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arXiv:2402.05352

Nonlinear functionals of master equation unravelings

Dustin Keys, Jan Wehr

Published Feb 8, 2024• 0 citations
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Abstract

Unravelings provide a probabilistic representation of solutions of master equations and a method of computation of the density operator dynamics. The trajectories generated by unravelings may also be treated as real -- as in the stochastic collapse models. While averages of linear functionals of the unraveling trajectories can be calculated from the master equation, the situation is different for nonlinear functionals, thanks to the corrections with nonzero expected values, coming from the Itô formula. Two types of nonlinear functionals are considered here: variance, and entropy. The corrections are calculated explicitly for two types of unravelings, based on Poisson and Wiener processes. In the case of entropy, these corrections are shown to be negative, expressing the localization introduced by the Lindblad operators.

DOI
10.48550/arxiv.2402.05352

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Unravelings provide a probabilistic representation of solutions of master equations and a method of computation of the density operator dynamics. The trajectories generated by unravelings may also be treated as real -- as in the stochastic collapse models. While averages of linear functionals of the unraveling trajectories can be calculated from the master equation, the situation is different for nonlinear functionals, thanks to the corrections with nonzero expected values, coming from the Itô formula. Two types of nonlinear functionals are considered here: variance, and entropy. The corrections are calculated explicitly for two types of unravelings, based on Poisson and Wiener processes. In the case of entropy, these corrections are shown to be negative, expressing the localization introduced by the Lindblad operators.
The Gorini-Kossakowski-Sudarshan-Lindblad master equation (ME) governs the density matrix of open quantum systems (OQSs). When an OQS is subjected to weak continuous measurement, its state evolves as a stochastic quantum trajectory, whose statistical average solves the ME. The ensemble of such trajectories is termed an unraveling of the ME. We propose a method to operationally distinguish unravelings produced by the same ME in different measurement scenarios, using nonlinear averages of observables over trajectories. We apply the method to the paradigmatic quantum nonlinear system of resonance fluorescence in a two-level atom. We compare the Poisson-type unraveling, induced by direct detection of photons scattered from the two-level emitter, and the Wiener-type unraveling, induced by phase-sensitive detection of the emitted field. We show that a quantum-trajectory-averaged variance is able to distinguish these measurement scenarios. We evaluate the performance of the method, which can be readily extended to more complex OQSs, under a range of realistic experimental conditions. Published by the American Physical Society 2024
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.
Stochastic unravelings of Lindblad-type master equations, such as stochastic Schrödinger equations, provide powerful tools to model open quantum systems and continuous measurement processes. The same master equation can be unraveled in different ways; while these unravelings differ at the level of quantum trajectories, by construction they all yield the same averaged dynamics for the density operator. A recent question of both foundational and practical relevance is whether such unravelings can be operationally distinguished, given that certain nonlinear quantities—such as covariances and higher-order moments of conditional expectation values—are unraveling dependent. We show that these quantities cannot be accessed unless the measurement scheme (i.e., the unraveling) is known in advance. This renders any operational protocol to distinguish unravelings fundamentally unfeasible. We further establish that assuming access to such nonlinear quantities without prior knowledge of the unraveling would enable superluminal signaling, violating relativistic causality.
Distinct monitoring or intervention schemes can produce different conditioned stochastic quantum trajectories while sharing the same unconditional (ensemble-averaged) dynamics. This is the essence of unravelings of a given Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation: any trajectory-ensemble average of a function that is linear in the conditional state is completely determined by the unconditional density matrix, whereas applying a nonlinear function before averaging can yield unraveling-dependent results beyond the average evolution. A paradigmatic example is resonance fluorescence, where direct photodetection (jump/Poisson) and homodyne or heterodyne detection (diffusive/Wiener) define inequivalent unravelings of the same GKSL dynamics. In earlier work, we showed that nonlinear trajectory averages can distinguish such unravelings, but observing the effect in that optical setting requires demanding experimental precision. Here we translate the same idea to a digital setting by introducing synthetic unravelings implemented as quantum circuits acting on one and two qubits. We design two unravelings - a projective measurement unraveling and a random-unitary "kick" unraveling - that share the same ensemble-averaged evolution while yielding different nonlinear conditional-state statistics. We implement the protocols on superconducting-qubit hardware provided by IBM Quantum to access trajectory-level information. We show that the variance across trajectories and the ensemble-averaged von Neumann entropy distinguish the unravelings in both theory and experiment, while the unconditional state and the ensemble-averaged expectation values that are linear in the state remain identical. Our results provide an accessible demonstration that quantum trajectories encode information about measurement backaction beyond what is fixed by the unconditional dynamics.
This paper studies a class of quantum stochastic differential equations, modeling an interaction of a system with its environment in the quantum noise approximation. The space representing quantum noise is the symmetric Fock space over L2R+. Using the isomorphism of this space with the space of square-integrable functionals of the Poisson process, the equations can be represented as classical stochastic differential equations, driven by Poisson processes. This leads to a discontinuous dynamical state reduction which we compare to the Ghirardi-–Rimini–Weber model. A purely quantum object, the norm process, is found, which plays the role of an observer {in the sense of Everett [H. Everett III, Rev. Mod. Phys. 29(3), 454 (1957)]}, encoding all events occurring in the system space. An algorithm introduced by Dalibard et al. [Phys. Rev. Lett. 68(5), 580 (1992)] to numerically solve quantum master equations is interpreted in the context of unraveling, and the trajectories of expected values of system observables are calculated.
Stochastic unravelings are a widely used tool to solve open quantum system dynamics, in which the exact solution is obtained via an average over a stochastic process on the set of pure quantum states. Recently, the generalized rate operator unraveling formalism was derived, allowing not only for the engineering of stochastic realizations, but also to unravel without reverse jumps even for some dynamics in which P-divisibility is violated, thus greatly improving the simulation efficiency. This is possible because the unraveling depends on an arbitrary nonlinear transformation which can incorporate the memory effects. In this work, a stochastic Schrödinger equation for this formalism is derived, both for the cases with and without reverse jumps. It is also shown that a failure of this method can be used to witness master equations leading to unphysical time evolutions, independently of the particular nonlinear transformation considered.