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arXiv:2207.11645 · Advanced Quantum Technologies

Maximum Entropy Methods for Quantum State Compatibility Problems

Shi‐Yao Hou, Zipeng Wu, Jinfeng Zeng, Ningping Cao, Chenfeng Cao, Youning Li, Bei Zeng

Published Oct 11, 2024• 0 citations

Abstract

Abstract Inferring a quantum system from incomplete information is a common problem in many aspects of quantum information science and applications, where the principle of maximum entropy (MaxEnt) plays an important role. The quantum state compatibility problem asks whether there exists a density matrix compatible with some given measurement results. Such a compatibility problem can be naturally formulated as a semidefinite programming (SDP), which searches directly for the existence of a . However, for large system dimensions, it is hard to represent directly, since it requires too many parameters. In this work, MaxEnt is applied to solve various quantum state compatibility problems, including the quantum marginal problem. An immediate advantage of the MaxEnt method is that it only needs to represent via a relatively small number of parameters, which is exactly the number of the operators measured. Furthermore, in case of incompatible measurement results, the method will further return a witness that is a supporting hyperplane of the compatible set. The method has a clear geometric meaning and can be computed effectively with hybrid quantum‐classical algorithms.

DOI: 10.1002/qute.202400172

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There exists, in general, a convex set of quantum state estimators that maximize the likelihood for informationally incomplete data. We propose an estimation scheme, catered to measurement data of this kind, to search for the exact maximum-likelihood-maximum-entropy estimator using semidefinite programming and a standard multi-dimensional function optimization routine. This scheme can be used to infer the expectation values of a set of entanglement witnesses that can be used to verify the entanglement of the unknown quantum state for composite systems. Next, we establish an alternative numerical scheme that is more computationally robust for the sole purpose of maximizing the likelihood and entropy.

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