Entropy exchange

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

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

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

${\displaystyle {\rho }_{QR}=|QR⟩⟨QR|,}$

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

and

${\displaystyle {\rho }_{Q^{\prime }R}=\varphi [{\rho }_{QR}],}$

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

• Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-00217-3. OCLC 844974180. 

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