Distance measures
Distance measurements are employed to measure the distance that quantitatively separate two quantum states.
Trace distance
The classical trace distance (also known as Kolmogorov or L_1 or total variation distance) computes the distance between 2 classical probability distributions. Considering 2 probability distributions \(\{p_x\}\) and \(\{q_x\}\) the classical trace distance is given by:
\[D(p_x, q_x) = \frac{1}{2} \sum_x |p_x - q_x|\]The quantum generalization of this metric is the quantum trace distance, which measure the distance between 2 quantum states \(\rho\) and \(\sigma\):
\[D(\rho, \sigma) = \frac{1}{2} Tr |\rho - \sigma|\]For further details, one may refer to [1] Part III. Chapter 9.
Fidelity
The classical fidelity computes the distance between 2 classical probability distributions \(\{p_x\}\) and \(\{q_x\}\):
\[F(p_x, q_x) = \sum_x \sqrt{p_x q_x}\]The fidelity is not a metric: \(F(p_x, q_x) = 1\) when \(\{p_x\}\) and \(\{q_x\}\) are identical.
The quantum fidelity between 2 quantum states \(\rho\) and \(\sigma\) is defined by:
\[F(\rho, \sigma) = Tr \left( \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}} \right)\]The fidelity and its square are both referenced as fidelity in the litterature.
For further details, one may refer to [1] Part III. Chapter 9.
References
- [1]M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information. Cambridge university press, 2010.