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
 * $$\rho_{QR}' = \phi[\rho_{QR}] \quad,$$

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