Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics
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Quantum
BiD-Informatics
IV
eds.
L.
Accardi,
W.
Freudenberg
and
M.
Ohya
© 2011
World
Scientific
Publishing
Co.
(pp.
61-89)
NEW
TYPES
OF
QUANTUM
ENTROPIES
AND
ADDITIVE
INFORMATION
CAPACITIES
VIACHESLAV P. BELAVKIN
School
of
Mathematics,
Nottingham
University, Nottingham,
NG'l2RD,
United
Kingdom
An
elementary
algebraic
approach
to
unified
quantum
information
theory
is given.
An
operational
meaning
of
entanglement
as
specifically
quantum
encoding
is dis-
closed.
The
general
relative
entropy
as
information
divergence
is
introduced
and
three
most
important
types.
of
relative
information,
namely,
Araki-Umegaki
(A-
type)
and
of
Belavkin-Staszewski
(B-type)
and
the
thermodynamical
(C-type)
are
shown.
The
true
quantum
entropy
different
from
the
von
Neumann
semiclassical
entropy
is
introduced
and
the
proper
quantum
conditional
entropy
is
shown.
The
general
quantum
mutual
information
via
entanglement
is
defined
and
the
corre-
sponding
types
of
quantum
channel
capacities
as
the
supremum
via
the
general-
ized
encodings
are
formulated.
The
additivity
problem
for
quantum
logarithmic
capacities
for
the
products
of
arbitrary
quantum
channels
under
the
appropriate
constraints
on
encodings
is
discussed.
It
is
proved
that
the
true
quantum
capacity,
which
is
achieved
on
the
standard
entanglement
as
the
optimal
quantum
encoding,
reclaims
the
additivity
property
of
the
logarithmic
quantum
channel
capacities
via
the
entanglement
on
the
products
of
quantum
input
states.
This
earlier
obtained
by
V.
P.
B.
result
for
quantum
logarithmic
information
of
A-type
is
extended
to
any
type
of
quantum
information
1.
Introduction
It
is
not
commonly known
that
quantum
information
is
already
almost
50
years old.
On
the
mathematical
side
it
was pioneered
by
Umegaki
and
de-
veloped
by
Ohya
and
other
Japanese
mathematicians.
On
the
physical
and
side
it
was pioneered
by
R L
Stratonovich
who, following
the
discovery
of
lasers
and
inspired
by
the
ideas
of
Shannon's
mathematical
theory
of
com-
munications
21,
published a series
of
papers
on
information
transmission
via
quantum
channels in
the
60's
and
even
completed
a
book
on
quantum
information.
In
particular,
he
was
the
first who
introduced
the
quantum
analog of
Shannon's
mutual
information
23
and
computed
it
for
entangled
quantum
Gaussian
variables
24,
predicted
the
classical
capacity
of
quantum
Gaussian
channels
25,
and
found,
jointly
with
Belavkin,
the
optimal
decod-
61
62
ing maximizing
the
capacity
for
such
channels
with
a
Gaussian
source
code
9,10.
Lebedev-Levitin
17
and
Holevo
15
were
among
the
followers
of
this
pioneering work
at
that
'prehistoric'
time
of
quantum
information.
Unlike classical channels,
quantum
channels
can
have several different
capacities
(e.g. for
sending
classical
information
or
quantum
information,
one-way
or
two-way
communication,
prior
or
via
entanglement,
etc.). (See
the
conceptions
of
coherent
information
20,
von
Neumann
mutual
12
and
entanglement
assisted
information
11).
Unfortunately,
most
of
these
at-
tempts
seam
not
satisfactory
because
the
defined
quantities
fail
to
preserve
such
naturally
conjectured
properties
of
informational
capacity
as
additiv-
ity
16,22
for
parallel
channels,
and
some do
not
have
even
the
monotonicity
property
for consecutive channels. Well,
until
recently
the
problem
of
uni-
fying all
these
capacities
within
a
general
information-theoretic
framework
remained
unsolved,
despite
the
innumerous
prediction
of
the
advantages
of
quantum
information.
The
comparison
of classical
and
quantum
capacities
and
the
advantages
and
disadvantages
of
quantum
information
can
be
rig-
orously proved only
within
a unifying
approach.
This
paper
contributes
to
such
unifying
approach
based
on
the
operational
entanglement
theory
to
quantum
channel
capacity
suggested
in
1,2,7,8.
By
enlargement
of
the
class
of
input
encodings,
including
the
encodings
via
entanglement
for one-way
communication
this
approach
shows
that
different
capacities
are
in fact
the
constraint
quantum
capacities
related
to
some
informational
divergences,
and
that
some
of
the
'capacities'
which do
not
have
the
natural
properties
of
the
quantum
divergence,
should
not
be
really considered as
the
capacities
of
quantum
channels.
Recently
tremendous
effort
has
been
made
to
prove
or
disprove
the
addi-
tivity
conjecture
16,22
for
the
Holevo semiclassical
capacity
giving
the
upper
bound
of
the
classical
information
capacity
of a
quantum
communication
channel.
Eventually
this
additivity
conjecture
has
been
steadily
disproved
13,14
proving
that
quasiclassical
constraint
of
quantum
capacity
is
inappro-
priate
for
the
additivity
of
quantum
channels
as
it
was
predicted
in
1,2.
In
this
paper
we
concentrate
on
extending
the
additivity
property
of
the
true,
unconstraint
quantum
entangled
capacity
first proved for
the
A-type
capacity
based
on
the
Araki-Umegaki
relative
entropy
in
1,2
and
also for
the
B-
type
capacity
based
on
Belavkin-Staszewski relative
entropy
in
5.
We
shall
also
introduce
a new,
thermodynamical
type
of
quantum
mutual
infor-
mation
and
corresponding
quantum
channel
capacities
via
entanglement
in
this
unifying
operational
approach.
In
defining
the
general
quantum
capac-
ity
we
shall
follow
an
axiomatic
approach
outlined
for
the
general
relative
63
entropy
by
Petz
19
and
Ruskai,18 see also,
6
however, we
shall
start
from
informational
divergences
and
will
treat
the
relative
entropies
and
informa-
tional
divergences differently
than
in.
18
,19
This
unifying
approach
has
the
natural
level
of
generality
based
on
von
Neumann
W*algebras
tak
en
in,3,8
however,
in
order
to
avoid
the
use
of
noncommutative
integration
theory
we
shall
restrict
the
level
of
generality
to
the
discretely
deco
mposable
algebras
as
it
was
done
in.
1
,7
The
paper
is
organized
as follows:
section
one
introduces
related
no-
tion
of
quantum
probability
and
information
theory,
such
as
quantum
state
and
quantum
entanglement;
section
two
introduces
the
general
quantum
divergence
and
new
types
of
quantum
relative
entropies
via
the
divergence,
section
three
introduces
the
entangled
quantum
mutual
information
and
the
true
quantum
entropies
of
three
types
achieved
via
entanglement;
section
four
introduc
es
quantum
channel
capacity
via
entanglement
encoding
and
shows
additivity
of
the
logarithmic
entangled
quantum
channel
capacity
of
any
type
;
the
final
section
contributes
to
conclusion
and
further
problems.
2.
The
Entanglement
States
and
Operations
2.1.
Normal
states,
densities
and
pairings
As
usual
in
quantum
information
theory, we
may
consider
only
simpl
e
quan-
tum
systems
described
by
matrix
algebras
Mn. However,
in
order
to
include
the
classical
random
codlings
and
k
eep
a closer link
with
classical informa-
tion
theor
y,
the
tensor
product
algebras
Ii-.
=
Mn®C
(J)
ofrandom
matrices
a
j
= a
(j)
E Mn
on
a
discrete
set
J
with
probability
distribution
Aj
2:
0,
L
Aj
=
1,
describing
a
partially
classical 'Alice'
as
a
purely
quantum
(if
IJI
=
1)
or
sem
i-quantum
(IJI
>
1)
, will also
be
allowed for
consideration.
More
generally,
the
semisimple
matrix
algebras
lB
=
EBjM
nj
with
nJ
=
dim
M
nj
,
describing
a
'decomposable
Bob'
as a
quantum
system
obeying
the
superselection
rule,
may
also
be
repres
ented
on
a Hilb
ert
space
~
,
say,
of
total
dimensionality
Lj
nj
:::;
00,
including
the
purely
classical
case
corresponding
to
all
nj
= 1, like
the
Schrodinger's
cat
no +
nl
= 2
as a classical
bit
corresponding
to
J =
{O,
I}.
Each
element
b E
lB
is
an
orthogonal
sum
(or series) b =
EBb
j
represented
as
the
block-diagonal
operator
b = [b
j
5i]
on
the
Hilbert
sum
~
=
EB~j
with
arbitrary
en
tries
b
j
E B
(~j)
:= M
nj
,
where
nj
=
dim
~j
.
Normally
the
quantum
states
64
c;
E
(3
on
such
algebra
IE
are
identified
with
the
linear functionals
C;
(b)
:=
LTr
[bjo-j]
==
T~
[bo-l
jEJ
giving
the
expectations
(b) =
T~
[bo-l
=
c;
(b)
of
b E
IE
considered
as
the
output
observables for a
quantum
system
(Bob). Here
o-j
=
0-;
2:
0 are
fJ-positive (i.e. semipositive
Hermitian)
matrices
normalized
to
the
proba-
bilities
Tr
[0-
j 1 = 7r
j,
L 7r j =
1,
defining
the
decomposable covariant density
operator
0-
=
tBjo-j
which is
in
one-to-one correspondence
C;
f---+
0-
==
~
with
the
state
C;.
In
order
to
avoid
the
consideration
of unphysical
operations
such as
partial
transposition,
and
to
deal only
with
completely positive
operations,
it
is convenient
to
describe
the
states
C;
not
by
the
decomposable
density
~perators
0-
= [o-i8;]
but
by
transposed
density
matrices
0-
=
[o-i8i]
=
0-.
Such decomposable densities
0-
=
tBjo-j
are
also semipositive trace-one
matrices
which
are
in
one-to-one correspondence
C;
f-+
~
:=
0-
with
the
states
C;
(b) =
(b,~)
given, say,
by
the
standard
transposition
~
=
0-
T
with
respect
to
the
usual
tensor
pairing
. I
(b,~)
:=
b~,nl~j,n
==
Tr
[b~]
,
where b =
[b~l,n8i]
==
b
T
is defined
in
a
normal
basis of each
fJj.
The
operators
~
=
0-
are
called contra-variant densities
of
the
states
C;
with
respect
to
the
tensor
pairing,
and
they
coincide
with
the
transposed
~
=
0-
for
the
symmetric
states
C;
(b) =
C;
(b
T).
The
contravariant
densities
are
proved
to
be
more
appropriate
for
the
operational
entanglement
theory
8,
they
can
be
defined (see
the
Appendix)
independent
of
the
basis,
and
they
were
introduced
also for
the
quantum
channels as infinite-dimensional Choi
operators
in
4.
2.2.
Examples:
the
standard
and
the
qubit
paring
If
the
transposition b
f---+
b is
taken
standard
b = b T
in
a basis
of
fJ,
then
jffi
=
IET
coincides
with
IE
only if
the
basis is com
partible
with
the
decomposition
IE
=
tBIEj
into
th!
full
matrix
algebras
IEj
= B (fJj). However,
it
is convenient
~ot
to
identify
IE
and
IE
even
in
the
full
matrix
case
IE
= B
(fJ),
considering
IE
as
an
opposite algebra
to
IE.
In
general
the
transposition
b
f---+
b
can
be
defined
not
related
to
any
specific
orthonormal
basis
of
fJ
as a linear
antimultiplicative
ac
=
ca
in-
vertible
isometry
IE
--+
IE,
commuting
with
the
antilinear
involution a
f---+
a*,
65
a**
= a
as
a*:=8:=8:*,
a=a
VaEE
in
the
usual
notation
identifying a f---+ a
with
the
inverse
transposition
E ----t
E.
The
contravariant
densities
<;-
=
~,
as f)-positive
trace-one
decomposable
matrices,
are
in one-to-one
correspondence
with
the
states
~
(b)
=
(b,
<;-)
with
respect
to
the
tilde-symmetric
pairing
(a, c) =
Tr
[ae]
= Tr [ea] = (c, a) .
The
general
tracial
pairing
(b,
<;-)
(see
the
appendix)
is fully
characterized
by
the
property
(ab, de) = (bc, ad)
Vb
E E,
dEE
and
strict
positivity
of
the
related
scalar
product
(ale) =
(8:,
c)
on
jjj given
by
the
complex
conjugation
a f---+
8:
into
E. More generally
than
the
usual
transposition
<;-T
in a specific basis
it
can
be
given
as
~
=
h-
1
<;-Th.
Here h is
a
Hermitian
invertible
element
h
of
E coinciding
with
the
transposed
h T =
sh
= h
up
to
a
central
signature
element s =
[sj
on
E c (E)
commuting
with
E
such
that
S2
=
11),
which
has
in
each
Ej
= B (f)j)
the
only two
possible values
sj
=
±1.
The
following
example
shows how
to
obtain
in
this
way
the
transposition
a =
h-
1
a
T
h
independent
on
basis
in
the
case
of
the
qubit
f)
=
((:2.
Quantum bit algebra E =
M2
of
2 x 2-
matrices
b =
8"b
:=
= b
o
1 +
b.
8"
[
bo
+ b
3
h -
ib
2
]
~
----t
b
1
+ ib
2
b
o
-
b
3
given in
the
Stock's
basis & =
(8"a)
of
M2
by
the
identity
8"0
= 1
and
----t
the
standard
Pauli
matrices
8"
= (8"1,8"2,8"3), is
naturally
paired
as in
the
Minkowski
space
3
(b,
<;-)
= b . q
:=
L
baqa
=
baqa
=
tr
(b~)
a=O
with
the
same
jjj = M2 by
the
normalized
trace
tr
(b) =
~Tr
[b]
= b
o
,
tr
(1)
=
1.
The
contravariant
density
matrices
<;-
=
qa<;-a
are
most
naturally
given in
the
dual
basis
<;-0
= 1 =
8"0
and
<;-a
= -8"a, a =
1,2,3
such
that
all
Pauli
matrices
<;-a
:=
8"a
=
~a
are
h-transposed,
8"a
=
h
-
l<;-~h
=
8"a,
to
<;-a
with
respect
to
the
antisymmetric
imaginary
Pauli
matrix
h =
8"2
=
h-
1
having
the
signature
s =
-1.
66
2.3.
Coupling
operators
From
now
on
we
shall consider
not
only finite
but
semifinite algebras A
with
involution A * = A,
identity
1 E A
and
a reference
trace
JL
as a linear
strictly
positive functional
JL
(?!)
==
(1,
?!)
/L
on
the
predual
space
AT
generated
by
the
positive functionals
on
A of
the
form a
f--+
(a,
r)
/L
such
that
JL
Car) = (a,r)/L =
JL
(ra)
Va
E
A,r
EAT n
A.
Note
that
the
trace
class
AT
with
respect
to
the
standard trace
JL
=
Trg
of
a Hilbert space
9 is a dense subspace
of
the
weak closure if
<:;;
8(g)
of
A
represented on g,
but
in
general
it
is
the
completion of
AT
n A
with
respect
to
the
dual
norm
to
the
usual
operator
norm
on
A.
The
entanglement
theory
for
the
general
semisimple algebras was developed
in
8,
but
here for simplicity we
can
consider only
the
discrete algebras A
with
normalized
or
nonnormalized
reference
traces
T 9 (a) =
JL
(a).
The
linear functionals
on
A of
the
form
i?
(a) = (a,?!)
/L
given by
?!
E
AT
are
called regular (normal if?!
:::::
0). Every
normal
state
i?
on
A
<:;;
8(g)
can
be
represented as
(1)
by
the
density
?!
=
x*
x E
AT
with
an
operator
x : 9
~
~
satisfying
the
condition
xAx*
<:;;
lB
given by a (decomposable) algebra
lB
<:;;
8(~).
Such
operator
x is said
to
be
coupling
the
state
i?
to
the
state
<;
(b) =
(b,~)
v
on
lB
given by
the
density
~
= x*x E
lBT
with
respect
to
the
trace
Both
states
i?
and
<;
are
the
margins
of
an
entangled state w defined
on
the
tensor
product
algebra A
Q9
lB
by
JL[ax*bxl =
w(a
Q9
b)
= v[bx*axl.
(2)
In
the
case 9 =
~
and
-:::
= ¢1/2 corresponding?! = ¢ this
entanglement
is
called
standard
for
lB
= A.
The
normal
state
i?
represented by
the
covariant
density
p =
xx*
= e as
the
margin
i?
(a)
=
Tg
(ap)
= v
(x*ax)
of
w is said
to
be
entangled
to
~
=
x*
x E
lBT
by
the
coupling
operator
x.
In
the
case 9 =
~
and
x = ¢1/2 corresponding?! = ¢
this
entanglement
is
67
called
standard
on
E = A. We shall see
that
it
is
maximal
entanglement
under
the
constraint
x*
x = g
of
the
given
input
state.
2.4.
Mixed
entangled
states
The
achieved
entanglement
defines
on
A
Q?I
E = B(g
Q?I
~)
a
pure
state
w(aQ?lb);=
(vi (aQ?lb)v)
==
v* (aQ?lb)v
in
terms
of
the
scalar
product
on
9
Q?I
~.
It
is given
by
a
unit
vector
v E 9
Q?I
~
obtained
by
a
partial
transposition
v = x
of
the
entangling
operator
x
such
that
where
J 9
denotes
an
isometric
complex
conjugation
on
g,
and
it
has
in
general
mixed
marginal
states
(a,
g)
= v*(a
Q?I
l~)v,
(b,~)
=
v*(lg
Q?I
b)v.
(3)
Mixed entangled states
can
be
obtained
by
using a stochastic entangling
operator as a row
x.
=
(Xi)
of
entangling
operators
Xi
: 9
----+
~
satisfying
the
constraints
The
partial
transposition
defines
the
components
Vi
=
~
of the
stochastic
state
vector
as
a row
v.
=
(Vi)
of
Vi
E 9
Q?I
~
such
that
W
:=
v.v!
E A
Q?I
E,
Trv!v.:=
L
V;Vi
=
1.
Given a
stochastic
state
vector
Vi
E 9
Q?I
~
we define a _mixed
compound
state
w : A
Q?I
E
----+
C
by
the
contravariant
density
w
:=
W
==
w as
w(a
Q?I
b)
=
T[(aQ?lb)w]
= (aQ?lb,w).
(4)
with
respect
to
the
product
trace
T = T 9
Q?I
T~.
Lemma
2.1.
The compound state (4) can
be
achieved
by
T~[x.ax!b]
= w(aQ?lb) = Tg[x.bx!a],
where
x.
= (Xi) is the row
of
Xi,
with
w(
aQ?ll~)
= (a,
g),
w(
19
Q?lb)
=
(b,~)
given
by
partial traces g = L xi
Xi,
¢ =
Li
xi
Xi
. Moreover, the operator
x.
as
the partially transposition
of
v.
=
(~)
is uniquely defied up
to
a
unitary transformation
U :
v.
f-+
v.U
on the
minimal
auxiliary space.
68
2.5.
Entanglement
as
operation
Let
us write
the
above entangled
state
w
on
A ®
Iffi
as
(5)
where
the
CP
map
7f(b)
=
L:i
X;bXi
with
values
in
AT
bounded
by
Ilbll
?!
is
dual
7f
=
7f
i
to
the
predual
CP
map
7fT
=
7fT
lA,
(6)
Remark
2.1.
The
entanglements
7f
and
7fT
can
be
written
in
terms
of
the
partial
traces
v
and
J-l
on
the
compound
density
operator
W =
L:
V;Vi
as
7f(b)
= v[(lg ® b)w]
==
(b,
w)v'
7fT
(a)
=
J-l[(a
®
l~)w]
==
(a,w)/L'
(7)
(8)
Definition
2.1. A
normal
CP
operation
7f
:
Iffi
-+
AT
normalized
to
a
state
density
?!
=
7f(
1~)
is called generalized entanglement
of
the
state
<;(b)
= (lg,7f(b))
to
Q(a)
=
(l~,7fT(a)).
It
is
proper
(or
true)
entanglement if
the
transposed
positive
map
7f
- :
~
---------
b
-+
7f
(b)
into
A is
not
CP
on
Iffi
(Equivalently, if
the
map
7f;
: a
-+
7fT
(a)
E
Iffi
is
not
CP.)
2.6.
The
standard
entanglement
The
entanglement
is called standard for
the
state
<;
on
Iffi
if
it
couples
<;
to
the
transposed
state
Q =
~
on
A =
iB
by
(9)
Obviously
pT
(a)
= e /
2
ae/
2
==
(j
(a), where
~
=
Q,
and
p =
pT
iff
Iffi
=
Iffi
and
<;
is symmetric,
<;
=
Q.
Given
<;,
the
standard entangled state is defined by
(10)
Theorem
2.1. Every entanglement
7f
on
Iffi
to the state?! E
AT
has a
decomposition
7f
= P 0
II,
(11)
69
where
IT
is normal
CP
contraction
JR
::'
A such that
19
2:
IT
(1~)
2:
E{2
on
the the
minimal
orthoprojector
Ee
E
It.
supporting the density operator
Q.
This decomposition is unique
by
the normalization condition
IT
( 1
~)
= E
{2'
Proof.
IT
can
be
found as a solution
to
the
equation
which is unique if Q is nondegenerate:
(12)
If
Q is dege
nerate,
one should reduce 9
to
the
Hilbert subspace
ge
= Ee9
given by Ee' 0
3.
Informational
Divergences
and
Relative
Entropies
3.1.
Entanglement
measures
The
measure of
entanglement
in
a
state
w is usually
determined
by
its
divergence
V
rs,
(w) =
inf
(V
(w;
'P)
:
'P
E
6d
from a reference subset 6
1
~
6
of
disentangled
states
on
M =
It.
®
JR,
the
convex
span
of
all
product
states
(2
®
<;
on
M, say.
The
simplest such divergence is given
by
the
trace
distance
1
V (w;
'P)
=
'2
(1,1& -
<PI)
==
VI, (w;
'P)
E
[0,1]
which is symmetric, V
(w;
'P)
= V
('P;
w).
Another
important
symmetric
divergence is defined by
quantum
Hellinger distance
V (w;
'P)
=
('P
+
w)
(1) -
2f
(w,
'P)
==
V
l2
(w; 'P),
(13)
I I I
where
f(w
,'P)
=
e[(<p2&<p2)1
2]
is
Uhlman's
symmetric
fidelity defined
with
respect
to
the
product
trace
e =
J1
® v as
f(w,'P)
=
~sup{e(c(+(*~)
:c~=&,(*(=<p}·