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Register a new userCompletely Positive Quantum Trajectories with Applications to Quantum State Smoothing
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Here, we are concerned with comparing estimation schemes for the quantum state under continuous measurement (quantum trajectories), namely quantum state filtering and, as introduced by us [Phys. Rev. Lett. 115, 180407 (2015)], quantum state smoothing. Unfortunately, the cumulative errors in the most typical simulations of quantum trajectories with a total time of simulation $T$ can reach orders of $T Delta t$. Moreover, these errors may correspond to deviations from valid quantum evolution as described by a completely positive map. Here we introduce a higher-order method that reduces the cumulative errors in the complete positivity of the evolution to of order $TDelta t^2$, whether for linear (unnormalised) or nonlinear (normalised) quantum trajectories. Our method also guarantees that the discrepancy in the average evolution between different detection methods (different `unravellings, such as quantum jumps or quantum diffusion) is similarly small. This equivalence is essential for comparing quantum state filtering to quantum state smoothing, as the latter assumes that all irreversible evolution is unravelled, although the estimator only has direct knowledge of some records. In particular, here we compare, for the first time, the average difference between filtering and smoothing conditioned on an event of which the estimator lacks direct knowledge: a photon detection within a certain time window. We find that the smoothed state is actually {em less pure}, both before and after the time of the jump. Similarly, the fidelity of the smoothed state with the `true (maximal knowledge) state is also lower than that of the filtered state before the jump. However, after the jump, the fidelity of the smoothed state is higher.
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We investigate the evolution of open quantum systems in the presence of initial correlations with an environment. Here the standard formalism of describing evolution by completely positive trace preserving (CPTP) quantum operations can fail and non-completely positive (non-CP) maps may be observed. A new classification of correlations between a system and environment using quantum discord is explored. However, we find quantum discord is not a symmetric quantity between exchange of systems and this leads to ambiguity in classifications - states which are both quantum and classically correlated depending on the order of the two systems. State preparation in quantum process tomography is investigated with regard to non-CP maps. In SQPT the preparation procedure can influence the complete-positivity of the reconstructed quantum operation if our system is initially correlated with an environment. We examine a recently proposed preparation procedures using projective measurements, and propose our own protocol that uses a single measurement followed by unitary rotations. The former can give rise to non-CP evolution while the later will always give rise to a CP map. State preparation in AAPT was found always to give rise to CP evolution. We examine the effect of statistical noise in process tomography and find it can result in the identification of a non-CP when the evolution should be CP. The variance of the distribution for reconstructed processes is found to be inversely proportional to the number of copies of a state used to perform tomography. Finally, we detail an experiment using currently available linear optics QC devices to demonstrate non-CP maps arising in SQPT.
Two long standing open problems in quantum theory are to characterize the class of initial system-bath states for which quantum dynamics is equivalent to (1) a map between the initial and final system states, and (2) a completely positive (CP) map. The CP map problem is especially important, due to the widespread use of such maps in quantum information processing and open quantum systems theory. Here we settle both these questions by showing that the answer to the first is all, with the resulting map being Hermitian, and that the answer to the second is that CP maps arise exclusively from the class of separable states with vanishing quantum discord.
We provide a general construction of quantum generalized master equations with memory kernel leading to well defined, that is completely positive and trace preserving, time evolutions. The approach builds on an operator generalization of memory kernels appearing in the description of non-Markovian classical processes, and puts into evidence the non uniqueness of the relationship arising due to the typical quantum issue of operator ordering. The approach provides a physical interpretation of the structure of the kernels, and its connection with the classical viewpoint allows for a trajectory description of the dynamics. Previous apparently unrelated results are now connected in a unified framework, which further allows to phenomenologically construct a large class of non-Markovian evolutions taking as starting point collections of time dependent maps and instantaneous transformations describing the microscopic interaction dynamics.
The wave-function Monte-Carlo method, also referred to as the use of quantum-jump trajectories, allows efficient simulation of open systems by independently tracking the evolution of many pure-state trajectories. This method is ideally suited to simulation by modern, highly parallel computers. Here we show that Krotovs method of numerical optimal control, unlike others, can be modified in a simple way, so that it becomes fully parallel in the pure states without losing its effectiveness. This provides a highly efficient method for finding optimal control protocols for open quantum systems and networks. We apply this method to the problem of generating entangled states in a network consisting of systems coupled in a unidirectional chain. We show that due to the existence of a dark-state subspace in the network, nearly-optimal control protocols can be found for this problem by using only a single pure-state trajectory in the optimization, further increasing the efficiency.
Quantum state smoothing is a technique to construct an estimate of the quantum state at a particular time, conditioned on a measurement record from both before and after that time. The technique assumes that an observer, Alice, monitors part of the environment of a quantum system and that the remaining part of the environment, unobserved by Alice, is measured by a secondary observer, Bob, who may have a choice in how he monitors it. The effect of Bobs measurement choice on the effectiveness of Alices smoothing has been studied in a number of recent papers. Here we expand upon the Letter which introduced linear Gaussian quantum (LGQ) state smoothing [Phys. Rev. Lett., 122, 190402 (2019)]. In the current paper we provide a more detailed derivation of the LGQ smoothing equations and address an open question about Bobs optimal measurement strategy. Specifically, we develop a simple hypothesis that allows one to approximate the optimal measurement choice for Bob given Alices measurement choice. By optimal choice we mean the choice for Bob that will maximize the purity improvement of Alices smoothed state compared to her filtered state (an estimated state based only on Alices past measurement record). The hypothesis, that Bob should choose his measurement so that he observes the back-action on the system from Alices measurement, seems contrary to ones intuition about quantum state smoothing. Nevertheless we show that it works even beyond a linear Gaussian setting.
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