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Master equation for cascade quantum channels: a collisional approach

Vittorio Giovannetti, G. Massimo Palma

Published Aug 16, 2016• 24 citations
OpenAlex

Abstract

It has been recently shown that collisional models can be used to derive a general form for the master equations which describe the reduced time evolution of a composite multipartite quantum system, whose components “propagate ” in an environmental medium which induces correlations among them via a cascade mechanism. Here we analyze the fundamental assumptions of this approach showing how some of them can be lifted when passing into a proper interaction picture representation.

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Master equations in the Lindblad form describe evolution of open quantum systems that are completely positive and simultaneously have a semigroup property. We analyze the possibility to derive this type of master equations from an intrinsically discrete dynamics that is modelled as a sequence of collisions between a given quantum system (a qubit) with particles that form the environment. In order to illustrate our approach we analyze in detail how the process of an exponential decay and the process of decoherence can be derived from a collision-like model in which particular collisions are described by SWAP and controlled-NOT interactions, respectively.

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It has been recently shown that collisional models can be used to derive a general form for the master equations which describe the reduced time evolution of a composite multipartite quantum system, whose components “propagate ” in an environmental medium which induces correlations among them via a cascade mechanism. Here we analyze the fundamental assumptions of this approach showing how some of them can be lifted when passing into a proper interaction picture representation.
We derive the general form of a master equation describing the reduced time evolution of a sequence of subsystems "propagating" in an environment which can be described as a sequence of subenvironments. The interaction between subsystems and subenvironments is described in terms of a collision model, with the irreversible dynamics of the subenvironments between collisions explicitly taken into account. In the weak coupling regime, we show that the collisional model produces a correlated Markovian evolution for the joint density matrix of the multipartite system. The associated Lindblad superoperator contains pairwise terms describing cross correlation between the different subsystems. Such a model can describe a broad range of physical situations, ranging from quantum channels with memory to photon propagation in concatenated quantum optical systems.
The interaction of a propagating pulse of quantum radiation with a localized quantum system can be described by a cascaded master equation with a distinct initially populated input and a finally populated output field mode [Kiilerich and M\o{}lmer, Phys. Rev. Lett. 123, 123604 (2019)]. By transformation to an appropriate interaction picture, we break the cascaded nature of the master equation and recover an effective time-dependent interaction with a lossless single mode and a supplementary lossy mode. The former closely represents the traveling pulse, while the latter constitutes a non-Markovian component in the exchange of quanta between the scatterer and the quantized field. The transformed master equation offers important insights into the system dynamics, and it permits numerically efficient solutions.
The quantum collision models are a useful method to describe the dynamics of an open quantum system by means of repeated interactions between the system and some particles of the environment, which are usually termed “ancillas”. In this paper, we review the main collision models for the dynamics of multipartite open quantum systems, which are composed of several subsystems. In particular, we are interested in models that are based on elementary collisions between the subsystems and the ancillas, and that simulate global and/or local Markovian master equations in the limit of infinitesimal timestep. After discussing the mathematical details of the derivation of a generic collision-based master equation, we provide the general ideas at the basis of the collision models for multipartite systems, we discuss their strengths and limitations, and we show how they may be simulated on a quantum computer. Moreover, we analyze some properties of a collision model based on entangled ancillas, derive the master equation it generates for small timesteps, and prove that the coefficients of this master equation are subject to a constraint that limits their generality. Finally, we present an example of such collision model with two bosonic ancillas entangled in a two-mode squeezed thermal state.
We study the Markovian dynamics of a collection of nnn quantum systems coupled to an irreversible environmental channel consisting of a stream of entangled qubits. Within the framework of repeated quantum interactions, we derive the master equation for the joint-state dynamics of the nnn quantum systems. We investigate the evolution of the joint state for two-qubit environments where the presence of antidiagonal coherences in the state of the bath qubits (in the local energy basis) is essential for preserving and generating entanglement between two remote quantum systems. However, maximally entangled bath qubits, such as Bell states, exhibit exceptional behavior, where the master equation does not have a unique steady state and can destroy entanglement between the systems. For the general case of nnn-qubit environments we show that antidiagonal coherences that arise from multibody entanglement in the bath qubits do not affect the composite system evolution in the weak-coupling regime.
Master equations in the Lindblad form describe evolution of open quantum systems that are completely positive and simultaneously have a semigroup property. We analyze the possibility to derive this type of master equations from an intrinsically discrete dynamics that is modelled as a sequence of collisions between a given quantum system (a qubit) with particles that form the environment. In order to illustrate our approach we analyze in detail how the process of an exponential decay and the process of decoherence can be derived from a collision-like model in which particular collisions are described by SWAP and controlled-NOT interactions, respectively.