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

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,

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