Quantum Steering Inequality with Tolerance for Measurement-Setting Errors: Experimentally Feasible Signature of Unbounded Violation

Quantum steering is a relatively simple test for proving that the values of quantum-mechanical measurement outcomes come into being only in the act of measurement. By exploiting quantum correlations, Alice can influence—steer—Bob’s physical system in a way that is impossible in classical mechanics, as shown by the violation of steering inequalities. Demonstrating this and similar quantum effects for systems of increasing size, approaching even the classical limit, is a long-standing challenging problem. Here, we prove an experimentally feasible unbounded violation of a steering inequality. We derive its universal form where tolerance for measurement-setting errors is explicitly built in by means of the Deutsch–Maassen–Uffink entropic uncertainty relation. Then, generalizing the mutual unbiasedness, we apply the inequality to the multisinglet and multiparticle bipartite Bell state. However, the method is general and opens the possibility of employing multiparticle bipartite steering for randomness certification and development of quantum technologies, e.g., random access codes.

Figure 1

Quantum steering scenario: Alice and Bob share a bipartite state $\rho$, either entangled (green) or separable (red), and Bob would like to verify which one this is. Alice performs a measurement using an untrusted device of one of her $N$ observables $A_x$, and communicates to Bob the outcome $a$ which she receives with probability $p(a|x)$. Bob applies his measurement ${\sigma }_x^a$ and checks violation of the steering inequality (8).

Figure 2

Quantum violation $V_Q^{\eta }$ of steering inequality (9) for the $\epsilon$-generalized MUBs with $\epsilon =0.1$, singlet state fidelity $F=0.98$, and $\sigma =0.5$ as a function of detection efficiency $\eta$.

Figure 3

Quantum violation $V_Q$ of steering inequality (9) for multiparticle Bell-singlet states $|{\psi }_n⟩$ (10) and rotation angle $\theta =\pi /2N$. The upper inset shows the optimal number of settings $N_{\text{opt}}$ for a given $n$ used in this computation. Gray area indicates a range of values of $V_Q$ for $N<N_{\text{opt}}$.

Figure 4

Quantum violation $V_{\mathrm{Q}}^{\eta }$ of steering inequality (9) computed for multiparticle Bell-singlet states $|{\psi }_n⟩$ (10), detection efficiency $\eta$, and the best $N$ found for given $n$ (see Fig. 3). White area indicates no violation.