Bures Metric - Quantum Fisher Information

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

^2 = \frac{1}{2} \mbox{tr}\left( \frac{d \rho}{d \theta^{\mu}} L_{\nu} \right) d \theta^{\mu} d\theta^{\nu},

where is the Symmetric Logarithmic Derivative operator (SLD) defined from

\frac{\rho L_{\mu} + L_{\mu} \rho}{2} = \frac{d \rho^{\,}}{d \theta^{\mu}}.

In this way, one has

^2 = \frac{1}{2} \mbox{tr}\left d \theta^{\mu} d\theta^{\nu}

where the quantum Fisher metric (tensor components) is identified as

J_{\mu \nu} = \mbox{tr}\left.

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

J_{\mu\nu}^{ } = 4 g_{\mu \nu}

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.

Read more about this topic:  Bures Metric

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