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AIP conference proceedings

Maximum entropy, minimum norm and the inference of entangled states

M. Casas

Published Jan 1, 2001• 0 citations
Publisher

Abstract

We present an approximate inference approach based on the Information Theory and the Pseudoinverse Formalism that allows to obtain all the accessible micro-states of a quantum system compatible with a given information. We assume that the Hamiltonian of the system is unknown and the only available information is restricted to a set of expectation values of noncommuting operators. With this information the density matrix is inferred and, exploring the Null space of the input operators, one can define the physical regions for which the density matrix is positive definite. The method is applied for the inference of entangled quantum states.

DOI
10.1063/1.1381907

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We present an approximate inference approach based on the Information Theory and the Pseudoinverse Formalism that allows to obtain all the accessible micro-states of a quantum system compatible with a given information. We assume that the Hamiltonian of the system is unknown and the only available information is restricted to a set of expectation values of noncommuting operators. With this information the density matrix is inferred and, exploring the Null space of the input operators, one can define the physical regions for which the density matrix is positive definite. The method is applied for the inference of entangled quantum states.
The inference of entangled states on the basis of incomplete prior information is usually affected by the problem of fake inferred entanglement. We show that different inferences schemes overestimate the amount of entanglement. However, the Maximum-Entropy Minimum-Norm method yields all the accessible states compatible with the prior information of the system. For bipartite systems the entanglement boundaries are explicitly determined when the knowledge of the system is restricted to the expectation value of a generic operator diagonal in the Bell basis.
The inference of entangled quantum states by recourse to the maximum entropy (MaxEnt) principle is considered in connection with the recently pointed out problem of fake inferred entanglement (Horodecki R et al 1999 Phys. Rev. A 59 1799). We show that there are operators Â, both diagonal and non-diagonal in the Bell basis, such that, when the expectation value  is taken as prior information, the problem of fake entanglement is not solved by adding a new constraint associated with the mean value of Â2 (unlike what happens when the partial information is given by the expectation value of a Bell operator). The fake entanglement generated by the MaxEnt principle is also studied quantitatively by comparing the entanglement of formation of the inferred state with that of the original one.
The inference of entangled quantum states by recourse to the maximum entropy (MaxEnt) principle is considered in connection with the recently pointed out problem of fake inferred entanglement (Horodecki R et al 1999 Phys. Rev. A 59 1799). We show that there are operators Â, both diagonal and non-diagonal in the Bell basis, such that, when the expectation value ⟨Â⟩ is taken as prior information, the problem of fake entanglement is not solved by adding a new constraint associated with the mean value of  2 (unlike what happens when the partial information is given by the expectation value of a Bell operator). The fake entanglement generated by the MaxEnt principle is also studied quantitatively by comparing the entanglement of formation of the inferred state with that of the original one.
Quantum computation has been growing rapidly in both theory and experiments.In particular, quantum computing devices with a large number of qubits have been developed by IBM, Google, IonQ, and others.The current quantum computing devices are noisy intermediate-scale quantum devices, and so approaches to validate quantum processing on these quantum devices are needed.One of the most common ways of validation for an n-qubit quantum system is quantum tomography, which tries to reconstruct a quantum system's density matrix by a complete set of observables.However, the inherent noise in the quantum systems and the intrinsic limitations pose a critical challenge to precisely know the actual measurement operators that make quantum tomography impractical in experiments.Here, we propose an alternative approach to quantum tomography, based on the maximal information entropy, that can predict the values of unknown observables based on the available mean measurement data.This can then be used to reconstruct the density matrix with high fidelity even though the results for some observables are missing.Of additional contexts, a practical approach to the inference of the quantum mechanical state using only partial information is also needed.
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.