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arXiv:1411.5994 · Physical Review Letters

Unbounded Violation of Quantum Steering Inequalities

Marcin Marciniak, A. Rutkowski, Zhi Yin, Michał Horodecki, Ryszard Horodecki

Published Oct 20, 2015• 55 citations
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Abstract

We construct steering inequalities that exhibit unbounded violation. The concept was to exploit the relationship between steering violation and the uncertainty relation. To this end, we apply mutually unbiased bases and anticommuting observables, known to exhibit the strongest uncertainty. In both cases, we are able to procure unbounded violations. Our approach is much more constructive and transparent than the operator space theory approach employed to obtain large violation of Bell inequalities. Importantly, using anticommuting observables we are able to obtain a dichotomic steering inequality with unbounded violation. Thus far, there is no analogous result for Bell inequalities. Interestingly, both the dichotomic inequality and one of our inequalities cannot be directly obtained from existing uncertainty relations, which strongly suggest the existence of an unknown kind of uncertainty relation.

DOI
10.1103/physrevlett.115.170401

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We construct steering inequalities that exhibit unbounded violation. The concept was to exploit the relationship between steering violation and the uncertainty relation. To this end, we apply mutually unbiased bases and anticommuting observables, known to exhibit the strongest uncertainty. In both cases, we are able to procure unbounded violations. Our approach is much more constructive and transparent than the operator space theory approach employed to obtain large violation of Bell inequalities. Importantly, using anticommuting observables we are able to obtain a dichotomic steering inequality with unbounded violation. Thus far, there is no analogous result for Bell inequalities. Interestingly, both the dichotomic inequality and one of our inequalities cannot be directly obtained from existing uncertainty relations, which strongly suggest the existence of an unknown kind of uncertainty relation.
In Junge and Palazuelos (2011 Commun. Math. Phys. 306 695–746 ) and Junge et al (2010 Commun. Math. Phys. 300 715–39 ) the operator space theory was applied to study bipartite Bell inequalities. The aim of the paper is to follow this line of research and use the operator space technique to analyze the steering scenario. We obtain a bipartite steering functional with unbounded largest violation of steering inequality, as well as constructing all ingredients explicitly. It turns out that the unbounded largest violation is obtained by a non maximally entangled state. Moreover, we focus on the bipartite dichotomic case where we construct a steering functional with unbounded largest violation of steering inequality. This phenomenon is different to the Bell scenario where only the bounded largest violation can be obtained by any bipartite dichotomic Bell functional.
In \cite{JP2011,JPPVW2010} the operator space theory was applied to study bipartite Bell inequalities. The aim of the paper is to follow this line of research and use the operator space technique to analyze the steering scenario. We obtain a bipartite steering functional with unbounded largest violation of steering inequality, as well as we can construct all ingredients explicitly. It turns out that the unbounded largest violation is obtained by non maximally entangled state. Moreover, we focus on the bipartite dichotomic case where we construct a steering functional with unbounded largest violation of steering inequality. This phenomenon is different to the Bell scenario where only bounded largest violation can be obtained by any bipartite dichotomic Bell functional.
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.
In this article, we study bipartite quantum steering using a general class of measurement operators known as the generalized equiangular measurement. Our approach allows for the construction of steering inequalities that are applicable not only to informationally complete measurements but also to incomplete or lossy scenarios, where the resulting quantum assemblages may be subnormalized. We develop a method to analytically bound the value of steering functionals under local hidden state (LHS) models by employing a generalized inequality for the eigenvalues of positive semidefinite operators, extending classical results such as the Samuelson inequality. This yields universal upper bounds on the LHS value in terms of the parameters defining the frame. Furthermore, we analyze the asymptotic behavior of these bounds in high-dimensional Hilbert spaces and demonstrate the possibility of unbounded quantum violations of steering inequalities. Our formalism encompasses previously known results for mutually unbiased bases and mutually unbiased measurements as special cases, showing that these are particular instances within a unified and more general framework.
Quantum steering, also called Einstein-Podolsky-Rosen steering, is the intriguing phenomenon associated with the ability of spatially separated observers to steer---by means of local measurements---the set of conditional quantum states accessible by a distant party. In the light of quantum information, all steerable quantum states are known to be resources for quantum information processing tasks. Here, via a quantity dubbed steering fraction, we derive a simple, but general criterion that allows one to identify quantum states that can exhibit quantum steering (without having to optimize over the measurements performed by each party), thus making an important step towards the characterization of steerable quantum states. The criterion, in turn, also provides upper bounds on the largest steering-inequality violation achievable by arbitrary finite-dimensional maximally entangled states. For the quantification of steerability, we prove that a strengthened version of the steering fraction is a convex steering monotone and demonstrate how it is related to two other steering monotones, namely, steerable weight and steering robustness. Using these tools, we further demonstrate the superactivation of steerability for a well-known family of entangled quantum states, i.e., we show how the steerability of certain entangled, but unsteerable quantum states can be recovered by allowing joint measurements on multiple copies of the same state. In particular, our approach allows one to explicitly construct a steering inequality to manifest this phenomenon. Finally, we prove that there exist examples of quantum states (including some which are unsteerable under projective measurements) whose steering-inequality violation can be arbitrarily amplified by allowing joint measurements on as little as three copies of the same state. For completeness, we also demonstrate how the largest steering-inequality violation can be used to bound the largest Bell-inequality violation and derive, analogously, a simple sufficient condition for Bell nonlocality from the latter.
In this work, we characterize the amount of steerability present in quantum theory by connecting the maximal violation of a steering inequality to an inclusion problem of free spectrahedra. In particular, we show that the maximal violation of an arbitrary unbiased dichotomic steering inequality is given by the inclusion constants of the matrix cube, which is a well-studied object in convex optimization theory. This allows us to find new upper bounds on the maximal violation of steering inequalities and to show that previously obtained violations are optimal. In order to do this, we prove lower bounds on the inclusion constants of the complex matrix cube, which might be of independent interest. Finally, we show that the inclusion constants of the matrix cube and the matrix diamond are the same. This allows us to derive new bounds on the amount of incompatibility available in dichotomic quantum measurements in fixed dimension.