Quantum estimation and stochastic control  


Studies in the foundations of noncommutative probability theory indicate that Bayesian decision theory is the only self-consistent approach to finite-sample size quantum estimation. Bayesian techniques have, however, been drastically underrepresented in the literature on quantum statistical inference.

We adopt a fundamental approach to developing Bayesian estimation theory for quantum systems, starting from discrete (finite-dimensional) systems, where rigorous results are possible. Emphasis is on the development of Monte Carlo algorithms suitable for conditional Bayesian simulation.

In order to optimally control a dynamical system, whether it be quantum, classical, or other (e.g. chemical, financial), it is necessary to have knowledge of its state at any given time. In deterministic open loop control theory, because the evolution of the system can be predicted exactly be solving the differential equations of motion, it is not necessary to estimate the state. However, in any real system subjected to noise or disturbances, the system trajectory will generally never precisely track that predicted by OCT, and a feedback control law is necessary, the input to which is the actual state of the system at time t.

In quantum mechanics, these disturbances are ubiquitous, due to the effects of environmental decoherence, which render the dynamics irreversible. Perhaps more fundamentally, measurement of a quantum system always disturbs it, rendering deterministic optimal feedback controllers very difficult to design. It is therefore important to develop state estimation techniques that provide the most information about the system for the smallest number of measurements.

In addition to the widespread applications of these techniques to quantum control and computation, they may offer evidence in favor of a subjectivist, rather than material interpretation of the quantum state, as a state of knowledge. This approach, dubbed "Bayesian quantum mechanics", has been vocally advanced most recently by P.B. Slater.

Perhaps the most fundamental problem in quantum statistical inference is the reconstruction of the density matrix of a quantum system on the basis of a limited number of quantum observations.  The papers below establish 1) the need for Bayesian estimation of the density matrix in practical quantum engineering applications; 2) an efficient, experimentally applicable methodology for Bayesian state estimation.



Asymptotic Efficiency and Finite Sample Performance of Frequentist Quantum State Estimation

R. Chakrabarti and A. Ghosh (2008)
[working paper]

We undertake a detailed study of the performance of maximum likelihood estimation (MLE) of the density matrix of finite-dimensional quantum systems in order to interrogate generic properties of frequentist quantum state estimation. Existing literature on frequentist quantum estimation has not rigorously examined the finite sample performance of the estimators and associated methods of statistical inference. While MLE is usually preferred on the basis of its asymptotic properties - it achieves the Cramer-Rao (CR) lower bound - the finite sample properties are often less than optimal. Moreover, in quantum estimation, there are multiple CR-type bounds, with the maximum asymptotic efficiency depending on the choice of measurements. The tightest quantum Cramer-Rao lower bound is not even asymptotically achievable in most circumstances.

Here, we compare the asymptotic and finite sample efficiencies of quantum state estimators and test statistics corresponding to various CR bounds for spin-1/2 (one qubit) and spin-1 systems. We show that, in each of these cases, the finite sample properties of MLE differ significantly from the corresponding asymptotic ones for experimentally realistic sample sizes, and that the relative finite sample variances across different measurement strategies are much closer to 1 than the corresponding asymptotic relative efficiencies. These results indicate that in order to fully exploit the information geometry of quantum states and achieve smaller reconstruction errors, the use of Bayesian state reconstruction methods - which, unlike frequentist methods, do not rely on asymptotic properties - is necessary, since the estimation error is typically lower due to the incorporation of prior knowledge.



Quantum Bayesian Reconstruction of General Mixed States: Markov Chain Monte Carlo Estimation

R. Chakrabarti and A. Ghosh

We use Bayesian statistical inference to reconstruct the density matrix of a quantum system for a range of specifications of the prior plausibility distribution. The literature on (numerical) Bayesian state estimation is scant and mostly limited to one-qubit systems, despite the fact that this approach is considerably more general than frequentist methodologies, lending itself readily to cases with incomplete observation levels and a finite number of measurements, and subsuming the more traditional maximum likelihood estimation (MLE) method. Moreover, the estimation error associated with the Bayesian approach is typically lower than that of frequentist MLE due to the incorporation of prior knowledge. Computational complexity is one of the factors inhibiting the proliferation of the Bayesian approach. We propose the use of Markov Chain Monte Carlo (MCMC) methods (tailored to the geometry of the state space) for computation of posterior quantities of interest, and implement numerical Bayesian state estimation for full-rank spin-1/2 (one qubit), spin 1 and spin-3/2 (two qubit) systems. This permits the use of flexible prior distributions in higher dimensions. We compare our results with those obtained using MLE, and explain the implications of the observed differences for state estimation experiments and the foundations of quantum statistical inference.




Next steps

The process by which the most likely values of the state variables of a system are determined are referred to as smoothing, filtering or prediction. For control applications, filtering (estimation of present state parameters based on sequential past measurements) and prediction (forecasting future evolution of the state) are the most important techniques.

In the classical control, financial econometrics and theoretical biology literature, it is well-known that Bayesian approaches to filtering and prediction have many practical advantages over traditional (frequentist) estimation. Since the Bayesian approach is particularly accurate for quantum systems, the development of quantum Bayesian filtering and prediction are essential for the developing the highest precision quantum regulators and controllers. This is my overarching future goal in this area, and it is synergistic with work on the quantum control of logical gates for quantum computation.


Please see the software development page for details on parallel Bayes estimation codes.