Entropy exchange

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

$S(\rho ,\varphi )\equiv S[Q^{\prime },R^{\prime }]=S({\rho }_{Q^{\prime }R})$

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

${\rho }_{QR}=|QR⟩⟨QR|,$

${\mathrm{T}\mathrm{r}}_R[{\rho }_{QR}]={\rho }_Q,$

and

${\rho }_{Q^{\prime }R}=({\varphi }_Q\otimes 1_R)[{\rho }_{QR}],$

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