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The Density Matrix Description of Matrix Shell Systems Associated with the Matrix Shell Model formalism

Debopam Ghosh

Published Aug 8, 2022• 0 citations
Publisher

Abstract

The article presents a mathematical framework to associate a Matrix Shell system (even type or of the odd type) with a unit trace, hermitian, positive definite or positive semi-definite matrix of order ‘2’ (Density matrix representation of single qubit quantum states). This framework therefore, allows any evolution scheme defined on the fundamental matrix space associated with the Matrix Shell system and consistent with the Matrix shell model formalism to be mapped to an evolution on or inside of the unit solid sphere, centered at the origin, in the three dimensional space( Bloch sphere representation of single qubit quantum states) Demonstration of the proposed mathematical framework is presented through its application on certain subsets of the Complex Matrix spaces M2×2(C) and M4×4(C) and on certain numerical examples from M3×3(C).

DOI
10.5281/zenodo.6974538

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The article presents a mathematical framework to associate a Matrix Shell system (even type or of the odd type) with a unit trace, hermitian, positive definite or positive semi-definite matrix of order ‘2’ (Density matrix representation of single qubit quantum states). This framework therefore, allows any evolution scheme defined on the fundamental matrix space associated with the Matrix Shell system and consistent with the Matrix shell model formalism to be mapped to an evolution on or inside of the unit solid sphere, centered at the origin, in the three dimensional space( Bloch sphere representation of single qubit quantum states) Demonstration of the proposed mathematical framework is presented through its application on certain subsets of the Complex Matrix spaces M2×2(C) and M4×4(C) and on certain numerical examples from M3×3(C).
Density matrices defined with respect to a finite basis set are considered as elements in a vector space. A basis set is introduced in the space whose elements are Hermitian matrices. By a suitable transformation of the basis, one component only will contribute to the trace, so the space of unit trace matrices is a translated linear subspace. The positivity constraint is then examined in terms of distance from a multiple of the unit matrix, which is in the interior of the set of density matrices. The boundary of this set is shown to lie between two concentric hyperspheres. The set whose elements are NNN-representable 1 matrices has similar properties.
A possible method of classifying all density matrices for a N-state quantum system into different types, using the sameness of the characteristic polynomial of density matrices belonging to a given type, is discussed. The method employs the SU(N) generator expansion of density matrices. It is demonstrated that density matrices belonging to a given type, although different, share many common properties including the same eigenvalues, the same shape of the region in the parametric space in which lie the allowed parameter values, and the same global averages of the pertinent observables. Furthermore, we discuss the time evolution of a given type of density matrices and show that, in the appropriately defined time-dependent comoving matrix basis, a density matrix of a given type belongs, during entire unitary evolution, to the same type. In this way one is able to group and organize the multitude of different density matrices with certain analogous properties, into a smaller number of types, thus systematizing and cataloguing different mixed and pure states of a N-state system. The general discussion is illustrated throughout with the special case of a N=4-state quantum system. Two- and three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method for several types of density matrices, providing some insight into the intricate and complex geometric structure of the space of density matrices.
The most general evolution of the density matrix of a quantum system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear convex set that may be viewed as supermatrices. The property of Hermiticity of density matrices renders an associated supermatrix Hermitian and hence diagonalizable. The positivity of the density matrix does not make the associated supermatrix positive though. If the map itself is positive, it is called completely positive and it has a simple parametrization. This is extended to all positive (not completely positive) maps. A general dynamical map that does not preserve the norm of the density matrices it acts on can be thought of as the contraction of a norm-preserving map of an extended system. The reconstruction of such extended dynamics is also given.
In the study of d-dimensional quantum channels (d⩾2), an assumption which includes many interesting examples, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of su(n) is a depolarizing channel for all n, but for most other representations this is not the case. Since the standard Bloch sphere only exists for the qubit representation of su(2), we develop a consistent generalization of Bloch’s technique. By representing the density matrix as a polynomial in Lie algebra generators, we determine a class of positive semidefinite matrices which represent quantum states for various channels defined by finite-dimensional representations of semisimple Lie algebras. We also give a general method for finding positive semidefinite matrices using Lie algebraic trace identities. This includes an analysis of channels based on the exceptional Lie algebra g2 and the Clifford algebra.
While in relativity theory space evolves over time into a single entity known as spacetime, quantum theory lacks a standard notion of how to encapsulate the dynamical evolution of a quantum state into a single "state over time". Recently it was emphasized in the work of Fitzsimons, Jones and Vedral that if such a state over time is to encode not only spatial but also temporal correlations which exist within a quantum dynamical process, then it should be represented not by a density matrix, but rather, by a pseudo-density matrix . A pseudo-density matrix is a hermitian matrix of unit trace whose marginals are density matrices, and in this work, we make use a factorization system for quantum channels to associate a pseudo-density matrix with a quantum system which is to evolve according to a finite sequence of quantum channels. We then view such a pseudo-density matrix as the quantum analog of a local patch of spacetime, and we make an in-depth mathematical analysis of such quantum dynamical pseudo-density matrices and the properties they satisfy. We also show how to explicitly extract quantum dynamics from a given pseudo-density matrix, thus solving an open problem posed in the literature.
An algebraic reformulation of the quantum search algorithm associated to a k‐valued oracle function, is introduced in terms of the so called oracle matrix algebra, by means of which a Bloch sphere like description of search is obtained. A parametric family of symmetric completely positive trace preserving (CPTP) maps, that formalize the presence of quantum noise but preserves the complexity of the algorithm, is determined. Dimensional reduction of representations of oracle Lie algebra is introduced in order to determine the reduced density matrix of subsets of qubits in database. The L1 vector‐induced norm of reduced density matrix is employed to define an index function for the quantum entanglement between database qubits, in the presence of non invariant noise CPTP maps. Analytic investigations provide a causal relation between entanglement and fidelity of the algorithm, which is controlled by quantum noise parameter.