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arXiv:2306.12122 · Physical Review Letters

Witnessing Quantum Incompatibility Structures in High-Dimensional Multimeasurement Systems

Xiaolin Zhang, Rui Qu, Zehong Chang, Yunlong Wang, Zhenyu Guo, Min An, Hong Xiu Gao, Fuli Li, Pei Zhang

Published Nov 8, 2024• 4 citations
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Abstract

Quantum incompatibility, referred as the phenomenon that some quantum measurements cannot be performed simultaneously, is necessary for various quantum information processing tasks, such as nonlocality and steering. When these applications come to high-dimensional multimeasurement scenarios, it is crucial and challenging to witness the incompatibility of measurements with complex structures. To address this problem, we propose a modified quantum state discrimination protocol that decomposes complex compatibility structures into pairwise ones and employs noise robustness to bound incompatibility structures. We then derive arithmetic bounds for arbitrary measurements and analytical bounds for mutually unbiased bases, and capture some quantum incompatibility structures where measurements are partly compatible and partly incompatible. Finally, we experimentally demonstrate our results and connect them with quantum steering, quantum simulability and quantum communications.

DOI
10.1103/physrevlett.133.190202

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Quantum incompatibility, referred as the phenomenon that some quantum measurements cannot be performed simultaneously, is necessary for various quantum information processing tasks, such as nonlocality and steering. When these applications come to high-dimensional multimeasurement scenarios, it is crucial and challenging to witness the incompatibility of measurements with complex structures. To address this problem, we propose a modified quantum state discrimination protocol that decomposes complex compatibility structures into pairwise ones and employs noise robustness to bound incompatibility structures. We then derive arithmetic bounds for arbitrary measurements and analytical bounds for mutually unbiased bases, and capture some quantum incompatibility structures where measurements are partly compatible and partly incompatible. Finally, we experimentally demonstrate our results and connect them with quantum steering, quantum simulability and quantum communications.
Quantum incompatibility, as an important quantum resource, is believed to play a key role in quantum information processing tasks.In order to detect this property, an entanglement-free incompatibility witness is proposed via constructing a combined quantum state discrimination task with prior-measurement and postmeasurement information strategies.Here we experimentally implement the witness of the incompatibility of mutually unbiased measurements under different noise levels in a qutrit system.Moreover, we estimate the robustness of incompatibility using collected data.Our work paves the path for understanding quantum incompatibility and exploring the features of the quantum resource.
Incompatible measurements are of fundamental importance to revealing the peculiar features of quantum theory, and are also useful resources in various quantum information tasks. In this work, we investigate the quantum incompatibility of mutually unbiased bases (MUBs) within the operational framework of quantum resource theory, and report an experimental validation via the task of state discrimination. In particular, we construct an experimentally friendly witness to detect incompatible MUBs, based on the probability of correctly discriminating quantum states. Furthermore, we prove that the noise robustness of MUBs can be retrieved from violating the above witness. Finally, we experimentally test the incompatibility of MUBs of dimensionality ranging from 2 to 4, and demonstrate that it is more robust to noise, as either the dimensionality of measurements or the number of MUBs increases. Our results may aid the exploration of the essential roles of incompatible measurements in both theoretical and practical applications in quantum information.
Quantum measurements based on mutually unbiased bases are commonly used in quantum information processing, as they are generally viewed as being maximally incompatible and complementary. Here we quantify precisely the degree of incompatibility of mutually unbiased bases (MUB) using the notion of noise robustness. Specifically, for sets of k MUB in dimension d, we provide upper and lower bounds on this quantity. Notably, we get a tight bound in several cases, in particular for complete sets of k=d+1 MUB (using the standard construction for d being a prime power). On the way, we also derive a general upper bound on the noise robustness for an arbitrary set of quantum measurements. Moreover, we prove the existence of sets of k MUB that are operationally inequivalent, as they feature different noise robustness, and we provide a lower bound on the number of such inequivalent sets up to dimension 32. Finally, we discuss applications of our results for Einstein-Podolsky-Rosen steering.
We demonstrate that quantum incompatibility can always be detected by means of a state discrimination task with partial intermediate information. This is done by showing that only incompatible measurements allow for an efficient use of premeasurement information in order to improve the probability of guessing the correct state. Thus, the gap between the guessing probabilities with pre- and postmeasurement information is a witness of the incompatibility of a given collection of measurements. We prove that all linear incompatibility witnesses can be implemented as some state discrimination protocol according to this scheme. As an application, we characterize the joint measurability region of two noisy mutually unbiased bases.
We demonstrate that quantum incompatibility can always be detected by means of a state discrimination task with partial intermediate information. This is done by showing that only incompatible measurements allow for an efficient use of premeasurement information in order to improve the probability of guessing the correct state. Thus, the gap between the guessing probabilities with pre- and postmeasurement information is a witness of the incompatibility of a given collection of measurements. We prove that all linear incompatibility witnesses can be implemented as some state discrimination protocol according to this scheme. As an application, we characterize the joint measurability region of two noisy mutually unbiased bases.
Quantum systems of high dimensions are attracting a lot of attention because they feature interesting properties when it comes to observing entanglement or other forms of correlations. In particular, their improved resistance to noise is favorable for experiments in quantum communication or quantum cryptography. However, witnessing this high-dimensional nature remains challenging, especially when the assumptions on the parties involved are weak, typically when one of them is considered as a black box. In this context, the concept of genuine high-dimensional steering has been recently introduced and experimentally demonstrated [Phys. Rev. Lett. 126, 200404 (2021)]; it allows for a one-sided device-independent certification of the dimension of a bipartite shared state by only using two measurements. Here I overcome this limitation by developing, for more than two measurements, universal bounds on the incompatibility robustness, turned into meaningful dimension certificates. Interestingly, even though the resulting bounds are quite loose, they still often offer an increased resistance to noise and could then be advantageously employed in experiments.