# Continuity of the quantum Fisher information

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**Continuity of the quantum Fisher information.** / Rezakhani, A. T.; Hassani, M.; Alipour, S.

Research output: Contribution to journal › Article › Scientific › peer-review

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*Physical Review A*, vol. 100, no. 3, 032317, pp. 1-14. https://doi.org/10.1103/PhysRevA.100.032317

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*Physical Review A*,

*100*(3), 1-14. [032317]. https://doi.org/10.1103/PhysRevA.100.032317

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TY - JOUR

T1 - Continuity of the quantum Fisher information

AU - Rezakhani, A. T.

AU - Hassani, M.

AU - Alipour, S.

PY - 2019/9/12

Y1 - 2019/9/12

N2 - 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.

AB - 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.

KW - RELATIVE ENTROPY

KW - DISTANCE

KW - BOUNDS

U2 - 10.1103/PhysRevA.100.032317

DO - 10.1103/PhysRevA.100.032317

M3 - Article

VL - 100

SP - 1

EP - 14

JO - Physical Review A

JF - Physical Review A

SN - 2469-9926

IS - 3

M1 - 032317

ER -

ID: 37157479