- Additivityupon combination of two subsystems which arestatistically uncorrelated. Let
the systems be described by Hilbert spacesHa andHb, with density operatorsρa andρb
respectively, then the full Hilbert space isHab=Ha⊗Hband the density matrix for the
combined system isρab=ρa⊗ρb. The entropy is then additive,
S(ρab) =S(ρa) +S(ρb) (16.41) - Subadditivityupon dividing a system with Hilbert spaceHaband density operatorρabinto
two subsystems with Hilbert spacesHaandHb, and density matricesρaandρbwhich are
statistically correlated. The full density operatorρabisnot the tensor product ofρaandρb,
in view of the non-trivial statistical correlations between the two subsystems. Instead, one
only has an inequality,
S(ρab)≤S(ρa) +S(ρb) (16.42)
with equality being attained iff there are no statistical correlations andρab=ρa⊗ρb. - Strong subadditivityupon dividing a system with density operatorρabcinto three sub-
systems with density operatorsρa,ρb, andρc, and their pairwise combinations with density
operatorsρab,ρbc, andρac, the entropy satisfies
S(ρabc) +S(ρb)≤S(ρab) +S(ρbc) (16.43)
or any rearrangement thereof. - ConcavityForλ 1 ,λ 2 ,···,λr≥0, andλ 1 +···+λr= 1, we have
S(λ 1 ρ 1 +···+λrρr)≥λ 1 S(ρ 1 ) +···+λrS(ρr) (16.44)
Properties 1, 2, 3, and 4 are straightforward. To prove property 5, we have to make careful use
of the definitions. Each density matrix is normalized,
TrHaρa= TrHbρb= 1 (16.45)and the corresponding entropies are defined by
S(ρa) = −kTrHa(
ρalnρa)S(ρb) = −kTrHb(
ρblnρb)
(16.46)The total entropy for the tensor product density operatorρab=ρa⊗ρbis then given by
S(ρab) = −kTrHaTrHb(
(ρa⊗ρb) ln(ρa⊗ρb))= −kTrHaTrHb(
(ρa⊗ρb)(lnρa⊗Ib)⊕(ρa⊗ρb)(Ia⊗lnρb))= −kTrHa(
ρalnρa)
−kTrHb(
ρblnρb)
(16.47)which proves the formula of 5. To prove 6, we first prove the following intermediate results.