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Physical Review A

Optimal state estimation of controllable quantum dynamical systems

Raj Chakrabarti, Anisha Ghosh

Published Mar 7, 2012• 7 citations
Publisher

Abstract

We present system-theoretic quantum state reconstruction methods that minimize estimation error by combining optimal quantum control with asymptotically efficient estimation. Introducing the notion of optimal observability of a quantum dynamical system---a concept that does not exist in classical control theory---we formulate and solve the Pareto optimal control problem of maximizing state estimation accuracy while minimizing the expenditure of available control and measurement resources. Necessary and sufficient conditions for optimal observability, based on the quantum optimal observability Gramian, are presented. We examine the finite sample efficiency of the estimation methodology for two- and three-level systems using ideal and noisy control fields, and demonstrate the advantages of state reconstruction schemes based on optimal observability theory for experimentally realistic sample sizes. These results indicate that the optimal control of quantum measurement bases can be used to minimize state reconstruction errors by fully exploiting the information geometry of quantum states.

DOI
10.1103/physreva.85.032305

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We present system-theoretic quantum state reconstruction methods that minimize estimation error by combining optimal quantum control with asymptotically efficient estimation. Introducing the notion of optimal observability of a quantum dynamical system---a concept that does not exist in classical control theory---we formulate and solve the Pareto optimal control problem of maximizing state estimation accuracy while minimizing the expenditure of available control and measurement resources. Necessary and sufficient conditions for optimal observability, based on the quantum optimal observability Gramian, are presented. We examine the finite sample efficiency of the estimation methodology for two- and three-level systems using ideal and noisy control fields, and demonstrate the advantages of state reconstruction schemes based on optimal observability theory for experimentally realistic sample sizes. These results indicate that the optimal control of quantum measurement bases can be used to minimize state reconstruction errors by fully exploiting the information geometry of quantum states.
This letter investigates parameter estimation in quantum systems that undergo dynamical evolution. Optimal control problems are formulated to maximize the information, about an unknown parameter, extracted by a given quantum measurement apparatus. This letter introduces the concept of “admissible controls”—control laws that do not depend on the unknown parameter they elicit. For scalar parameter estimation in unital quantum systems interrogated by binary measurements, this letter derives a necessary and sufficient condition on quantum measurement operators so that an information maximizing control law is admissible. When the admissibility condition is satisfied, it is shown that the resulting optimal control problem may be solved using well-established techniques.
Knowing and guessing, these are two essential epistemological pillars in the theory of quantum-mechanical measurement. As formulated quantum mechanics is a statistical theory. In general, a priori unknown states can be completely determined only when measurements on infinite ensembles of identically prepared quantum systems are performed. But how one can estimate (guess) quantum state when just incomplete data are available (known)? What is the most reliable estimation based on a given measured data? What is the optimal measurement providing only a finite number of identically prepared quantum objects are available? These are some of the questions we address. We present several schemes for a reconstruction of states of quantum systems from measured data: (1) We show how the maximum entropy (MaxEnt) principle can be efficiently used for an estimation of quantum states on incomplete observation levels. (2) We show how Bayesian inference can be used for reconstruction of quantum states when only a finite number of identically prepared systems are measured. (3) We describe the optimal generalized measurement of a finite number of identically prepared quantum systems which results in the estimation of a quantum state with the highest fidelity.
A fundamental difficulty in demonstrating quantum state tomography is that the required resources grow exponentially with the system size. For pure states and nearly pure states, the task of tomography can be more efficient. We proposed two methods for state reconstruction, by (1) minimizing entropy and (2) maximizing likelihood. The algorithm of compressed sampling is employed to solve the optimization problem. Experiments are demonstrated considering 4-qubit photonic states. The results show that (1) much fewer measurements than the standard tomography are sufficient to obtain high fidelity, and (2) the method of maximizing likelihood is more accurate and noise robust than the original reconstruction method of compressed sampling. Furthermore, the physical meaning of the methods of minimizing entropy and maximizing likelihood is clear.
Measurements of quantum states form a key component in quantum-information processing. It is therefore an important task to compare measurements and furthermore decide if a measurement strategy is optimal. Entropic quantities, such as the quantum Fisher information, capture asymptotic optimality but not optimality with finite resources. We introduce a framework that allows one to conclusively establish if a measurement is optimal in the non-asymptotic regime. Our method relies on the fundamental property of expected errors of estimators, known as risk, and it does not involve optimisation over entropic quantities. The framework applies to finite sample sizes and lack of prior knowledge, as well as to the asymptotic and Bayesian settings. We prove a no-go theorem that shows that only classical states admit optimal measurements under the most common choice of error measurement: least squares. We further consider the less restrictive notion of an approximately optimal measurement and give sufficient conditions for such measurements to exist. Finally, we generalise the notion of when an estimator is inadmissible (i.e. strictly worse than an alternative), and provide two sufficient conditions for a measurement to be inadmissible.
Bosonic quantum systems provide a hardware-efficient platform for quantum information processing but remain challenging to characterise due to their large Hilbert space and the high measurement cost of state tomography. Existing approaches estimate the fidelity with respect to a single target state, making them unsuitable for applications in which physically equivalent states differ by phase space translations, rotations, or other transformations. Here, we introduce an adaptive reconstruction technique that estimates the fidelity with respect to a family of bosonic states while reconstructing the underlying Wigner function from a small number of measurements. The method combines a physics-informed parametric model with Bayesian inference, bootstrap, and active learning to iteratively select the most informative phase space sampling points. We implement the approach on a circuit quantum electrodynamics platform and benchmark it on Schrödinger cat states with amplitudes α∈[1,3]α\in[1,3]α∈[1,3]. The reconstruction yields reproducible fidelity estimates within a few minutes, remains robust to substantial displacements and rotations in phase space despite using a mismatched prior, and is sensitive to subtle state imperfections. We further compare the adaptive strategy with existing Wigner function sampling protocols experimentally, demonstrating the advantage of adaptive sampling for measurement-efficient fidelity estimation with respect to a family of cat states. Finally, we incorporate the reconstructed fidelity into the figure of merit used in a proof-of-principle closed-loop quantum optimal control experiment, demonstrating the applicability of the method to autonomous optimisation of bosonic quantum states.
We study the relations between classical information and the feasibility of accurate manipulation of quantum system dynamics. We show that if an efficient classical representation of the dynamics exists, optimal control problems on many-body quantum systems can be solved efficiently with finite precision. In particular, one-dimensional slightly entangled dynamics can be efficiently controlled. We provide a bound for the minimal time necessary to perform the optimal process given the bandwidth of the control pulse, which is the continuous version of the Solovay-Kitaev theorem. Finally, we quantify how noise affects the presented results.