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Journal of Nanchang University

The Matrix Trace Method For Spectral Resolution of Bipartite Systems

Gong Ren-shan

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

By using the generators of su(N) algebra the most general expression for the density matrix of bipartite systems has been given.Furthermore,the successive iteration formulas for calculating any power of density matrix are derived.Then,we proposed the matrix trace method for spectral resolution of density matrix and all the correlative formulas are given.At the end,we forecast still possible applications in the theory of local invariants in quantum mechanics and quantum information.

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By using the generators of su(N) algebra the most general expression for the density matrix of bipartite systems has been given.Furthermore,the successive iteration formulas for calculating any power of density matrix are derived.Then,we proposed the matrix trace method for spectral resolution of density matrix and all the correlative formulas are given.At the end,we forecast still possible applications in the theory of local invariants in quantum mechanics and quantum information.
By using the most general expression for the density matrix of bipartite systems,which has been given by virtue of the generatiors of su(N) algebra,and the matrix trace method for spectral resolution of density matrix,proposed by author before,it presents a complete set of successive iteration formulas for calculating all the local invariants for bipartite systems having a generic density matrix with full rank.Then,it shows that the complete set of local invariants proposed by S.Albeverio et al can be simplified.
The methods of quantum chemistry and solid state theory to solve the many-body problem are reviewed. We start with the definitions of reduced density matrices, their properties (contraction sum rules, spectral resolutions, cumulant expansion, NNN-representability), and their determining equations (contracted Schrödinger equations) and we summarize recent extensions and generalizations of the traditional quantum chemical methods, of the density functional theory, and of the quasi-particle theory: from finite to extended systems (incremental method), from density to density matrix (density matrix functional theory), from weak to strong correlation (dynamical mean field theory), from homogeneous (Kimball-Overhauser approach) to inhomogeneous and finite systems. Measures of the correlation strength are discussed. The cumulant two-body reduced density matrix proves to be a key quantity. Its spectral resolution contains geminals, being possibly the solutions of an approximate effective two-body equation, and the idea is sketched of how its contraction sum rule can be used for a variational treatment.
The matrix element of a general many-body operator between two Slater determinants is calculated explicitly. For this, a split-and-pair method is introduced that provides a convenient expression of Wick's theorem and simplifies many-body calculations. The same method is used to determine the generating function of the matrix elements of many-body operators. The split-and-pair method allows also for the diagonalization of the density correlation operators $:n({x}_{1})\ensuremath{\cdots}n({x}_{k}):$, where $n(x)={\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(x)\ensuremath{\psi}(x)$ is the density operator. The relation between the split-and-pair method and quantum group theory is clarified.
Taking partial traces (PTrs) for computing reduced density matrices, or related functions, is a ubiquitous procedure in the quantum mechanics of composite systems. In this paper, we present a thorough description of this function and analyze the number of elementary operations (ops) needed, under some possible alternative implementations, to compute it on a classical computer. As we note, it is worthwhile doing some analytical developments in order to avoid making null multiplications and sums, what can considerably reduce the ops. For instance, for a bipartite system [Formula: see text] with dimensions [Formula: see text] and [Formula: see text] and for [Formula: see text], while a direct use of PTr definition applied to [Formula: see text] requires [Formula: see text] ops, its optimized implementation entails [Formula: see text] ops. In the sequence, we regard the computation of PTrs for general multipartite systems and describe Fortran code provided to implement it numerically. We also consider the calculation of reduced density matrices via Bloch’s parametrization with generalized Gell Mann’s matrices.
Reduced density matrices are a powerful tool in the analysis of entanglement structure, approximate or coarse-grained dynamics, decoherence, and the emergence of classicality. It is straightforward to produce a reduced density matrix with the partial-trace map by tracing out part of the quantum state, but in many natural situations this reduction may not be achievable. We investigate the general problem of identifying how the quantum state is reduced given a restriction on the observables. For example, in an experimental setting, the set of observables that can actually be measured is usually modest (compared to the set of all possible observables) and their resolution is limited. In such situations, the appropriate state-reduction map can be defined via a generalized bipartition, which is associated with the structure of irreducible representations of the algebra generated by the restricted set of observables. One of our main technical results is a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. We demonstrate the viability of this approach with two examples of limited-resolution observables. The definition of quantum state reductions can also be extended beyond algebras of observables. To accomplish this task we introduce a more flexible notion of bipartition, the partial bipartition, which describes coarse grainings preserving information about a limited set (not necessarily algebra) of observables. We describe a variational method to choose the coarse grainings most compatible with a specified Hamiltonian, which exhibit emergent classicality in the reduced state space. We apply this construction to the concrete example of the one-dimensional Ising model. Our results have relevance for quantum information, bulk reconstruction in holography, and quantum gravity.