capacity per unit cost
著者
日合 文雄
journal or
publication title
Journal of mathematical physics
volume
48
number
9
page range
092102
year
2007
URL
http://hdl.handle.net/10097/46856
doi: 10.1063/1.2779138Limit relation for quantum entropy and channel capacity
per unit cost
Imre Csiszára兲
Alfréd Rényi Institute of Mathematics, P.O. Box 127, H-1364 Budapest, Hungary Fumio Hiaib兲
Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan
Dénes Petzc兲
Alfréd Rényi Institute of Mathematics, P.O. Box 127, H-1364 Budapest, Hungary
共Received 10 July 2007; accepted 13 August 2007; published online 13 September 2007兲 In a thermodynamic model, Diósi et al. 关Int. J. Quantum Inf. 4, 99–104 共2006兲兴 arrived at a conjecture stating that certain differences of von Neumann entropies converge to relative entropy as system size goes to infinity. The conjecture is proven in this paper for density matrices. The analytic proof uses the quantum law of large numbers and the inequality between the Belavkin-Staszewski and Umegaki relative entropies. Moreover, the concept of channel capacity per unit cost is intro-duced for classical-quantum channels. For channels with binary input alphabet, this capacity is shown to equal the relative entropy. The result provides a second proof of the conjecture and a new interpretation. Both approaches lead to generalizations of the conjecture. © 2007 American Institute of Physics.
关DOI:10.1063/1.2779138兴
I. INTRODUCTION
It was conjectured by Diósi et al.4that the von Neumann entropy of a quantum state equal to a mixture,
Rnª
1
n共丢
丢共n−1兲+丢丢丢共n−2兲+ ¯ +丢共n−1兲丢兲,
exceeds the entropy of a component asymptotically by the Umegaki relative entropy S共储兲, that is,
S共Rn兲 − 共n − 1兲S共兲 − S共兲 → S共储兲, 共1兲
as n→⬁. Here andare density matrices acting on a finite dimensional Hilbert space. Recall that S共兲=−Trlog and
S共储兲 =
再
Tr共log− log兲 if supp艋 supp+⬁ otherwise.
冎
Concerning the background of quantum entropy quantities, we refer to Refs.12and14. The set of bounded linear operators on a Hilbert spaceH is denoted by B共H兲. When H is d dimensional, d finite, B共H兲 is identified as usual with the set Md共C兲 of d⫻d matrices with complex entries.
a兲Electronic mail: [email protected] b兲Electronic mail: [email protected] c兲Electronic mail: [email protected]
JOURNAL OF MATHEMATICAL PHYSICS 48, 092102共2007兲
48, 092102-1
In Ref.4, a composite system consisting of n molecules has been considered, originally each in a quantum state, and interaction with environment changed the state of one molecule to. Irreversibility has been introduced via a completely positive mapM acting as
M共丢丢共n−1兲兲 =1
n共丢
丢共n−1兲+丢丢丢共n−2兲+ ¯ +丢共n−1兲丢兲, 共2兲
interpreted as total randomization over the n subsystems共molecules兲. A thermodynamical argu-ment showed that the thermodynamical entropy of the system increased by S共储兲. This moti-vated the conjecture that the increase of the “informatic entropy,” given by the left-hand side of Eq.共1兲, also equals S共储兲, at least in the limit n→⬁.
The quantum formulation includes the case where both and are diagonal matrices. This will be referred to as the classical case. If andcommute, then in an appropriate basis both of them will be diagonal. Apparently, no exact proof of Eq. 共1兲 has been published even for the classical case, although for that case a heuristic proof was offered in Ref.4.
In this paper, first an analytic proof of Eq.共1兲 is given, see Theorem 1, using an inequality between the Umegaki and the Belavkin-Staszewski relative entropies and the law of large numbers in the quantum case. The idea is based on the identity
Rn=共1/2兲丢n
冉
1n共X丢I
丢共n−1兲+ I丢X丢I丢共n−2兲+ ¯ + I丢共n−1兲丢X兲
冊
共1/2兲丢n,
where X =−1/2−1/2. The limit of the term in the middle can be computed by the共quantum兲 law
of large numbers. For readers not familiar with the required tools, the arguments are simplified to the classical case, where the ordinary law of large numbers is used, see Theorem 2.
In the second part of this paper, we recognize that S共Rn兲−共n−1兲S共兲−S共兲 is a particular
Holevo quantity or a quantum mutual information. The Holevo capacity of classical-quantum channels is well understood.5,8,10 Channel capacity per unit cost has been studied in classical information theory, see primarily Ref. 15, but not in quantum information theory. An indirect approach to capacity per unit cost is possible via the concept of capacity with constrained inputs, which is available for classical-quantum channels.7We take a direct approach which—as in the classical case15—appears preferable.
We will consider 共memoryless兲 classical-quantum channels with binary input alphabet X =兵0,1其 which assigns to 共classical兲 input sequences 共x1, x2, . . . , xn兲苸Xn output quantum states x1丢x2丢¯丢xn, where 0= and1=, assuming that the cost of an input sequence is the
number of characters 0 in it. Considerations similar to Ref.15simultaneously provide a proof of Eq. 共1兲, and the result that the capacity per unit cost of the above channel equals S共储兲, see Theorems 3 and 4.
We note that our analytic proof of Eq.共1兲requires the assumption that supp艋supp, while the proof given in Sec. III does not共neither does the simplified version of the analytic proof to the classical case兲. It is remarkable that the two proofs lead to different generalizations of Eq.共1兲. The second proof is based on a purely information theoretic interpretation, nevertheless, the result of Theorem 3 admits also a thermodynamical interpretation as in Ref.4, see the discussion after the proof of Theorem 3.
II. AN ANALYTIC PROOF OF THE CONJECTURE
Assuming that supp艋supp for the support projections of and, one can simply com-pute
S共Rn储丢n兲 = Tr共Rnlog Rn− Rnlog丢n兲 = − S共Rn兲 − 共n − 1兲Trlog− Trlog.
S共Rn储丢n兲 = − S共Rn兲 + 共n − 1兲S共兲 + S共储兲 + S共兲,
holds. It follows that conjecture共1兲is equivalent to the statement
S共Rn储丢n兲 → 0 as n → ⬁,
when supp艋supp.
Recall the Belavkin-Staszewski relative entropy
SBS共储兲 = Tr共log共1/2−11/2兲兲 = − Tr共共−1/2−1/2兲兲,
if supp艋supp, where 共t兲ª−t log t, see Refs. 1 and 12. 关The equality of the above two expressions is easily seen from the fact that Xf共X*X兲= f共XX*兲X for a matrix X and for a polynomial
f.兴 It was proved by Hiai and Petz that
S共储兲 艋 SBS共储兲, 共3兲
see Ref.6 or Proposition 7.11 in Ref.12.
Theorem 1: If supp艋supp, then S共Rn兲−共n−1兲S共兲−S共兲→S共储兲 as n→⬁.
Proof: We want to use the quantum law of large numbers, see Proposition 1.17 in Ref. 12. Assume that andare d⫻d density matrices and we may suppose that is invertible. Due to the GNS construction with respect to the limit⬁of the product states n on the n-fold tensor
product Md共C兲丢n, n苸N, defined byn共A兲=Tr丢nA, all finite tensor products Md共C兲丢n are
em-bedded into a von Neumann algebraM acting on a Hilbert space H. If␥ denotes the right shift and Xª−1/2−1/2, then R nis written as Rn=共1/2兲丢n
冉
1 n兺
i=0 n−1 ␥i共X兲冊
共1/2兲丢n. By inequality 共3兲, we get 0艋 S共Rn储丢n兲 艋 SBS共Rn储丢n兲 = − Tr共丢n共共−1/2兲丢nRn共−1/2兲丢n兲兲 =冓
⍀,冉
1 n兺
i=0 n−1 ␥i共X兲冊
⍀冔
, 共4兲 where⍀ is the cyclic vector in the GNS construction.The law of large numbers, see Proposition 1.17 in Eq. 共12兲, gives 1
n
兺
i=0 n−1␥i共X兲 → I,
in the strong operator topology in B共H兲, since共X兲=Tr−1/2−1/2= 1.
Since the continuous functional calculus preserves the strong convergence共simply due to the approximation by polynomials on a compact set兲, we obtain
冉
1 n兺
i=0n−1
␥i共X兲
冊
→共I兲 = 0 strongly.This shows that upper bound共4兲converges to 0 and the proof is complete. 䊐
By the same proof, one can obtain that for
Rᐉ,nª 1
n −ᐉ + 1共
丢ᐉ丢丢共n−ᐉ兲+丢丢ᐉ丢丢共n−ᐉ−1兲+ ¯ +丢共n−ᐉ兲丢丢ᐉ兲,
the limit relation
S共Rᐉ,n兲 − 共n − ᐉ兲S共兲 − ᐉS共兲 → ᐉS共储兲 共5兲
holds as n→⬁ when ᐉ is fixed.
In the next theorem, we treat the classical case in a matrix language.The proof includes the case where supp艋suppis not true. Those readers who are not familiar with the tools used in the proof of the previous theorem are suggested to follow the arguments below.
Theorem 2: Assume that and are commuting density matrices. Then S共Rn兲 − 共n − 1兲S共兲 − S共兲 → S共储兲
as n→⬁.
Proof: We may assume that = Diag共1, . . . ,ᐉ, 0 , . . . , 0兲 and= Diag共1, . . . ,d兲 are d⫻d
diagonal matrices,1, . . . ,ᐉ⬎0 and ᐉ⬍d. 共We may consider ,in a matrix algebra of bigger
size if is invertible.兲 If supp艋supp, then ᐉ+1=¯ =d= 0; this will be called the regular
case. When supp艋supp is not true, we may assume thatd⬎0 and we refer to the singular
case.
The eigenvalues of Rn correspond to elements共i1, . . . , in兲 of 兵1, ... ,d其n,
1
n共i1i2¯in+i1i2i3¯in+ ¯ +i1¯in−1in兲. 共6兲
We divide the eigenvalues in three different groups as follows: 共a兲 A corresponds to 共i1, . . . , in兲苸兵1, ... ,d其nwith 1艋i1, . . . , in艋ᐉ;
共b兲 B corresponds to 共i1, . . . , in兲苸兵1, ... ,d其nwhich contains exactly one d;
共c兲 C is the rest of the eigenvalues. If eigenvalue共6兲 is in group A, then it is
共i1/i1兲 + ¯ + 共in/in兲 n i1i2¯in. First, we compute
兺
苸A共兲 =i1兺
,. . .,in 冉
共i1/i1兲 + ¯ + 共in/in兲 n i1¯in冊
.Below, the summations are over 1艋i1, . . . , in艋ᐉ,
兺
i1,. . .,in 冉
共i1/i1兲 + ¯ + 共in/in兲 n i1¯in冊
= −兺
i1,. . .,in冉
共i1/i1兲 + ¯ + 共in/in兲 n i1¯in冊
log共i1¯in兲 + Qn = −1 n兺
k=1 n冉
兺
i1,. . .,in i1i2¯inlogik+兺
i1,. . .,in i2i1¯inlogik + ¯ +兺
i1,. . .,in ini1¯in−1logik冊
+ Qn = −1 n兺
k=1 n冉
共n − 1兲兺
ik iklogik+兺
ik iklogik冊
+ Qn=共n − 1兲S共兲 −兺
i=1 ᐉ ilogi+ Qn, whereQnª
兺
i1,. . .,in共i1¯in兲
冉
共i1/i1兲 + ¯ + 共in/in兲
n
冊
.Note that Qnis equal to the expected value of
冉
X1+ ¯ + Xnn
冊
for independent and identically distributed random variables X1, X2, . . . defined on the product probability space共兵1,2, ... ,ᐉ其,共1,2, . . . ,ᐉ兲兲N, where X
ntakes the valuei/ion a sequence in
兵1,2, ... ,ᐉ其N whose nth component is equal to i.
By the strong law of large numbers,
X1+ ¯ + Xn n → E共X1兲 =
兺
i=1 ᐉ冉
i i冊
i=兺
i=1 ᐉ i almost surely.Since 共共X1+¯ +Xn兲/n兲 is uniformly bounded, the Lebesgue bounded convergence theorem
implies that Qn→
冉
兺
i=1 ᐉ i冊
, as n→⬁.In the regular case, all nonzero eigenvalues are in group A; hence,
S共Rn兲 − 共n − 1兲S共兲 − S共兲 = −
兺
i=1 ᐉ ilogi+兺
i=1 ᐉ ilogi+ Qn= S共储兲 + Qn.As兺i=1ᐉ i= 1 implies Qn→0, the statement follows.
Next, we consider the singular case, when the contributions of the eigenvalues in A is
兺
苸A共兲 = 共n − 1兲S共兲 + O共1兲,
and we turn to eigenvalues in B. If an eigenvalue corresponding to共i1, . . . , in兲苸兵1, ... ,d其n is in
group B and i1= d, then the eigenvalue is
1
ndi2¯in.
Summation of such eigenvalues gives
−
兺
i2,. . .,in冉
di2¯in n冊
log冉
di2¯in n冊
= − d n i2,. . .,i兺
n 共i2¯in兲log共i2¯in兲 − d n log d n =d n共n − 1兲S共兲 − d n log d n .When ij= d for some 2艋 j艋n, we get the same quantity, so this should be multiplied with n,
兺
苸B共兲 = d共n − 1兲S共兲 − dlog
d n .
It follows that
S共Rn兲 − 共n − 1兲S共兲 − S共兲 艌
兺
苸A共兲 +苸B
兺
共兲 − 共n − 1兲S共兲 − S共兲 艌 d共n − 1兲S共兲 + dlog n+ O共1兲 → + ⬁,
as n→⬁. 䊐
III. CHANNEL CAPACITY PER UNIT COST
A classical-quantum channel with classical input alphabetX transfers the input x苸X into the output W共x兲⬅xwhich is a density matrix acting on a Hilbert spaceK. We restrict ourselves to the
case whenX is finite and K is finite dimensional.
If a classical random variable X is chosen to be the input, with probability distribution P =兵p共x兲:x苸X其, then the corresponding output is the quantum state Xª兺x苸Xp共x兲x. When a
measurement is performed on the output quantum system, it gives rise to an output random variable Y which is jointly distributed with the input X. If a partition of unity兵Fy: y苸Y其 in B共K兲
describes the measurement, then
Prob共Y = y兩X = x兲 = TrxFy 共x 苸 X,y 苸 Y兲. 共7兲
The Holevo bound says that the classical mutual information
I共X ∧ Y兲 ª H共Y兲 − H共Y兩X兲 = H共X兲 − H共X兩Y兲
共expressed by the classical Shannon entropy H兲 satisfies9,10 I共X ∧ Y兲 艋 I共X,W兲 ª S共X兲 −
兺
x苸Xp共x兲S共x兲. 共8兲
This bound is a simple consequence of the monotonicity of relative entropy under state transformation12,13but has been proved earlier than monotonicity. Here I共X,W兲 is called a Holevo quantity or a classical-quantum mutual information, and it satisfies the identity
兺
x苸X
p共x兲S共x储兲 = I共X,W兲 + S共X储兲, 共9兲
where is an arbitrary density matrix.
When the channel W :X→B共K兲 is used to transfer sequences x=共x1, x2, . . . , xn兲苸Xn in a
memoryless manner, a sequence x =共x1, x2, . . . , xn兲苸Xnis transferred into the quantum state
W丢n共x兲 =xªx1丢x2丢 ¯ 丢xn. 共10兲
Formally, this defines a new channel W丢n:Xn→B共Kn兲 called the nth extension of W.
A code of block length n is defined by a subset An傺Xn called the code word set and by a
measurement 兵Fy: y苸Bn其 called the decoder. For technical convenience, the set Bn of possible
decoding results may be different from An, and only An傺Bn is required. The probability of error
is Prob共X⫽Y兲, where the input random variable X is uniformly distributed on Anand the output
random variable Y is as in Eq.共7兲 with x and y replaced by x and y.
The essential observation is that S共Rn兲−共n−1兲S共兲−S共兲 in the conjecture equals the Holevo
quantity I共X,W丢n兲 for the nth extension of the channel W with input alphabet X=兵0,1其,
0=,
1= and with X uniformly distributed on those length-n binary sequences that contain exactly
one 0. More generally, we shall consider Holevo quantities
I共A,0,1兲 ª S
冉
1 兩A兩x兺
苸A x冊
− 1 兩A兩x兺
苸A S共x兲, 共11兲The concept related to the conjecture we study is the channel capacity per unit cost which is defined next for simplicity only in the case where X=兵0,1其, the cost of a character 0苸X is 1, while the cost of 1苸X is 0. Given such channel and ⬎0, a number R⬎0 is called an -achievable rate per unit cost if for every␦⬎0 and for any sufficiently large T there exists a code of block length n⬎T with at least eT共R−␦兲code words such that each of the code words contains at
most T 0’s and the error probability is at most. The largest R which is an -achievable rate per unit cost for every⬎0 is the channel capacity per unit cost.
The next theorem is our main result of this section.
Theorem 3: Let the classical-quantum channel W :X=兵0,1其→B共K兲 be defined by W共0兲
=0⬅and W共1兲=1⬅. Let An傺兵0,1其n, n = 1 , 2 , . . ., be sets such that for some natural number
ᐉ and for some real number c⬎0,
(a) each element x =共x1, x2, . . . , xn兲苸An contains at mostᐉ copies of 0;
(b) log兩An兩/log n→c as n→⬁; (c) c共An兲 ª 1 兩An兩x
兺
苸An 兩兵i:xi= 0其兩 → c as n → ⬁. Then lim n→⬁ I共An,,兲 = cS共储兲.The proof of the theorem is divided into lemmas.
Lemma 1: For an arbitrary A傺兵0,1其n,
I共A,0,1兲 艋 c共A兲S共0储1兲
holds.
Proof: Let c共x兲ª兩兵i:xi= 0其兩 for x苸A. Since I共A,0,1兲=I共X,W丢n兲, we can use identity共9兲to
get an upper bound 1 兩A兩x
兺
苸A S共x储1丢 n兲 = 1 兩A兩x兺
苸A c共x兲S共0储1兲 = c共A兲S共0储1兲 for I共A,0,1兲. 䊐Lemma 2: If A傺兵0,1其n is a code word set of a code whose probability of error does not exceed a given 0⬍⬍1, then
共1 − 兲log兩A兩 − log 2 艋 I共A,0,1兲.
Proof: For the input and output random variables corresponding to the given code, the
clas-sical mutual information I共X∧Y兲 is bounded above by I共X,W丢n兲=I共A,
0,1兲, see Eq.共8兲. Since
the error probability Prob共X⫽Y兲 does not exceed , the Fano inequality 共see, e.g., Ref.3兲 gives
H共X兩Y兲 艋 log兩A兩 + log 2.
Therefore,
I共X ∧ Y兲 = H共X兲 − H共X兩Y兲 艌 共1 − 兲log兩A兩 − log 2,
and the proof is complete. 䊐
We need the direct part of the so-called quantum Stein lemma obtained in Ref. 6, see also Refs.2,5,11, and14.
Lemma 3: Given arbitrary density matricesandin B共K兲,⬎0, and 0⬍R⬍S共储兲, if N sufficiently large, there is a projection E苸B共K丢N兲 such that
␣关E兴 ª Tr丢N共I − E兲 ⬍ and
关E兴 ª Tr丢N
E⬍ e−NR.
Here E 关or the measurement 共E,I−E兲兴 is interpreted as a test of the null hypothesis that the state is丢N, against the alternative hypothesis that it is丢N. This test incorrectly accepts the null
hypothesis 共error of the first kind兲 with probability␣关E兴, and incorrectly rejects it 共error of the second kind兲 with probability关E兴.
Lemma 4: Assume that ⬎0, 0⬍R⬍S共0储1兲, and ᐉ is a positive integer. If n is large enough, then for any set Anof sequences x苸兵0,1其nthat contain at mostᐉ copies of 0, there exists a code with error probability smaller than whose code words are the N-fold repetitions xN
=共x,x, ... ,x兲 of the sequences x苸An, where N is the smallest integer satisfying
N艌 1 R log
2n .
Proof: We follow the probabilistic construction in Ref.15. The output states corresponding to input sequences of length nN are density matrices acting on the Hilbert spaceK丢Nn⬅共K丢n兲丢N. We
decompose this Hilbert space into an N-fold tensor product in a different way. For each 1艋i 艋n, let Kibe the tensor product of the factors i , i + n , i + 2n , . . . , i +共N−1兲n. So K丢Nnis identified
withK1丢K2丢¯丢Kn.
We construct a decoder to the code word set in the lemma as follows. For each 1艋i艋n, we test the null hypothesis that the ith component of the actually chosen x苸An is 0, against the
alternative that it is 1, based on the channel outputs at time instances i , i + n , . . . , i +共N−1兲n. More exactly, let the projection Ei苸B共Ki兲 be a test of the null hypothesis丢n against the alternative 丢n. According to the quantum Stein lemma共Lemma 3兲, applied with =/2ᐉ and the given 0
⬍R⬍S共储兲, for N sufficiently large, there exists a test Eisuch that the probability of error of the
first kind is smaller than , while the probability of error of the second kind is smaller than e−NR艋/2n. The projections E
iand I − Eiform a partition of unity in the Hilbert spaceKi, and the n-fold tensor product of these commuting projections will give a partition of unity inK丢Nn. For
y苸兵0,1其n, set F
yª丢i=1 n
Fy
i, where Fyi= Eiif yi= 0 and Fyi= I − Eiif yi= 1, and let the decoder be
the measurement 兵Fy: y苸兵0,1其n其. Thus, the result of the decoding will be an arbitrary 0–1
se-quence in兵0,1其n.
The error probability should be estimated, Prob共Y ⫽ X兩X = x兲 =
兺
y:y⫽x Trx丢 N Fy=兺
y:y⫽x兿
i=1 n Trx丢i N Fy i艋兺
i=1 n兺
y:yi⫽xi兿
j=1 n Trx丢j N Fy j 艋兺
i=1 n Trx丢i N共I − F xi兲. If xi= 0, then Trx丢i N共I − F xi兲 = Tr0丢 N共I − E i兲 =␣共Ei兲 艋 2ᐉ, and if xi= 1, Trx丢i N共I − F xi兲 = Tr1丢 N Ei=共Ei兲 艋 e−RN艋 2n.As xi= 0 holds for at most ᐉ indices, it follows that the probability of error of this code is
Proof of Theorem 3: Since Lemma 1 gives the upper bound,
lim sup
n→⬁
I共An,0,1兲 艋 cS共储兲,
it remains to prove that
lim inf
n→⬁
I共An,0,1兲 艌 cS共储兲.
By Lemma 4, the set兵xN: x苸A
n其 with N given there is the code word set of a code with error
probability smaller than. According to Lemma 2 and Eq.共11兲, this implies 共1 − 兲log兩An兩 − log 2 艋 S共XN兲 −
1 兩An兩x
兺
苸AnS共xN兲,
where X is uniformly distributed on An and XNdenotes its N-fold repetition.
From the subadditivity of the von Neumann entropy, we have
S共XN兲 艋 NS共X兲
and
S共xN兲 = NS共x兲
holds due to the additivity for product. It follows that 共1 − 兲log兩An N − 1 N艋 S共X兲 − 1 兩An兩x
兺
苸An S共x兲 = I共An,0,1兲.From the choice of N in Lemma 4, we have
Rlog兩An
log n
log n
log n + log 2 − log艋 log兩An
N ,
and the lower bound is arbitrarily close to cR. Since R⬍S共0储1兲 was arbitrary, the proof is
complete. 䊐
Assume that Anis the set of all x苸兵0,1其ncontaining exactlyᐉ 0’s for a fixed natural number
ᐉ. Then c共An兲=ᐉ and from the Stirling formula, one can easily check log兩An兩 /log n→ᐉ.
Conse-quently, Theorem 3 proves that
S共Rn共ᐉ兲兲 − 共n − ᐉ兲S共兲 − ᐉS共兲 → ᐉS共储兲 共12兲
holds as n→⬁ when ᐉ is fixed and
Rn共ᐉ兲 ª
冉
n ᐉ冊
−1兺
x苸An x1丢x2丢 ¯ 丢xn 共0=,1=兲.In particular, whenᐉ=1, conjecture共1兲 is proven in full generality. We have two generalizations 关Eqs.共5兲and共12兲兴 of Eq.共1兲, which are similar but different.
We note that Eq.共12兲admits a thermodynamical interpretation, analogous of that of Eq.共1兲in Ref. 4, sketched in the Introduction. Indeed, suppose that interaction with the environment changes the state not of 1 but ᐉ molecules to and irreversibility is introduced again by total randomization. The new state of the system will be Rn共ᐉ兲 above and Eq.共12兲says that “informatic
entropy” increases byᐉ times the relative entropy 共in the limit as system size goes to infinity兲.
Theorem 4: The capacity per unit cost of the channel with a binary input alphabet and
W共0兲=0, W共1兲=1is equal to the relative entropy S共0储1兲.
Proof: Assume that R⬎0 is an -achievable rate per unit cost. For every␦⬎0 and T⬎0, there
is a code A傺兵0,1其n for which we get by Lemmas 1 and 2,
TS共0储1兲 艌 c共A兲S共0储1兲 艌 I共A,0,1兲 艌 共1 − 兲log兩A兩− log 2 艌 共1 − 兲T共R −␦兲 − log 2.
Since T is arbitrarily large and,␦ are arbitrarily small, R艋S共0储1兲 follows.
Let An be the set of x苸兵0,1其n containing exactly one 0, and consider the N-times repeated
code words given in Lemma 4. Then each of the n code words contains exactly N 0’s. For every
R⬍S共0储1兲 and ,␦⬎0, if N is chosen as in Lemma 4, we have
n艌 2e NR =e N␦ 2 e N共R−␦兲⬎ eN共R−␦兲
as long as n is so large that N satisfieseN␦/ 2⬎1. This implies that R is an -achievable rate per
unit cost for every⬎0. Hence, the result follows. 䊐
ACKNOWLEDGMENTS
The first author 共I.C.兲 was partially supported by the Hungarian Research Grants OTKA T046376 and T068258. The second author 共F.H.兲 was partially supported by Grant-in-Aid for Scientific Research 共B兲17340043. The third author was partially supported by the Hungarian Research Grant T068258.
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