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arXiv:quant-ph/0202117 · Physical Review A

Non-Markovian stochastic Schrödinger equations: Generalization to real-valued noise using quantum-measurement theory

Jay Gambetta, Howard M. Wiseman

Published Jul 29, 2002• 82 citations
arXivHTMLPDF

Abstract

Do stochastic Schr\"odinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system on average obeys a master equation, the answer is yes. Markovian stochastic Schr\"odinger equations generate quantum trajectories for the system state conditioned on continuously monitoring the bath. For a given master equation, there are many different unravelings, corresponding to different sorts of measurement on the bath. In this paper we address the non-Markovian case, and in particular the sort of stochastic Schr\"odinger equation introduced by Strunz, Di\'osi, and Gisin [Phys. Rev. Lett. 82, 1801 (1999)]. Using a quantum-measurement theory approach, we rederive their unraveling that involves complex-valued Gaussian noise. We also derive an unraveling involving real-valued Gaussian noise. We show that in the Markovian limit, these two unravelings correspond to heterodyne and homodyne detection, respectively. Although we use quantum-measurement theory to define these unravelings, we conclude that the stochastic evolution of the system state is not a true quantum trajectory, as the identity of the state through time is a fiction.

DOI
10.1103/physreva.66.012108

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Do stochastic Schr"odinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system on average obeys a master equation, the answer is yes. Markovian stochastic Schr"odinger equations generate quantum trajectories for the system state conditioned on continuously monitoring the bath. For a given master equation, there are many different unravelings, corresponding to different sorts of measurement on the bath. In this paper we address the non-Markovian case, and in particular the sort of stochastic Schr"odinger equation introduced by Strunz, Di'osi, and Gisin [Phys. Rev. Lett. 82, 1801 (1999)]. Using a quantum-measurement theory approach, we rederive their unraveling that involves complex-valued Gaussian noise. We also derive an unraveling involving real-valued Gaussian noise. We show that in the Markovian limit, these two unravelings correspond to heterodyne and homodyne detection, respectively. Although we use quantum-measurement theory to define these unravelings, we conclude that the stochastic evolution of the system state is not a true quantum trajectory, as the identity of the state through time is a fiction.
Do non-Markovian diffusive stochastic Schroedinger equations (SSEs) for open quantum systems have a physical interpretation? In a recent paper we investigated this question using the orthodox interpretation of quantum mechanics. We found that the solution of a non-Markovian SSE at time t represents the state the system would be in at that time if a measurement was performed on the environment at that time, and yielded a particular result. However, the linking of solutions at different times to make a trajectory is, we concluded, a fiction. In this paper we show that such trajectories be interpreted using a non-standard interpretation of quantum mechanics, namely hidden variables theory. The simplest example is a non-Markovian SSE, which we call a position unraveling, which results from a description of the bath using Bohmian mechanics. We find that the noise function z(t) appearing in the non-Markovian SSE can be interpreted as a weighted sum of trajectories for the positions of the bath modes. The solution of the non-Markovian SSE is the state of the system 'conditioned' on values of the bath positions. It also determines the guiding wave for these positions. We also discuss previously used unravelings which have the Markovian limits of homodyne and heterodyne detection.
Do diffusive non-Markovian stochastic Schr\"odinger equations (SSEs) for open quantum systems have a physical interpretation? In a recent paper [Phys. Rev. A 66, 012108 (2002)] we investigated this question using the orthodox interpretation of quantum mechanics. We found that the solution of a non-Markovian SSE represents the state the system would be in at that time if a measurement was performed on the environment at that time, and yielded a particular result. However, the linking of solutions at different times to make a trajectory is, we concluded, a fiction. In this paper we investigate this question using the modal (hidden variable) interpretation of quantum mechanics. We find that the noise function z(t)z(t)z(t) appearing in the non-Markovian SSE can be interpreted as a hidden variable for the environment. That is, some chosen property (beable) of the environment has a definite value z(t)z(t)z(t) even in the absence of measurement on the environment. The non-Markovian SSE gives the evolution of the state of the system ``conditioned'' on this environment hidden variable. We present the theory for diffusive non-Markovian SSEs that have as their Markovian limit SSEs corresponding to homodyne and heterodyne detection, as well as one which has no Markovian limit.
A large class of non-Markovian quantum processes in open systems can be formulated through time-local master equations which are not in Lindblad form. It is shown that such processes can be embedded in a Markovian dynamics which involves a time-dependent Lindblad generator on an extended state space. If the state space of the open system is given by some Hilbert space H\mathcal{H}H, the extended state space is the triple Hilbert space $\mathcal{H}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$ which is obtained by combining the open system with a three-state system. This embedding is used to derive an unraveling for non-Markovian time evolution by means of a stochastic process in the extended state space. The process is defined through a stochastic Schr\"odinger equation which generates genuine quantum trajectories for the state vector conditioned on a continuous monitoring of an environment. The construction leads to a continuous measurement interpretation for non-Markovian dynamics within the framework of the theory of quantum measurement.
An important and well established area of quantum optics is the theory of Markovian stochastic Schr\"odinger equations (or by another name quantum trajectory theory). Recently stochastic Schr\"odinger equations have been developed for non-Markovian systems. In this paper we extend the current known stochastic Schr\"odinger equations for non-Markovian systems to include the position unraveling. We also discuss and illustrate that this stochastic Schr\"odinger equation can have an interpretation under both the orthodox and the de Broglie-Bohm hidden variable interpretation of quantum mechanics. We conclude that only the de Broglie-Bohm hidden variable theory provides a continuous-in-time interpretation of the non-Markovian stochastic Schr\"odinger equation.
By using a condition of average trace preservation we rederive a general class of non-Markovian Gaussian diffusive unravelings [L. Di\'osi and L. Ferialdi, Phys. Rev. Lett. 113, 200403 (2014)], here valid for arbitrary non-Hermitian system operators and noise correlations. The conditions under which the generalized stochastic Schr\"odinger equation has the same symmetry properties (invariance under unitary changes of operator base) as a microscopic system-bath Hamiltonian dynamics are determined. While the standard quantum diffusion model (standard noise correlations) always shares the same invariance symmetry, the generalized stochastic dynamics can be mapped with an arbitrary bosonic environment only if some specific correlation constraints are fulfilled. These features are analyzed for different non-Markovian unravelings equivalent on average. Results based on quantum measurement theory that leads to specific cases of the generalized dynamics [J. Gambetta and H. M. Wiseman, Phys. Rev. A 66, 012108 (2002)] are studied from the perspective of the present analysis.
It is well known that the state operator of an open quantum system can be generically represented as the solution of a time-local equation — a quantum master equation. Unraveling in quantum trajectories offers a picture of open system dynamics dual to solving master equations. In the unraveling picture, physical indicators are computed as Monte Carlo averages over a stochastic process valued in the Hilbert space of the system. This approach is particularly adapted to simulate systems in large Hilbert spaces. We show that the dynamics of an open quantum system generically admits an unraveling in the Hilbert space of the system described by a Markov process generated by ordinary stochastic differential equations for which rigorous concentration estimates are available. The unraveling can be equivalently formulated in terms of norm-preserving state vectors or in terms of linear “ostensible” processes trace preserving only on average. We illustrate the results in the case of a two level system in a simple boson environment. Next, we derive the state-of-the-art form of the Diósi-Gisin-Strunz Gaussian random ostensible state equation in the context of a model problem. This equation provides an exact unraveling of open systems in Gaussian environments. We compare and contrast the two unravelings and their potential for applications to quantum error mitigation.