System of integral equations for partial density matrices

Dmitri M Gitman, A. E. Arinshtein

Apostol Vourdas

The set M of d × d Hermitian matrices (observables) is studied as a partially ordered set with the Löwner partial order. Upper and lower sets in it, define the concept of cumulativeness (used mainly with scalar quantities) in the context of Hermitian matrices. Partial order and topology are intimately related to each other and the set M of Hermitian matrices is also studied as a topological space, where open and closed sets are the upper and lower sets. It is shown that the set M of Hermitian matrices is a T 0 topological space, and its subset D of density matrices is Hausdorff totally disconnected topological space. These ideas are a prerequisite for studying quantum Choquet integrals with Hermitian matrices (as opposed to classical Choquet integrals with scalar quantities). Capacities (non-additive probabilities), cumulative quantities that involve Hermitian matrices, and Möbius transforms that remove the overlaps between non-commuting observables, are used in quantum Choquet integrals. An application of the formalism is to find a density matrix, with given expectation values with respect to n (non-commuting) observables. Examples of calculations of such a density matrix (with quantified errors in its expectation values), are presented.

K. Husimi, Yoshihiro Kitano, Toshiyuki Nishiyama

I. Knežević, David K. Ferry

Evolution of a system, coupled to its environment and influenced by external driving fields, is an old problem that remains of interest. In this paper, we derive an equation of motion for the reduced system density matrix, which is time convolutionless and free of the partial trace with respect to the environment states. This new approach uses an extension of the projection-operator technique, which incorporates an isomorphism between the system's Liouville space and the unit eigenspace of the projection operator induced by the uniform environment density matrix. Numerical application of the present approach is particularly useful in large externally driven systems, as the partial-trace-free equation is given in terms of submatrices significantly smaller than the matrices in the conventional time-convolutionless approaches, which alleviates the computational burden. We also show that all time-convolutionless approaches, conventional or partial-trace-free, are based upon a hidden underlying assumption of time reversibility of the system's evolution. This feature puts significant constraints on applicability of time-convolutionless approaches when employing approximations that yield time irreversibility. Also, we investigate the application of the approach in the description of far-from-equilibrium systems.

Christian Käding, Mario Pitschmann

Density matrices are powerful mathematical tools for the description of closed and open quantum systems. Recently, methods for the direct computation of density matrix elements in scalar quantum field theory were developed based on thermo field dynamics (TFD) and the Schwinger-Keldysh formalism. In this article, we provide a more detailed discussion of these methods and derive expressions for density matrix elements of closed and open systems. At first, we look at closed systems by discussing general solutions to the Schrödinger-like form of the quantum Liouville equations in TFD, showing that the dynamical map is indeed divisible, deriving a path integral-based expression for the density matrix elements in Fock space, and explaining why perturbation theory enables us to use the last even in situations where all initial states in Fock space are occupied. Subsequently, we discuss open systems in the same manner after tracing out environmental degrees of freedom from the solutions for closed systems. We find that, even in a general basis, the dynamical map is not divisible, which renders the dynamics of open systems non-Markovian. Finally, we show how the resulting expressions for open systems can be used to obtain quantum master equations, and comment on the artificiality of time integrals over density matrices that usually appear in many other master equations in the literature but are absent in ours.

Karl Blum