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IEEE Conference on Decision and Control (CDC)

Information control and online estimation for unital quantum systems

Maison Clouâtré, Stefano Maranò, Moe Z. Win

Published Dec 9, 2025• 0 citations
Publisher

Abstract

Information control and online estimation are studied in this paper, focusing on unital quantum systems. To infer the value of an unknown parameter characterizing a quantum state, control is applied to the quantum system and the resulting state is measured by a fixed apparatus. The control is selected with the aim of maximizing the Fisher information, about the unknown parameter, extracted from the fixed apparatus. We introduce the notion of "controlled Fisher information (CFI)" to describe the maximum Fisher information attainable by control. A method for computing the CFI is designed and an analytic upper bound on the CFI is derived. Necessary and sufficient conditions for the admissibility (i.e., parameter independence) of controls achieving the CFI are given. For scenarios in which the admissibility conditions are not met, we devise an online joint estimation and control algorithm that, with high probability, achieves the CFI after a sufficient number of measurements.

DOI
10.1109/cdc57313.2025.11312256

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Information control and online estimation are studied in this paper, focusing on unital quantum systems. To infer the value of an unknown parameter characterizing a quantum state, control is applied to the quantum system and the resulting state is measured by a fixed apparatus. The control is selected with the aim of maximizing the Fisher information, about the unknown parameter, extracted from the fixed apparatus. We introduce the notion of "controlled Fisher information (CFI)" to describe the maximum Fisher information attainable by control. A method for computing the CFI is designed and an analytic upper bound on the CFI is derived. Necessary and sufficient conditions for the admissibility (i.e., parameter independence) of controls achieving the CFI are given. For scenarios in which the admissibility conditions are not met, we devise an online joint estimation and control algorithm that, with high probability, achieves the CFI after a sufficient number of measurements.
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.
We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101–102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439–3443]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481–4490]. In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states.
We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101–102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439–3443]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481–4490]. In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states. 1. Introduction. Quantum
Quantum control plays a crucial role in quantum inference tasks such as parameter estimation and state discrimination. Despite this, the fields of quantum control and quantum inference have remained largely disjoint. We bridge these fields, revealing that (i) quantum control determines the fundamental performance limits of quantum estimation and state discrimination, and (ii) control of statistical information measures enables performance approaching these limits. This thesis begins by establishing an axiomatic formalization of quantum control theory and developing a mathematical framework for analyzing, approximating, and controlling the evolution of quantum systems. A self-contained treatment of optimal quantum control is presented. This treatment exploits the unique structure of quantum dynamics rather than merely borrowing notions from classical control. Quantum control systems are shown to be universally contractive, which plays an important role in the treatment of optimal quantum control and in the derivation of various approximation results. Next, the thesis proceeds by addressing statistical inference problems, with emphasis on parameter estimation and state discrimination. Controlled information measures are introduced, providing more sensible alternatives to existing information measures for quantum inference, e.g., quantum Fisher information. Controlled information measures are shown to exhibit useful properties, e.g., data-processing inequalities. Methods for deriving achievability and converse bounds on controlled information measures are developed, and adaptive control techniques for information maximization are proposed. This thesis leverages a cross-pollination of control theory, statistical inference, and quantum information towards the development of an application-focused theory of quantum inference.
As we enter the era of quantum technologies, quantum estimation theory provides an operationally motivating framework for determining high precision devices in modern technological applications. The aim of any estimation process is to extract information from an unknown parameter embedded in a physical system such as the estimation converges to the true value of the parameter. According to the Cramér-Rao inequality in mathematical statistics, the Fisher information in the case of single-parameter estimation procedures, and the Fisher information matrix in the case of multi-parameter estimation, are the key quantities representing the ultimate precision of the parameters specifying a given statistical model. In quantum estimation strategies, it is usually difficult to derive the analytical expressions of such quantities in a given quantum state. This review provides comprehensive techniques on the analytical calculation of the quantum Fisher information as well as the quantum Fisher information matrix in various scenarios and via several methods. Furthermore, it provides a mathematical transition from classical to quantum estimation theory applied to many freedom quantum systems. To clarify these results, we examine these developments using some examples. Other challenges, including their links to quantum correlations and saturating the quantum Cramér-Rao bound, are also addressed.
Quantum metrology comprises a set of techniques and protocols that utilize quantum features for parameter estimation which can in principle outperform any procedure based on classical physics. We formulate the quantum metrology in terms of an optimal control problem and apply Pontryagin's maximum principle to determine the optimal protocol that maximizes the quantum Fisher information for a given evolution time. As the quantum Fisher information involves a derivative with respect to the parameter which one wants to estimate, we devise an augmented dynamical system that explicitly includes gradients of the quantum Fisher information. The necessary conditions derived from Pontryagin's maximum principle are used to quantify the quality of the numerical solution. The proposed formalism is generalized to problems with control constraints, and can also be used to maximize the classical Fisher information for a chosen measurement.