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arXiv:quant-ph/0308091 · Physical Review A

Measurable entanglement criterion for two qubits

Hoshang Heydari, Gunnar G. Bjork, Luis Lorenzo Sánchez-Soto

Published Dec 18, 2003• 4 citations
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Abstract

We propose a directly measurable criterion for the entanglement of two qubits. We compare the criterion with other criteria, and we find that for pure states, and some mixed states, it coincides with the state's concurrence. The measure can be obtained with a Bell-state analyzer and the ability to make general local unitary transformations. However, the procedure fails to measure the entanglement of a general mixed two-qubit state.

DOI
10.1103/physreva.68.062314

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We propose a directly measurable criterion for the entanglement of two qubits. We compare the criterion with other criteria, and we find that for pure states, and some mixed states, it coincides with the state's concurrence. The measure can be obtained with a Bell-state analyzer and the ability to make general local unitary transformations. However, the procedure fails to measure the entanglement of a general mixed two-qubit state.
We propose a measure of two-qubit entanglement that is invariant under local unitary transformations, and which is based on local measurements covariances. It measures the Hilbert-Schmidt distance between the state and the product state obtained by multiplying the local density matrices. The measure has the benefit that the experimentalist need not have any a priori knowledge of the state to make the measurements. For pure states, the measure provides the state's concurrence directly, without resorting to state tomography. For statistically mixed states, the measure provides bounds for the concurrence. The two-qutrit case is also studied.
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We consider mixed states of two qubits and show under which global unitary operations their entanglement is maximized. This leads to a class of states that is a generalization of the Bell states. Three measures of entanglement are considered: entanglement of formation, negativity, and relative entropy of entanglement. Surprisingly all states that maximize one measure also maximize the others. We give a complete characterization of these generalized Bell states and prove that these states for fixed eigenvalues are all equivalent under local unitary transformations. Furthermore we characterize all nearly entangled states closest to the maximally mixed state and derive a lower bound on the volume of separable mixed states.