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arXiv:1808.01198 · Entropy

Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems

Ana C. S. Costa, Roope Uola, Otfried Gühne

Published Oct 5, 2018• 35 citations
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Abstract

The effect of quantum steering describes a possible action at a distance via local measurements. Whereas many attempts on characterizing steerability have been pursued, answering the question as to whether a given state is steerable or not remains a difficult task. Here, we investigate the applicability of a recently proposed method for building steering criteria from generalized entropic uncertainty relations. This method works for any entropy which satisfy the properties of (i) (pseudo-) additivity for independent distributions; (ii) state independent entropic uncertainty relation (EUR); and (iii) joint convexity of a corresponding relative entropy. Our study extends the former analysis to Tsallis and Rényi entropies on bipartite and tripartite systems. As examples, we investigate the steerability of the three-qubit GHZ and W states.

DOI
10.3390/e20100763

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The effect of quantum steering describes a possible action at a distance via local measurements. Whereas many attempts on characterizing steerability have been pursued, answering the question as to whether a given state is steerable or not remains a difficult task. Here, we investigate the applicability of a recently proposed method for building steering criteria from generalized entropic uncertainty relations. This method works for any entropy which satisfy the properties of (i) (pseudo-) additivity for independent distributions; (ii) state independent entropic uncertainty relation (EUR); and (iii) joint convexity of a corresponding relative entropy. Our study extends the former analysis to Tsallis and Rényi entropies on bipartite and tripartite systems. As examples, we investigate the steerability of the three-qubit GHZ and W states.
Abstract EPR steering is an asymmetric quantum correlation, as such, it is an important resource for quantum information protocols, like those used in quantum key distribution that are of tremendous relevance nowadays. The security of the aforementioned protocols can be shown by using entropic uncertainty measures, which are based on quantum information quantities, among which the entropies are found. In this work, we propose an entropic measure of steering, starting from the detection of steering given in a previous entropic uncertainty relation-based criterion, as well as using a generalised entropy, the Tsallis entropies. We define the entropic measure of steering in terms of an indicator, a quantity depending on the entropies of the measurements of the system, which assigns a value to the steering content of quantum states, while also comprising the asymmetry property of steering. We investigate the properties of the proposed quantity as a measure of steering and its asymmetry and show how it can be used for specific examples. For instance, a class of noisy two-qubit states that includes the Werner states is used to demonstrate the potential of such a proposal. Furthermore, a comparison is also made with the detection of steering in these states using existing steering criteria and the amount of steering a geometric measure for the Werner state gives.
Einstein-Podolsky-Rosen steering is a kind of powerful nonlocal quantum resource in quantum information processing such as quantum cryptography and quantum communication. Many criteria have been proposed in the past few years to detect steerability, both analytically and numerically, for bipartite quantum systems. We propose effective criteria for tripartite steerability and genuine tripartite steerability of three-qubit quantum states by establishing connections between the tripartite steerability (resp. genuine tripartite steerability) and the tripartite entanglement (resp. genuine tripartite entanglement) of certain corresponding quantum states. From these connections, tripartite steerability and genuine tripartite steerability can be detected without using any steering inequalities. The "complex cost" of determining tripartite steering and genuine tripartite steering can be reduced by detecting the entanglement of the newly constructed states in the experiment. Detailed examples are given to illustrate the power of our criteria in detecting the (genuine) tripartite steerability of tripartite states.
We establish a general connection between entropic uncertainty relations, Einstein-Podolsky-Rosen steering, and joint measurability. Specifically, we construct steering inequalities from any entropic uncertainty relation, given that the latter satisfies two natural properties. We obtain steering inequalities based on R\'enyi entropies. These turn out to be tight in many scenarios using max- and min-entropies. Considering steering tests with two noisy measurements, our inequalities exactly recover the noise threshold for steerability. This is the case for any pair of qubit two-outcome measurements as well as for pairs of mutually unbiased bases in any dimension. This shows that easy-to-evaluate quantities, such as entropy, can optimally witness steering, despite the fact that they are coarse-grained representations of the underlying statistics.
We propose a general framework for constructing universal steering criteria that are applicable to arbitrary bipartite states and measurement settings of the steering party. The same framework is also useful for studying the joint measurement problem. Based on the data-processing inequality for an extended Rényi relative entropy, we then introduce a family of steering inequalities, which detect steering much more efficiently than those inequalities known before. As illustrations, we show unbounded violation of a steering inequality for assemblages constructed from mutually unbiased bases and establish an interesting connection between maximally steerable assemblages and complete sets of mutually unbiased bases. We also provide a single steering inequality that can detect all bipartite pure states of full Schmidt rank. In the course of study, we generalize a number of results intimately connected to data-processing inequalities, which are of independent interest.