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arXiv:quant-ph/0003058 · Physical Review A

Local and nonlocal properties of Werner states

Tohya Hiroshima, Satoshi Ishizaka

Published Sep 13, 2000• 46 citations
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Abstract

We consider a special kind of mixed states---a Werner derivative, which is the state transformed by nonlocal unitary---local or nonlocal---operations from a Werner state. We show the following: (i) The amount of entanglement of Werner derivatives cannot exceed that of the original Werner state. (ii) Although it is generally possible to increase the entanglement of a single copy of a Werner derivative by local quantum operations and classical communication, the maximal possible entanglement cannot exceed the entanglement of the original Werner state. The extractable entanglement of Werner derivatives is limited by the entanglement of the original Werner state.

DOI
10.1103/physreva.62.044302

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We consider a special kind of mixed states---a Werner derivative, which is the state transformed by nonlocal unitary---local or nonlocal---operations from a Werner state. We show the following: (i) The amount of entanglement of Werner derivatives cannot exceed that of the original Werner state. (ii) Although it is generally possible to increase the entanglement of a single copy of a Werner derivative by local quantum operations and classical communication, the maximal possible entanglement cannot exceed the entanglement of the original Werner state. The extractable entanglement of Werner derivatives is limited by the entanglement of the original Werner state.
Linden, Massar, and Popescu have recently given an optimization argument to show that a single two-qubit Werner state, or any other mixture of the maximally entangled Bell states, cannot be purified by local operations and classical communications. We generalize their result and give a simple explanation. In particular, we show that no purification scheme using local operations and classical communications can produce a pure singlet from any mixed state of two spin- 1/21/21/2 particles. More generally, no such scheme can produce a maximally entangled state of any pair of finite-dimensional systems from a generic mixed state. We also show that the Werner states belong to a large class of states whose fidelity cannot be increased by such a scheme.
Entanglement and Bell nonlocality are used to describe quantum inseparabilities. Bell-nonlocal states form a strict subset of entangled states. A natural question arises concerning how much territory Bell nonlocality occupies entanglement for a general two-qubit entangled state. In this work, we investigate the relation between entanglement and Bell nonlocality by using lots of randomly generated two-qubit states, and give out a constraint inequality relation between the two quantum resources. For studying the upper or lower boundary of the inequality relation, we discover maximally (minimally) nonlocal entangled states, which maximize (minimize) the value of the Bell nonlocality for a given value of the entanglement. Futhermore, we consider a special kind of mixed state transformed by performing an arbitrary unitary operation on werner state. It is found that the special mixed state's entanglement and Bell nonlocality are related to ones of a pure state transformed by the unitary operation performed on the Bell state.
It has been suggested that there may exist quantum correlations that go beyond entanglement. The existence of such correlations can be revealed by information theoretic quantities such as quantum discord, but not by the conventional measures of entanglement. We argue that a state displays quantumness, that can be of local and nonlocal origin. Information theoretic measures not only characterize the nonlocal quantumness, but also the local quantumness, such as the “local superposition”. This can be a reason, why such measures are nonzero, when there is no entanglement. We consider a generalized version of the Werner state to demonstrate the interplay of local quantumness, nonlocal quantumness and classical mixedness of a state.
Maximally entangled states, Werner state and maximally entangled mixed states (MEMS) have been created and fully characterized by a novel high brilliance universal source of entangled photon pairs with striking spatial characteristics. Mixed states of any structure, spanning a 2 x 2 Hilbert space may be created by this source. The non local properties of the generated entanglement have been tested by standard Bell measurements. Tunable Werner states and Maximally Entangled Mixed States (MEMS) have been created by an original patchwork technique and investigated by quantum tomography. The entropic and nonlocal properties of these states have been also undertaken.
It has been suggested that there may exist quantum correlations that go beyond entanglement. The existence of such correlations can be revealed by quantum discord, but not by the conventional measure of entanglement. We argue that a state displays quantumness that can be of and nonlocal origin. The physical quantity such as the quantum discord probes not only the nonlocal quantumness but also the quantumness, such as the local superposition. This can be a reason why such measures are non-zero when there is no entanglement. We consider a generalized version of the Werner state to demonstrate the interplay of quantumness, nonlocal quantumness, and classical mixedness of a state.