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arXiv:quant-ph/0407111 · Physical Review A

Operator-sum representation of time-dependent density operators and its applications

D. M. Tong, L. C. Kwek, C. H. Oh, Jing‐Ling Chen, Libo Ma

Published May 24, 2004• 27 citations
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Abstract

We show that any arbitrary time-dependent density operator of an open system can always be described in terms of an operator-sum representation (Kraus representation) regardless of its initial condition and the path of its evolution in the state space, and we provide a general expression of Kraus operators for arbitrary time-dependent density operator of an $N$-dimensional system. Moreover, applications of our result are illustrated through several examples.

DOI
10.1103/physreva.69.054102

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We show that any arbitrary time-dependent density operator of an open system can always be described in terms of an operator-sum representation (Kraus representation) regardless of its initial condition and the path of its evolution in the state space, and we provide a general expression of Kraus operators for arbitrary time-dependent density operator of an NNN-dimensional system. Moreover, applications of our result are illustrated through several examples.
Abstract The dynamical evolution of open systems can be conveniently expressed in the operator sum representation in terms of Kraus operators. Here we introduce an expression for the canonical Kraus representation in terms of the eigenstates |γ〉 and eigenvalues Eγ of the total Hamiltonian, thus making a stronger connection to the tools in chemistry. The method is exact, and hence valid for non‐Markovian evolution of the system to all orders of the coupling to the bath. Significantly, it provides a starting point for developing approximate Kraus operators based upon perturbative approximations to |γ〉 and Eγ. As an example, the explicit form of these operators is derived for a d‐level system interacting with a single‐mode thermal bath.
The necessity and utility of considering the interaction of a quantum system with its environment when describing its time evolution have been recognized in several branches of physics and of other sciences. The Kraus' representation is a general and succinct approach to describe such open system dynamics in a wide range of relevant physical scenarios. In this article, by abdicating from the generality of the formalism of quantum operations and with this avoiding its associated complications, we show in a simple manner how we can obtain the Kraus' representation using basically the closed system (system plus environment) unitary dynamics and the partial trace function. The example of a two-level atom interacting with the vacuum of the electromagnetic field is regarded for the sake of instantiating this formalism, which is then applied to study the time evolution of the atom's quantum coherence.
For any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, a general procedure is given for constructing the corresponding linear map from the initial state to the state at time t, including its Kraus-type representations. Formally, this is equivalent to solving the master equation. For an N-dimensional Hilbert space it requires (i) solving a first order N 2×N 2 matrix time evolution (to obtain the completely positive map), and (ii) diagonalizing a related N 2×N 2 matrix (to obtain a Kraus-type representation). Conversely, for a given time-dependent linear map, a necessary and sufficient condition is given for the existence of a corresponding master equation, where the (not necessarily unique) form of this equation is explicitly determined. It is shown that a “best possible” master equation may always be defined, for approximating the evolution in the case that no exact master equation exists. Examples involving qubits are given.
Already in the case of finite dimensional Hilbert spaces [Formula: see text] the general form of density matrices ρ is not known. The main reason for this lack of knowledge is the nonlinear constraint for these matrices. We propose a representation of density matrices on finite dimensional Hilbert spaces in terms of finitely many independent parameters. For dimensions 2, 3, and 4 we write down this representation explicitly. As a further application of this representation we study the time dependence of density matrices ρ(t) which in our case is implemented through time dependence of the independent parameters. Under obvious differentiability assumptions the explicit form of [Formula: see text] is determined. As a special case we recover, for instance, the Lindblad form.
We propose a way to understand the evolution of an open quantum system using a description that dispenses a continuous evolution in time, by discrete operators entangled states, in its most direct and fundamental way. We show that the successive application of these operators in very small time intervals reproduce continuous evolution. It describes and compares the temporal evolution of an open quantum system of three levels, for which the Lindblad equation is solved to obtain the density matrix function of time, a method is developed to find Kraus operators time dependent and also finds constant Kraus operators differentials operate computationally $ n $ times evolving in discrete-time $ Δt = τ= t / n $. It is seen in the example of atom cavity inside a reservoir that by calculating the distance and the relative error we define is a relationship with the evolution of discrete steps and physical variables as $ ω$ which presuppose a natural discretization found time.