Talk:Trace distance

The trace distance (also called the variational or Kolmogorov distance) \delta:\mathcal{S}(\mathcal{H})\times\mathcal{S}(\mathcal{H})\rightarrow \mathbb{R} is one of the most natural distance measures on \mathcal{S}(\mathcal{H}). It is intimately related to the problem of distinguishing two states in the following way: The value \frac{1}{2}+\frac{1}{2}\delta(\rho,\sigma) is the average success probability when distinguishing (by a measurement) two states ρ and σ which are given with equal a priori probability.

Mathematically, it is defined as follows:

\delta(\rho,\sigma)=\frac{1}{2}\mathrm{tr}(|\rho-\sigma|)


The connection between this definition in terms of the eigenvalues of the operator ρ − σ goes back to Helstrom [1].

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