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Journal of Physics Conference Series

Extremal Density Matrices for the Expectation Value of a Qudit Hamiltonian

Octavio Castaños, Armando Figueroa, Julio A. López-Saldívar, Ramón López–Peña

Published May 26, 2017• 4 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/839/1/012012

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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.
We establish a procedure to find the extremal density matrices for any finite Hamiltonian of a qudit system. These extremal density matrices provide an approximate description of the energy spectra of the Hamiltonian. In the case of restricting the extremal density matrices by pure states, we show that the energy spectra of the Hamiltonian is recovered for d=2d=2d=2 and 333. We conjecture that by means of this approach the energy spectra can be recovered for the Hamiltonian of an arbitrary finite qudit system. For a given qudit system Hamiltonian, we find new inequalities connecting the mean value of the Hamiltonian and the entropy of an arbitrary state. We demonstrate that these inequalities take place for both the considered extremal density matrices and generic ones.
We establish a novel procedure to analyze the entanglement properties of extremal density matrices depending on the parameters of a finite dimensional Hamiltonian. It was applied to a general 2-qubit Hamiltonian which could exhibit Kramers degeneracy. This is done through the extremal density matrix formalism, which allows to extend the conventional variational principle to mixed states. By applying the positive partial transpose criterion in terms of the Correlation and Schlienz-Mahler matrices on the extremal density matrices, we demonstrate that it is possible to reach both pure and mixed entangled states, changing properly the parameters of the Hamiltonian. For time-reversal invariant Hamiltonians, the extremal pure states can be entangled or not and we prove that they are not time-reversal invariants. For extremal mixed states we have in general 5 possible cases: three of them are entangled and the other two separable.
Abstract The stationary Schrödinger equation can be cast in the form , where H is the system’s Hamiltonian and is the system’s density matrix. We explore the merits of this form of the stationary Schrödinger equation, which we refer to as SSE , applied to many-body systems with symmetries. For a nondegenerate energy level, the solution of the SSE is merely a projection on the corresponding eigenvector. However, in the case of degeneracy is non-unique and not necessarily pure. In fact, it can be an arbitrary mixture of the degenerate pure eigenstates. Importantly, can always be chosen to respect all symmetries of the Hamiltonian, even if each pure eigenstate in the corresponding degenerate multiplet spontaneously breaks the symmetries. This and other features of the solutions of the SSE can prove helpful by easing the notations and providing an unobscured insight into the structure of the eigenstates. We work out the SSE for a general system of spins 1/2 with Heisenberg interactions, and address simple systems of spins 1. Eigenvalue problem for quantum observables other than Hamiltonian can also be formulated in terms of density matrices. As an illustration, we provide an analytical solution to the eigenproblem , where is the total spin of N spins 1/2, and is chosen to be invariant under permutations of spins. This way we find an explicit form of projections to the invariant subspaces of .
Along with an elementary description of the well-known density matrix formalism and of its application to the description of quantum-statistical ensembles, this chapter includes the proofs of a few propositions that are especially useful for understanding most of the developments. In the new formulation, ensembles are directly described by density matrices, observables are still described by Hermitean operators, and the rules giving the statistical distribution and the mean values of the predicted measurement results are, respectively, expressions. When stated in this manner, the rules of quantum mechanics apply directly not only to pure states but also to mixtures. This is a distinct advantage. In the case of mixtures defined by their density matrices, the new formulation also has another merit. If two proper mixtures are described by the same density matrix, any state vector representing a subensemble of one of these mixtures can be expressed as a linear combination of the state vectors representing the subensembles of the other mixture.