In mathematics, in the area of quantum information geometry, the Bures metric or Helstrom metric 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 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.
Quantum Fisher information
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 is the Symmetric Logarithmic Derivative operator (SLD) defined from
In this way, one has
where the quantum Fisher metric (tensor components) is identified as
The definition of the SLD implies that the quantum Fisher metric is 4 times the Bures metric. In other words, given that are components of the Bures metric tensor, one has
The actual computation of the Bures metric is not evident from the definition, so, some formulas were developed for that purpose. Dittmann obtained the following formulas for the quadratic form of the Bures metric, valid for 2x2 and 3x3 systems, respectively
Another important formula is the one found by Hübner. This formula is written in terms of the eigenvectors and eigenvalues of the density matrix and reads
The state of a two-level system can be parametrized with three variables as
with . The components of the Bures metric in this parametrization can be calculated as
The Bures measure can be calculated by taking the square root of the determinant to find
which can be used to calculate the Bures volume as
- Donald Bures, (1969) Trans. Am. Math. Soc. 135, p.199.
- C.W. Helstrom, (1967) "Minimum mean-squared error of estimates in quantum statistics", Phys. Lett. A 25 pp.101-102.
- Paolo Facchi, Ravi Kulkarni, V. I. Man'ko, Giuseppe Marmo, E. C. G. Sudarshan, Franco Ventriglia "Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics" (2010), Physics Letters A 374 pp. 4801. DOI: 10.1016/j.physleta.2010.10.005
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- ^ Braunstein, Samuel L. and Caves, Carlton M., Statistical distance and the geometry of quantum states, Phys. Rev. Lett.,72, 3439, 1994.
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- Hübner, M., Computation of Uhlmann's parallel transport for density matrices and the Bures metric on three-dimensional Hilbert space, Phys. Lett. A, 179, 4-5, 1993.
- ^ Paul B Slater, Applications of quantum and classical Fisher information to two-level complex and quaternionic and three-level complex systems , J. Math. Phys. 37, 2682, 1996.
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- ^ Unfortunately, some authors use a different definition as