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arXiv:1312.6910 · Communications in Theoretical Physics

Quantum Fisher Information for Density Matrices with Arbitrary Ranks

Jing Liu, Xiao-Xing Jing, Wei Zhong, Xiao-Guang Wang

Published Jan 1, 2014• 127 citations
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Abstract

We provide a new expression of the quantum Fisher information (QFI) for a general system. Utilizing this expression, the QFI for a non-full rank density matrix is only determined by its support. This expression can bring convenience for an infinite-dimensional density matrix with a finite support. Besides, a matrix representation of the QFI is also given.

DOI
10.1088/0253-6102/61/1/08

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We provide a new expression of the quantum Fisher information (QFI) for a general system. Utilizing this expression, the QFI for a non-full rank density matrix is only determined by its support. This expression can bring convenience for an infinite-dimensional density matrix with a finite support. Besides, a matrix representation of the QFI is also given.
The quantum Fisher information matrix is a central object in multiparameter quantum estimation theory.It is usually challenging to obtain analytical expressions for it because most calculation methods rely on the diagonalization of the density matrix.In this paper, we derive general expressions for the quantum Fisher information matrix that bypass matrix diagonalization and do not require the expansion of operators on an orthonormal set of states.Additionally, we can tackle density matrices of arbitrary rank.The methods presented here simplify analytical calculations considerably when, for example, the density matrix is more naturally expressed in terms of nonorthogonal states, such as coherent states.Our derivation relies on two matrix inverses that, in principle, can be evaluated analytically even when the density matrix is not diagonalizable in closed form.We demonstrate the power of our approach by deriving novel results in the timely field of discrete quantum imaging: the estimation of positions and intensities of incoherent point sources.We find analytical expressions for the full estimation problem of two point sources with different intensities and for specific examples with three point sources.We expect that our method will become standard in quantum metrology.
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 ρθρ_θρθ​ 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.
Quantum Fisher information matrix (QFIM) is a cornerstone of modern quantum metrology and quantum information geometry. Apart from optimal estimation, it finds applications in description of quantum speed limits, quantum criticality, quantum phase transitions, coherence, entanglement, and irreversibility. We derive a surprisingly simple formula for this quantity, which, unlike previously known general expression, does not require diagonalization of the density matrix, and is provably at least as efficient. With a minor modification, this formula can be used to compute QFIM for any finite-dimensional density matrix. Because of its simplicity, it could also shed more light on the quantum information geometry in general.
The quantum Fisher information (QFI) sets a fundamental bound on the attainable precision when estimating multiple parameters simultaneously. Incompatibility among the individually optimal measurements can, in some cases, imply that the precision limit set by the QFI is not attainable. In certain special cases, including pure states and full-rank states, the exact conditions for when the precision limit set by the QFI can be saturated are known. However, general conditions for the attainability of the QFI with measurements on individual copies of the quantum state have long been sought. Indeed, this was recently stated as one of the five problems in quantum information theory highlighted by [P. Horodecki et al, PRX Quantum 3, 010101 (2022)]. In this work we prove that exact attainability with individual measurements is NP-hard to decide, even for a restricted set of real, constant-rank quantum models. The source problem for our proof is the NP-hard problem of deciding whether a given bipartite density matrix is separable or not. Our result shows that the longstanding difficulty in obtaining general conditions for QFI attainability reflects a fundamental computational obstruction, rather than merely a limitation of existing mathematical techniques: unless P=NP, no efficiently computable necessary-and-sufficient criterion can exist in general.