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Quantum Information and Computation

Entanglement and nonlocality for a mixture of a pair-coherent state

Stefano Mancini, P. Tombesi

Published Mar 1, 2003• 7 citations
Publisher

Abstract

We consider a bipartite continuous variables quantum mixture coming from phase randomization of a pair-coherent state. We study the nonclassical properties of such a mixture. In particular, we quantify its degree of entanglement, then we show possible violations of Bell's inequalities. We also consider the use of this mixture in quantum teleportation. Finally, we compare this mixture with that obtained from a pair-coherent state by single photon loss.

DOI
10.26421/qic3.2-2

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We consider a bipartite continuous variables quantum mixture coming from phase randomization of a pair-coherent state. We study the nonclassical properties of such a mixture. In particular, we quantify its degree of entanglement, then we show possible violations of Bell's inequalities. We also consider the use of this mixture in quantum teleportation. Finally, we compare this mixture with that obtained from a pair-coherent state by single photon loss.
We consider the probability that a bipartite quantum state contains phase-conjugate-state (PCS) pairs and/or identical-state pairs as signatures of quantum entanglement. While the fraction of the PCS pairs directly indicates the property of a maximally entangled state, the fraction of the identical-state pairs negatively determines antisymmetric entangled states such as singlet states. We also consider the physical limits of these probabilities. This imposes fundamental restrictions on the pair appearance of the states with respect to the local access of the physical system. For continuous-variable system, we investigate similar relations by employing the pairs of phase-conjugate coherent states. We also address the role of the PCS pairs for quantum teleportation in both discrete-variable and continuous-variable systems.
We investigate teleportation of bipartite pure entanglement through two equally noisy quantum channel states, namely, a mixture of Bell states. There is a particular mixed-state channel for which all pure entangled states in a known biorthogonal product basis remain entangled after this inexact teleportation, which occurs for the duration of the entanglement. The Werner state channel lacks both these features. The relation of these noisy channels to the violation of Bell's inequality and quantum two-entropy inequality is studied.
Quantum coherence and quantum entanglement are two different manifestations of the superposition principle. In this paper, we show that the right choice of basis to be used to estimate coherence is the separable basis. The quantum coherence estimated using the Bell basis does not represent the coherence in the system, since there is a coherence in the system due to the choice of the basis states. We first compute the entanglement and quantum coherence in the two-qubit mixed states prepared using the Bell states and one of the states from the computational basis. The quantum coherence is estimated using the [Formula: see text]-norm of coherence, the entanglement is measured using the concurrence and the mixedness is measured using the linear entropy. Then we estimate these quantities in the Bell basis and establish that coherence should be measured only in separable basis, whereas entanglement and mixedness can be measured in any basis. We then calculate the teleportation fidelity of these mixed states and find the regions where the states have a fidelity greater than the classical teleportation fidelity. We also examine the violation of the Bell-CHSH inequality to verify the quantum nonlocal correlations in the system. The estimation of the above-mentioned quantum correlations, teleportation fidelity and the verification of Bell-CHSH inequality is also done for bipartite states obtained from the tripartite systems by the tracing out of one of their qubits. We find that for some of these states’ teleportation is possible even when the Bell-CHSH inequality is not violated, signifying that nonlocality is not a necessary condition for quantum teleportation.
We analyze nonclassical correlations between outcomes of measurements conducted on two spatial radiation modes. These correlations cannot be simulated with statistical mixtures of coherent states or, more generally, with non-negative phase-space functions of quantum states and measurements. We argue that nonclassical correlations are naturally related to Bell nonlocality, the former being a more general class of quantum correlations. Indeed, it is known that local realistic as well as noncontextual models correspond to non-negative solutions to a system of linear equations for the joint probability distributions of all observables. We demonstrate that nonclassical correlations correspond to a particular solution to this system, which may have negative values even if local realism is not violated. A modification of Bell inequalities enables us to test such correlations. At the same time, our approach leads to a formulation of Bell inequalities applicable also to continuous variables. The results are illustrated with two-mode squeezed vacuum states and with hybrid entangled states (Schr\"odinger-Cat states), one mode being analyzed by balanced and the other one by unbalanced homodyne detection.
The violation of Bell’s inequality in quantum mechanics implies that there exist nonlocality and entanglement. When the density matrix of a composite system cannot be written as a convex combination of the product of the density matrices of its subsystems, we say there exists entanglement. For pure states, the existence of entanglement always leads to the violation of Bell’s inequality. However, in the case of the mixed states, there may be entanglement, but Bell's inequality is not violated and in other words, the nonlocality is not manifested. In addition to Bell's inequality, quantum teleportation is also a manifestation of nonlocality. Quantum teleportation using entangled states is more successful than quantum teleportation with separable states. Therefore, the corresponding fidelity of teleported state with the initial state (in short, the fidelity) of the former is always greater than the fidelity of the latter. In this paper, for Werner's state, we will show that in a range of the related parameter, while the CHSH inequality is violated, the fidelity, which indicates the amount of success of quantum teleportation, is lower than the upper bound of the corresponding fidelity for states that can be simulated with a local hidden variable theory. Meanwhile, we will see that for Gisin's state with hidden nonlocality, filtering, which leads to the appearance of nonlocality and more specifically leads to the violation of the CHSH inequality, also increases the fidelity.
We consider a continuous-variable Clauser-Horne Bell-type inequality to study nonlocality in four-mode continuous-variable systems, which goes beyond two-photon states and can be applied to mixed as well as to states with fluctuating photon number. We apply the inequality to a wide variety of states such as pure and mixed Gaussian states (including squeezed thermal states) and non-Gaussian states. We consider beam splitters as a model for leakage and show that the inequality is able to detect nonlocality of noisy Gaussian states as well. Finally, we investigate nonlocality in pair-coherent states and entangled coherent states, which are prominent examples of nonclassical, non-Gaussian states.