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.
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.
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
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.
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.
Every open-system dynamics can be associated to infinitely many stochastic pictures, called unravelings, which have proved to be extremely useful in several contexts, both from the conceptual and the practical point of view. Here, focusing on quantum-jump unravelings, we demonstrate that there exists inherent freedom in how to assign the terms of the underlying master equation to the deterministic and jump parts of the stochastic description, which leads to a number of qualitatively different unravelings. As relevant examples, we show that a fixed basis of post-jump states can be selected under some definite conditions, or that the deterministic evolution can be set by a chosen time-independent non-Hermitian Hamiltonian, even in the presence of external driving. Our approach relies on the definition of rate operators, whose positivity equips each unraveling with a continuous-measurement scheme and is related to a long known but so far not widely used property to classify quantum dynamics, known as dissipativity. Starting from formal mathematical concepts, our results allow us to get fundamental insights into open quantum system dynamics and to enrich their numerical simulations.
Stochastic unravelings provide a useful way to represent open quantum system dynamics in terms of pure state realizations, and have been widely studied both from a fundamental and from a computational point of view. They were initially formulated for Markovian dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad master equation. However, due to recent technological and experimental development, most physical relevant dynamics present temporal correlations beyond the Markov approximation. Such correlations cause decay rates to turn temporarily negative, thus requiring the generalization of stochastic unravelings from Markovian to non-Markovian scenarios. Indeed, many unraveling techniques have been introduced in this regime, and a comprehensive review of the different jump methods is currently missing. In this work, we provide an overview of widely used quantum jump unraveling techniques for non-Markovian systems and also discuss them in terms of their numerical efficiency, divisibility requirements, Hilbert space extension, and measurement interpretation.
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.