Quantification of concurrence via weak measurement

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.

Figure 1

(a) Stereographical representation of the weak value of the state ${\rho }_A$. Following Ref. [29], the weak measurement of ${\sigma }_x$ on the state ${\rho }_A$, followed by postselection on $|0\rangle$ or $|1\rangle$, may be interpreted as stereographic projections of the qubit state ${\rho }_A$ on the two ${\mathbb{R}}^2$ planes that intersect the north and south pole of the Bloch sphere. For our purposes, the absolute values ${|}_{\langle 0|}{\langle {\sigma }_x^A\rangle }_{}^w|$ and ${|}_{\langle 1|}{\langle {\sigma }_x^A\rangle }_{}^w|$ are particularly important because they may be used to measure the distance between ${\rho }_A$ and the maximally mixed state $\frac{1}{2}{\mathit{1}}_A$, which in turn is related to the amount of entanglement. (b) Concurrence $C\left({\mathrm{\Psi }}_{AB}\right)$ in terms of ${|}_{\langle 0|}{\langle {\sigma }_x^A\rangle }_{}^w|$ and ${|}_{\langle 1|}{\langle {\sigma }_x^A\rangle }_{}^w|$. The concurrence is fully determined by these variables except for the point $(0,0)$, which corresponds to the black dashed line in panel (a). The white region equals the canceled area ${|}_{\langle 0|}{\langle {\sigma }_x^A\rangle }_{}^w{\left|\right|}_{\langle 1|}{\langle {\sigma }_x^A\rangle }_{}^w|>1$.

Figure 2

Weak measurement setup for determining concurrence in the two-photon polarization state $|{\mathrm{\Psi }}_{AB}\rangle$. The initial pointer state is prepared as the Laguerre-Gaussian (LG) mode using a mode converter. The weak interaction between eigenvectors of ${\sigma }_x$, $|\pm \rangle =\left(\right|0\rangle \pm |1\rangle )/\sqrt{2}$, can be implemented using a polarization Sagnac interferometer (PBS, polarization beam splitter, HWP, half waveplate).