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arXiv:2311.10552 · Entropy

Analysis of Quantum Steering Measures

Lucas Maquedano, Ana C. S. Costa

Published Mar 14, 2024• 4 citations
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Abstract

The effect of quantum steering describes a possible action at a distance via local measurements. In the last few years, several criteria have been proposed to detect this type of correlation in quantum systems. However, there are few approaches presented in order to measure the degree of steerability of a given system. In this work, we are interested in investigating possible ways to quantify quantum steering, where we based our analysis on different criteria presented in the literature.

DOI
10.3390/e26030257

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The effect of quantum steering describes a possible action at a distance via local measurements. In the last few years, several criteria have been proposed to detect this type of correlation in quantum systems. However, there are few approaches presented in order to measure the degree of steerability of a given system. In this work, we are interested in investigating possible ways to quantify quantum steering, where we based our analysis on different criteria presented in the literature.
In the last few years, several criteria to identify Einstein-Podolsky-Rosen steering have been proposed and experimentally implemented. On the operational side, however, the evaluation of the steerability degree of a given state has shown to be a difficult task and only a few results are known. In this work, we propose a measure of steering that is based on the maximal violation of well-established steering inequalities. Applying this approach to two-qubit states, we managed to derive simple closed formulas for steering in the two- and three-measurement scenarios. We also provide closed formulas for quantifiers of Bell nonlocality in the respective scenarios. Finally, we show that our measures of steering verify the entanglement-steering-nonlocality hierarchy and reproduce results reported so far.
In the last few years, several criteria to identify Einstein-Podolsky-Rosen steering have been proposed and experimentally implemented. On the operational side, however, the evaluation of the steerability degree of a given state has shown to be a difficult task and only a few results are known. In this work, we propose a measure of steering that is based on the maximal violation of well-established steering inequalities. Applying this approach to two-qubit states, we managed to derive simple closed formulas for steering in the two- and three-measurement scenarios. We also provide closed formulas for quantifiers of Bell nonlocality in the respective scenarios. Finally, we show that our measures of steering verify the entanglement-steering-nonlocality hierarchy and reproduce results reported so far.
In quantum mechanics, if one party performs a measurement on one system, different outcomes can lead to different states for another system. This phenomenon is known as quantum steering. The thesis begins with some general results about steering: a classification of which states permit the most powerful type of steering, and the implications of this for quantum correlations. The first main topic is steering in two-qubit systems. It turns out that steering provides an excellent way to visualize two-qubit states, leading to a novel criteria for entanglement and a better understanding of separable states. Oddly, the structure of steering can be more subtle in separable states than it can be in entangled ones. Returning to general quantum systems, I then turn to the EPR paradox, and its generalisation using Local Hidden State models. I show that the lack of such models can be used to quantify the amount of entanglement shared by two parties, even when one of them does not trust their measuring devices. Finally, the desire to understand steering without invoking "spooky action-at- a-distance" leads to the idea that quantum states are states of knowledge. I explore some de Finetti theorems that help to make sense of this idea, but then show a significant roadblock the most natural formalisation of it. The main results in chapters 3 and 4 appear in [JPJR13] and [Pus13] respectively. The result in chapter 6 was improved in collaboration with my supervisors and published in [PBR12].
Quantum nonlocality is a prominent discrepancy when we compare quantum physics with classical physics, and it manifests in different forms, one of which is quantum steering. Quantum steering refers to a correlation produced from a bipartite entangled state between the local measurement outcomes of one party and the postmeasurement states of the other. Combining with uncertainty relation, local measurements can be used to distinguish whether a quantum state is steerable. Thus, this method can be treated as an effective steering criterion. In this work, we certify this steering criterion based on the local uncertainty relation by an all-optical setup with a mixed state which consists of two Bell-like states. Furthermore, we also testify that the quantum steering criterion based on the local uncertainty relation is not only superior to the linear steering criterion but also superior to the steering criterion based on the entropic uncertainty relation in our scenario. Our efforts may promote the use of this criterion in quantum information processing tasks.
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.