ON ENTANGLED QUANTUM CAPACITY
VIACHESLAV P BELAVKIN
Abstract. The pure quantum entanglement is generalized to the case of
mixed compound states to include the classical and quantum correspondences
as particular cases. We show that the classical-quantum encodings can be
treated as classical entanglements. The mutual information of the entangled
states leads to two di¤erent types of entropy for a given quantum state: the
von Neumann entropy, which is achieved as the supremum of the information
over all c-entanglements, and the true quantum entropy, which is achieved at
the standard entanglement. The q-capacity, de…ned as the supremum over all
entanglements, doubles the c-capacity. The conditional q-entropy is positive,
and q-information and q-capacity of a quantum channel are additive.
1. Introduction
The entanglements as speci…cally quantum (q-) correlations, are used for quantum information processes in quantum computation, quantum teleportation, and
quantum cryptography [2, 3, 4]. In this paper, we develope the operational approach
to quantum entanglement and study the mathematical structure of quantum entangled states to provide a comparison of di¤erent quantum information processings.
We show that any compound state can be achieved by a generalized entanglement, and the classically (c-) entangled states of c-q encodings and q-c decodings
can be achieved by d-entanglements, the diagonal c-entanglements for these disentangled states. The pure orthogonal disentangled compound states are most
informative among the c-entangled states in the sense that the maximum of mutual
information over all c-entanglements is achieved on the extreme c-entangled states
as the von Neumann entropy S (%) of a given normal state %. Thus the maximum of
mutual entropy over all classical couplings, described by c-entanglements of a classical probe systems A to the system B, is bounded by the c-capacity C = log rankB,
where rankB is the dimensionality of a maximal Abelian subalgebra A B.
We prove that the truly entangled states are most informative in the sense that
the maximum of mutual entropy over all entanglements to the quantum system B is
achieved by an extreme entanglement of the probe system A = B, called standard
for a given %. The mutual information for such extreme q-compound state de…nes
another type of entropy, the q-entropy H (%)
2S (%). The maximum of mutual
entropy over all quantum couplings, described by true quantum entanglements of
probe systems A to the system B is bounded by Cq = log dim B.
Date : July 2, 2000.
1991 Mathematics Subject Classi…cation. Quantum Information.
Key words and phrases. Entanglements, Compound States, Dimensional Entropy and Quantum Information.
This work was partially supported by the Royal Society grant for UK–Japan collaboration.
Published in: Quantum Communication, Computing, and Measurement 3. Ed Tombesi and
Hirota, Kluwer/Plenum, 2001, 325— 333.
1
2
VIACHESLAV P BELAVKIN
In this short paper we consider the case of a simple algebra B for which some
results are rather obvious and given without proofs. Some proofs are given in the
paper [1], others will be published elsewhere for a more general case of decomposable
algebra B to include the classical discrete systems as a particular quantum case.
2. Compound States and Entanglements
Let H denote a separable Hilbert space of quantum system, and B = L (H) be
the algebra of all linear operators B : H ! H with the Hermitian adjoints B y on
H. A normal state can be expressed as
(1)
y
% (B) = trG
B = trB ;
B 2 B.
Here G is another separable Hilbert space, is a Hilbert-Schmidt operator from G
to H and y is its adjoint H ! G. This is called the amplitude operator (just
amplitude if G is one dimensional space C, = 2 H with y = k k2 = 1 in
which case y is the functional y from H to C). Moreover the density operator
y
=
in (1) is uniquely de…ned as a probability operator in B, i.e. as a positive
trace one operator 2 B. We denote by B
B the predual space identi…ed
with the Banach space T (H) of all trace class operators in H, and the probability
operators (densities) PA 2 A , PB 2 B of the states &, % on di¤erent algebras
A, B will be usually denoted by di¤erent letters , corresponding to their Greek
variations &, %.
In general, G is not one dimensional, the dimensionality dim G must be not less
than rank , the dimensionality of the range ran
H of the density operator :
We shall always equip G with an isometric involution J = J y , J 2 = I, having the
properties of complex conjugation on G,
X
X
J
8 j 2 C; j 2 G
j j =
jJ j;
with respect to which J = J for the positive and so self-adjoint operator =
y
= y on G. The latter can also be expressed as the symmetricity property
~& = & of the state & (A) = trA given by the real and so symmetric density operator
= = ~ on G with respect to the complex conjugation A = JAJ and the tilda
operation (G-transponation) A~ = JAy J on the algebra A = L (G).
Given the amplitude operator , one can de…ne not only the states % by =
y
and & by = y on the algebras B = L (H) and A = L (G) but also a pure
entanglement state $ on the algebra A B of all bounded operators on the tensor
product Hilbert space G H by
$ (A B) = trH B A~ y = trG A~ y B :
This state is pure as it is given by an amplitude 2 G H de…ned as
(
y
)
y
=
J ;
and % and & are the marginals of $:
(2)
$ (I
B) = trH B ;
8 2 G; 2 H;
$ (A
I) = trG A :
As follows from the next theorem, any compound state
$ (A
B) =
y
(A
B) ;
A 2 A; B 2 B
given on L (G H) by an amplitude 2 G H with
a unique entanglement of its marginal states & and %.
y
= 1, can be achieved by
ON ENTANGLED QUANTUM CAPACITY
Theorem 2.1. Let $ : A
(3)
3
B ! C be a compound state
$ (A
y
B) = trF
(A
B) ;
de…ned by an amplitude operator : F ! G H on a separable Hilbert space F into
the tensor product Hilbert space G H with tr y = 1. Then this state is achieved
by an entanglement
(4)
$ (A B) = trF H (I B) A~ y = trG A~ y (I B)
of the states (2) with = y and = trF y , where is an operator G ! F H.
The entangling operator is uniquely de…ned up to a unitary transformation U of
the minimal space by F = ran y ~ U = , where
(5)
(
y
y
) ~ = (J
)
J ;
8 2 F; 2 G; 2 H;
is given by the isometric involutions J in the spaces G and F.
Note that the entangled state (4) is written as
$ (A
B) = trH B (A) = trG A
where the operator (A) = trF
space B for any A 2 L (G), and
|
(6)
(B) = J
A~
y
y
I
|
(B) ;
2 B, bounded by kAk 2 B , is in the predual
By
J=
y
^B)
(I
is in A as a trace-class operator in G, bounded by kBk 2 A . The linear map
|
is written in the Steinspring form [5] of the normal completely positive map
| (B), while
B 7! ^
: A ! B is written in the Kraus form [6] of the normal
A~
completely positive map A 7!
2
F
` (N):
(A) =
X
in the canonical orthonormal basis jki of
(hkj
I) A~
y
(jki
I) :
k
Thus the linear maps | and are dual to each other, || = , both are positive, but they are not completely positive but co-positive as the compositions of the
transponation A 7! A~ and the completely positive (CP) maps. In terms of the compound density operator ! = y for the entangled state $ (A B) = tr (A B) !
they can be written simply as
(7)
(A) = trG (A
|
I) !;
(B) = trH (I
B) !:
De…nition 2.1. The completely co-positive map | : B ! A (or its dual map
: A ! B ), normalized as trG | (I) = 1 (or trH (I) = 1) is called the quantum
entanglement of the state % on B to = | (I) (or of & on A to = (I) ). The
entanglement = q = | by
(8)
q
(B) =
1=2
~
B
1=2
;
B2B
of the state & = % on the algebra A = B is called standard for the system (B; %).
The standard entanglement de…nes the standard compound state
~ 1=2
$q (A B) = trH B 1=2 A~ 1=2 = trH A 1=2 B
on the algebra B B. This state is pure, given by the amplitude = j
y
j 1=2 i = ~ with = 1=2 and ~ de…ned in (5) as (
) ~= y J .
1=2
i, where
4
VIACHESLAV P BELAVKIN
3. Quantum Entropy via Entanglements
As we shall prove in this section, the most informative for a quantum system
(B; %) is the standard entanglement 0 = q = |0 of the probe system A0 ; & 0 =
(B; %), described in (8).
Let us consider the entangled mutual entropy and quantum entropies of states
by means of the di¤erent types of compound states. To de…ne the quantum mutual
entropy, we need to apply a quantum version of the relative entropy to compound
state on the algebra A B, called also the information divergency of the state $
with respect to a reference state '. It can be de…ned by the density operators !;
of these states as
I ($; ') = tr! (ln !
(9)
ln ) :
It has a positive value I ($; ') 2 [0; 1] if the states are equally normalized, say
(as usually) tr! = 1 = tr , and it can be …nite only if the state $ is absolutely
continuous with respect to the reference state ', i.e. i¤ $ (E) = 0 for the maximal
null-orthoprojector E = 0. The most important property of the information
divergence I is its monotonicity property [7, 8], i. e. nonincrease of the divergency
I ($0 ; '0 ) after the application of the pre-dual K of a normal completely positive
unital map K : M ! M0 to the states $0 and '0 on a von Neumann algebra M0 :
(10)
$ = $0 K; ' = '0 K ) I ($; ')
I ($0 ; '0 ) :
The mutual information IA;B ( ) = IB;A ( | ) in a compound state $ achieved
by an entanglement : B ! A , or by : A ! B with the marginals
& (A) = $ (A
I) = trG A ; % (B) = $ (I
B) = trH B
is de…ned as the relative entropy (9) of the state $ on M = A
the product state ' = & %:
(11)
IA;B ( ) = tr ! (ln !
ln (
I)
ln (I
B with respect to
)) :
The following proposition follows from the monotonicity property (10) of the
relative entropy on M = A B with respect to the predual K (! 0 ) = $0 (K I)
to the ampliation K I of a normal completely positive unital map K : A ! A0 .
Proposition 3.1. Let 0 : A0 ! B be an entanglement of a state & 0 (A) =
tr 0 (A), A 2 A0 = L (G0 ), to (B; %) with
= 0 (I) on B, where 0 = |0 ,
and = 0 K be an entanglement de…ned as the composition of 0 with a normal
completely positive unital map K : A ! A0 . Then IA;B ( )
IA0 ;B 0 . In
particular, for any c-entanglement
to (B; %) there existsP
a not less informative dP
entanglement d (A) = hnjAjni (n) with the same =
(n), and the standard
entanglement q (A) = 1=2 A~ 1=2 of & 0 = % on A0 = B is the maximal one in this
sense.
Note that any extreme d-entanglement decomposed into pure normalized states
n = (n) =tr (n) is maximal among all c-entanglements in the sense IA;B ( d )
IA;B ( c ). This is because tr n ln n = 0, and therefore the information gain
X
IA;B ( d ) =
(n) tr n (ln n ln )
n
ON ENTANGLED QUANTUM CAPACITY
5
with a …xed d (I) = achieves its supremum trH ln at any such extreme
d-entanglement 1d . Thus the supremum of the information gain (11) over all centanglements to the system (B; %) is the von Neumann entropy
(12)
SB (%) =
trH ln :
It is P
achieved on any extreme d , for example given by a Schatten decomposition
= n jnihnj (n). The maximal value ln rankB of the von Neumann entropy is
1=2
de…ned by rankB = (dim B) , i.e. by dim H.
De…nition 3.1. The maximal mutual entropy
(13)
HB (%) = sup IA;B ( ) = IB;B (
q) ;
(I)=
~ 1=2 for a …xed
achieved on A0 = B by the standard q-entanglement q (B) = 1=2 B
state % (B) = trH B , is called q-entropy of the state %. The di¤ erence
HBjA ($) = HB (%)
IA;B ($)
is called the q-conditional entropy on B with respect to A.
Obviously, HBjA ($) is positive in contrast to SB (%) IA;B ($) which can achieve
also the negative value
(14)
SB (%)
HB (%) = tr ln
in the case the standard entanglement as the following theorem states.
Theorem 3.2. The q-entropy for the simple algebra B = L (H) is given by
(15)
HB (%) =
2trH ln = 2SB ( ) ;
It is positive, HB (%) 2 [0; 1], and if B is …nite dimensional, it is bounded, with the
1
maximal value HB (% ) = ln dim B which is achieved on the tracial
= (dim H) I,
2
where dim B = (dim H) .
4. Quantum Channel and its Q-Capacity
A noisy quantum channel sends pure input states %1 on the algebra B 1 = L (H1 )
into mixed ones % =
(%1 ) given by the predual
=
jB1 to a normal com1
pletely positive unital map : B ! B ,
(B) = trF1 Y y BY;
B2B
where Y is a linear operator from H1 F+ to H with trF+ Y y Y = I, and F+ is a
separable Hilbert space of quantum noise in the channel. Each input mixed state
%1 is transmitted into an output state % = %1 given by the density operator
( 1) = Y
1
I+ Y y 2 B
for each density operator 1 2 B 1 , where I + is the identity operator in F+ .
The input entanglements 1 = A ! B 1 dual to 1 : B 1 ! A will be denoted as
1
= = |1 . They de…ne the quantum-quantum correspondences (q-encodings) of
probe systems (A; &) with the density operator = | I 1 , to the input B 1 ; %1
of the channel with = (I).
Let Kq1 be the set of all normal completely positive maps : A ! B1 with any
probe algebra A, normalized as tr (I) = 1, and Kq (%1 ) be the subset of 2 Kq1
with (I) = %1 . Each
2 Kq1 can be decomposed as 1q K, where 1q = 0 is
6
VIACHESLAV P BELAVKIN
the standard entanglement on A0 ; & 0 = B 1 ; %1 , and K is a normal unital CP
map A ! B1 . FurtherPlet Kc1 be the set of all c-entanglements described by the
combinations (A) = n & n (A) 1 (n)of the primitive maps A 7! & n (A) 1 (n), and
Kd1 be thePsubset of the diagonalizing entanglements , i. e. the decompositions
(A) =
n hnjAjni 1 (n). As in the …rst case Kc (%1 ) and Kd ( 1 ) denote the
subsets corresponding to a …xed (I) = %1 , and each Kc (%1 ) can be represented as
= 1d K, where 1d = 0 is a pure d-entanglement A0 = B 1 ! B1 normalized as
tr 1d (B) = %1 (B) for all B 2 B1 , by a proper choice of the CP map K : A ! B 1 .
Furthermore
let Ko1 and Ko (%1 ) be the subsets with orthogonal 1 (n) (and …xed
P
2 Ko (%1 ) can also be represented as = 1o K, where
n 1 (n) = 1 ). Each
1
0
1
o = 0 is a pure o-entanglement on A = B corresponding to the same %1 .
Now, let us maximize the entangled mutual entropy for a given quantum channel
(and a …xed input state %1 ) by means of the above four types entanglements
. The mutual entropy (11) was de…ned in the previous section by the density
operators ! of the corresponding compound state $ on A B, and the productstate ' = & % of the marginals &, % for $. In each case
$ = $01 (K
);
' = '01 (K
);
where K is a CP map A ! A0 = B 1 , $01 is one of the corresponding extreme
compound states $q1 , $c1 = $d1 , $o1 on B 1 B 1 , and '01 = 0 %1 .
Proposition 4.1. The entangled mutual informations achieve the following maximal values
(16)
sup
2Kq (%1 )
Ic (%1 ; ) =
(17)
IA;B (
sup
2Kc (%1 )
sup
2Ko (%1 )
IA;B (
IA;B (
1
q
) = Iq (%1 ; ) := IB1 ;B
1
d
) = sup IB1 ;B
1
d
;
= Id (%1 ; ) ;
) = Io (%1 ; ) := sup IB1 ;B
1
o
1
o
;
where 1 are the corresponding extremal input entangled states on B 1
tr 1 (B) = %1 (B) for all B 2 B 1 . They are ordered as
(18)
Iq (%1 ; )
Ic (%1 ; ) = Id (%1 ; )
B 1 with
Io (%1 ; ) :
We shall denote the information Ic (%1 ; ) = Id (%1 ; ) simply as I1 (%1 ; ).
De…nition 4.1. The suprema
Cq ( ) = sup IA;B (
2Kq1
(19)
sup IA;B (
1
2Kd
) = sup Iq (%1 ; ) ;
%1
) = C1 ( ) := sup I1 (%1 ; ) ;
%1
Co ( ) = sup IA;B (
1
2Ko
) = sup Io (%1 ; ) ;
%1
are called the q-, c- or d-, and o-capacities respectively for the quantum channel
de…ned by a normal unital CP map : B ! B1 .
Obviously the capacities (19) satisfy the inequalities
Co ( )
C1 ( )
Cq ( ) :
ON ENTANGLED QUANTUM CAPACITY
7
Theorem 4.2. Let (B) = Y y BY be a unital CP map B ! B1 describing a
quantum deterministic channel. Then
I1 (%1 ; ) = Io (%1 ; ) = S (%1 ) ;
Iq (%1 ; ) = H (%1 ) ;
and thus in this case
C1 ( ) = Co ( ) = ln rankB 1 ;
Cq ( ) = ln dim B 1
In the general case d-entanglements can be more informative than o-entanglements
as it can be shown on an example of a quantum noisy channel for which
I1 (%1 ; ) > Io (%1 ; ) ;
C1 ( ) > Co ( ) :
5. Additivity of Entangled Information.
In order to consider block entanglements let us introduce the product systems
(B n ; % n ) in the tensor product H n = nl=1 Hl of identical spaces Hl = H, and
n
: B n 7! B n , the preduals of the normal unital CP
the product channels
product maps
n
1
:B
n
! Bn =
n
1
l=1 Bl ;
where Bl = B for all l. Let us denote
Inq (%1 ; ) = Iq %1 n ;
n
n
B
n
=
(B)
Cqn ( ) = Cq
;
n
;
n
the suprema of the mutual information IA;B n ( n ) over the set Kq %1 n of
all input entanglements
: A ! B n with any probe algebra A, normalized as
n
(I) = 1 , and with any such normalization respectively, 2 Kqn . It is easily
seen, by applying the monotonicity property (10) with respect to the normal unital
CP map K : A ! A0 = B n in the decomposition = nq K, where
n
q
A0 =
n 1=2
1
n 1=2
A~0
1
that the quantities Inq (%1 ; ), Cqn ( ) are additive:
Inq (%1 ; ) = nIq (%1 ; ) ;
;
A0 2 B n ;
Cqn ( ) = nCq ( ) .
This additivity cannot be proved but super-additivity for the quantities
In1 (%1 ; ) = I1 %1 n ;
n
C1n ( ) = C1
;
n
corresponding to the c-entanglements (or d-entanglements) which are usually used
to describe the classical-quantum encodings. This implies that the quantities In =
In1 =n; Cn = C1n =n have the limits
I (%1 ; ) = lim In ((%1 ; ))
n!1
I1 (%1 ; ) ;
C ( ) = lim Cn ( )
n!1
C1 ( )
which are usually taken as the bounds of classical information and capacity [9, 10]
for a quantum channel . As it has been recently proved in [11] under a certain
regularity condition, the upper bound C ( ) is indeed asymptotically achievable
by long block classical-quantum encodings. Note that the c-quantities In , Cn are
not easy to evaluate for each n, but they all are bounded by the corresponding
q-quantities Iq , Cq , and
I (%1 ; )
Iq (%1 ; ) ;
C( )
Cq ( ) :
In the same way one can de…ne the smallest quantities
Ino (%1 ; ) = Io %1 n ;
n
;
Con ( ) = Co
n
8
VIACHESLAV P BELAVKIN
corresponding to the o-entanglements (orthogonal encodings), and establish the
superadditivity for these quantities. One can prove that the information I and the
capacity C can be achieved via such the orthogonal block encodings:
1
1
lim Ino (%1 ; ) = I (%1 ; ) ;
lim Con ( ) = C ( ) :
n!1 n
n!1 n
In order to measure the ”real”entangled information of quantum channels ”without the classical part”, another quantity, the ”coherent information”was introduced
in [3]. It is de…ned in our notations as
Is (%1 ; ) = Iq (%1 ; )
S (%1 ) ;
Cs1 ( ) = sup Is (%1 ; )
%1
if these quantities are positive, otherwise they are taken to be zero. The supremum
Csn ( ) = Cs ( n ) of Is (%n1 ; n ) over all states %n1 2 n1 B 1 in general is not
additive but superadditive, and the coherent capacity is de…ned as the limit
Cs ( ) = lim Csn ( ) =n:
n!1
Obviously this capacity has the bounds
Cs1 ( )
Cs ( )
Cq ( )
as Is (%n1 ; n ) Iq (%n1 ; n ) 0 for each input state %1 .
Thus the entangled information Iq (%1 ; ) for a single channel corresponding to
the standard entanglement 1 and the entangled capacity Cq ( ) = Cq1 ( ) give
upper bounds of all other capacities and are good analogs of the corresponding
classical quantities.
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Department of Mathematics, University of Nottingham, NG7 2RD Nottingham, UK