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

Abstract

Quantum steering is a relatively simple test for quantumness of correlations, 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 inaccessible in classical world, leading to violation of some 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 provide experimentally feasible signature of unbounded violation of a steering inequality. We derive its universal form where tolerance for measurement-setting-errors is explicitly build-in by means of the Deutsch-Maassen-Uffink uncertainty relation. Then, generalizing the mutual unbiasedness, we apply the inequality to the multi-singlet and multi-particle bipartite Bell-state. However, the method is general and opens the possibility of employing multi-particle bi-partite steering for randomness certification and development of quantum technologies, e.g. random access codes. In their famous paper, Einstein, Podolsky and Rosen (EPR) highlighted the phenomenon of entanglement [1]: it is possible to see perfect correlations between measurement outcomes obtained by two observers, Alice and Bob, at distant locations, while for each observer his/her outcomes appear to be statistically random. These are the EPR correlations. Validation of entanglement requires designing a specific experimental scenario where measurements on a quantum state give outcomes which violate a classical inequality. The inequality can be constructed , for example, on the basis of probability distribution satisfying the Kolmogorov axioms. Its unbounded violation is equivalent to observation of the EPR correlations which become more and more pronounced when size of a system increases, reaching even classical limit of macroscopic population. This is very challenging to accomplish in the paradigm of Bell-nonlocality testing: if specific observables with (2 log 2 d) d settings and d possible outcomes are used, bipartite quantum states with local Hilbert space dimension d can violate a Bell inequality by a factor of O √ d log 2 d [2], later improved to O √ d log d [3-5]. Thus, an unbounded violation of a Bell inequality requires exponentially many observables (or equivalently, settings). According to the monogamy relation [6], this scaling can be improved only up to the linear one (see the Supplementary Information). How-* present results are still far from this limit and this makes them purely academic as far as the experimental perspective is concerned [7]. In case of quantum steering, the task has been found to be less difficult: violation of a steering inequality by a factor of O √ d requires d + 1 observables in the form of mutually unbiased bases (MUBs) [8]. However, this scenario necessitates the complementarity relation among the bases to be fulfilled exactly, which is experimentally impossible to attain. Here we provide a gen...

Maximal violation of steering inequalities and the matrix cube

“…The authors show that increasing the dimension of the quantum systems, the violation of certain steering inequalities can become arbitrarily large. Experimental demonstrations of this fact was proposed in [44,49]. Our work continues this line of investigation by making the connection to inclusions of free spectrahedra, in particular the matrix cube, which is a well studied object in convex optimization theory [7].…”

Device-independent verification of Einstein-Podolsky-Rosen steering

“…Finally, it is pointed out that there exist possible ways to improve the performance of our DI steering protocol and avoid the potential loopholes. For example, the delicate methods proposed in [15,25,43] may help to tolerate more worse transmission loss and measurement eciency. Moreover, the resource ecient method used in [44] could improve the success probability of the partial BSM, and the self-testing could be more noise robust by adopting other techniques [10].…”

Experimental verification of multidimensional quantum steering

“…See Ref. [32] for detailed discussions about the issue of imprecise experimental verification and how to overcome the problem by including the tolerance for measurement-setting errors. It also can be analyzed in an manner similar to self-testing multipartite entangled states [33].…”