Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics
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70
Although
these
divergences look very
natural,
it
is usually difficult
to
com-
pute
them,
unless
the
operators
rp
and
&
commute,
in
which case
they
both
can
be
written
as
VZr
(W;4?)
=
(4?-w)(1)
-w
(lr
(!)),
with
h/2
(q)
= 2
(01
-1)
for r =
1/2
and
1
h
(q)
=
"2
(q
- 1 +
Iq
-
11)
==
(q
-
lL
\lq;::O:
O.
(14)
(15)
Note
that
the
operator
function
ll/2
is
the
square
root
case
of
the
REmyi
logarithm
1 1
lr
(q)
= -
(e
r1nq
-
1)
= -
(qr
- 1),
lo
(q)
= lnq
(16)
r r
or
r-Iogarithm,
which is well defined for
any
0
S;
r < 1
as
a
smooth,
strictly
monotone
and
concave
operator
function
of
q > 0 including
the
limiting
case r
~
0
when
lo
= lim
lr
is
the
natural
logarithm
lo
= In.
It
can
be
naturally
extended
to
a
proper
concave function
on
lR
by
l
(q)
=
-00
on
q
S;
0,
has
finite
strictly
negative values for 0 < q < 1
with
l (1) = 0
and
the
normalized derivative
l~
(1) = 1
at
q = 1
and
is
strictly
positive if q >
O.
However,
in
the
case r = 1
the
REmyi
"logarithm"
l
(q)
=
q-l
is
not
concave
but
only affine,
corresponding
to
the
trivial
divergence d (w,
4?)
= 0
in
(14)
if
h is replaced
by
this
l.
This
is why
in
the
case r = 1 we redefine
the
divergence
by
another
monotone
concave function (15) which is, however,
not
strictly
monotone
and
concave
and
is
not
smooth
at
q =
l.
The information divergence V
(w;
4?)
of w from
4?
is usually defined
as
a positive negaentropy V
z
=
-Sz
by
the
semifinite relative entropy
(17)
Here w
(ill)
= w
(m)
and
l is usually
taken
to
be
the
Renyi
logarithm
(16),
r E
[0,
1[
for which In
(rpj&)
is usually
understood
as
In
rp
- In &.
3.2.
The
general
information
divergences
The
general information divergence V (w;
4?)
of
states
on
a
matrix
algebra
M
can
be
defined like a
distance
to
have only positive values, however, unlike
the
distance
it
is
not
assumed
to
be
symmetric
and
satisfying
the
triangular
inequality,
and
usually is allowed
to
have also
the
infinite value
+00,
say,
for some
central
states
w i-
4?
on
an
infinite dimensional M
S;;;
lIl\
(~).
The
71
reference
state
cp,
or
weight as
unnormalized
state,
is
said
to
be
tracial,
or
central,
if
the
density
operator
rp
commutes
with
all & E
SM.
Obviously
it
is
the
case if
rp
= 1
corresponding
to
the
standard
trace
cp
(m)
=
(m,1)
=
Tt
[m]
which majorizes
as
1
:::>
&
any
state
w normalized as
e [&]
:=
(1,&)
= 1
'Vw
E SM.
with
respect
to
the
trace
e
on
M
T
•
If
the
state
w is
not
too
different from
the
reference
cp
in
the
sense
that
it
is
dominated
by
the
equally normalized
cp,
i.e.
if
cp
(1) = w (1)
and
w is
majorized
by
ACP
for a A > 0, a finite
information
divergence V
(w;
cp)
of
w from
cp
is usually defined as a positive
negaentropy
VI
=
-SI
by
the
semifinite relative entropy (17).
More generally, we shall define a positive l-divergence
VI
as
dual
to
a
relative
l-entropy
in
the
sense
VI
(w,
cp)
+ Sl (w,
cp)
=
(cp
-
w)
(rz)
with
rl
= l' (1) (18)
for a
suitable
contrast
function
l.
One
can
always
take
l (q) = e
r1nq
- 1
==
rlr
(q) having q = l' (1) =
r1,
or
the
Renyi
logarithm
l =
IT)
0
~
r < 1,
normalized
by
l~
(1)
:=
Oqlr
(1)
=
1,
but
it
can
be
any
operator-monotone
concave function l :
JR.+
----+
JR.,
positive
on
(1, (0), negative
on
(0,1)
and
smooth
at
q =
1.
Note
that
the
latter
condition
excludes
the
function (15)
corresponding
to
the
trace
distance
V = d
1
for which
the
entropy
S is
not
well-defined
by
(18)
but
can
be
taken
zero Sh (w,
cp)
= 0 for
any
w
~
cp
by
choosing
the
subderivative
rl
E
l~
(1) as
rl
=
1.
The
above
properties
of
the
contrast
function
are
derived from
the
fol-
lowing
divergence
axioms
if
they
hold
on
the
whole cone Jt
of
w,
or
at
least
on
the
convex
subset
Jt
o
=
{w:::>
0:
W (1)
~
I}
containing
0 E Jt.
By
UCP
we
denote
the
properties
of
unitality
A (1) = 1
and
complete
positivity
[A
(bibk)]
:::>
0 for a linear
normal
map
A :
lIlS
----+
M.
(1)
V(w;
cp)
:::>
0
with
V = 0
{?
W =
cpo
(Strict
positivity
and
distin-
guishability)
(2)
V
(EBAiwi,
EBAiCPi)
=
2:
Ai
V
(Wi,
CPi)
'VAi
:::>
0,
2:
Ai
=
1.
(Direct
affinity)
(3)
V(w
0
A;
cp
0
A)
~
V(w;
cp)
for
any
UCP
map
(Operational
monotonicity. )
In
addition,
we say
that
V(
W;
cp)
is
semifinite
if
(4)
V
(w;
cp)
<
00
if
W
~
ACP
for a positive
A;
that
is
smooth
(differentiable)
if
(5)
the
function d (s, t) = V (w +
s{);
cp
+ M) is
smooth.
72
that
it
is
negaentropy
defining
the
entropy
S =
(cp
-
w)
(r) - V
if
(6)
S(w;
cp)
::;
S(w;
CPl)
for every
cP
::;
CPl
and
an
r >
O.
Not
e
that
in
the
commutative
case
[iV,
cp]
= 0
the
above axioms define
the
l-entropy
(17)
and
the
corresponding
divergence
uniquely
up
to
the
choice
of
the
function
l
similat
to
the
entropic
function g in
18,19
if
they
hold
on
the
whole cone
.it
of
w ,
or
at
least
on
the
convex
subset
.ito
=
{w
;::::
0 : w (1)
::;
I}
containing
0
E.it.
Given a
relativ
e
entropy
S,
the
function
l
can
be
easily
deduced
as l
(q)
= S (1;
q)
corresponding
to
the
case
w = 0
EB
1,
cP
= 0
EB
q,
and
V
(w;
cp)
is uniquely defined for
the
completely
decomposable
w =
EBAi
and
cP
=
EB
Aiqi
due
to
the
direct
additivity
by
such
l.
In
future
we
shall
always choose r = 1
by
normalizing
l
such
that
rz
= l' (1) = 1.
3.3.
The
relative
l-entropies
of
types
A&B
If
cp
and
iV
do
not
commute,
cp/iV
is
not
uniquely
defined.
In
the
logarithmic
case l = In
the
naive convention In
(cp
/
iV)
= In
cp
- In
iV
gives
the
Araki-
Umegaki
relative
entropy
SLA)
(w;
cp)
=
e[iV~
(In
cp
-In
iV)iV~]
= w [(In
cp
-In
iv)].
(19)
Note
that
for
the
noncommuting
cp
and
iv
this
convention does
not
lead
to
the
natural
generalization
of
classical
formula
SIn
(w;
cp)
= ¢ (h (iv
jcp))
in
terms
of
logarithmic
entropic
function
h (p) =
-p
In p for
the
Radon-
Nikodym
(RN)
density
p
of
w
with
respect
to
cpo
One
can
take
this
conven-
tion
to
produce
also
the
relative
l-entropy
of
the
A-type
for
any
contrast
function
l,
although
it
may
look even less
natural
as
in
the
case of
A-type
REmyi
entropy
siA)(w;
cp)
= e
[iv~
(er(ln.p-lnib)
-
l)iV~]
=
w(er(ln.p-lnib)
- 1).
corresponding
to
l
(q)
=
qr
-
1.
Moreover,
if
cp
is
not
dominated
by
w,
this
entropy
is ill-defined, while h (iv /
cp)
may
be
still well-defined as a function
1 1
of
the
positive
bounded
operator
cp
- 2
ivcp
- 2 for
any
w
dominated
by
cpo
Therefore,
it
is
more
natural
to
define
the
B-type
relative
l-entropy
(20)
1 1
in
terms
of
the
weight ¢ = e 0 7r
<p'
7r
<p
(p) =
cp
2
pcp
2
and
the
contravariant
1 1
RN-density
iV<p
=
cp
-
2
iVcp-2
of w
with
respect
to
the
weight
cpo
Here
73
h
(p)
= pl
(p
-1)
is l-entropic function, say
defined as a positive
and
concave
on
[0,1]
for
any
contrast
function l,
with
h (0) = 0 = h (1),
having
a
maximum
p~l'
(Po)
> 0
at
the
unique
solution
Po
of
the
equation
pl'
(p)
+ l
(p)
= 0 for a
smooth
l.
The
B-type
of
relative
entropy
was
introduced
by
Belavkin-Staszewski in 1986 for
the
case l = In.
One
can
rewrite
it
in
the
form similar
to
(19) as
where
In(&0-
1
)&
=
&In(0-
1
&)
is
understood
as
(22)
Ohya
and
Petz
proved
that
the
Belavkin-Staszewski divergence gives
better
distinction
of
W relative
to
tp
than
Araki-Umegaki divergence
in
the
sense
that
V9;l
2':
VIC:
l
,
and
that
it
satisfies all required axioms above.
Note
that
the
Hellinger divergences
vi"
1
(w;
tp) of A&B-
type
can
be
1/2
written
as
(13)
in
terms
of
the
corresponding
nonsymmetric
A&B-type
fidelities
fU2
(w;
tp),
and
f;Al = w (er(ln<p-lnwl) , f;B) = ¢
[(0-
1
/
2
&0-
1
/
2
fJ
define
the
relative lr-entropies
of
the
corresponding
type
as
3.4.
The
entropy
increase,
its
concavity
and
additivity
The
relative
entropy
5,
defined
by
the
general divergence as
5 (w;
tp)
=
tp
(1) - w (1) - V (w;
tp)
,
has
almost all
properties
of
V
except
the
positivity
(1), unless tp
2':
w.
Indeed,
by
taking
the
standard
normalization
r = 1
in
the
definition
(6) of
S we have 5 (w;
tp)
:S
tp
(1) - w (1) < 0 if
(tp
-w)
(1) < 0 since
V
(w;
tp)
2':
O.
Thus,
the
relative
entropy
5,
unlike
the
divergence
V,
can
be
used
to
distinguish only
the
states
with
equal
normalization
w (1) = tp (1)
when
5 =
-V.
74
The
property
(2)
of
the
general divergence V is equivalent
to
the
direct
affinity
S(EBA(Wi,EBAi'Pi)
=
LAiS(Wi,'Pi)
VAi
~
O'LAi
= 1
of
the
general relative
entropy
S:
L
Ai
S
(Wi;
'Pi)
= L
Ai
('Pi
- Wi) (1) - L
Ai
V
(Wi;
'Pi)
=
(EBAi'Pi
-
EBAiWi)
(r) - V
(EBAiWi'
EBAi'Pi)'
The
monotonicity
property
(3) implies also
the
entropy
semi increase
S(W
0
A;
'P
0
A)
~
S(w;
'P)
Vw,
'P
for
every
normal
UCP
map
A :
lE
-+
M equivalent
to
the
semidecrease of
the
divergence V
under
the
coarsgraining
A
due
to
('P
0 A - W 0
A)
(1)
=
('P
-
w)
0 A
(1)
=
('P
-
w)
(1).
In
particular,
by
taking
A as
the
embedding
b f---+ 1 ® b of
lE
into
M =
EBiEI
lEi
==
A ®
lE
corresponding
to
the
identical copies
lEi
=
lE
indexed
by
a finite
set
I
with
the
abelian
A =
reI
of
the
diagonal
I x
I-matrices,
we
obtain
by
the
direct
affinity
of
S
that
the
general relative
entropy
S
must
be
jointly
concave
(but
not
convex as
V):
S
(L
AiWi, L Ai'Pi)
~
S
(EBAiwi,
EBAi'Pi)
= L
Ai
S
(Wi,
'Pi)
for
any
Ai
~
0, L
Ai
= 1
and
states
(weights) Wi,
'Pi
on
lEo
The
additional
properties
(4) - (6) have
similar
formulations
in
terms
of
S.
The
logarithmic
case is defined
by
another
additional
joint
additivity
property
'P
= ®'Pi' W = ®Wi
=}
S (w;
'P)
= L S (Wi;
'Pi)
(23)
of
the
entropy
with
respect
to
products
of
weights
'Pi
and
states
Wi
on
Mi'
Here
®'Pi
= ®i=l
'Pi
and
similar
W is
the
product
state
on
M = ®i=l Mi
of
Wi.
One
can
easily see
this
for A
and
B
types
entropy,
(24)
where
e[wln(wi/rPi)]
=
edwdn(wdc,Oi)]
for
each
i
due
to
the
product
trace
e = ®e
i
and
the
normalizations
e
j
[Wj]
= 1
of
each
Wj
with
respect
to
the
trace
e
j
on
the
pre
dual
space
of
M
j
.
The
logarithmic
additivity
can
be
similarly
formulated
in
terms
of
VjS!
only
for
the
equally
normalized
Wi
and
'Pi'
75
3.5.
A
new
type
of
relative
l-entropy
C
Since
the
joint
convexity
of
V
(w;
<p)
implies its convexity
as
a function
w
f-7
Vrp
(ftr)
:=
V(w;<p)
of
only
w,
the
general divergence as a
proper
convex function
Vrp
:
ftr
f-7]-
00,
00],
extended
to
any
ftr
=
&*
by
Vrp
(&) =
00
if &
rf-
Mi,
is uniquely defined
by
the
Legendre-Fenchel
transform
Vrp
(&) =
s~p
{Vz,rp
(m)
-
(m,
&) : m =
m*
EM}
(25)
of
the
inverse Legendre-Fenchel
transform
(ILFT)
Vrp
(m)
=
inf
{(m,
ftr)
+
Vrp
(w) :
ftr
=
ftr*
E M
T
} .
(26)
w
The
ILFT
image
Vrp
of
Vrp
is also well defined
as
a
proper
but
concave
function
Vrp
(m)
E [-oo,oo[
of
any
Hermitian
operator
m from
the
dual
algebra
M.
If
cp
commutes
with
m,
one
can
explicitly find
the
optimal
covariant density
ftr
also
commuting
with
cp
and
evaluate
for
any
type
of
l-divergence V
z
the
transform
Vz,rp
(m)
of l-divergence V
z
on such m E M
independently
of
the type
(.) for each
smooth
contrast
function l.
Thus,
for
the
Ri"myi
logarithm
l = lr
one
can
find
that
V~)
(m)
=
<p
(1-
(1
+
~m)
-t),
where t =
~
-1,
(27)
and
in
the
limit r
--+
0
corresponding
to
l = In
it
is
One
can
interpret
this
Vrp
(m),
which is usually finite
on
m > 0,
as
a free
energy
coinciding
in
first
order
with
the
total
energy
<p
(m)
if m is considered
as a
mass
operator
(relativistic energy), otherwise
Vrp
(m)
could
be
-00
as
a
proper
concave function
on
not
positive m =
m*
.
However,
in
the
noncommutative
case we have implicitly two different
types
of
"free energy"
Vz(A)
and
Vr
B
)
which
do
not
coincide
with
the
above
,rp ,rp
V~
explicitly
evaluated
for
the
function I = lr.
The
Legendre-Fenchel
transform
VZ,rp
(w)
:=
sup
{Vz,rp
(m)
- w
(m)
: m =
m*
EM},
(29)
m
of
the
classically
evaluated
Vz,rp
and
analytically
extended
to
noncommuting
cp
and
m,
say, by (27)
or
(28) for I = lr
and
I = In, defines implicitly
the
informational
divergence
of
a new,
thermodynamical
type
C.
76
One
can
also
take
a
sharper
divergence
V(C)
(w;
<p)
coinciding
on
the
state
space
(3
=
{w
2:
0 : w (1) =
I}
with
V (w;
<p)
and
equal
00
outside of
(3.
It
is defined as
the
Legendre-Fenchel
transform
of
the
constraint
ILFT
v(C)
(m;
<p)
=
i~,f
{ w
(m)
+
V(C)
(w;
<p)
: w
2:
0,
w (1) = 1 }
and
can
be
equivalently defined as
V(C)
=
(<p
-
w)
(1)
-S(C)
by
the
reduced
relative
entropy
s(C)
(w;
<p)
=
inf
{S
(m;
<p)
+ w
(m)
: m =
m*
E M} , (30)
m
where S(m;<p) = <p(1) -
V(m+
1;<p). We shall call such l-entropy
or
l-
divergence
type
C if
it
is given by
V(C)
(m;
<p)
=
<p
(1) -
SI(C)
(m
- 1)
evaluated for all
iIi
commuting
with
rp
by
the
constraint
transform
of
the
entropy
siC)
written
with
the
help
of
Lagrangian
multiplier,
as
SI(C)
(m;
<p)
=
sup
{
SI(C)
(w;
<p)
- w
(m
-
,1)
+ , } .
(31)
'U7
and
analytically
extended
to
any
m E M for which
it
makes sense, otherwise
VI(C)
(m;
<p)
=
00.
Thus
,
the
reduced lr-divergence
vi~)
of
type
C is defined
as
the
transform
(25)
of
the
constraint
REmyi
energy
vi~~
(m;
<p)
=
<p
(1
-
(1
+
ms
-=-
~1
)
-8)
+ r
(m;
<p),
where,
= r
(m;
<p)
is
determined
by
the
normalization condition
w~
(1) = 1
of
the
optimal
state
w~
given m
and
<po
In
the
logarithmic case r
can
be
found explicitly as r = 1 - In
<p
(e-
m
)
such
that
the
constraint
energy
V[;)
(m;
<p)
=
<p
(1) -
Sl~C)
(m
-
1;
<p),
Sl~C)
(m;
<p)
=
In<p
(e
-
m
)
(32)
coincides in
the
first
order
with
the
mean
energy
<p
(m)
/<p
(1) + c
up
to
the
additive
constant
c =
<p
(1) - 1 - In
<p
(1) which is zero for
the
normalized
reference
state
<po
3.6.
Other
new
entropy
types
D&E
Unlike in
the
case
of
types
A
and
B
the
optimal
state
w~C)
resolving
the
variational problem (26) for divergence
of
the
type
C
can
be
found explicitly
77
even for
noncommuting
rp
and
iii
at
least in
the
logarithmic
case
lIn.
Indeed,
it
is given
as
&(e)
-1
1
(8-1)
A
-8md
free
- e
'Pe
s,
o
A
(e)
_
r-111
(8-1)
A
-8md
'W
nor
- e e
'Pe
s
o
by
solving
the
dual
problem
(25) respectively for free (28)
or
constraint
(32)
by
normalization
logarithmic energy using
the
ordering
index
method
for
noncommuting
variation
8m
of
m.
This
implicitly defines
the
entropic
function hi:) (& :
rp)
of
C-
type
relative density for
the
state
'W
with
respect
to
'P
presenting
the
solution
Sl~e)
('W;
'P)
= e
(rp
~
hi:) (& :
rp)
'P
~)
=
'P
(hi:)
(&
:
rp))
. (33)
of
the
variational
problem
(30) coinciding
on
('3
with
unreduced
logarith-
mic relative
entropy
S<p
('W).
It
must
satisfy
the
stationarity
differential
equation
rp
~
hr;:)
(&
:
rp)
rp
~
= iii -
,1
in
terms
of
the
variational
derivative
hr;:)
(&
:
rp)
= 8hi:) (& :
rp)
/8&
re-
solving (31) for (33).
Similarly
we
can
define
another
two
interesting
logarithmic relative en-
tropies
of
type
D
and
E in
the
form
Sl~)
('W;
'P)
= e
(rp
~
hf;!
(& :
rp)
'P
~)
=
'P
(
hf;!
(& :
rp))
.
They
are
determined
respectively
by
the
logarithmic
entropic
functions
hCJ;)
(& :
rp)
and
h~)
(& :
rp)
as
the
solutions
of
the
stationarity
differential
equation
equations
rp~h~(&:rp)rp~
=iii-,1
defined
by
the
variational
derivatives
h~
(& :
rp)
=
8hf;!
(& :
rp)
/8&
in
terms
of
the
RN
derivative
&SD)
=
&<p
(D-type)
and
of
the
exponential
relative
density
&SE)
=
exp
[In
&
-lnrp]
(E-type):
rp~hCJ;)
(&:
rp)
rp~
=
h'
(&SD)),
&SD)
=
rp-~&rp-~,
rp~
h~)
(&
:
rp)
rp~
=
h'
(&SE)) =
h'
(exp
[In
&
-lnrp]).
Here
h'
(p)
=
-In
p
-1
is
the
derivative of
the
logarithmic
entropic
function
h (p) =
-plnp.
Unfortunately,
the
free (28)
or
constraint
(32)
logarithmic
energy
of
types
D
and
E
in
general is difficult
to
evaluated
explicitly if
rp
does
not
78
commute
with
ill,
however
the
optimal
states
resolving
the
corresponding
minimization
problems
(25)
can
be
easily found from
the
stationarity
dif-
ferential
equation with
I = 1 for
the
free case
and
I = r for
the
normalized
case:
,(E)
_
In
",,-iii
W
free
- e ,
,(El
_
r-lln",,-iii
W
nor
- e e .
4.
Quantum
Mutual
Information
and
Encodings
4.1.
Entangled
mutual
information
Here
we
define
symmetric
mutual
information
m a
quantum
compound
state
W achieved
by
an
entanglement
7f
:
lBl
---7
AT' or, equivalently, by
7fT
: A
---7
lBlT'
as
the
general
information
divergence
of
the
entangled
state
W
with
respect
to
the
product
state
cp
=
12
Q9
C;
on
M = A
Q9
lBl:
(34)
In
particular,
I.t,k
(7f)
=
vU
(Wi
12
Q9
c;)
is
the
usual
logarithmic
quantum
mutual
information
of
a
particular
type.
Theorem
4.1.
Let A :
lBl
---7
A~
be
an entanglement
of
the state
c;(b)
=
(1go,A(b)) on
lBl
to
(AO,
12°)
with
AO
<;;;
8(gO) as a
CP
map such
as?/
=
A(1~),
and
7f
=
KT
0 A
be
the entanglement to the state
12
=
(!oK
on A
<;;;
8 (g) defined as the composition
of
A through a channel
KT
:
A~
---7
AT
as
the predual to a normal UCP map K : A
---7
A
0
•
Then the following
monotonicity
holds
I
('l()
(.)
()
A
IE
7f
~
I
Ao
IE
A .
, ,
Proof:
This
simply follows from
the
monotonicity
of
V.
(35)
It
is
interesting
to
compute
and
compare
the
different
types
of
entangled
mutual
informations Vz
(Wi
12
Q9
c;)
for
the
particular
types
of
the
contrast
function l. In
particular,
to
compute
the
RE'myi
and
the
usual
logarithmic
informations
vU
(Wi
12
Q9
c;)
for
the
special, say
quantum
Gaussian
entan-
gled
states
wand
to
compare
them
with
the
wellknown
Gaussian
entangled
information
of
the
type
A.
4.2.
The
proper
quantum
entropies
Applying
the
monotonicity
property
for
the
standard
entanglement
A =
7fJ
==
a of
C;
to
12°
=
~
decomposing every
7fT
by
the
Theorem
1 as
7fT
=
aoK
79
we
obtain
immediately
the
explicit
solution
A =
iffi,
1TT
=
a-
:=
1T~
to
the
optimization
problem
sup
IA,lIl\(1T)
= IlIl\,j(a-)
==
7-{lIl\(C;).
7r:/-L07r=c;
(36)
Definition
4.1.
The
maximal
quantum
information
I
~(1T
) =
7-{lIl\(c;)
=
I~
(1T~)
7-{e·)
= Ie·)
lIl\,lIl\
<;
lIl\,lIl\
<;,
In In
(37)
over all
entanglements
1TT
E
Kq(c;)
of
any
(A,a)
to
(lffi,c;),
achieved
on
A
0
=
iffi
by
the
standard
quantum
entanglement
1T
T (a) =
e/
2
ae/
2
==
a-
(a),
is
named
as entangled,
or
proper quantum entropy
of
the
state
c;.
The
positive difference
(38)
is called
the
conditional proper quantum entropy
7-{lIl\IA
=
7-{lIl\
-
IA,lIl\
of
the
entanglement
1T
:
lffi
--->
AT
.
The
semiclassical entropy
SlIl\
(c;)
is defined
the
solution
of
the
extremum
problem
(36)
under
the
additional
constraint
1T
T E
Kc
(c;)
that
A is
an
Abelian
(the
diagonal)
algebra
A = C
(I)
indexed
by
a discrete
set:I
:
sup
{IA,lIl\(1T):
A = C (I)}
==
SlIl\(c;).
7r:V07r=C;
It
is achieved
on
a classical
state
a (a) = L
aipi
such
that
/W1T
= L
PiC;i
=
C;
is given
by
pure
optimal
states
C;i
on
lffi.
If
1T
T E
Kc
(c;),
then,
obviously,
(39)
otherwise
this
semiclassical
conditional
entropy
can
be
negative
if
1TT
does
not
satisfy
the
semiclassical
constraint
Kc
(c;).
Naturally,
7-{lIl\
(c;)
2-
SlIl\
(c;)
and
7-{lIl\IA
(1T)
2-
SlIl\lA
(1T).
Note
that
SI~A)
(c;)
is
the
usual
von
Neumann
entropy
which is achieved
as
the
solution
of
the
semiclassical
extremum
problem
(39)
at
any
decomposition
C;
= LPiC;i
into
the
pure
states
C;i
as
st)(c;)
=
S(c;)
+
sup
I:>iV
[~i
ln~il
=
S(c;)
voc;1.=l
Thus,
7-{}:)
2-
S(
c;),
in
particular,
7-{}:)
=
2S
in
the
case
of
full
matrix
algebra
lffi
= B
(f)).
It
is
interesting
also
to
compare
the
entropies 7-{J')
and
r
SL)
of
the
other
types.