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arXiv:quant-ph/0606119 · Physical Review A

Quantification of entanglement via uncertainties

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

Published Mar 12, 2007• 28 citations
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Abstract

We show that entanglement of pure multiparty states can be quantified by means of quantum uncertainties of certain basic observables through the use of a measure that was initially proposed by Klyachko et al. [Appl. Phys. Lett. 88, 124102 (2006)] for bipartite systems.
DOI
10.1103/physreva.75.032315

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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 present a significantly improved scheme of entanglement detection inspired by local uncertainty relations for a system consisting of two qubits. Developing the underlying idea of local uncertainty relations, namely correlations, we demonstrate that it is possible to define a measure which is invariant under local unitary transformations and which is based only on local measurements. It is quite simple to implement experimentally and it allows entanglement quantification in a certain range for mixed states and exactly for pure states, without first obtaining full knowledge (e.g., through tomography) of the state.
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We present a measure of quantum correlation of an ensemble of multiparty states. It is based on the idea of minimal entropy production in a measurement whose elements are locally distinguishable. It is shown, for pure ensembles, to be a relative entropy distance from a set of ensembles. For pure bipartite ensembles, which span the whole bipartite Hilbert space, the measure is bounded below by the average entanglement. We naturally obtain a monotonicity axiom for any measure of quantum correlation of ensembles for single systems. We evaluate the measure for certain cases. Subsequently we use this measure to propose a complementarity relation between our measure and the accessible information obtainable about the ensemble under local operations. The measure is well defined even for the case of a single system, where the complementarity relation is seen to be yet another face of the ``Heisenberg uncertainty relation.''