Detector entanglement: Quasidistributions for Bell-state measurements

Measurements in the quantum domain can exceed classical notions. This concerns fundamental questions about the nature of the measurement process itself, as well as applications, such as their function as building blocks of quantum information processing protocols. In this paper we explore the notion of entanglement for detection devices in theory and experiment. A method is devised that allows one to determine nonlocal quantum coherence of positive-operator-valued measures via negative contributions in a joint distribution that fully describes the measurement apparatus under study. This approach is then applied to experimental data for detectors that ideally project onto Bell states. In particular, we describe the reconstruction of the aforementioned entanglement quasidistributions from raw data and compare the resulting negativities with those expected from theory. Therefore, our method provides a versatile toolbox for analyzing measurements regarding their quantum-correlation features for quantum science and quantum technology.

Figure 1

Ideal entanglement quasidistributions $Q_k$ for $k\in \{0,x,y,z\}$ [Eq. (2)] for Bell-projection POVM elements [Eq. (3)]. Positive contributions ($+\frac{1}{3}$) display classical detector correlations while negativities ($-\frac{1}{6}$) are an unambiguous certification of POVM entanglement. Local projectors $|w_{\pm }\rangle \langle w_{\pm }|$ (left Bloch-sphere plots) are eigenstates of the Pauli matrices ${\sigma }_w$ for $w\in \{x,y,z\}$, allowing for a decomposition of the POVM elements according to Eq. (1).

Figure 2

Sketch of the detection scheme that implements a BSM. A comprehensive characterization of our realization can be found in Ref. [39]. Because of the final detection in the $D-A$ basis, the POVM elements are labeled as ${\mathrm{\Pi }}_{AA}, {\mathrm{\Pi }}_{AD}, {\mathrm{\Pi }}_{DA}$, and ${\mathrm{\Pi }}_{DD}$ throughout this work, indicating between which detectors coincidences have been recorded.

Figure 3

Raw data in the form of the total number of measured coincidences from both detectors (top plot) and the resulting relative coincidences for each POVM element ${\mathrm{\Pi }}_k$. Axes label the input product states for horizontal ($H$), vertical ($V$), diagonal ($D$), antidiagonal ($A$), right-circular ($R$), and left-circular ($L$) polarization for realizing the detector tomography. Note that bars for relative coincidences are filled to the value $\frac{1}{4}$ to easily distinguish between above- and below-average count rates, i.e., the deviation from uniformity of the total counts distributed among the four individual POVM elements.

Figure 4

Reconstructed ${\mathrm{\Pi }}_k$ in terms of the real (left column) and imaginary (right column) parts in the computational basis $|k^{\left(A\right)}\rangle \langle l^{\left(A\right)}|\otimes |k^{\left(B\right)}\rangle \langle l^{\left(B\right)}|$ for $k^{\left(A\right)},l^{\left(A\right)},k^{\left(B\right)},l^{\left(B\right)}\in \{0,1\}$. Imaginary contributions are comparably small. Real parts show relations to measurements in terms of Bell-state projectors. No corrections for imperfections have been carried out to determine the decomposition shown.

Figure 5

Transformation to the standard form of the correlation matrix $C_{AA}={\left[{\pi }_{w,w^{\prime }|AA}\right]}_{w,w^{\prime }\in \{0,x,y,z\}}$ that describes the POVM element ${\mathrm{\Pi }}_{AA}$ as an example. The initial matrix (i) is transformed such that ${\pi }_{w,0|AA}={\pi }_{0,w^{\prime }|AA}=0$ holds true for (ii), thus removing local correlations that are given via an identity in one subsystem. Then the fully diagonal form in (iii) is obtained by local rotations, resulting in coefficients ${\pi }_{w,w^{\prime }|AA}=0$ for $w\ne w^{\prime }$.

Figure 6

Local state transformation on the Bloch sphere as obtained from the diagonalization steps in Fig. 5 for ${\mathrm{\Pi }}_{AA}$, but in inverse order from left to right. Also compare Eqs. (12) and (13) in this context. The first transformation constitutes a rotation and the second one is an invertible but not orthogonality-preserving Lorentz boost.

Figure 7

The left column shows the reconstructed quasidistributions for all POVM elements, including a one-standard-deviation error margin (black bars). The right column shows the corresponding local states for the decomposition according to Eq. (1). In comparison with the ideal cases (Fig. 1), one can observe here the same general structure of POVM elements. In theory, the maximal negativity is $-\frac{1}{6}\approx -0.17$. Here we find the highest negativities as $-0.20\pm 0.06$ (note the error margin), $-0.14\pm 0.01, -0.13\pm 0.02$, and $-0.14\pm 0.02$ for $AA, AD, DA$, and $DD$, respectively.

Figure 8

Quasidistributions for POVMs that combine two Bell-like states and are therefore expected to be separable. Our reconstruction correctly reveals this feature.