entropy exchange

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

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

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

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

:$\backslash mathrm\{Tr\}\_R[\backslash rho\_\{QR\}]\; =\; \backslash rho\_Q\; \backslash quad,$

and

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

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