Editing Quantum channel (section)

== Channel fidelity ==
Another measure of how well a quantum channel preserves information is called '''channel fidelity''', and it arises from [[fidelity of quantum states]]. Given two pure states <math>|\psi\rangle</math> and <math>|\phi\rangle</math>, their fidelity is the probability that one of them passes a test designed to identify the other: <math display="block">F(|\psi\rangle, |\phi\rangle) = |\langle \psi | \phi \rangle|^2.</math> This can be generalized to the case where the two states being compared are given by density matrices:<ref>{{cite journal|first=R. |last=Jozsa |author-link=Richard Jozsa |doi=10.1080/09500349414552171 |title=Fidelity for mixed quantum states |journal=Journal of Modern Optics |date=1994 |volume=41 |number=12 |pages=2315–2323|bibcode=1994JMOp...41.2315J }}</ref><ref>{{cite journal|first1=C. A. |last1=Fuchs |first2=C. M. |last2=Caves |author-link2=Carlton Caves |title=Mathematical techniques for quantum communication theory |journal=Open Systems & Information Dynamics |volume=3 |number=3 |pages=345–356 |year=1995 |arxiv=quant-ph/9604001 |doi=10.1007/BF02228997}}</ref> <math display="block">F(\rho, \sigma) = \left(\mathrm{tr} \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}}\right)^2.</math>

The channel fidelity for a given channel is found by sending one half of a maximally entangled pair of systems through that channel, and calculating the fidelity between the resulting state and the original input.<ref>{{cite journal|first1=Dennis |last1=Kretschmann |first2=Reinhard F. |last2=Werner |author-link2=Reinhard F. Werner |year=2004
|title=''Tema con variazioni'': quantum channel capacity |journal=New Journal of Physics |volume=6 |issue=1 |page=26 |doi=10.1088/1367-2630/6/1/026 |arxiv=quant-ph/0311037|bibcode=2004NJPh....6...26K }}</ref>