states as a resource, and thus we demonstrate an efficient way of information distribution.
Basing on the analogy between the coherent states of light and separable states of NNN bosons, we demonstrate that the violation Cauchy-Schwarz inequality for any-order correlation function signals the entanglement among the constituent particles. Rather than restricting to the correlations between the positions of particles, we consider the broadest set of measurements allowed by quantum mechanics. Our result is general -- it applies to any quantum system of bosons, even when the number of particles is not fixed, provided that there is no coherence between different number states. We also demonstrate that the compact expression for the separable state of bosons can be used to relate some known metrological quantities to the particle entanglement in a very simple way.
In [Piani et al., PRL106 (2011) 220403], an activation protocol was introduced which maps the general non-classical (multipartite) correlations between given systems into bipartite entanglement between the systems and local ancillae by means of a potentially highly entangling interaction. Here, we study how this activation protocol can be used to entangle the starting systems themselves via entanglement swapping through a measurement on the ancillae. Furthermore, we bound the relative entropy of quantumness (a naturally arising measure of non-classicality in the scheme of Piani et al. above) for a special class of separable states, the so-called classical–quantum states. In particular, we fully characterize the classical–quantum two-qubit states that are maximally non-classical.