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arXiv:1611.00149 · Physical Review A

Quantification of concurrence via weak measurement

Mikko Tukiainen, H. Kobayashi, Yutaka Shikano

Published May 1, 2017• 10 citations
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Abstract

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.

DOI
10.1103/physreva.95.052301

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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 revisit the problem of detection of entanglement of an unknown two-qubit state using minimal resources. Using weak values and just two copies of an arbitrary two-qubit state, we present a protocol where a postselection measurement in the computational basis provides enough information to identify if the state is entangled or not. Our protocol enables complete state identification with a single-setting postselection measurement on two copies of the state. It follows that by restricting to pure states, the global interaction required for determining the weak values can be realized by local operations. We further show that our protocol is robust against errors arising from inappropriate global interactions applied during weak value determination.
This paper proposes a method to measure directly the concurrence of an arbitrary two-qubit pure state based on a generalized Grover quantum iteration algorithm and a phase estimation algorithm. The concurrence can be calculated by applying quantum algorithms to two available copies of the bipartite system, and a final measurement on the auxiliary working qubits gives a better estimation of the concurrence. This method opens new prospects of entanglement measure by the application of quantum algorithms. The implementation of the protocol would be an important step toward quantum information processing and more complex entanglement measure of the finite-dimensional quantum system with an arbitrary number of qubits.
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.
Numerous work had been done to quantify the entanglement of a two-qubit quantum state, but it can be seen that previous works were based on joint measurements on two copies or more than two copies of the quantum state under consideration. In this work, we show that a single copy and two measurements are enough to estimate the entanglement quantifier like entanglement negativity and concurrence. To achieve our goal, we establish a relationship between the entanglement negativity and the minimum eigenvalue of the structural physical approximation of the partial transpose of an arbitrary two-qubit state. The derived relation makes it possible to estimate entanglement negativity experimentally by Hong-Ou-Mandel interferometry with only two detectors. Also, we derive the upper bound of the concurrence of an arbitrary two-qubit state and have shown that the upper bound can be realized in experiment. We will further show that the concurrence of i) an arbitrary pure two-qubit state and ii) a particular class of mixed states, namely, rank-2 quasi-distillable mixed states, can be exactly estimated with two measurements.