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Register a new userQuantum state smoothing as an optimal estimation problem with three different cost functions
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Quantum state smoothing is a technique to estimate an unknown true state of an open quantum system based on partial measurement information both prior and posterior to the time of interest. In this paper, we show that the smoothed quantum state is an optimal state estimator; that is, it minimizes a risk (expected cost) function. Specifically, we show that the smoothed quantum state is optimal with respect to two cost functions: the trace-square deviation from and the relative entropy to the unknown true state. However, when we consider a related risk function, the linear infidelity, we find, contrary to what one might expect, that the smoothed state is not optimal. For this case, we derive the optimal state estimator, which we call the lustrated smoothed state. It is a pure state, the eigenstate of the smoothed quantum state with the largest eigenvalue.
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In this paper, we study the quantum-state estimation problem in the framework of optimal design of experiments. We first find the optimal designs about arbitrary qubit models for popular optimality criteria such as A-, D-, and E-optimal designs. We also give the one-parameter family of optimality criteria which includes these criteria. We then extend a classical result in the design problem, the Kiefer-Wolfowitz theorem, to a qubit system showing the D-optimal design is equivalent to a certain type of the A-optimal design. We next compare and analyze several optimal designs based on the efficiency. We explicitly demonstrate that an optimal design for a certain criterion can be highly inefficient for other optimality criteria.
Rather than point estimators, states of a quantum system that represent ones best guess for the given data, we consider optimal regions of estimators. As the natural counterpart of the popular maximum-likelihood point estimator, we introduce the maximum-likelihood region---the region of largest likelihood among all regions of the same size. Here, the size of a region is its prior probability. Another concept is the smallest credible region---the smallest region with pre-chosen posterior probability. For both optimization problems, the optimal region has constant likelihood on its boundary. We discuss criteria for assigning prior probabilities to regions, and illustrate the concepts and methods with several examples.
We derive the maximum fidelity attainable for teleportation using a shared pair of d-level systems in an arbitrary pure state. This derivation provides a complete set of necessary and sufficient conditions for optimal teleportation protocols. We also discuss the information on the teleported particle which is revealed in course of the protocol using a non-maximally entangled state.
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.
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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