# Entropy exchange

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

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

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

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

and

${\displaystyle \rho _{QR}'=\phi [\rho _{QR}]\quad ,}$

where in the above equation ${\displaystyle \phi }$ acts on ${\displaystyle Q}$ leaving ${\displaystyle R}$ unchanged.