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arXiv:1909.03963 · Physical Review A

Quantum steering of Bell-diagonal states with generalized measurements

H. Chau Nguyen, Otfried Gühne

Published Apr 27, 2020• 9 citations
arXivHTMLPDF

Abstract

The phenomenon of quantum steering in bipartite quantum systems can be reduced to the question of whether or not the first party can perform measurements such that the conditional states on the second party can be explained by a local hidden-state model. Clearly, the answer to this depends on the measurements which the first party is able to perform. We introduce a local hidden-state model explaining the conditional states for all generalized measurements on Bell-diagonal states of two qubits. More precisely, it is known for the restricted case of projective measurements and Bell-diagonal states characterized by the correlation matrix $T$ that a local hidden-state model exists if and only if ${R}_{T}=2\ensuremath{\pi}{N}_{T}\left|det(T)\right|\ensuremath{\ge}1$, where ${N}_{T}$ is defined by an integral over the Bloch sphere. For generalized measurements described by positive operator-valued measures we construct a model if ${R}_{T}\ensuremath{\ge}6/5$. Our work paves the way for a systematic study of steerability of quantum states with generalized measurements beyond the highly symmetric Werner states.

DOI
10.1103/physreva.101.042125

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The phenomenon of quantum steering in bipartite quantum systems can be reduced to the question of whether or not the first party can perform measurements such that the conditional states on the second party can be explained by a local hidden-state model. Clearly, the answer to this depends on the measurements which the first party is able to perform. We introduce a local hidden-state model explaining the conditional states for all generalized measurements on Bell-diagonal states of two qubits. More precisely, it is known for the restricted case of projective measurements and Bell-diagonal states characterized by the correlation matrix TTT that a local hidden-state model exists if and only if RT=2\ensuremathπNT∣det(T)∣\ensuremath≥1{R}_{T}=2\ensuremath{\pi}{N}_{T}\left|det(T)\right|\ensuremath{\ge}1RT​=2\ensuremathπNT​∣det(T)∣\ensuremath≥1, where NT{N}_{T}NT​ is defined by an integral over the Bloch sphere. For generalized measurements described by positive operator-valued measures we construct a model if RT\ensuremath≥6/5{R}_{T}\ensuremath{\ge}6/5RT​\ensuremath≥6/5. Our work paves the way for a systematic study of steerability of quantum states with generalized measurements beyond the highly symmetric Werner states.
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.
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.
Not all entangled states can exhibit quantum steering, and determining whether a given entangled state is steerable is a crucial problem in quantum information theory. The main challenge lies in verifying the existence of a local hidden-state (LHS) model capable of reproducing all post-measurement assemblages generated by arbitrary measurements. To address this, we propose a machine learning-based framework that employs batch sampling of measurements and gradient-based optimization to construct an optimal LHS model. We validate our method by analyzing the steerability of two-qubit Werner and two-qutrit isotropic states. For Werner states, our approach saturates the analytical visibility bounds under three Pauli measurements, arbitrary projective measurements (PVMs), and arbitrary positive operator-valued measurements (POVMs). For isotropic states, we achieve the known analytical bounds under arbitrary PVMs. We further investigate the steerability of this class of states under arbitrary POVMs, and our results suggest that POVMs can offer an advantage over PVMs in revealing the steerability of such states.
It is a crucial feature of quantum mechanics that not all measurements are compatible with each other. However, if measurements suffer from noise they may lose their incompatibility. Here, we consider the effect of white noise and determine the critical visibility such that all qubit measurements, i.e., all positive operator-valued measures (POVMs), become compatible, i.e., jointly measurable. In addition, we apply our methods to quantum steering and Bell nonlocality. We obtain a tight local hidden state model for two-qubit Werner states of visibility 1/2. This determines the exact steering bound for two-qubit Werner states and also provides a local hidden variable model that improves on previously known models. Interestingly, this proves that POVMs are not more powerful than projective measurements to demonstrate quantum steering for these states.
We address the problem of quantum nonlocality with positive operator valued measures (POVM) in the context of Einstein–Podolsky–Rosen quantum steering. We show that, given a candidate for local hidden state (LHS) ensemble, the problem of determining the steerability of a bipartite quantum state of finite dimension with POVMs can be formulated as a nesting problem of two convex objects. One consequence of this is the strengthening of the theorem that justifies choosing the LHS ensemble based on symmetry of the bipartite state. As a more practical application, we study the classic problem of the steerability of two-qubit Werner states with POVMs. We show strong numerical evidence that these states are unsteerable with POVMs up to a mixing probability of within an accuracy of 10 −3 .
We investigate the steerability of two-qubit Bell-diagonal states under projective measurements by the steering party. In the simplest nontrivial scenario of two projective measurements, we solve this problem completely by virtue of the connection between the steering problem and the joint-measurement problem. A necessary and sufficient criterion is derived together with a simple geometrical interpretation. Our study shows that a Bell-diagonal state is steerable by two projective measurements iff it violates the Clauser-Horne-Shimony-Holt (CHSH) inequality, in sharp contrast with the strict hierarchy expected between steering and Bell nonlocality. We also introduce a steering measure and clarify its connections with concurrence and the volume of the steering ellipsoid. In particular, we determine the maximal concurrence and ellipsoid volume of Bell-diagonal states that are not steerable by two projective measurements. Finally, we explore the steerability of Bell-diagonal states under three projective measurements. A simple sufficient criterion is derived, which can detect the steerability of many states that are not steerable by two projective measurements. Our study offers valuable insight on steering of Bell-diagonal states as well as the connections between entanglement, steering, and Bell nonlocality.