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arXiv:2003.00460 · Physical Review A

Necessary and sufficient condition for the reduced dynamics of an open quantum system interacting with an environment to be linear

Iman Sargolzahi

Published Aug 11, 2020• 8 citations
arXivHTMLPDF

Abstract

The dynamics of a closed quantum system, under a unitary time evolution $U$, is obviously linear. But, the reduced dynamics of an open quantum system $S$, interacting with an environment $E$, is not linear, in general. Dominy et al. [Quantum Inf. Process. 15, 465 (2016)] considered the case that the set $\mathcal{S}={{\ensuremath{\rho}}_{SE}}$ of possible initial states of the system-environment is convex, and also possesses another property, which they called $U$-consistency. They have shown that, under such circumstances, the reduced dynamics of the system $S$ is linear. Whether the Dominy-Shabani-Lidar framework is the most general one is the subject of this paper. We assume that the reduced dynamics is linear and show that this leads us to their framework. In other words, the reduced dynamics of the system is linear if and only if it can be formulated within the Dominy-Shabani-Lidar framework.

DOI
10.1103/physreva.102.022208

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•Jan 1, 2017•0 citations
The dynamics of a closed quantum system, under a unitary time evolution UUU, is obviously linear. But, the reduced dynamics of an open quantum system SSS, interacting with an environment EEE, is not linear, in general. Dominy et al. [Quantum Inf. Process. 15, 465 (2016)] considered the case that the set S=\ensuremathρSE\mathcal{S}={{\ensuremath{\rho}}_{SE}}S=\ensuremathρSE​ of possible initial states of the system-environment is convex, and also possesses another property, which they called UUU-consistency. They have shown that, under such circumstances, the reduced dynamics of the system SSS is linear. Whether the Dominy-Shabani-Lidar framework is the most general one is the subject of this paper. We assume that the reduced dynamics is linear and show that this leads us to their framework. In other words, the reduced dynamics of the system is linear if and only if it can be formulated within the Dominy-Shabani-Lidar framework.
Consider an open quantum, system interacting with its environment. Whether the reduced dynamics of the system can be given by a linear map or not is an important question in the theory of open quantum systems. Dominy, Shabani and Lidar have proposed a general framework for linear Hermitian reduced dynamics. In addition, it has been shown that their framework is, in fact, the most general one: The reduced dynamics of the system is linear if and only if it can be formulated within the Dominy-Shabani-Lidar framework. This result has been given in a rather abstract way. Here, we want to give another proof for it, in a more illustrative manner.
In the simplest approximation, the reduced dynamics of a quantum system SSS interacting with its environment EEE is considered to be given by a completely positive map. But, in general, this is not the case. In fact, the reduced dynamics of the system in not even linear, in general. Whether the reduced dynamics is linear or not is determined by two factors: the set of possible initial states of the system-environment $\mathcal{S}=\left\lbrace ρ_{SE} \right\rbrace $, and the joint system-environment unitary evolution UUU. When UUU is factorisable as U=US⊗UEU=U_S\otimes U_EU=US​⊗UE​, then we can choose S=D\mathcal{S}=\mathcal{D}S=D, where D\mathcal{D}D is the set of all system-environment density operators. In other words, when UUU is factorisable, the reduced dynamics of the system SSS is linear (in fact unitary) for arbitrary initial state of the system-environment ρSEρ_{SE}ρSE​. We show that this result cannot be generalized to any non-factorisable UUU: For any non-factorisable unitary evolution of the whole system-environment UUU, the set S\mathcal{S}S must be chosen as a proper subset of D\mathcal{D}D to achieve linear reduced dynamics. As a byproduct, considering a convex set of possible initial states of the system-environment S\mathcal{S}S such that TrE S=TrE D\mathrm{Tr}_{E} \ \mathcal{S}=\mathrm{Tr}_{E} \ \mathcal{D}TrE​ S=TrE​ D, we show that when the reduced dynamics of the system, for one system-environment unitary evolution U1U_1U1​, is positive, but not completely positive, this implies that reduced dynamics is not linear for another U2U_2U2​.
Consider an open quantum system S , interacting with its environment E. Whether the reduced dynamics of the system can be given by a linear map, or not, is an important subject, in the theory of open quantum systems. Dominy, Shabani and Lidar have proposed a general framework for linear Hermitian reduced dynamics. They have considered the case that both the system and the environment are finite dimensional. Their framework can be generalized to include the case that the environment is infinite dimensional too. In this paper, after demonstrating this generalization, we discuss the role of the convexity of the set, of possible initial states of the system-environment, in their framework. Next, we give a proof for the existence of the operator sum representation, for arbitrary linear Hermitian map. This proof enables us to prove the Choi-Jamiołkowski and the Jamiołkowski isomorphisms, straightforwardly.
The reduced dynamics of an open quantum system SSS, interacting with its environment EEE, is not completely positive, in general. In this paper, we demonstrate that if the two following conditions are satisfied, simultaneously, then the reduced dynamics is completely positive: (1) the reduced dynamics of the system is linear, for arbitrary system-environment unitary evolution UUU; and (2) the reduced dynamics of the system is linear, for arbitrary initial state of the system ${\ensuremath{\rho}}_{S}$.
We study the influence of the preparation of an open quantum system on its reduced time evolution. In contrast to the frequently considered case of an initial preparation where the total density matrix factorizes into a product of a system density matrix and a bath density matrix the time evolution generally is no longer governed by a linear map nor is this map affine. Put differently, the evolution is truly nonlinear and cannot be cast into the form of a linear map plus a term that is independent of the initial density matrix of the open quantum system. As a consequence, the inhomogeneity that emerges in formally exact generalized master equations is in fact a nonlinear term that vanishes for a factorizing initial state. The general results are elucidated with the example of two interacting spins prepared at thermal equilibrium with one spin subjected to an external field. The second spin represents the environment. The field allows the preparation of mixed density matrices of the first spin that can be represented as a convex combination of two limiting pure states, i.e., the preparable reduced density matrices make up a convex set. Moreover, the map from these reduced density matrices onto the corresponding density matrices of the total system is affine only for vanishing coupling between the spins. In general, the set of the accessible total density matrices is nonconvex.
A quantum system which evolves in time while interacting with an external environ- ment is said to be an open quantum system (OQS), and the influence of the environment on the unperturbed unitary evolution of the system generally leads to non-unitary dynamics. This kind of open system dynamical evolution has been typically modeled by a Standard Prescription (SP) which assumes that the state of the OQS is initially uncorrelated with the environment state. It is here shown that when a minimal set of physically motivated assumptions are adopted, not only does there exist constraints on the reduced dynamics of an OQS such that this SP does not always accurately describe the possible initial cor- relations existing between the OQS and environment, but such initial correlations, and even entanglement, can be witnessed when observing a particular class of reduced state transformations termed purity extractions are observed. Furthermore, as part of a more fundamental investigation to better understand the minimal set of assumptions required to formulate well defined reduced dynamical quantum maps, it is demonstrated that there exists a one-to-one correspondence between the set of initial reduced states and the set of admissible initial system-environment composite states when G-consistency is enforced. Given the discussions surrounding the requirement of complete positivity and the reliance on the SP, the results presented here may well be found valuable for determining the ba- sic properties of reduced dynamical maps, and when restrictions on the OQS dynamics naturally emerge.