Entropy exchange

In quantum mechanics, and especially quantum information processing, the entropy exchange of a quantum operation [math]\displaystyle{ \phi \, }[/math] acting on the density matrix [math]\displaystyle{ \rho_Q \, }[/math] of a system [math]\displaystyle{ Q \, }[/math] is defined as

[math]\displaystyle{ S(\rho,\phi) \equiv S[Q',R'] = S(\rho_{QR}') }[/math]

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

[math]\displaystyle{ \rho_{QR} = |QR\rangle\langle QR| \quad, }[/math]
[math]\displaystyle{ \mathrm{Tr}_R[\rho_{QR}] = \rho_Q \quad, }[/math]

and

[math]\displaystyle{ \rho_{QR}' = \phi[\rho_{QR}] \quad, }[/math]

where in the above equation [math]\displaystyle{ \phi }[/math] acts on [math]\displaystyle{ Q }[/math] leaving [math]\displaystyle{ R }[/math] unchanged.

## References

* 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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