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arXiv:2502.20055

The likelihood operator and Fisher information in quantum probability

Kalyan B. Sinha, Ritabrata Sengupta

Published Feb 27, 2025• 0 citations
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Abstract

We study the problem of Quantum Likelihood Operators (LO) and their connection with quantum Fisher information (QFI). It is observed that the present approaches to this problem tacitly assume commutativity of parametrised density matrix $ρ_θ$ and its derivative, which, in general, need not be true, and this has nontrivial consequences in QFI. As examples, we discuss the parametrised two-level system exhaustively, and, as a further example, the one-mode coherent states of an infinite-dimensional system.

DOI
10.48550/arxiv.2502.20055

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In quantum theory, there are various possibilities of defining logarithmic-derivative operators, each giving rise to a different Likelihood operator, and also to quantum Fisher information (QFI). In each case, the Cramér–Rao (CR) bound is derived from the first principles. It is also noted that if one does not make (often tacitly made) an assumption that the density operator [Formula: see text] and its derivative commute, then they lead to different QFI. Applications to one-mode coherent states (an infinite-dimensional quantum system) and to two-level quantum systems are studied.
In estimating an unknown parameter of a quantum state the quantum Fisher information (QFI) is a pivotal quantity, which depends on the state and its derivate with respect to the unknown parameter. We prove the continuity property for the QFI in the sense that two close states with close first derivatives have close QFIs. This property is completely general and irrespective of dynamics or how states acquire their parameter dependence and also the form of parameter dependence---indeed this continuity is basically a feature of the classical Fisher information that in the case of the QFI naturally carries over from the manifold of probability distributions onto the manifold of density matrices. We demonstrate that, in the special case where the dependence of the states on the unknown parameter comes from one dynamical map (quantum channel), the continuity holds in its reduced form with respect to the initial states. In addition, we show that, when one initial state evolves through two different quantum channels, the continuity relation applies in its general form. A situation in which such a scenario can occur is an open-system metrology where one of the maps represents the ideal dynamics, whereas the other map represents the real (noisy) dynamics. In the making of our main result, we also introduce a regularized representation for the symmetric logarithmic derivative which works for general states even with incomplete rank, and it features continuity similar to the QFI.
The quantum Fisher information (QFI) is a fundamental quantity of interest in many areas from quantum metrology to quantum information theory. It can in particular be used as a witness to establish the degree of multiparticle entanglement in quantum many-body systems. In this work, we use polynomials of the density matrix to construct monotonically increasing lower bounds that converge to the QFI. Using randomized measurements we propose a protocol to accurately estimate these lower bounds in state-of-the-art quantum technological platforms. We estimate the number of measurements needed to achieve a given accuracy and confidence level in the bounds, and present two examples of applications of the method in quantum systems made of coupled qubits and collective spins.
Open quantum system dynamics traditionally described by a quantum master equation can be effectively reformulated in the Liouville space of vectorised density matrices. Despite an extensive use of the Liouville space framework to describe quantum states evolution, its application to quantum metrology is not well explored. Several papers have studied a Liouville-space generalisation of the quantum Fisher information, called the dissipative quantum Fisher information (DQFI). We perform a comparative study between the quantum Fisher information (QFI) in Hilbert space and DQFI, deriving an explicit relation between the two quantities for qubit, qudit, and harmonic oscillator systems. The derived relations are nontrivial, and depend on the state's purity. We show that for a single qubit, the QFI can be efficiently recovered from the DQFI via a simple mapping, while in higher dimensions the expression becomes convoluted and does not convert to a compact form. We find examples where the DQFI is neither an upper nor lower bound for the Hilbert space QFI, and suggest that the informational content of the DQFI itself cannot be straightforwardly interpreted. Our results clarify the limitations of the Liouville space based approach and structural differences with the QFI in the context of quantum parameter estimation problem.