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arXiv:0804.2246 · Physical Review Letters

Novel Schemes for Directly Measuring Entanglement of General States

Jianming Cai, Wei Song

Published Nov 7, 2008• 28 citations
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Abstract

An intrinsic relation between maximally entangled states and entanglement measures is revealed, which plays a role in establishing connections for different entanglement quantifiers. We exploit the basic idea and propose a framework to construct schemes for directly measuring entanglement of general states. In particular, we demonstrate that rank-1 local factorizable projective measurements, which are achievable with only one copy of an entangled state involved at a time in a sequential way, are sufficient to directly determine the concurrence of an arbitrary two-qubit entangled state.

DOI
10.1103/physrevlett.101.190503

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An intrinsic relation between maximally entangled states and entanglement measures is revealed, which plays a role in establishing connections for different entanglement quantifiers. We exploit the basic idea and propose a framework to construct schemes for directly measuring entanglement of general states. In particular, we demonstrate that rank-1 local factorizable projective measurements, which are achievable with only one copy of an entangled state involved at a time in a sequential way, are sufficient to directly determine the concurrence of an arbitrary two-qubit entangled state.
We demonstrate that any pure bipartite state of two qubits may be decomposed into a superposition of a maximally entangled state and an orthogonal factorizable one. Although there are many such decompositions, the weights of the two superposed states are, remarkably, unique. We propose a measure of entanglement based on this decomposition. We also demonstrate that this measure is connected to three measures of entanglement previously set forth: maximal violation of Bell's inequality, concurrence, and two-particle visibility.
Since entanglement is not an observable per se, measuring its value in practice is a difficult task. Here we propose a protocol for quantifying a particular entanglement measure, namely, concurrence, of an arbitrary two-qubit pure state via a single fixed measurement setup by exploiting so-called weak measurements and the associated weak values together with the properties of the Laguerre-Gaussian modes. The virtue of our technique is that it is generally applicable for all two-qubit systems and does not involve simultaneous copies of the entangled state. We also propose an explicit optical implementation of the protocol.
We propose an entanglement measure for pure M ⊗ N bipartite quantum states. We obtain the measure by generalizing the equivalent measure for a 2 ⊗ 2 system, via a 2 ⊗ 3 system, to the general bipartite case. The measure emphasizes the role Bell states have, both for forming the measure and for experimentally measuring the entanglement. The form of the measure is similar to the generalized concurrence. In the case of 2 ⊗ 3 systems, we prove that our measure, which is directly measurable, equals the concurrence. It is also shown that, in order to measure the entanglement, it is sufficient to measure the projections of the state onto a maximum of M ( M − 1) N ( N − 1)/2 Bell states.
It is shown that, despite strong nonlinearity, entanglement of formation of a two-qubit state can be measured without prior state reconstruction. Collective measurements on a small number of copies are provided that allow one to determine quantum concurrence via estimation of only four parameters. It is also pointed out that another entanglement measure based on so-called "negativity" can also be measured in a similar way. The result is related to the general problem: What kind of information can be extracted efficiently from an unknown quantum state?