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Physical Review A

Universal local measure of bipartite entanglement

Alexander A. Klyachko, Barış Öztop, A.S. Shumovsky

Published Jun 14, 2006• 0 citations
OpenAlex

Abstract

We show that the complete information about the amount of entanglement carried by a bipartite pure state of any dimension (qubits, qutrits, etc), is determined by a local quantity that can be measured for either party of the system. This quantity is expressed in terms of averages of certain basic observables. In particular case of two entangled photon beams, this corresponds to the measurement of three Stokes parameters for either beam.

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We show that the complete information about the amount of entanglement carried by a bipartite pure state of any dimension (qubits, qutrits, etc), is determined by a local quantity that can be measured for either party of the system. This quantity is expressed in terms of averages of certain basic observables. In particular case of two entangled photon beams, this corresponds to the measurement of three Stokes parameters for either beam.
For a given pure state of a composite quantum system we analyze the product of its projections onto a set of locally orthogonal separable pure states. We derive a bound for this product analogous to the entropic uncertainty relations. For bipartite systems the bound is saturated for maximally entangled states and it allows us to construct a family of entanglement measures, we shall call collectibility. As these quantities are experimentally accessible, the approach advocated contributes to the task of experimental quantification of quantum entanglement, while for a three-qubit system it is capable to identify the genuine three-party entanglement.
We consider entanglement-swapping schemes with general (rather than maximally) entangled bipartite states of arbitary dimension shared pairwise between three or more parties in a chain. The intermediate parties perform generalized Bell measurements with the result that the two end parties end up sharing an entangled state which can be converted into maximally entangled states. We obtain an expression for the average amount of maximal entanglement concentrated in such a scheme and show that in a certain reasonably broad class of cases this scheme is provably optimal and that, in these cases, the amount of entanglement concentrated between the two ends is equal to that which could be concentrated from the weakest link in the chain.