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Physical Review A

Inferring the density matrix for a system of an unknown Hamiltonian

A. Rigó, M. Casas, Angel Ricardo Plastino

Published Apr 1, 1998• 6 citations
Publisher

Abstract

An approximate inference approach is advanced in order to determine the density matrix in cases in which the pertinent Hamiltonian is unknown, and the available expectation values pertain to noncommuting observables. The inference of both pure and mixed states is addressed, and on an equal footing, without facing infinities in the pure state limit. Information-theoretical tools are employed.

DOI
10.1103/physreva.57.2319

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An approximate inference approach is advanced in order to determine the density matrix in cases in which the pertinent Hamiltonian is unknown, and the available expectation values pertain to noncommuting observables. The inference of both pure and mixed states is addressed, and on an equal footing, without facing infinities in the pure state limit. Information-theoretical tools are employed.
An algebraic procedure to find extremal density matrices for the expectation value of a finite Hamiltonian matrix is established. The extremal density matrices for pure states provide a complete description of the system, that is, its corresponding energy spectrum and projectors. For density matrices representing mixed states, one gets the most probable eigenstates that yield extremal mean values of the energy. The procedure uses mainly the stationary solutions of the von Neumann equation of motion, the orbits of the Hamiltonian, and the positivity conditions of the density matrix. The method is illustrated for matrix Hamiltonians of dimensions d = 2 and d = 3.
An algebraic procedure to find extremal density matrices for any Hamiltonian of a qudit system is established. The extremal density matrices for pure states provide a complete description of the system, that is, the energy spectra of the Hamiltonian and their corresponding projectors. For extremal density matrices representing mixed states, one gets mean values of the energy in between the maximum and minimum energies associated to the pure case. These extremal densities give also the corresponding mixture of eigenstates that yields the corresponding mean value of the energy. We enhance that the method can be extended to any hermitian operator.
Suppose you receive a sequence of qubits where each qubit is guaranteed to be in one of two pure states, but you do not know what those states are. Your task is to determine the states. This can be viewed as a kind of quantum state learning -- or quantum state estimation. A problem is that, without more information, all that can be determined is the density matrix of the sequence and, in general, density matrices can be decomposed into pure states in many different ways. To solve the problem, additional information, either classical or quantum, is required. We show that if an additional copy of each qubit is supplied -- that is, one receives pairs of qubits, both in the same state, rather than single qubits -- the task can be accomplished. This is possible because the mixed two-qubit state has a unique decomposition into pure product states. For illustration, we simulate numerically the symmetric, informationally complete measurement of a sequence of qubit pairs and show that the unknown states and their respective probabilities of occurrence can be inferred from the data with high accuracy. Finally, we propose an experiment that employs a product measurement and can be realized with existing technology, and we demonstrate how the data tell us the states and their probabilities. We find that it is enough to detect a few thousand qubit pairs.
We provide a brief overview of approaches for calculating the density of states of quantum systems and random matrix Hamiltonians using the tools of free probability theory. For a given Hamiltonian of a quantum system or a generic random matrix Hamiltonian, which can be written as a sum of two non-commutating operators, one can obtain an expression for the density of states of the Hamiltonian from the known density of states of the two component operators by assuming that these operators are mutually free and by using the free additive convolution. In many examples of interacting quantum systems and random matrix models, this procedure is known to provide a reasonably accurate approximation to the exact numerical density of states. We review some of the examples that are known in the literature where this procedure works very well, and also discuss some of the limitations of this method in situations where the free probability approximation fails to provide a sufficiently accurate description of the exact density of states. Subsequently, we describe a perturbation scheme that can be developed from the subordination formulas for the Cauchy transform of the density of states and use it to obtain approximate analytical expressions for the density of states in various models, such as the Rosenzweig-Porter random matrix ensemble and the Anderson model with on-site disorder.