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Conditional entropy in algebraic statistical physics
REPORTS
Vol. 20 (1984)
CONDITIONAL
ENTROPY V. P.
Institute
ON
MATHEMATICAL
IN ALGEBRAIC
BELAVKIN **
of Physics, Nicholas (Received
and P.
Copernicus December
No. 3
PNYSICS
STATISTICAL
PHYSICS *
STASZEWSKI
University,
Toruri, Poland
3, 1980)
The notion of conditional entropy as entropy of conditional state on C*-algebra .d with respect to its C*-subalgebra XI, c .N’ is introduced. It is proved that for a compatible state (r on .rr’ ‘(which admits the conditional expectation of UmegakiTakesaki) the mean conditional entropy in an a priori state 0, on ,d, is equal to the difference of the entropy of the state D on .cl and the entropy of the state rrl on ~1~. The conditional entropy enables us to define the input-output information of a quantum communication channel in analogy to the classical Shannon formula.
Introduction In this paper we introduce the notion of conditional entropy as entropy of conditional state on a C*-algebra AZ?with respect to its C*-subalgebra d1 c .d. The conditional state is a generalization of conditional expectation of UmegakiTakesaki ([7]) and enables to describe completely positive dynamical maps and non-ideal communication channels A: ~2, + .nl, in terms of its restriction to a C*subalgebra .d, c .d. The conditional entropy as a measure of unicertainty connected with the conditional state can be treated as a measure of stochasticity of the dynamical map ([9]) or measure of imperfection of the communication channel described by this conditional’ state in the case in which the C*-algebra .c/ is generated by its C*-subalgebras .c/, and XI,, i.e. .d = ,cJI v .c/~. In the commutative case the introduced conditional entropy becomes the Boltzmann-Gibbs-Shannon conditional entropy (with opposite sign) averaged with respect to an-a priori state o1 on d, and it is equal to the difference of the entropy of the joint state on ,d = LX!*v JY, and the entropy of the state ol. Here we will show that the latter holds also in the general case for the states o * Supported in part by Polish Ministry of Science, Higher Education MR.I.7. ** On leave of absence from M.I.E.M., Moscow, USSR. [3731
and Technology,
Project
V. P. BELAVKIN
374
and P. STASZEWSKI
on .cul compatible with .rJ, c .d, this means in the case of W*-algebras .nC, ~dr and normal B the existence for every element ue.d of the conditional expectation E(a) of Umegaki-Takesaki ([7]). In the case of zero a priori entropy the mean conditional entropy is simply the relative entropy of the state o related to a state ‘p defined by Naudts ([S]). Note that with the help of the notion of conditional entropy we can define the input-output information of a quantum communication channel in analogy to the classical Shannon formula ([ 11) as the difference of the entropy of the restriction of the state o to the output subalgebra %d, c .d and mean conditional entropy related to the input subalgebra &, c .d. 1. Conditional state on C*-algebra Let .d be a C*-algebra, ,d, c .d its C*-subalgebra and let D denote a state on algebra .4. Let us denote by .
Vu, E rd, )
= 0, (a:&(a)a,),
(1.1)
where ?YI stands for the extension of the state or = ~1~~ to the normal state on .J1. In the sequel this conditional expectation we shall call Umegaki-Takesaki conditional expectation though these authors always considered the case &(a)~.&~. DEFINITION1.1 A state (T 011 .d will be called compatible
with a suhalgehra .c/~
if (1.1) holds for every a~.d. LEMMA 1.1. Let (T be a state on .d being compatible with s&algebra the map .d 3u -+ E(U) is defined up to the two-sided ideal
I,, and has the following (a) ~(&a, +lza2)
= {uE.c~,:
a, (a:aal) = 0, Vu, ExJ1j
.rll.
Then
(1.2)
properties = ll,e(~~)+&~(u~)(mod~,),
(b) ~(a*) = s(u)*, E(u*a) >, 0, VuE,d(mod8,), (c) &(araa;) = are(u (mod@,), (d) E(~*a) 2 E(a)*&(a) (mod@,), where .for
Proof:
(mod@,) means that 5(afaaI) means that a, -u2 E In.
UE ~7, a 2 0
a, = a, (mode)
Let us first remark
3 0, Va, ~.d.
In particular,
that the condition
ar (a:&
= 0,
vu, E 22,
(1.3)
CONDITIONAL
occurring
ENTROPY
in (1.2) is equivalent
IN ALGEBRAIC
STATISTICAL
PHYSICS
375
to
al (a:au;)
= 0,
Vu,,
(1.3’)
a; E .d, .
The implication (1.3’) * (1.3) is trivial. To prove the opposite it is sufficient to write condition (1.3) in terms of linear combinations u1 = a; f a;’ and u1 = a; + iuy and next consider the appropriate linear combi’nations of a, (((I; + a;‘)*u (a; f a:))
and
01 ((a; + iu;‘)*a (a; f ia;‘)).
From (1.3’) it is actually easy to verify that .f,, is a two-sided ideal. The remaining part of the proof follows immediately from (1.2) and Definition 1.1. n In order to get a necessary and sufficient condition for a state compatible with ~1, let us introduce the following notion of conditional C*-algebra ~1.
o to be state on
DEFINITION 1.2. Let .c/ be a C*-algebra, .ciI c .r/ its C*-subalgebra, (TVstate on -r/l and let zl:.c/, -+B(.Hc,) be the cyclic representation of .‘/I with respect to ol. We shall call the map Z: .ti + B(.Y?‘,,) a conditional stute on xl with respect to .d, and crl (shortly (rdl, cl) conditional sfute on .sJ), if it fulfils the following conditions (i) E^is linear positive (ii) 2 is modular,
of norm
one,
i.e. E^(a~uul) = x1 (a,)*.C(a)n, (a,),
It is easy to verify validity
of the following
LEMMA 1.2. Let E^be a (..r/, , crl) conditional
(a’) ;(a,)
Vu, ~.c3/~, a~.o/
lemma. state on .d.
We obtain
= 7c1(a,), Vu, E xl,,
(b’) <(a*) = ;(a)*, (c’, c(u:au;)
= 7tl (a,)*C(u)7-c, (a; 1, VU,) u; E .Tll)
(d’) ;(a*~) 3 ;(u)*;(u). Note that E^(l) = I for the identity 1 of .d or of some extension of .d if .d does not contain the identity. Moreover, every conditional state c^ can be extended to the normal conditional state C: .cJ ---*B (X’,,) on the W*-enveloping algebra .
V. P. BELAVKIN
376
and P. STASZEWSKI
an injection of the Hilbert space .3y01 of cyclic representation rcr with respect to c1 = (~1~~~into X,. (Ff denotes the Hermitian conjugate of F.) THEOREM
exists a (.c/,,
1.1. For any state a on .d and any’ C*-suhalgebra sI, c J@’ there aI) conditional state Z: .r/ -+ B(.W,,) such that a = a<, oC, where a<,
denotes the state on B(.Y,,) corresponding to the cyclic vector
(1.4)
Proof:
The map (1.4) is obviously linear and positive, moreover, its extension the W*-enveloping algebra .r/ satisfies the condition F(e) = F+%(e)F = I, hence E^has norm one. Because z(a,)F = F7cI (a,), we have
c = F+tiF = F+F
to
E^(a:aaJ = F+n(a~aa,)F
= F+n(a,)*x(a)z(a,)F
= z1 (aI)*F+7t(a)Fx,
(aI) = n, (a,)*E(a)7r, (al)
that is the modularity condition, hence (1.4) defines a (dI, aI) conditional state. Denote by (a,~a~),, = a1 (aTa\) and (ala’), = a(a*a’) the scalar products in XO1 and X,,, respectively. We obtain atI (6(a)) = (
= at,
(E^(afaa;))
from which we obtain
= a ( a:aa;)
= orI (c^‘(a:aa;))
Z(a) = E’(a), VaE .r/.
=
where rcn,(.~,)
Vu,,
n
THEOREM 1.2. A state a on C*-algebra .d is compatible d, c .d if and only if (zI1, al) conditional state t: ~2 -+ B(X’,,) satisfies the condition
(iii) i(&)
(aI(E’(a)a;),,,
with C*-subalgebrq corresponding to a
c 7cn,(dr), denotes the W*-algebra
of the representation
n, i.e. the weak closure
of 711(&I). Proof: to
the
because c(a)E.c/l
Let K1 be an extension of the representation rcl : .cll -+ B(:c,,,) W*-enveloping algebra ~7~. Assume (iii). Then we obtain E^(a)EE, (xzl), en, (J,
) = nl (.dl) and hence for every a E ..G/ there exists an element such that n,(s(a)) =6(a). Taking .5(a) of the form (1.4) and taking a(a~aa,) into account that acl o n, = 01, we obtain that for any aI E.c/, = orI (xl (a,)*?lI (s(a))x, (a,)) = a<, (El (a:s(a)a,)) = O, (a:s(a)al), otherwise the state a is compatible. Assume now that a is compatible with &, c ~2. Putting .?(a) = E1 (E(a)), VaE.d and taking into account that .1,,, = ker 7L1we obtain from Lemma 1.1 that
the map at,E^(a)Exl (.d,) satisfies conditions (i), (ii) of Definition 1.2 and a<, 0; = a<, o nI o E = Or o E = a. From this fact and from Theorem 1.1 we infer that E^
CONDITIONAL
ENTROPY
IN ALGEBRAIC
STATISTICAL
= %i OE is the uniquely detined conditional state corresponding which the condition (ii) is obviously fulfilled. m A conditional conditional
state
satisfying
state. In the following
PHYSICS
377
to the state o’, for
the condition (iii) will be called compatible we will denote the conditional state E^by c.
Remark. As follows from [3], the modularity (i) and (iii) and can be omitted in the definition
It is well known that the cyclic representation
condition (ii) is a consequence of of compatible conditional state. rci: ~/i ---fB(N’,,)
is irreducible
for every pure state 0i on SZ!,. Then rc1(.‘3c,) = B(XO1) 2 E(.v/), hence we obtain the following COROLLARY. Every (olr .&,) conditional state with respect to a pure state o1 is compatible, or in other words, every stare IJ on .cli for which CT, = ~1,~~ is pure on subalgebra ,d, c ._cI?is compatible with _d,.
2. Entropy of conditional
state
Let cp be a state on C*-algebra .cy’ and let z: .c/’ -+ B(X) denote the cyclic representation of algebra .4ul with respect to the state cp. A state o on .d will be called dffferentiable with respect to q, if it has the form a(a) = <+x(a)<, where 4 E-N is a vector for which there exists a closable defined in .X by the equation Q(<)~u) = R(U)<, where
la) = ja’ 3 u(mod 41,
(2.1) operator VUE.~/.
Q(<) densely (2.2)
In this case there exists one and only one vector 5 for which the operator e(t) is positive and self-adjoint ([S]). We shall call such a vector a positive vector. An P = e(t)’ ~(5) is called the density operator of the state CTwith respect to cp and ~(5) = P112 for positive 5, which we denote by (da/dq)‘12. It was proved in
operator
[S] that a state on W*-algebra .4 is differentiable with respect to a normal faithful state cp on .d if and only if it is almost majorised ([S], [2]) by cp. In the commutative case this means that the probability measure corresponding to g is absolutely continuous with .respect to the Bore1 measure corresponding to cp. According to [S] we define the entropy of the state cr differentiable with respect to cp in the following way y@+’ = lim 5’ ln(PE,)<, 610
(2.3)
whenever this limit exists, Ed = E ([S, 6- ‘I), where E(dA) is the spectral measure of the density operator P. As an example consider the algebra _ti = B(X) of all bounded linear operators in the Hilbert space * with a normal state p(A) = Tr(@A), AE.&‘, where @
378
V. P. BELAVKIN
and P. STASZEWSKI
denotes the density operator. Assume that with respect to 40. In the case of invertible
o(A) = Tr(CA) and is differentiable @ we obtain (for detail see [lo])
,Y@+’= TrClnW’,Z,
(2.4)
i.e. the relative quantum entropy (C’s]) of the state C with respect to @. In the commutative case we consider .d = Y”(h4, cp) i.e. the algebra of all essentially bounded measurable functions on a localisable measure space. Because the predual of _F (M. cp) is the space 9’ (M, qp) of all integrable functions ([6]) the normal states on IP”(M, cp) are positive normalised measures r~ on M, absolutely continuous with respect to cp. According to [S], for 0 absolutely continuous with respect to cp
i.e. it becomes the Boltzmann-Gibbs-Shannon entropy (with opposite the most general form of relative classical entropy ([S]).
sign). It is
Remark. The relative = Tr ClnX also in the case density operator C = @X, respect to the inner product X = @- ’ Z. It is obvious
entropy (2.4) is well-defined by the formula YiP of noninvertible @ for every differentiable 0 having the where X is an essentially self-adjoint operator with ((x. /3)@= (@(x, ,!I), which for invertible @ takes the form that X is positive with respect to (x, y),; (q, XII)@ = (@II, Xv) = (ye, @Xv]) = (v], Z:v])> 0 for all VE V(X) c .Y? due to the positivity of C. Note that 1nX # lnC-In@, except for the case of commuting C and @. Hence our relative entropy differs from the Araki’s relative entropy [ll], which is welldefined only in the case of faithful states and in this example takes the form ~~~‘,Si= TrZ(lnC-In@).
Let cpl be a state on C*-algebra .r/r c .c/ and let n,: .cll + R(.iY ,) denote the cyclic representation of ,d, with respect to cpl. Let moreover F: 9?‘l -+ Z, let F+ stand for its conjugate. Our next lemma generalizes Theorem 1.1. L~~Mti’2.1. The map C: ai-tFf7t(a)F is (.rll, cpl) cor?diriona/ state on .?/ if and only if the operator F sati?jies the @lowing conditions 1” F+F = I, 2” Fq (q) = n(u,)F, Vu, EA’~.
Proofi Condition 1’ is equivalent to the demand that E(U) = F+z(a)F is linear condition (cf. (ii) from positive of norm one. Condition 2“ is the modularity Definition 1.2), because E(U~UU~)= F+n(u:aa,)F
= F+~-c(u,)*~(u)~~ (a,)F
= zn, (al)*F+n(a)Fn:,
(al) = ~1 (al)*~(a)zl
(a,),
vu, E &, .
CONDITIONAL
Conversely,
ENTROPY
IN ALGEBRAIC
from the modularity
(?r(a)Fu;IFrr,
(u,)ur)r
condition
STAilSTICAL
PHYSICS
a(a*)nr (al) = t:(a*ar) VUG.d, vu,,
= (7l(u)Fu;~7r(u,)Fu,),,
379
we obtain u; EJYr,
from which condition 2’ follows immediately. = The notion of a state differentiable with respect to state cp can be generalized to the case of conditional state in the following way: DEFINITION 2.1. A conditional respect
to u state cp on C*-algebra
state i:: -F/ + B(.H ,) is called tl@renriuhle .d if and only if it is of the form
E(U) = F+rc(u)F,
wit/~
(2.6)
where F denotes an operator from .N 1 into X for which there exists a closable operator q(F), densely defined in .H by the equation VUEd,
~(F)laa, > = n (u)Fla, >1
e
one operutor F denoted selfudjoint and
by (dcldcp)“2
(2.7)
Q = Q(F)+ Q(F) the density operator
We shall call the positive self-adjoint operator of conditional state E with respect to cp. LEMMA 2.2. For every d@rentiuhle
vu, Ed,.
conditional
stute there exists one and only
such that the operator
g(F)
is positive and
g(F) = Q”‘.
(2.8)
Proqf’: Let F’ denote a closable operator defined by (2.7) and let Q(F’) = UQ112 denote the polar decomposition ([6]) of g(F’). Using (2.7) one can obtain g(F’)lr(u) 3 x(u)g(F’), VUE.~. This condition is necessary and sufficient for the operator @(F’) to be affiliated with the commutant rc(&‘)’ ([a]). From this fact and
from (2.7) we obtain = U+F’.
Suppose
g(U’F’) __
= U’@(F’), ~
that g(F’) = g(F),
consequently
e(F)
LEMMA 2.3. The conditional state ;: differentiable with respect lf and only if the operutor F sutisjies the following conditions g(nJF
= U+UQ’/2
for F
then g(F’) = Q(F) which implies F” = F. H
= Fe1 (VI)>
to cp is computihle
(2.9)
e(nJ+F
= Fe, (VI)+ 3
(2.10)
e1 (rllh
>I
(2.11)
where
and the operators
g(nI)
are defined
=
711 (~lhl~
uniquely
YI= n(u)Fla,
on the vectors )I
(2.12)
V. P. BELAVKIN
380 in the jbllowiry
manner v--v~~~)~
Proof!
and P. STASZEWSKI
The condition
(2.13)
= ~(a)Fel(vll)laA.
of compatibility
s(a)el = e,&(a),
(iii) is equivalent
to the condition
VeI ErrI (.cy’r)‘, Va E d.
Let us define a subset of .%r in the following a; = IyIr: El~r(~r) -
(2.14)
way (2.15)
bounded).
It is easy to verify that zl (.r/,)’ and a; are isomorphic, be written in the form
hence condition
(2.14) can
a(a)er (R) = el (vlr)s(a).
(2.16)
Taking into account that the differentiable conditional can express condition (2.16) in the form
state is of the form (2.6) one
(Vl~(a)Fer (rl&r>1
= (n(Met
(~r)+arlr)r,
val,
a; E,c*/r, vaE,d,
V++I~a;,
where q is of the form (2.12). This condition means that for every q, ~a’, there exists an operator Q(v,) defined at least on vectors q by (2.9). The conjugate operator I’ is defined in the following way (2.17) Because
of e(q,)n(a)Flal)l ~(a)e(M%r)~
= 7-c(a)t_J(q,)Fla,),,
= n(a)Fer
from which follows (2.9). Analogously, The converse implication, namely (2.16), is immediate. We have s(a)er (vlr) = F+n(a)Fer
from (2.13) we obtain
(vJaIhy
t/a1 ELdl, Va-4
from (2.17) one can obtain (2.10). (2.9) (2.10) + condition of compatibility
(VI) = F+n(a)e(vlN
= F+e(r&(a)F
= el (MF++)F
= el (rl&(a).
n
We shall call an operator F: Z1 + A? positive if it maps every positive vector qI E PI into positive vector Z3 q = Fql. The condition of positivity of F can be written in the form e(FVI)+ = e(Fvl,) z 0, From (2.9), (2.13) using the modularity obtain
v?1: e1 hl)’ = ei (11,) 3 0. condition 2’ of Lemma 2.1 and (2.7) we
CONDITIONAL
ENTROPY
IN ALGEBRAIC
STATISTICAL
381
PHYSICS
hence (2.17)
@(FrlI) = Q(rl,)@(~) and
(2.18)
e(FVI)+ = @(U’@(VI)‘. For positive operators
@I(ur) = I+
and we obtain
e(qr) = a’
2 0, because
(VlQ(rlI)q) = (x(a)Fa,Ie~rl,)n(a)Fa,), = (4~+~(4*~(4k?, = (%ls(a*4el
(YlI)%)I
(IlI)4),
b 0, (2.19)
vu, E.41, VUE.P/.
According to the condition of compatibility (2.16) the positive operators E(U*U) and er(qr) commute, hence their product is a positive operator. For every differentiable conditional state there exists one and only one operator F = (d&/d@” operator
such that
Q(F) is positive
and selfadjoint
@WV,) = e(r&(U is positive
(Lemma
2.2). Then
the (2.20)
= @(@(VI)
too.
THEOREM 2.1. Let o denote a state on C*-algebra .3 differentiable with respect to the state cp on ,d and compatible with C*-subalgebra ~9, c .d. Let u1 = 01.~~ be differentiuhle with respect to q1 on AJ? Then there exists the compatible (?tiI, yl) conditional state E: .c3 + B(H,) such that G = orI OE, differentiable with respect to q and the density operator P of the state (r relative to cp is a commutative product, RQ 2 QR, of conditional density operator Q and positive self-udjoint operator R, ,for which RF = FP,, where P, is the density operator of c1 with respect to ql.
Proof: Let < = (da/dq)1’2, ;I = (do,/dq,)‘!2 and +I (a) = Fg+l~(u)F~, be the compatible conditional state corresponding to CI, where the operator Frl: XO1 + .X0 is defined by F;rr-c~ (451
(2.21)
= ~(45.
From (2.21) it follows that F,, fulfils the condition .!;I Ftl = E,,, where E,, is orthoprojector onto Hi”r. Let us verify that F,, is posrttve. Let q1 = rcn,(b,)<, . The condition of positivity can be formulated as follows ~(FQII) From
2 0,
(2.21) (2.11) and (2.13) we obtain
vr1:
e1 (VI)
2
0.
(2.22)
382
V. P. BELAVKIN
and P. STASZEWSKI
which implies positivity of FgI, because positivity of er (qr) implies positivity of e(qr), cf. (2.19). Let F, be a positive operator defined on the orthocomplement .#,, of the space Xrl with range F,.X,, c P,, the orthocomplement of N,. Let FI satisfy conditions (2.9), (2.10) and FfF, = E,,, where E,, is the orthoprojection on SII. Then the operator F = F,, @ F, defined on X1 is positive, satisfies (2.9) and (2.10) and F+F = I. This means that E(U) = F+n(a)F is the compatible conditional state with respect to (pr, differentiable with respect to cp and CJ(a) = <5rlE(a)51>1 = fl<* o&(a). From (2.20) we get e(<) = e(<&(F) for 5 = (do/dq)“‘,
cl = (da,/dq,)‘iZ
and F = (d~/dq)‘/‘.
= RQ c QR, where P = Q({)~, R = Q(<~)~, Q = e(F)‘. Now we can define the conditional differentiable with respect to a state.
(2.23)
c e(F)e(s’r)
entropy
From (2.23) we obtain
P
n
as the entropy
of conditional
state
DEFINITION2.2. Let r(u) = F+n(u)F
be an (.r/r, cpr) conditional state on .r/ differentiable with respect to cp and let Q denote the conditional density operator. We define the conditionally-relative entropy of conditional state E relative to cp as the operator affiliated with the cornmutant n,(.d,)’ in the following way (2.24)
=YCiq= lim Ff (ln(QE,))F, 610 whenever this limit exists in strong operator E(dA) denotes the spectral measure of Q.
topology.
E = E([6, X1]),
where
LEMMA 2.4. Let E be a compatible conditional state differenti ‘-le with respect to state cp. Ifthe entropy .Yq exists, then it is ufhliuted with the centre of the von Neumann ulgebru n1 (all)“.
Because the conditional density operator Q is affiliated with 7t(.d)‘, the operator F+,f(Q)E belongs to 7c1(~1~) for any real-valued bounded function f: In fact, from condition 2- of Lemma 2.1 we obtain Proof
F+f(QFn, (a11 = F+f (Qb(a,F From
this formula
and Lemma
2.3 we get
F+J-(QP’QI (~1) = F+f (Qk(vI)F because Q&q,) = e(qr)Q nzl(.r/l)“, which means
= F++,)f (QF = ~1 (a,)P+f (QF. = F+e(rl,V(Q)F
= el (rl,F’+l‘(QF,
for any ql E a; (cf. (2.20)). Hence, F+.f(Q)Fe7z1 (.rdr)’ chat .YE/a is affiliated with z, (.d,)’ n x1 (.c/~)” n
THEOREM 2.2. Let CTbe a state on .r/ differentiable with respect to the state cp to be differentiable with respect to cp,. and compatible with d,. Assume o, = CT~,&~ Let moreover E be a compatible (‘dI, q,) conditional state, d#erentiuble with respect
CONDITIONAL
ENTROPY
IN
ALGEBRAIC
STATISTICAL
383
PHYSICS
to cp,fbr which CT= oil ox. Then the entrap!. .Y” O’of’the stute o dijjkrentiahle with respect to q is the sum y0 = .
= lnQE,+lnRE,.
Consequently, <+(lnPE,)< Taking
into account
= <‘(lnRE,+lnQE.,)<.
that < = F<, and RF = FP,.
5,“(lnPE&
FfF
= I, we obtain
= <:(lnP,E,,+F+(lnRE,)F)<,,
from which (2.25) follows directly.
=
One can notice that the introduced notion of conditional entropy and the result of Theroem 2.2 enable us to define the input-output information of a communication channel in the general case. Take .d = %d, v .rLZ where .cJ,, .c/~ denote C*-subalgebras describing the input and output of a communication channel described in terms of conditional expectation compatible with .~/i . Generalizing the famous Shannon classical formula (cf. for instance [l]) we obtain the following expression for the input-output information
i.e. the difference of an output priori state on the input.
entropy
and the mean conditional
entropy
in an a
Acknowledgments
We wish to thank Professor R. S. Ingarden for his interest in this work. One of us (V.P.B.) wishes to express his gratitude to Professor R. S. Ingarden for the kind hospitality received during his stay in Torun. REFERENCES [l] [2] [3]
Billingsley, P.: Ergo&c Theory urul Injtirmution, Wiley, New York 1965. Dixmier, J.: I!XS dg&res d’opirateurs dam fespace Hilherrien, Gauthier-Villars, Evans, D. E., J. T. Lewis: Communications of the Dublin Institute for Advanced 24, Dublin
1977.
Paris 1969. Studies A, No.
384 [4] [S] [6] [7] [S] [9] [lo] [11]
V. P. BELAVKIN
and P. STASZEWSKI
Gudder, S., J.-P. Marchand: Rep. Math. Phys. 12 (1977), 317. Naudts, J.: Commun. Math. Phys. 37 (1974), 175. Sakai, S.: C*-algebras and W*-algebras, Springer-Verlag, Berlin 1971. Takesaki, M.: .I. Funcrional Analysis 9 (1972), 306. Wehrl, A.: Rev. Mod. Phys. 50 (19X0), 221. Belavkin, V. P., Staszewski P.: On rhe Entropy of Dynamical Maps, unpublished. Belavkin, V. P., P. Staszewski: Ann. Inst. H. PoincarC 37, 1 (1982), 51. Araki, H.: Publ. RIMS Kyoto Univ. 11 (1975/76), 809.
Vol. 20 (1984)
CONDITIONAL
ENTROPY V. P.
Institute
ON
MATHEMATICAL
IN ALGEBRAIC
BELAVKIN **
of Physics, Nicholas (Received
and P.
Copernicus December
No. 3
PNYSICS
STATISTICAL
PHYSICS *
STASZEWSKI
University,
Toruri, Poland
3, 1980)
The notion of conditional entropy as entropy of conditional state on C*-algebra .d with respect to its C*-subalgebra XI, c .N’ is introduced. It is proved that for a compatible state (r on .rr’ ‘(which admits the conditional expectation of UmegakiTakesaki) the mean conditional entropy in an a priori state 0, on ,d, is equal to the difference of the entropy of the state D on .cl and the entropy of the state rrl on ~1~. The conditional entropy enables us to define the input-output information of a quantum communication channel in analogy to the classical Shannon formula.
Introduction In this paper we introduce the notion of conditional entropy as entropy of conditional state on a C*-algebra AZ?with respect to its C*-subalgebra d1 c .d. The conditional state is a generalization of conditional expectation of UmegakiTakesaki ([7]) and enables to describe completely positive dynamical maps and non-ideal communication channels A: ~2, + .nl, in terms of its restriction to a C*subalgebra .d, c .d. The conditional entropy as a measure of unicertainty connected with the conditional state can be treated as a measure of stochasticity of the dynamical map ([9]) or measure of imperfection of the communication channel described by this conditional’ state in the case in which the C*-algebra .c/ is generated by its C*-subalgebras .c/, and XI,, i.e. .d = ,cJI v .c/~. In the commutative case the introduced conditional entropy becomes the Boltzmann-Gibbs-Shannon conditional entropy (with opposite sign) averaged with respect to an-a priori state o1 on d, and it is equal to the difference of the entropy of the joint state on ,d = LX!*v JY, and the entropy of the state ol. Here we will show that the latter holds also in the general case for the states o * Supported in part by Polish Ministry of Science, Higher Education MR.I.7. ** On leave of absence from M.I.E.M., Moscow, USSR. [3731
and Technology,
Project
V. P. BELAVKIN
374
and P. STASZEWSKI
on .cul compatible with .rJ, c .d, this means in the case of W*-algebras .nC, ~dr and normal B the existence for every element ue.d of the conditional expectation E(a) of Umegaki-Takesaki ([7]). In the case of zero a priori entropy the mean conditional entropy is simply the relative entropy of the state o related to a state ‘p defined by Naudts ([S]). Note that with the help of the notion of conditional entropy we can define the input-output information of a quantum communication channel in analogy to the classical Shannon formula ([ 11) as the difference of the entropy of the restriction of the state o to the output subalgebra %d, c .d and mean conditional entropy related to the input subalgebra &, c .d. 1. Conditional state on C*-algebra Let .d be a C*-algebra, ,d, c .d its C*-subalgebra and let D denote a state on algebra .4. Let us denote by .
Vu, E rd, )
= 0, (a:&(a)a,),
(1.1)
where ?YI stands for the extension of the state or = ~1~~ to the normal state on .J1. In the sequel this conditional expectation we shall call Umegaki-Takesaki conditional expectation though these authors always considered the case &(a)~.&~. DEFINITION1.1 A state (T 011 .d will be called compatible
with a suhalgehra .c/~
if (1.1) holds for every a~.d. LEMMA 1.1. Let (T be a state on .d being compatible with s&algebra the map .d 3u -+ E(U) is defined up to the two-sided ideal
I,, and has the following (a) ~(&a, +lza2)
= {uE.c~,:
a, (a:aal) = 0, Vu, ExJ1j
.rll.
Then
(1.2)
properties = ll,e(~~)+&~(u~)(mod~,),
(b) ~(a*) = s(u)*, E(u*a) >, 0, VuE,d(mod8,), (c) &(araa;) = are(u (mod@,), (d) E(~*a) 2 E(a)*&(a) (mod@,), where .for
Proof:
(mod@,) means that 5(afaaI) means that a, -u2 E In.
UE ~7, a 2 0
a, = a, (mode)
Let us first remark
3 0, Va, ~.d.
In particular,
that the condition
ar (a:&
= 0,
vu, E 22,
(1.3)
CONDITIONAL
occurring
ENTROPY
in (1.2) is equivalent
IN ALGEBRAIC
STATISTICAL
PHYSICS
375
to
al (a:au;)
= 0,
Vu,,
(1.3’)
a; E .d, .
The implication (1.3’) * (1.3) is trivial. To prove the opposite it is sufficient to write condition (1.3) in terms of linear combinations u1 = a; f a;’ and u1 = a; + iuy and next consider the appropriate linear combi’nations of a, (((I; + a;‘)*u (a; f a:))
and
01 ((a; + iu;‘)*a (a; f ia;‘)).
From (1.3’) it is actually easy to verify that .f,, is a two-sided ideal. The remaining part of the proof follows immediately from (1.2) and Definition 1.1. n In order to get a necessary and sufficient condition for a state compatible with ~1, let us introduce the following notion of conditional C*-algebra ~1.
o to be state on
DEFINITION 1.2. Let .c/ be a C*-algebra, .ciI c .r/ its C*-subalgebra, (TVstate on -r/l and let zl:.c/, -+B(.Hc,) be the cyclic representation of .‘/I with respect to ol. We shall call the map Z: .ti + B(.Y?‘,,) a conditional stute on xl with respect to .d, and crl (shortly (rdl, cl) conditional sfute on .sJ), if it fulfils the following conditions (i) E^is linear positive (ii) 2 is modular,
of norm
one,
i.e. E^(a~uul) = x1 (a,)*.C(a)n, (a,),
It is easy to verify validity
of the following
LEMMA 1.2. Let E^be a (..r/, , crl) conditional
(a’) ;(a,)
Vu, ~.c3/~, a~.o/
lemma. state on .d.
We obtain
= 7c1(a,), Vu, E xl,,
(b’) <(a*) = ;(a)*, (c’, c(u:au;)
= 7tl (a,)*C(u)7-c, (a; 1, VU,) u; E .Tll)
(d’) ;(a*~) 3 ;(u)*;(u). Note that E^(l) = I for the identity 1 of .d or of some extension of .d if .d does not contain the identity. Moreover, every conditional state c^ can be extended to the normal conditional state C: .cJ ---*B (X’,,) on the W*-enveloping algebra .
V. P. BELAVKIN
376
and P. STASZEWSKI
an injection of the Hilbert space .3y01 of cyclic representation rcr with respect to c1 = (~1~~~into X,. (Ff denotes the Hermitian conjugate of F.) THEOREM
exists a (.c/,,
1.1. For any state a on .d and any’ C*-suhalgebra sI, c J@’ there aI) conditional state Z: .r/ -+ B(.W,,) such that a = a<, oC, where a<,
denotes the state on B(.Y,,) corresponding to the cyclic vector
(1.4)
Proof:
The map (1.4) is obviously linear and positive, moreover, its extension the W*-enveloping algebra .r/ satisfies the condition F(e) = F+%(e)F = I, hence E^has norm one. Because z(a,)F = F7cI (a,), we have
c = F+tiF = F+F
to
E^(a:aaJ = F+n(a~aa,)F
= F+n(a,)*x(a)z(a,)F
= z1 (aI)*F+7t(a)Fx,
(aI) = n, (a,)*E(a)7r, (al)
that is the modularity condition, hence (1.4) defines a (dI, aI) conditional state. Denote by (a,~a~),, = a1 (aTa\) and (ala’), = a(a*a’) the scalar products in XO1 and X,,, respectively. We obtain atI (6(a)) = (
= at,
(E^(afaa;))
from which we obtain
= a ( a:aa;)
= orI (c^‘(a:aa;))
Z(a) = E’(a), VaE .r/.
=
where rcn,(.~,)
Vu,,
n
THEOREM 1.2. A state a on C*-algebra .d is compatible d, c .d if and only if (zI1, al) conditional state t: ~2 -+ B(X’,,) satisfies the condition
(iii) i(&)
(aI(E’(a)a;),,,
with C*-subalgebrq corresponding to a
c 7cn,(dr), denotes the W*-algebra
of the representation
n, i.e. the weak closure
of 711(&I). Proof: to
the
because c(a)E.c/l
Let K1 be an extension of the representation rcl : .cll -+ B(:c,,,) W*-enveloping algebra ~7~. Assume (iii). Then we obtain E^(a)EE, (xzl), en, (J,
) = nl (.dl) and hence for every a E ..G/ there exists an element such that n,(s(a)) =6(a). Taking .5(a) of the form (1.4) and taking a(a~aa,) into account that acl o n, = 01, we obtain that for any aI E.c/, = orI (xl (a,)*?lI (s(a))x, (a,)) = a<, (El (a:s(a)a,)) = O, (a:s(a)al), otherwise the state a is compatible. Assume now that a is compatible with &, c ~2. Putting .?(a) = E1 (E(a)), VaE.d and taking into account that .1,,, = ker 7L1we obtain from Lemma 1.1 that
the map at,E^(a)Exl (.d,) satisfies conditions (i), (ii) of Definition 1.2 and a<, 0; = a<, o nI o E = Or o E = a. From this fact and from Theorem 1.1 we infer that E^
CONDITIONAL
ENTROPY
IN ALGEBRAIC
STATISTICAL
= %i OE is the uniquely detined conditional state corresponding which the condition (ii) is obviously fulfilled. m A conditional conditional
state
satisfying
state. In the following
PHYSICS
377
to the state o’, for
the condition (iii) will be called compatible we will denote the conditional state E^by c.
Remark. As follows from [3], the modularity (i) and (iii) and can be omitted in the definition
It is well known that the cyclic representation
condition (ii) is a consequence of of compatible conditional state. rci: ~/i ---fB(N’,,)
is irreducible
for every pure state 0i on SZ!,. Then rc1(.‘3c,) = B(XO1) 2 E(.v/), hence we obtain the following COROLLARY. Every (olr .&,) conditional state with respect to a pure state o1 is compatible, or in other words, every stare IJ on .cli for which CT, = ~1,~~ is pure on subalgebra ,d, c ._cI?is compatible with _d,.
2. Entropy of conditional
state
Let cp be a state on C*-algebra .cy’ and let z: .c/’ -+ B(X) denote the cyclic representation of algebra .4ul with respect to the state cp. A state o on .d will be called dffferentiable with respect to q, if it has the form a(a) = <+x(a)<, where 4 E-N is a vector for which there exists a closable defined in .X by the equation Q(<)~u) = R(U)<, where
la) = ja’ 3 u(mod 41,
(2.1) operator VUE.~/.
Q(<) densely (2.2)
In this case there exists one and only one vector 5 for which the operator e(t) is positive and self-adjoint ([S]). We shall call such a vector a positive vector. An P = e(t)’ ~(5) is called the density operator of the state CTwith respect to cp and ~(5) = P112 for positive 5, which we denote by (da/dq)‘12. It was proved in
operator
[S] that a state on W*-algebra .4 is differentiable with respect to a normal faithful state cp on .d if and only if it is almost majorised ([S], [2]) by cp. In the commutative case this means that the probability measure corresponding to g is absolutely continuous with .respect to the Bore1 measure corresponding to cp. According to [S] we define the entropy of the state cr differentiable with respect to cp in the following way y@+’ = lim 5’ ln(PE,)<, 610
(2.3)
whenever this limit exists, Ed = E ([S, 6- ‘I), where E(dA) is the spectral measure of the density operator P. As an example consider the algebra _ti = B(X) of all bounded linear operators in the Hilbert space * with a normal state p(A) = Tr(@A), AE.&‘, where @
378
V. P. BELAVKIN
and P. STASZEWSKI
denotes the density operator. Assume that with respect to 40. In the case of invertible
o(A) = Tr(CA) and is differentiable @ we obtain (for detail see [lo])
,Y@+’= TrClnW’,Z,
(2.4)
i.e. the relative quantum entropy (C’s]) of the state C with respect to @. In the commutative case we consider .d = Y”(h4, cp) i.e. the algebra of all essentially bounded measurable functions on a localisable measure space. Because the predual of _F (M. cp) is the space 9’ (M, qp) of all integrable functions ([6]) the normal states on IP”(M, cp) are positive normalised measures r~ on M, absolutely continuous with respect to cp. According to [S], for 0 absolutely continuous with respect to cp
i.e. it becomes the Boltzmann-Gibbs-Shannon entropy (with opposite the most general form of relative classical entropy ([S]).
sign). It is
Remark. The relative = Tr ClnX also in the case density operator C = @X, respect to the inner product X = @- ’ Z. It is obvious
entropy (2.4) is well-defined by the formula YiP of noninvertible @ for every differentiable 0 having the where X is an essentially self-adjoint operator with ((x. /3)@= (@(x, ,!I), which for invertible @ takes the form that X is positive with respect to (x, y),; (q, XII)@ = (@II, Xv) = (ye, @Xv]) = (v], Z:v])> 0 for all VE V(X) c .Y? due to the positivity of C. Note that 1nX # lnC-In@, except for the case of commuting C and @. Hence our relative entropy differs from the Araki’s relative entropy [ll], which is welldefined only in the case of faithful states and in this example takes the form ~~~‘,Si= TrZ(lnC-In@).
Let cpl be a state on C*-algebra .r/r c .c/ and let n,: .cll + R(.iY ,) denote the cyclic representation of ,d, with respect to cpl. Let moreover F: 9?‘l -+ Z, let F+ stand for its conjugate. Our next lemma generalizes Theorem 1.1. L~~Mti’2.1. The map C: ai-tFf7t(a)F is (.rll, cpl) cor?diriona/ state on .?/ if and only if the operator F sati?jies the @lowing conditions 1” F+F = I, 2” Fq (q) = n(u,)F, Vu, EA’~.
Proofi Condition 1’ is equivalent to the demand that E(U) = F+z(a)F is linear condition (cf. (ii) from positive of norm one. Condition 2“ is the modularity Definition 1.2), because E(U~UU~)= F+n(u:aa,)F
= F+~-c(u,)*~(u)~~ (a,)F
= zn, (al)*F+n(a)Fn:,
(al) = ~1 (al)*~(a)zl
(a,),
vu, E &, .
CONDITIONAL
Conversely,
ENTROPY
IN ALGEBRAIC
from the modularity
(?r(a)Fu;IFrr,
(u,)ur)r
condition
STAilSTICAL
PHYSICS
a(a*)nr (al) = t:(a*ar) VUG.d, vu,,
= (7l(u)Fu;~7r(u,)Fu,),,
379
we obtain u; EJYr,
from which condition 2’ follows immediately. = The notion of a state differentiable with respect to state cp can be generalized to the case of conditional state in the following way: DEFINITION 2.1. A conditional respect
to u state cp on C*-algebra
state i:: -F/ + B(.H ,) is called tl@renriuhle .d if and only if it is of the form
E(U) = F+rc(u)F,
wit/~
(2.6)
where F denotes an operator from .N 1 into X for which there exists a closable operator q(F), densely defined in .H by the equation VUEd,
~(F)laa, > = n (u)Fla, >1
e
one operutor F denoted selfudjoint and
by (dcldcp)“2
(2.7)
Q = Q(F)+ Q(F) the density operator
We shall call the positive self-adjoint operator of conditional state E with respect to cp. LEMMA 2.2. For every d@rentiuhle
vu, Ed,.
conditional
stute there exists one and only
such that the operator
g(F)
is positive and
g(F) = Q”‘.
(2.8)
Proqf’: Let F’ denote a closable operator defined by (2.7) and let Q(F’) = UQ112 denote the polar decomposition ([6]) of g(F’). Using (2.7) one can obtain g(F’)lr(u) 3 x(u)g(F’), VUE.~. This condition is necessary and sufficient for the operator @(F’) to be affiliated with the commutant rc(&‘)’ ([a]). From this fact and
from (2.7) we obtain = U+F’.
Suppose
g(U’F’) __
= U’@(F’), ~
that g(F’) = g(F),
consequently
e(F)
LEMMA 2.3. The conditional state ;: differentiable with respect lf and only if the operutor F sutisjies the following conditions g(nJF
= U+UQ’/2
for F
then g(F’) = Q(F) which implies F” = F. H
= Fe1 (VI)>
to cp is computihle
(2.9)
e(nJ+F
= Fe, (VI)+ 3
(2.10)
e1 (rllh
>I
(2.11)
where
and the operators
g(nI)
are defined
=
711 (~lhl~
uniquely
YI= n(u)Fla,
on the vectors )I
(2.12)
V. P. BELAVKIN
380 in the jbllowiry
manner v--v~~~)~
Proof!
and P. STASZEWSKI
The condition
(2.13)
= ~(a)Fel(vll)laA.
of compatibility
s(a)el = e,&(a),
(iii) is equivalent
to the condition
VeI ErrI (.cy’r)‘, Va E d.
Let us define a subset of .%r in the following a; = IyIr: El~r(~r) -
(2.14)
way (2.15)
bounded).
It is easy to verify that zl (.r/,)’ and a; are isomorphic, be written in the form
hence condition
(2.14) can
a(a)er (R) = el (vlr)s(a).
(2.16)
Taking into account that the differentiable conditional can express condition (2.16) in the form
state is of the form (2.6) one
(Vl~(a)Fer (rl&r>1
= (n(Met
(~r)+arlr)r,
val,
a; E,c*/r, vaE,d,
V++I~a;,
where q is of the form (2.12). This condition means that for every q, ~a’, there exists an operator Q(v,) defined at least on vectors q by (2.9). The conjugate operator I’ is defined in the following way (2.17) Because
of e(q,)n(a)Flal)l ~(a)e(M%r)~
= 7-c(a)t_J(q,)Fla,),,
= n(a)Fer
from which follows (2.9). Analogously, The converse implication, namely (2.16), is immediate. We have s(a)er (vlr) = F+n(a)Fer
from (2.13) we obtain
(vJaIhy
t/a1 ELdl, Va-4
from (2.17) one can obtain (2.10). (2.9) (2.10) + condition of compatibility
(VI) = F+n(a)e(vlN
= F+e(r&(a)F
= el (MF++)F
= el (rl&(a).
n
We shall call an operator F: Z1 + A? positive if it maps every positive vector qI E PI into positive vector Z3 q = Fql. The condition of positivity of F can be written in the form e(FVI)+ = e(Fvl,) z 0, From (2.9), (2.13) using the modularity obtain
v?1: e1 hl)’ = ei (11,) 3 0. condition 2’ of Lemma 2.1 and (2.7) we
CONDITIONAL
ENTROPY
IN ALGEBRAIC
STATISTICAL
381
PHYSICS
hence (2.17)
@(FrlI) = Q(rl,)@(~) and
(2.18)
e(FVI)+ = @(U’@(VI)‘. For positive operators
@I(ur) = I+
and we obtain
e(qr) = a’
2 0, because
(VlQ(rlI)q) = (x(a)Fa,Ie~rl,)n(a)Fa,), = (4~+~(4*~(4k?, = (%ls(a*4el
(YlI)%)I
(IlI)4),
b 0, (2.19)
vu, E.41, VUE.P/.
According to the condition of compatibility (2.16) the positive operators E(U*U) and er(qr) commute, hence their product is a positive operator. For every differentiable conditional state there exists one and only one operator F = (d&/d@” operator
such that
Q(F) is positive
and selfadjoint
@WV,) = e(r&(U is positive
(Lemma
2.2). Then
the (2.20)
= @(@(VI)
too.
THEOREM 2.1. Let o denote a state on C*-algebra .3 differentiable with respect to the state cp on ,d and compatible with C*-subalgebra ~9, c .d. Let u1 = 01.~~ be differentiuhle with respect to q1 on AJ? Then there exists the compatible (?tiI, yl) conditional state E: .c3 + B(H,) such that G = orI OE, differentiable with respect to q and the density operator P of the state (r relative to cp is a commutative product, RQ 2 QR, of conditional density operator Q and positive self-udjoint operator R, ,for which RF = FP,, where P, is the density operator of c1 with respect to ql.
Proof: Let < = (da/dq)1’2, ;I = (do,/dq,)‘!2 and +I (a) = Fg+l~(u)F~, be the compatible conditional state corresponding to CI, where the operator Frl: XO1 + .X0 is defined by F;rr-c~ (451
(2.21)
= ~(45.
From (2.21) it follows that F,, fulfils the condition .!;I Ftl = E,,, where E,, is orthoprojector onto Hi”r. Let us verify that F,, is posrttve. Let q1 = rcn,(b,)<, . The condition of positivity can be formulated as follows ~(FQII) From
2 0,
(2.21) (2.11) and (2.13) we obtain
vr1:
e1 (VI)
2
0.
(2.22)
382
V. P. BELAVKIN
and P. STASZEWSKI
which implies positivity of FgI, because positivity of er (qr) implies positivity of e(qr), cf. (2.19). Let F, be a positive operator defined on the orthocomplement .#,, of the space Xrl with range F,.X,, c P,, the orthocomplement of N,. Let FI satisfy conditions (2.9), (2.10) and FfF, = E,,, where E,, is the orthoprojection on SII. Then the operator F = F,, @ F, defined on X1 is positive, satisfies (2.9) and (2.10) and F+F = I. This means that E(U) = F+n(a)F is the compatible conditional state with respect to (pr, differentiable with respect to cp and CJ(a) = <5rlE(a)51>1 = fl<* o&(a). From (2.20) we get e(<) = e(<&(F) for 5 = (do/dq)“‘,
cl = (da,/dq,)‘iZ
and F = (d~/dq)‘/‘.
= RQ c QR, where P = Q({)~, R = Q(<~)~, Q = e(F)‘. Now we can define the conditional differentiable with respect to a state.
(2.23)
c e(F)e(s’r)
entropy
From (2.23) we obtain
P
n
as the entropy
of conditional
state
DEFINITION2.2. Let r(u) = F+n(u)F
be an (.r/r, cpr) conditional state on .r/ differentiable with respect to cp and let Q denote the conditional density operator. We define the conditionally-relative entropy of conditional state E relative to cp as the operator affiliated with the cornmutant n,(.d,)’ in the following way (2.24)
=YCiq= lim Ff (ln(QE,))F, 610 whenever this limit exists in strong operator E(dA) denotes the spectral measure of Q.
topology.
E = E([6, X1]),
where
LEMMA 2.4. Let E be a compatible conditional state differenti ‘-le with respect to state cp. Ifthe entropy .Yq exists, then it is ufhliuted with the centre of the von Neumann ulgebru n1 (all)“.
Because the conditional density operator Q is affiliated with 7t(.d)‘, the operator F+,f(Q)E belongs to 7c1(~1~) for any real-valued bounded function f: In fact, from condition 2- of Lemma 2.1 we obtain Proof
F+f(QFn, (a11 = F+f (Qb(a,F From
this formula
and Lemma
2.3 we get
F+J-(QP’QI (~1) = F+f (Qk(vI)F because Q&q,) = e(qr)Q nzl(.r/l)“, which means
= F++,)f (QF = ~1 (a,)P+f (QF. = F+e(rl,V(Q)F
= el (rl,F’+l‘(QF,
for any ql E a; (cf. (2.20)). Hence, F+.f(Q)Fe7z1 (.rdr)’ chat .YE/a is affiliated with z, (.d,)’ n x1 (.c/~)” n
THEOREM 2.2. Let CTbe a state on .r/ differentiable with respect to the state cp to be differentiable with respect to cp,. and compatible with d,. Assume o, = CT~,&~ Let moreover E be a compatible (‘dI, q,) conditional state, d#erentiuble with respect
CONDITIONAL
ENTROPY
IN
ALGEBRAIC
STATISTICAL
383
PHYSICS
to cp,fbr which CT= oil ox. Then the entrap!. .Y” O’of’the stute o dijjkrentiahle with respect to q is the sum y0 = .
= lnQE,+lnRE,.
Consequently, <+(lnPE,)< Taking
into account
= <‘(lnRE,+lnQE.,)<.
that < = F<, and RF = FP,.
5,“(lnPE&
FfF
= I, we obtain
= <:(lnP,E,,+F+(lnRE,)F)<,,
from which (2.25) follows directly.
=
One can notice that the introduced notion of conditional entropy and the result of Theroem 2.2 enable us to define the input-output information of a communication channel in the general case. Take .d = %d, v .rLZ where .cJ,, .c/~ denote C*-subalgebras describing the input and output of a communication channel described in terms of conditional expectation compatible with .~/i . Generalizing the famous Shannon classical formula (cf. for instance [l]) we obtain the following expression for the input-output information
i.e. the difference of an output priori state on the input.
entropy
and the mean conditional
entropy
in an a
Acknowledgments
We wish to thank Professor R. S. Ingarden for his interest in this work. One of us (V.P.B.) wishes to express his gratitude to Professor R. S. Ingarden for the kind hospitality received during his stay in Torun. REFERENCES [l] [2] [3]
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