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International Journal of Theoretical Physics

Purity of Density Operators for Non-Orthogonal Quantum States with Bounds and Applications to Quantum Fisher Information

Orhan Amirov, Necati Çeli̇k, Mesut Karabacak

Published Jul 15, 2026• 0 citations
Publisher
DOI
10.1007/s10773-026-06401-6

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The quantum Fisher information (QFI) quantifies the ultimate precision of estimating a parameter from a quantum state and can be regarded as a reliability measure of a quantum system as a quantum sensor. However, estimation of the QFI for a mixed state is in general a computationally demanding task. In this paper we present a variational quantum algorithm called variational quantum Fisher information estimation (VQFIE) to address this task. By estimating lower and upper bounds on the QFI, based on bounding the fidelity, VQFIE outputs a range in which the actual QFI lies. This result can then be used to variationally prepare the state that maximizes the QFI, for the application of quantum sensing. In contrast to previous approaches, VQFIE does not require knowledge of the explicit form of the sensor dynamics. We simulate the algorithm for a magnetometry setup and demonstrate the tightening of our bounds as the state purity increases. For this example, we compare our bounds with literature bounds and show that our bounds are tighter.
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.
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.