Entropy exchange

In quantum mechanics, and especially quantum information processing, the entropy exchange of a quantum operation $\phi \,$ acting on the density matrix $\rho _{Q}\,$ of a system $Q\,$ is defined as

$S(\rho ,\phi )\equiv S[Q',R']=S(\rho _{QR}')$

where $S(\rho _{QR}')\,$ is the von Neumann entropy of the system $Q\,$ and a fictitious purifying auxiliary system $R\,$ after they are operated on by $\phi \,$. Here,

$\rho _{QR}=|QR\rangle \langle QR|\quad ,$

$\mathrm {Tr} _{R}[\rho _{QR}]=\rho _{Q}\quad ,$

and

$\rho _{QR}'=(\phi _{Q}\otimes 1_{R})[\rho _{QR}]\quad ,$

where in the above equation $(\phi _{Q}\otimes 1_{R})$ acts on $Q$ leaving $R$ unchanged.

References

• Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-00217-3. OCLC 844974180.