Reference-frame-independent Einstein-Podolsky-Rosen steering

Protocols for testing or exploiting quantum correlations—such as entanglement, Bell nonlocality, and Einstein-Podolsky-Rosen steering—generally assume a common reference frame between two parties. Establishing such a frame is resource intensive and can be technically demanding for distant parties. While Bell nonlocality can be demonstrated with high probability for a large class of two-qubit entangled states when the parties have one or no shared reference direction, the degree of observed nonlocality is measurement-orientation dependent and can be arbitrarily small. In contrast, we theoretically prove that steering can be demonstrated with $100\%$ probability for a larger class of states, in a rotationally invariant manner, and experimentally demonstrate rotationally invariant steering in a variety of cases. We also show, by comparing with the steering inequality of Cavalcanti et al. [J. Opt. Soc. Am. B 32, A74 (2015)], that the steering inequality we derive is the optimal rotationally invariant one for the case of two settings per side and maximally mixed local qubit states.

Figure 1

Entangled photon pairs at 820 nm were produced via SPDC in a Sagnac interferometer consisting of a polarizing beam splitter (PBS), a dual-coated half-wave plate (HWP), two mirrors (M), and a periodically poled KTP crystal (ppKTP). The polarization of the pump is controlled by a Glan Taylor (GT) prism followed by a HWP. Different measurement settings are performed by rotating half- and quarter-wave plates (QWP) relative to the PBSs. Long pass (LP) filters, a dichroic mirror (DI), and an additional bandpass (BP) filter in Bob's line remove 410-nm pump photons copropagating with the 820-nm photons, before photons are coupled into single-mode fibers and detected by single-photon counting modules and counting electronics.

Figure 2

Poincaré (Bloch) spheres contain vectors showing one of the eigenstates of the three relevant directions (blue, red, and green) in the experiments we performed. In each of the experiments, measurements are made along two or three of these directions. (a) Bob uses the same three measurement directions in each of the $n=3$ experiments, while using only the red and blue directions for experiments with $n=2$. (b) Alice's directions, in the case where Alice and Bob share a reference direction [shown as green in (a)]. We test the invariance of the $m=n=2$ RIS inequality under rotations in the plane (grey), as the blue and red settings are rotated through $90{}^{\circ }$ in steps (blue and red dots). (c) Alice's directions for $m=2$ (blue and red dots) when her plane of measurement directions is tilted by $\mathrm{\Phi }={64}^{\circ }$ and the settings are rotated in that plane, while maintaining local orthogonality. (d) Same as (c), but with $\mathrm{\Phi }={90}^{\circ }$. (e) Alice's directions for testing the $m=n=3$ RIS inequality (blue and red dots, and green axis), corresponding to measurement triads strongly misaligned with respect to Bob's directions in (a). (f) Nonorthogonal measurement directions for Alice.

Figure 3

Steering parameter vs rotation angle $\alpha$ (degrees) in Alice's measurement plane, for experiments with $m=n=2$ measurement directions (cases in Secs. 5a and 5b of the main text). The plane tilt angle takes on the values (a) $\mathrm{\Phi }=0^{\circ }$, (b) $\mathrm{\Phi }={64}^{\circ }$, and (c) $\mathrm{\Phi }={90}^{\circ }$ [see Figs. 2–2 respectively]. For all angles, we calculated the theoretically expected curves for the RIS inequality (2) (blue) and the CFFW inequality (4) (red) for the Werner state. The Werner parameter $W$ of the closest Werner state was calculated from the average tomographic data. The RIS data are represented by blue squares, and the CFFW data by red circles. The error bars are too small to be seen. Data points in the upper white region imply steering of Bob by Alice.