Bures metric


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.

Definition

The metric may be defined as
where is Hermitian 1-form operator implicitly given by
which is a special case of a continuous Lyapunov equation.
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.

Bures distance

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 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
which holds as long as and have the same rank. In cases where they do not have the same rank, there is an additional term on the right hand side.
is the Symmetric Logarithmic Derivative operator defined from
In this way, one has
where the quantum Fisher metric 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
As it happens with the classical Fisher information metric, the quantum Fisher metric can be used to find the Cramér–Rao bound of the covariance.

Explicit formulas

The actual computation of the Bures metric is not evident from the definition, so, some formulas were developed for that purpose. For 2x2 and 3x3 systems, respectively, the quadratic form of the Bures metric is calculated as
For general systems, the Bures metric can be written in terms of the eigenvectors and eigenvalues of the density matrix as
as an integral,
or in terms of Kronecker product and vectorization,
where the overbar denotes complex conjugate, and denotes conjugate transpose.

Two-level system

The state of a two-level system can be parametrized with three variables as
where is the vector of Pauli matrices and is the Bloch vector satisfying.
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

Three-level system

The state of a three-level system can be parametrized with eight variables as
where are the eight Gell-Mann matrices and the 8-dimensional Bloch vector satisfying certain constraints.