Bures metric

In mathematics, in the area of quantum information geometry, the Bures metric (named after Donald Bures) or Helstrom metric (named after Carl W. Helstrom) defines an infinitesimal distance between density matrix operators defining quantum states. It is a quantum generalization of the Fisher information metric, and is identical to the Fubini–Study metric when restricted to the pure states alone.

The metric may be defined as

where ${\displaystyle G}$ is Hermitian 1-form operator implicitly given by

Some of the applications of the Bures metric include that given a target error, it allows the calculation of the minimum number of measurements to distinguish two different states and the use of the volume element as a candidate for the Jeffreys prior probability density for mixed quantum states.

The Bures distance is the finite version of the infinitesimal square distance described above and is given by

where the fidelity function is defined as

Another associated function is the Bures arc also known as Bures angle, Bures length or quantum angle, defined as

which is a measure of the statistical distance between the quantum states.

The Bures metric can be seen as the quantum equivalent of the Fisher information metric and can be rewritten in terms of the variation of coordinate parameters as

where ${\displaystyle L_{\mu }}$ is the Symmetric Logarithmic Derivative operator (SLD) defined from

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