In quantum mechanics , and especially quantum information processing , the entropy exchange of a quantum operation
ϕ
{\displaystyle \phi \,}
acting on the density matrix
ρ
Q
{\displaystyle \rho _{Q}\,}
of a system
Q
{\displaystyle Q\,}
is defined as
S
(
ρ
,
ϕ
)
≡
S
[
Q
′
,
R
′
]
=
S
(
ρ
Q
R
′
)
{\displaystyle S(\rho ,\phi )\equiv S=S(\rho _{QR}')}
where
S
(
ρ
Q
R
′
)
{\displaystyle S(\rho _{QR}')\,}
is the von Neumann entropy of the system
Q
{\displaystyle Q\,}
and a fictitious purifying auxiliary system
R
{\displaystyle R\,}
after they are operated on by
ϕ
{\displaystyle \phi \,}
. Here,
ρ
Q
R
=
|
Q
R
⟩
⟨
Q
R
|
,
{\displaystyle \rho _{QR}=|QR\rangle \langle QR|\quad ,}
T
r
R
[
ρ
Q
R
]
=
ρ
Q
,
{\displaystyle \mathrm {Tr} _{R}=\rho _{Q}\quad ,}
and
ρ
Q
R
′
=
(
ϕ
Q
⊗
1
R
)
[
ρ
Q
R
]
,
{\displaystyle \rho _{QR}'=(\phi _{Q}\otimes 1_{R})\quad ,}
where in the above equation
(
ϕ
Q
⊗
1
R
)
{\displaystyle (\phi _{Q}\otimes 1_{R})}
acts on
Q
{\displaystyle Q}
leaving
R
{\displaystyle R}
unchanged.
References
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