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arXiv:1603.07861 · Physical Review Letters

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

A. Rutkowski, Adam Buraczewski, Paweł Horodecki, Magdalena Stobińska

Published Jan 9, 2017• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.118.020402

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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.
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.
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.
Recent developments, such as the experimental realization of large-scale cluster states, have built a valid case for continuous-variable quantum optics as a promising platform for quantum information processing. The capability of creating non-Gaussian states is key to building a universal quantum computer and achieving a quantum computational advantage. On the other hand, quantum correlations are also at the core of current developments in quantum technologies. Yet, quantum correlations in non-Gaussian states are still poorly understood for continuous-variable systems. In this contribution we will focus on quantum steering, where Alice and Bob each share a part of bipartite quantum state and perform local measurements on their respective subsystem. Quantum steering from Alice to Bob occurs when Bob can exploit information from Alice’s measurements to infer the outcome of his observables’ measurement more precisely than allowed by classical correlations. The paradigmatic example for this phenomenon is found when Alice and Bob both measure field quadratures. In this case, Bob can construct conditional variances that violate Heisenberg’s inequality. This violation, known as Reid’s criterion, is a signature of quantum steering that relies purely on Gaussian features of the state. More generally, we speak of Gaussian steering when we can violate steering inequality using only information from the state’s covariance matrix. For non-Gaussian states, the covariance matrix only offers limited information about the state, and many properties remain under the radar. Therefore, certification protocols of quantum steering for non-Gaussian states are scarce and generally highly demanding from an experimental point of view. In this contribution, we use a recently established connection between quantum steering and the (quantum) Fisher information to develop a new protocol for detection of quantum steering in non-Gaussian. This protocol relies exclusively on homodyne measurements.
Quantum steering, as a manifestation of nonlocal quantum correlations, plays a crucial role in enabling various quantum information processing tasks. However, practical implementations are often hindered by significant challenges arising from imperfect or untrusted measurement devices. This study investigates the impact of measurement inaccuracies on quantum steering, with a particular focus on errors in the untrusted party's measurement devices. We first analyze how such errors affect the evaluation of steering inequalities, and then derive bipartite steering inequalities based on correlation matrices under imperfect measurements. Our findings show that even small measurement errors can significantly compromise the certification of quantum steerability, an effect that becomes particularly pronounced as the system dimension increases. Furthermore, by extending the proposed steering inequality to a modified tripartite scenario via correlation matrices, we demonstrate that the influence of measurement imperfections is far more severe in multipartite quantum steering than in the bipartite case. Our results underscore the critical need to account for measurement imperfections in experimental quantum steering and provide a theoretical framework for characterizing and mitigating these effects in high-dimensional quantum systems.
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.