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arXiv:1601.07458 · International Journal of Modern Physics C

Computing partial traces and reduced density matrices

Jonas Maziero

Published Aug 2, 2016• 24 citations

Abstract

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.

DOI: 10.1142/s012918311750005x

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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.

Coherence vectors and correlation matrices are important functions frequently used in physics. The numerical calculation of these functions directly from their definitions, which involves Kronecker products and matrix multiplications, may seem to be a reasonable option. Notwithstanding, as we demonstrate in this paper, some algebraic manipulations before programming can reduce considerably their computational complexity. Besides, we provide Fortran code to generate generalized Gell-Mann matrices and to compute the optimized and unoptimized versions of associated Bloch’s vectors and correlation matrix in the case of bipartite quantum systems. As a code test and application example, we consider the calculation of Hilbert-Schmidt quantum discords.

The partial trace is commonly introduced in quantum mechanics as an algebraic operation used to define reduced states of composite systems. However, the probabilistic origin of this operation goes systematically unnoticed in the literature. Here, we show that the partial trace emerges naturally from the requirement of consistency between the Born rule for measurement probabilities and the classical marginalization of probability mass functions. Starting from the classical marginalization rule relating joint and marginal probability distributions, we impose that the reduced density operator of a subsystem must reproduce the local measurement statistics derived from the global state. We show that this requirement directly leads to the standard expression of the partial trace. From this perspective, the reduced density operator appears not as an ad hoc algebraic construction, but as a natural consequence of the probabilistic structure of quantum mechanics.

The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ2 and Ŝz operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.

The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ2 and Ŝz operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.

Partial trace is a very important mathematical operation in quantum mechanics. It is not only helpful in studying the subsystems of a composite quantum system but also used in computing a vast majority of quantum entanglement measures. Calculating partial trace becomes computationally very intensive with increasing number of qubits as the Hilbert space dimension increases exponentially. In this paper, we discuss about our new method of partial tracing which is based on set theory and it is more efficient. The proposed method of partial tracing overcomes all the limitations of the other well-known methods such as being computationally intensive and being limited to low dimensional Hilbert spaces. We give a detailed theoretical description of our method and also provide an explicit example of the computation. The merits of the new method and other key ideas are discussed.