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arXiv:2301.09326 · Physical Review A

Quantum conditional entropies and steerability of states with maximally mixed marginals

Komal Kumar, Nirman Ganguly

Published Mar 10, 2023• 4 citations
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Abstract

Quantum steering is an asymmetric correlation which occupies a place between entanglement and Bell nonlocality. In the paradigmatic scenario involving the protagonists Alice and Bob, the entangled state shared between them is said to be steerable from Alice to Bob if the steering assemblage on Bob's side does not admit a local hidden state (LHS) description. Quantum conditional entropies, on the other hand, provide for another characterization of quantum correlations. Contrary to our common intuition, conditional entropies for some entangled states can be negative, marking a significant departure from the classical realm. Quantum steering and quantum nonlocality, in general, share an intricate relation with quantum conditional entropies. In the present contribution, we investigate this relationship. For a significant class, namely the two-qubit Weyl states, we show that negativity of conditional R\'enyi 2-entropy and conditional Tsallis 2-entropy is a necessary and sufficient condition for the violation of a suitably chosen three settings steering inequality. With respect to the same inequality we find an upper bound for the conditional R\'enyi 2-entropy, such that the general two-qubit state is steerable. Moving from a particular steering inequality to local hidden-state descriptions, we show that some two-qubit Weyl states which admit a LHS model possess nonnegative conditional R\'enyi 2-entropy. However, the same does not hold true for some non-Weyl states. Our study further investigates the relation between nonnegativity of conditional entropy and LHS models in two-qudits for the isotropic and Werner states. There we find that, whenever these states admit a LHS model, they possess a nonnegative conditional R\'enyi 2-entropy. We then observe that the same holds true for a noisy variant of the two-qudit Werner state.

DOI
10.1103/physreva.107.032206

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Quantum steering is an asymmetric correlation which occupies a place between entanglement and Bell nonlocality. In the paradigmatic scenario involving the protagonists Alice and Bob, the entangled state shared between them is said to be steerable from Alice to Bob if the steering assemblage on Bob's side does not admit a local hidden state (LHS) description. Quantum conditional entropies, on the other hand, provide for another characterization of quantum correlations. Contrary to our common intuition, conditional entropies for some entangled states can be negative, marking a significant departure from the classical realm. Quantum steering and quantum nonlocality, in general, share an intricate relation with quantum conditional entropies. In the present contribution, we investigate this relationship. For a significant class, namely the two-qubit Weyl states, we show that negativity of conditional R'enyi 2-entropy and conditional Tsallis 2-entropy is a necessary and sufficient condition for the violation of a suitably chosen three settings steering inequality. With respect to the same inequality we find an upper bound for the conditional R'enyi 2-entropy, such that the general two-qubit state is steerable. Moving from a particular steering inequality to local hidden-state descriptions, we show that some two-qubit Weyl states which admit a LHS model possess nonnegative conditional R'enyi 2-entropy. However, the same does not hold true for some non-Weyl states. Our study further investigates the relation between nonnegativity of conditional entropy and LHS models in two-qudits for the isotropic and Werner states. There we find that, whenever these states admit a LHS model, they possess a nonnegative conditional R'enyi 2-entropy. We then observe that the same holds true for a noisy variant of the two-qudit Werner state.
The two-qubit maximally steerable mixed state (MSMS), defined as one that violates to the most degree a steering inequality for any fixed linear entropy, is derived mathematically here based on the two-setting (three-setting) linear steering inequality and the two-setting Clauser-Horne-Shimony-Holt--like steering inequality. Interestingly, the form of the MSMS based on such different steering inequalities obtained here is identical to the two-qubit maximally nonlocal mixed state (MNMS). It is clearly shown that finding any two-qubit state, of which the state mixedness exceeds 2/32/32/3, thus violating the three-setting steering inequalities, is impossible. The violation of the inequalities with the Werner state and with the maximally entangled mixed state, respectively, as well as the relations between their optimal violation and the linear entropy, is also discussed comprehensively. In particular, within the range $\ensuremath{\varepsilon}(\ensuremath{\rho})\ensuremath{\in}[0,2/3]$ of the fixed linear entropy, the Werner state reaches the same violation as the MSMS does of the three-setting linear steering inequality, but not of the two-setting steering inequalities (for the latter inequalities the violation with the Werner state is generally less than that with the MSMS). For the MEMS, the optimal violation is always lower than that of the MSMS for any fixed linear entropy.
Einstein-Podolsky-Rosen (EPR) steering is an intermediate type of quantum nonlocality which sits between entanglement and Bell nonlocality. A set of correlations is Bell nonlocal if it does not admit a local hidden variable (LHV) model, while it is EPR nonlocal if it does not admit a local hidden variable--local hidden state (LHV-LHS) model. It is interesting to know what states can generate EPR-nonlocal correlations in the simplest nontrivial scenario, that is, two projective measurements for each party sharing a two-qubit state. Here we show that a two-qubit state can generate EPR-nonlocal full correlations (excluding marginal statistics) in this scenario if and only if it can generate Bell-nonlocal correlations. If full statistics (including marginal statistics) is taken into account, surprisingly, the same scenario can manifest the simplest one-way steering and the strongest hierarchy between steering and Bell nonlocality. To illustrate these intriguing phenomena in simple setups, several concrete examples are discussed in detail, which facilitates experimental demonstration. In the course of study, we introduce the concept of restricted LHS models and thereby derive a necessary and sufficient semidefinite-programming criterion to determine the steerability of any bipartite state under given measurements. Analytical criteria are further derived in several scenarios of strong theoretical and experimental interest.
Quantum steering can be detected via the violation of steering inequalities, which provide sufficient conditions for the steerability of quantum states. Here we discuss the converse problem, namely, ensuring that an entangled state is unsteerable and hence Bell local. We present a simple criterion, applicable to any two-qubit state, that guarantees that the state admits a local hidden state model for arbitrary projective measurements. Specifically, we construct local hidden state models for a large class of entangled states, which thus cannot violate any steering or Bell inequality. In turn, this leads to sufficient conditions for a state to be only one-way steerable and provides the simplest possible example of one-way steering. Finally, by exploiting the connection between steering and measurement incompatibility, we give a sufficient criterion for a continuous set of qubit measurements to be jointly measurable.
The bipartite state is said to be steerable iff it does not have single system description, i.e., the bipartite state can not be explained by local hidden state model [Phys. Rev. Lett. {\bf 98}, 140402 (2007)]. Several steering inequalities have been derived using different local uncertainty relations to verify the control on the state of one subsystem by other. Here, we derive complementarity kind of relations between coherences measured on mutually unbiased bases using various coherence measures like the l1l_1l1​-norm, relative entropy and skew information. Using these relations, we formulate steering inequalities to check whether one subsystem has control to the coherence of the other entangled subsystem. According to our results, all pure entangled states are maximally steerable but for mixed entangled states, coherence steerability needs stronger non-local correlation than the existing state steering inequalities. We also show that our steering criteria is monogamous.
For a bipartite entangled state shared by two observers, Alice and Bob, Alice can affect the postmeasured states left to Bob by choosing different measurements on her half. Alice can convince Bob that she has such an ability if and only if the unnormalized postmeasured states cannot be described by a local-hidden-state (LHS) model. In this case, the state is termed steerable from Alice to Bob. By converting the problem to construct LHS models for two-qubit Bell diagonal states to the one for Werner states, we obtain the optimal models given by Jevtic et al. [J. Opt. Soc. Am. B 32, A40 (2015)], which are developed by using the steering ellipsoid formalism. Such conversion also enables us to derive a sufficient criterion for unsteerability of any two-qubit state.
Phenomena that slip beyond the grasp of our classical intuition reveal uniquely quantum effects that deepen our understanding of the physical world and enable advances in information processing, particularly in quantum communication and computation. One such phenomenon is quantum steering, whereby measurements performed by one party influence the conditional states of another when the two share an entangled quantum system. If the observed correlations cannot be explained by a local hidden state model, the state is said to be steerable. In this work, we investigate the typicality of this behavior: given a generic two-qubit state and mmm Haar-random projective measurements, what is the probability of observing steering? We derive analytical expressions for the steering probability PS\mathcal{P}_SPS​ of Werner states in two- and three-setting scenarios, the latter restricted to coplanar projective measurements on the Bloch sphere. For larger numbers of settings and various random states ensembles, we perform numerical analyses showing that PS\mathcal{P}_SPS​ increases systematically with the number of measurements and substantially exceeds the corresponding probabilities associated with Bell nonlocality. Our results demonstrate that random states with minimal environmental coupling exhibit a high probability of steering for finite mmm and approach genuine typicality, PS=100%\mathcal{P}_S=100\%PS​=100%, as the number of settings increases. We provide a detailed characterization of PS\mathcal{P}_SPS​ across different state ensembles and specific families, including Bell-diagonal and Werner states, identifying those with the greatest non-classical potential and highlighting their relevance for protocols in which steering serves as a key resource.