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arXiv:2309.12290 · Physical Review Letters

Compatibility of Generalized Noisy Qubit Measurements

Martin J. Renner

Published Jun 17, 2024• 16 citations
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Abstract

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.

DOI
10.1103/physrevlett.132.250202

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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.
Whether positive operator-valued measures (POVMs) provide advantages in demonstrating Bell nonlocality has remained unknown, even in the simple scenario of Einstein-Podolsky-Rosen steering with noisy singlet state, known as Werner states. Here we resolve this long-standing open problem by constructing a local hidden state model for Werner states with any visibility r≤1/2 under general POVMs, thereby closing the so-called Werner gap. This construction is based on an exact measurement compatibility model for the set of all noisy POVMs and also provides a local hidden variable model for a larger range of Werner states than previously known.
We study when a noisy qubit channel renders all positive-operator-valued measures (POVMs) jointly measurable. For every unital qubit channel, we derive the exact incompatibility-breaking criterion and prove that projective measurements already determine the boundary for arbitrary POVMs. Through the steering--joint-measurability correspondence, this result gives the exact POVM-steering boundary for all two-qubit states with maximally mixed marginals, with the Werner states recovered as a special case. For nonunital qubit channels, we construct a generally asymmetric parent POVM and obtain an explicit sufficient incompatibility-breaking condition, which in turn yields a sufficient unsteerability criterion for arbitrary two-qubit states.
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 ${R}_{T}=2\ensuremath{\pi}{N}_{T}\left|det(T)\right|\ensuremath{\ge}1$, 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 ${R}_{T}\ensuremath{\ge}6/5$. Our work paves the way for a systematic study of steerability of quantum states with generalized measurements beyond the highly symmetric Werner states.
We address the problem of characterizing the steerability of quantum states under restrictive measurement scenarios, i.e., the problem of determining whether a quantum state can demonstrate steering when subjected to NNN measurements of kkk outcomes. We consider the cases of either general positive operator-valued measures (POVMs) or specific kinds of measurements (e.g., projective or symmetric). We propose general methods to calculate lower and upper bounds for the white-noise robustness of a ddd-dimensional quantum state under different measurement scenarios that are also applicable to the study of the noise robustness of the incompatibility of sets of unknown qudit measurements. We show that some mutually unbiased bases, symmetric informationally complete measurements, and other symmetric choices of measurements are not optimal for steering the isotropic states and provide candidates for the most incompatible sets of measurements in each case. Finally, we provide numerical evidence that nonprojective POVMs do not improve over projective ones for this task.
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 relation between the incompatibility of quantum measurements and quantum nonlocality. We show that a set of measurements is not jointly measurable (i.e., incompatible) if and only if it can be used for demonstrating Einstein-Podolsky-Rosen steering, a form of quantum nonlocality. Moreover, we discuss the connection between Bell nonlocality and joint measurability, and give evidence that both notions are inequivalent. Specifically, we exhibit a set of incompatible quantum measurements and show that it does not violate a large class of Bell inequalities. This suggests the existence of incompatible quantum measurements which are Bell local, similarly to certain entangled states which admit a local hidden variable model.