We define an ideal optimal quantum measurement as that measurement on the apparatus for which the average algorithmic information in the measurement record is minimized. We apply the definition to a chaotic system subject to continuous (Markov) quantum nondemolition measurements. For optimized measurements the average information in the record is much closer to the von Neumann entropy than in the nonoptimized case, but increases more quickly in the chaotic region than in the regular region.
We define an ideal optimal quantum measurement as that measurement on the apparatus for which the average algorithmic information in the measurement record is minimized. We apply the definition to a chaotic system subject to continuous (Markov) quantum nondemolition measurements. For optimized measurements the average information in the record is much closer to the von Neumann entropy than in the nonoptimized case, but increases more quickly in the chaotic region than in the regular region.
Using the method of quantum trajectories we show that a known pure state can be optimally monitored through time when subject to a sequence of discrete measurements. By modifying the way that we extract information from the measurement apparatus we can minimize the average algorithmic information of the measurement record, without changing the unconditional evolution of the measured system. We define an optimal measurement scheme as one which has the lowest average algorithmic information allowed. We also show how it is possible to extract information about system operator averages from the measurement records and their probabilities. The optimal measurement scheme, in the limit of weak coupling, determines the statistics of the variance of the measured variable directly. We discuss the relevance of such measurements for recent experiments in quantum optics.
It is shown that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a quantum-classical channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. As a result the entropy reduction is defined for arbitrary a priori states (not only for states with finite von Neumann entropy) and its analytical properties is studied in detail by using general properties of the quantum mutual information. By using this approach it is shown that the entropy reduction of an efficient quantum measurement is a nonnegative lower semicontinuous concave function on the set of all a priori states having continuous restrictions to subsets on which the von Neumann entropy is continuous. Monotonicity and subadditivity of the entropy reduction are also easily proved by this method. A simple continuity condition for the entropy reduction and for the mean posteriori entropy considered as functions of a pair (a priori state, measurement) is obtained. A characterization of an irreducible measurement (in the Ozawa sense) which is not efficient is considered in the Appendix.
The theory of continuous quantum measurement allows to reconstruct the state ρt of a system from a continuous stochastic measurement record It. However, this truly continuous-time signal It is never available in practice. In experiments, one generally has access to its digitization, i.e., to a series of time averages Ik over finite intervals of duration Δt. In this letter, we take this digitization seriously and define ρˉn as the best Bayesian estimate of the quantum state given (only) a digitized record (I1,…,In). We show that ρˉn+1 can be computed recursively from In+1 and ρˉn using an exact formula. The latter can be evaluated numerically exactly, or used as the basis for a perturbative expansion into successive powers of Δt. This allows reconstructing quantum trajectories in regimes of coarse Δt where existing methods fail, estimating parameters at fixed Δt without bias, and directly sampling digitized quantum trajectories with schemes of arbitrarily high order.
Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement which excludes in principle the singular direct observability continual case. Quantum theory of time continuous measurements and quantum prediction theory, developed by the author on the basis of an independent-increment model for quantum noise and nondemolition causality principle in the 80's, solves this problem allowing continual quantum predictions and reducing many quantum information problems like problems of quantum feedback control to the classical stochastic ones. Using explicit indirect observation models for diffusive and counting measurements we derive quantum filtering (prediction) equations to describe the stochastic evolution of the open quantum system under the continuous partial observation. Working in parallel with classical indeterministic control theory, we show the Markov Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these two kinds of measurements. The resulting filtering and Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities and differences with the corresponding classical nonlinear filtering and control problems and demonstrates microduality between quantum filtering and classical control.
When one performs a continuous measurement, whether on a classical or quantum system, the measurement provides a certain average rate at which one becomes certain about the state of the system. For a quantum system this is an average rate at which the system is projected onto a pure state. We show that for a standard kind of continuous measurement, for a qubit this rate may be increased by applying unitary operations during the measurement (that is, by using Hamiltonian feedback), in contrast to the equivalent measurement on a classical bit, where reversible operations cannot be used to enhance the rate of entropy reduction. We determine the optimal feedback algorithm and discuss the Hamiltonian resources required.