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Error exponents for quantum channels with constrained inputs
Vol. 46 (2000)
REPORTS ON MATHEMATICAL PHYSICS
No. 3
ERROR EXPONENTS FOR QUANTUM CHANNELS WITH CONSTRAINED INPUTS A. S. HOLEVO Steldov Mathematical Institute, Gubkina 8, Moscow 117966, Russia (e-mall: [email protected]) M. SOHMA Matsushita Research Institute Tokyo Inc., 3-10-1, Higashi-mita, Tama-ku, Kawasaki, Kanagawa 214-8501, Japan (e-mail: [email protected])
and O. HIROTA Research Institute for Quantum Communication, Tamagawa University, Machida, Tokyo 194, Japan (e-mal: [email protected])
(Received August 28, 2000)
Dedicated to Professor R. S. lngarden on the occasion of his eightieth birthday The purpose of this work is the study of the error probability for quantum channels with infinite (continuous) alphabets and additive constraints onto the input signal. We derive the lower bounds for the reliability function of such channels in terms of quantum Gallager's functions and compute them for quantmn Gaussian channels. Keywords: quantum communication channel, reliability function, Ganssian state.
1.
Introduction The importance of considering channels with infinite (continuous) alphabets and with constrained inputs was clear from the beginning of quantum communications [6, 7, 15] and was confirmed by a recent outburst of interest in the so called "continuous variables" systems in quantum optics, see e.g. [1, 4]. Recent results concern the classical capacity of such channels, see e.g. [11, 13]. However, as is well known in information theory, a much more detailed and practically applicable description of the asymptotical channel performance is given by the reliability function, which is essentially the speed of exponential decay of the error probabilities at different rates. In the paper [2], quantum analogs of the random coding and the expurgation [3431
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A. S. HOLEVO, M. SOI'iMA and O. HIROTA
lower bounds for the reliability function were given for the case of pure signal states represented by rank-one density operators, and in [14] the expurgation bound was generalized to arbitrary signal states. These papers dealt with the case of finite input alphabet. In the present paper we use the tools from the classical information theory [5] to generalize these bounds to quantum channels with infinite alphabets and constrained inputs. As an application, we compute them for particularly important case of quantum Gaussian channels, by using analytical tools developed in [9]. For the readers convenience we start with repeating some basic notions, see [11]. We take as the input alphabet A an arbitrary Borel subset in a finite-dimensional Euclidean space E. We assume that the channel is given by weakly continuous mapping x --* Sx from the input alphabet A to the set of density operators in a Hilbert space 7-/. (The weak continuity means continuity of all matrix elements (~lSx~b); ~ , ~ ~ ~). We assume that the states Sx have a finite von Neumann entropy H (Sx) = - Tr Sx log Sx 1, moreover sup H(Sx) < oo. xEA
The input is described by an a priori probability distribution zr = Jr(dx) on A. We assume that a continuous positive function f on E is fixed and consider the set Pl of probability distributions rr on A satisfying L f ( x ) T r ( d x ) < E.
(1)
Note that the alphabet A may be discrete; in that case the integrals should be simply replaced by sums. Let us consider also the product channel in the Hilbert space H®n = 9g ® . . . ® with the input alphabet A n consisting of words w = (xl . . . . . xn) of length n, and the density operator Sw = Sx, ® . . . ® corresponding to the word w. We call by code of size M a sequence (wl, X l ) . . . . . (wM, XM), where w k are words of length n, satisfying the additive constraint f (xl) + . . . + f (xn) < nE,
(2)
and {Xk} is a family of positive operators in ~®n, satisfying ~ 1 Xj < I. Defining Xo = I - ~_,j Xj, we have a resolution of identity in ~®" [11]. An output k > 1 means the decision that the word w t was transmitted, while the output 0 is interpreted as the evasion of any decision. The conditional probability to make a decision in favour of a message w k provided that the codeword w j was transmitted is TrSwiXk, in particular, the probability 1 Throughout the paper we use the natural logarithms.
ERROR EXPONENTS FOR QUANTUM CHANNELS WITH CONSTRAINED INPUTS
345
to make a wrong decision is equal to
Pj (•, X) = 1 - TrSwj Xj. The quality of the code (~V, X) is measured by the error probabilities Pmax(W, X) = max Pj (W, X) I<_j<_M
or 1
M
x) =
Pj(W, x),
j=l
which are known to be essentially equivalent from the point of view of information theory [5]. We shall be interested in the speed of the exponential decay of the error probabilities when n ~ oo and M = e nR for different rates R below the classical capacity of the channel C given by the expression [11]
:u ?(fa Sx ' XOfa 'Sx' x'1 The computation of the quantity C is itself an interesting and nontrivial problem for a general quantum Gaussian channel (see [13] for the case of squeezed-state channel). In Section 2 we derive the two upper bounds for the error probabilities in terms of the quantum Gallager functions /z(zr, s, p),/2(a', s, p), which are the most important and detailed analytical characteristics of the channel performance. In Section 3 we give some new formulae for quantum Gaussian states, which are then applied in Section 4 for computation of the Gallager functions for an arbitrary quantum Gaussian channel. The optimization of the resulting bounds involves Legendre transforms of the Gallager functions, which in general cannot be performed analytically. We consider it only for the case of a channel without squeezing in one mode. Finally, we were able to evaluate the reliability function at the zero rate, which is another important quantity characterizing the channel. 2.
The upper bounds for the error probability
Let us consider first the channel with the pure signal states Sx, let Jr be an a priori distribution satisfying the restriction (1), and let f be a function satisfying the condition for the central limit theorem for random variables f(xi), namely
A f (x)27r(dx) < oo. Following the basic Shannon idea of random coding, we consider the random ensemble of M codewords of the length n, in which the codewords are independent,
346
A. S. HOLEVO, M. SOHMA and O. HIROTA
identically distributed (i.i.d.), and each codeword has the probability distribution P(w
=
(X1 . . . . .
Xn) ~ B) = fB rr(dxO.., rr(dxn).
To account for the constraint (2), one introduces the modified ensemble of i.i.d. codewords with the probability distribution
n,~
Xs,n
f(ik)
ar(dxl)... ~:(dxn),
(3)
where
Xs,~(Y) =
nE-8<_y<_nE,
1
if
0,
otherwise,
(4)
and v~,~ = P ( E n - ~5 <_ ~-,~k=lf(Xk) < nE). It is easy to see that the corresponding expectations of any positive random variable ¢ depending on one word w = (Xl . . . . . xn) satisfy a Markov-type inequality Ivl~~ <
\ vn2 /
M exp{p ~ [ f ( x D k=l
- E]} ~,
(5)
where p > 0 [5]. THEOREM 1. Let Sx be pure states, then the following random coding bound holds for the codes of size M = e nR with R < C:
(eP'~ 2 Ivl~mxf P(W, X) _< 2 - -
where /z(zr, s, p) = - logTr
exp{-n[/z(n, s, p) - sR]},
(fa)1, ePtffx)-E1SxTr(dx )
,
(6)
(7)
and 0 < s < 1, 0 < p, 0 < 8 are arbitrary parameters. By the central limit theorem, the quantity vn,8 satisfies limn~oo ~/rffvn,s > 0 [5], thus adding only o(n) to the exponential in (6).
Proof: The bound (6) is obtained in the same way as Proposition 1 in [2], that is by evaluating the expectation of the average error probability using random, independently chosen codewords, but with the modified codeword distribution. To prove this, we use the estimate obtained in Theorem 2 of [11]. It was shown that for an ensemble of independent identically distributed codewords M inf ab(W, X) < 2(M - 1)STr(MSw) TM, X
0 < s < 1.
(8)
ERROR EXPONENTS FOR QUANTUM CHANNELS WITH CONSTRAINED INPUTS
347
Using this inequality with the modified probability distribution 15~, and then evaluating the right-hand side of the resulting inequality with the help of (5), we obtain
(ep,~l+s Msinf P(W, X) < 2 ( M - 1) s - \v,,8,/
n Tr(M exp{p E [ f ( x k ) -- E]}Sw) l+s.
(9)
k=l
Taking into account that M exp{p E l f ( x \ )
- E]}Sw =
exp(p[f(x) - E])Sx~r(dx)
,
k=l
the bound (6) follows. [] There is a conjecture [2] that the random coding bound for arbitrary density operators is obtained by replacing (7) with 1 \ l+s , /z(rr, x, p) = - logTr fA ePtf(x)-elSxr-~Jr(dx))
(10)
which is a natural noncommutative generalization of the classical bound [5]. THEOREM 2. Let Sx be arbitrary density operators. Then the following expurgated
bound holds rninPmax(W'X)
R + 2 1 ° g 2et'85] Vn., } / / '/
(11)
where /2(zr, s, p) = - s log
fafa epty~x)+/~y)-zel (
~ rr(dx)rc(dy).
Tr
(12)
Proof: We obtain this by modifying the proof of Theorem in Section 3 of [14]. Indeed, according to the relation (28) from [14] there exists a code (W, X) of the size M for which Pj(W,X)<
4(M-1)M~
Tr
,
j=l
. . . . . M;
l#j.
(13)
Using the analog of the inequality (5) for two words and the fact that the words are i.i.d., we find
< ( eAe , ~ -2 [ f a f \ vn,~j whence (11) follows.
e "tf
348
A. S. HOLEVO, M. S O H M
and O. HIROTA
We shall denote by Pc(M, n) any of the minimal error probabilities
rain Pr x(W, X),
rnini'(W, X).
W,X
W,X
Introducing the reliability function 1
1
E(R) = lim sup - log n~oo n Pe(e nR, n)' which characterizes the exponential rate of convergence of the error probability, we get the lower bound for E(R) in the case of pure state channel,
E(R) > max{Er(R), Eex(R)}, where
Er(R) ---- max (max max Iz(Tr, s, p) - sR),
(14)
E~.( R) = max(max max ~z(Tr, s, p) - s R).
(15)
O
0
For arbitrary states the bound Er(R) is still unavailable, and we content ourselves with Eex(R). We call #(Jr, s, p), ~z(rr, s, p) the quantum Gallager functions. The quantity = max max/2(a', 1, p) 0
is the cut-off rate of the channel, giving an idea of its performance at intermediate rates. Another interesting characteristic of the channel is the value E ( + 0 ) of the reliability function at the zero rate, which we can evaluate both from below and above. THEOREM 3.
-min
rt ~7~1
trEIDI J J
Proof: The first inequality follows from the lower bound for E ( + 0 ) given by Eex(0) = maxmax max/2(zr, s, p), l
0_
(17)
by letting s --~ cx~, p ~ 0. As a first step for obtaining an upper bound for E(+0), we recall the following inequality in the proof of Proposition 2 from [14],
Pmax(W, X) > max -
w~w' "4
Tr
(18)
ERROR EXPONENTS FOR QUANTUM CHANNELS w r r H CONSTRAINED INPUTS
349
where w, w' are codewords from W = (w I . . . . . toM). Generalizing the proof of Proposition 3 from [2], we can estimate the logarithm of the right-hand side of (18) as follows, maxlogTr
>- M ( M
W~W !
--
1)
n
-
- 1)
M(M
logTr V/~wV/~w,
E W,WtE'V~
rain E l°gTr S x / ~ w t k ) ~ l
t
n
-M(M_
l) E k i k j l o g T r
(19)
v/-ffiiv~j
LJ
> - M-
min ~
1 {~i}E7~d1
>
rain
- M-
2riIrj log Tr
(20)
J,j
1 ~'1
log Tr
~ (dx) ~r(dy).
Here w(k) is the k-th component of the codebloek w and ki = ~riM is the number of codeblocks w with Sw(l~,t) = Si, where kopt is the value of k which minimizes Eww, logTr I In addition, ~,¢ is the set of probability distributions with the power constraint f(xi)Tri < E.
(21)
i
Thus the upper bound for E ( + 0 ) is given by (16). 3.
The quantum
Gaussian
[]
channel
We are going to apply the results of the previous section to the quantum Gaussian channel in one mode, which is described below. However, we do this in the way easy to generalize to the multimode case (see [13]). To define the quantum Gaussian state, we consider the unitary operator in 7-[ for a vector z = [Zq, zp] t, V(z) = exp i (Zqq + Zpp),
(22)
where q and p are operators satisfying the Heisenberg CCR [q, p] = ifiI.
(23)
DEFINITION. The density operator S is called Gaussian, if its quantum characteristic function has the form T r S V (z) = exp [imt z - l zt otz] , where m is a column vector and a is a real symmetric matrix.
(24)
350
A. S. HOLEVO, M. SOHMA and O. HIROTA
For simplicity we introduce the column vector R = [q, p]t, by which (22) can be rewritten as V ( z ) = exp(iRtz). Then we can show that m = TrSR,
i t~ - ~ A = TrRSR t.
(25)
The mean m = [mq, mp] t can be an arbitrary vector; the necessary and sufficient condition on the correlation matrix ot is the generalized Robertson uncertainty relation c~ - (i/2)A > 0, where
A =
[0 :]
(26)
-h is the commutation matrix. Let So be the Gaussian density operator with the mean m = 0 and the correlation matrix oc Consider the family of operators S m = Z~(m)SoZ~(m) t, where 7~(m) = e x p ~ ( m p q - mqp) is the unitary displacement operator in 7-t. Then the density operator Sm has the characteristic function (24). In an optical communication system, So describes background noise, comprising quantum noise, and m is the classical signal. Thus the mapping m --~ Sm gives a quantum Gaussian channel. The natural input power constraint is given by (1) with
f(m)
=
1 2 -1-m 2) = ~m 1 t m. ~(mq
(27)
In preparation for obtaining lower bounds for the reliability function, we shall calculate the characteristic function of the s-th power of the Gaussian density operator. LEMMA. For any s > 0, TrSSmV (z) = .Ms(or) exp [imtz - ½ztt~ff, (or)z],
(28)
where the functions .Ms, ~s are defined below by the relations (39). Proof: Consider the density operator Sm with the correlation matrix ot and the mean vector m. Then there exists the linear symplectic transformation £ such that £ A £ t = A and ~t = £ot£. t = )d, where I is the identity matrix [13]. For this transformation we can find the unitary operator U satisfying
v(r.'z) = u t v ( z ) U .
(29)
Applying the unitary operator U and the displacement operator D ( - m ) to S, we obtain the Gaussian density operator S0 with the correlation matrix t~ = ~.I and the mean vector m = 0, 7So = U D ( - m ) S m 2 9 ( - m ) t U t. (30)
ERROR EXPONENTS FOR QUANTUM CHANNELS WITH CONSTRAINED INPUTS
351
Then the density operator So has the spectral decomposition
~n~(~./~'l--1/2~n
1
50 -- )./h + 1/2 _ \ X / h + 1/2,1 In)(nl,
(31)
which allows us to calculate the s-th power of $0 as follows, -
S~) =
(
1
)' ~
A./h + 1/2
(~./h - 1/2~ ~n
,,--o \A.Ih '+ 1/2,1
+ 1/2 = \~./h + 1 / 2 ]
(2./h + 1/2)'
(32)
In)(nl In)(nl],
(33)
where
with
~. = ~.gs ( ~ ) ,
(34)
(d + 1/2)' + ( d - 1/2) ~ gs(d) = - ~ . (d + 1/2), - (d - 1/2) ~"
(35)
1
This implies that up to a constant factor, 5g is a Gaussian density operator with the correlation matrix ~I. We shall use the matrix A - l t ~ which is diagonalizable and has purely imaginary eigenvalues +i ~. It is important that under the linear transformations it changes as the matrix of an operator. We shall also need the matrix function abs(.), which is defined as follows: for a diagonalizable matrix M = Tdiag(mj)T -1, we put absM = Tdiag(Imjl)T -1, see [12] for details. We can rewrite (34) as ~ [ .-~ otgs ( a b s ( A - l o t ) )
(36)
.
Going back to the initial basis, we obtain
1
I
it
]
TrSSV(z) = fs(X/h--------~exp imtz - ~z Otgs (abs(A-lo0) z ,
(37)
where
L(d)=(d+l/2)S-(d-1/2)
s.
(38)
Then we set 1
Afs(t~) = [detfs(abs(A-la))] - : , and (28) follows.
~s(u) = gs(abs(A-lo0),
(39) []
352
A.S. HOLEVO,M. SOHMAand O. HIROTA
This result can be easily extended to the mulfirnode Gaussian state, by using the fact that the operator A-lot is diagonalizable as explained in [12, 13]. If we restrict our attention to one mode, then detot = dett~ = ~2,
abs(A_lot) _ x / ~ o t t,
(4o)
and
JV's(ot)= fs ( dVrd-~/h)-1 ,
(41)
~s(ot)=gs ( d~/'d-~et~/h)I.
(42)
The case s = 1/2 considered in [9] has special features. The functions 1
(d + 1/2) 1/2 + (d - 1/2) 1/2
gl/2(d) = 2---d" (d + 1/2)1/2 - (d - 1/2)1/2, fl/2(d) = (d + 1/2) 1/2 - (d - 1/2) 1/2 are then related by the equation
2dgl/2(d)fl/2(d)2= 1.
(43)
In Appendix we give the proof of the following formula for the the two Gaussian states Sm~, S,n2: Tr
= exp - ~ ( m l - m 2 ) t o t - l ( m l
-
fidelity between
m2) ,
(44)
which will be used in the next section to evaluate the zero-rate value of the reliability function. This formula also has an independent interest, because of the importance of the fidelity in many questions of quantum information theory.
4.
Evaluating reliability function of a quantum Gaussian channel
1. Computation of the Gallager functions. By an appropriate generalization of can take the a priori distribution Gaussian
the argument in the classical case, we
2~r dex/-d-~-~exp where the trace of E satisfies the relation power constraint.
mtE-lm
din,
(45)
l(~qq .~_ ~pp): E following from the
ERROR EXPONENTS FOR QUANTUM CHANNELS WITH CONSTRAINED INPUTS
353
The Gallager function /z(~r, s, p) is given by [
/
"
1
\ l+s
Iz(zr, s,p) =-logTr~JaeP[½mtm-E]Sm~rc(dm) )
•
Using (28),-we obtain /z(Tr, s, p) = (1 + s ) p E - ( 1
+ s) log [det(I - pI2)-1/2A/'r~7+(u) ]
_ log [)V-l+s (G.~+ (or)or + ( ~ - 1 - p / ) - l ) ] .
(46)
On the other hand, using (28) and the formula for the product of two Gaussian states (see Appendix), we have
[1:,
Tr S v / ~ S v ~ 2 = V r S q ~ I Sv~2V(0) =exp
-
m2)t(2~½(oOoO-l(ml
-
m2)
]
,
(47)
and hence
1, × exp [p(lm'm + in n-
S
= 2spE + ~ log det [(I -
p~g)(/-
p ~ + 2 ( 2 s ~ (~)ot) -1E)].
(48)
2. Lower bounds for the reliability function. Now let us turn to the optimization problems (14) and (15), which involve the Legendre transform, and for this the concavity in s of the functions lz(zr, s, p(s, E)), ~z(zr, s, p(s, E)), where p(s, E) is the maximizing value of p for fixed s, is essential. Although it has not been shown that /z(zr, s, p) provides the random coding bounds for mixed state signals, we would assume it in this section. If we consider only pure state signals, the following statements are valid strictly. We shall restrict ourselves to the case where So is a Ganssian state without squeezing, which is a simple example of the mixed Gaussian state. Then ot = ~.I O. > hi2), and the optimum a priori distribution is given by the Gaussian state with the correlation matrix ~ = diag[E, E]. Eq. (46) is simplified as /z(zr, s, p) ---- (1 + s)log f__~ (X/h) + p(1 + s)E + log [A(s, p)l+s _ B(s, p)l+s], (49) where
A(s, p) = ((~.lh)gr~+ (klh) + 1/2)(1 - pE) + EIh, B(s, p) = ((~.lh)g~+~ (X/h) - 1/2)(1 - pE) + Ell~.
(50)
354
A.S. HOLEVO, M. SOHMA and O. HIROTA
Trying to maximize /~(Jr, s, p) with respect to p we obtain the equation 2(1/h - (p)~/h)gr~+~ (~./h))(a(s, p)S _ B(s, p)S) = p ( a ( s , p)S + B(s, p)S).
(51)
This equation can be solved explicitly only for s = 0, 1. Thus, contrary to the classical case [5], the maximum in (14) in general can be found only numerically. The concavity of the function /z0r, s, p(s, E)) can be shown analytically in the case of pure states; we conjecture it holds in general, and we postpone the study of this question to a subsequent publication. For s = 0 we have p = 0 and C = ~--~Iz(zr, O, O) = [(Ns + N + 1) log(Ns + N + 1) - (Ns + N ) log(Ns + N)]
- [ ( N + 1) log(N + 1) - N logN], where N = L / h - 1/2 and Ns = E / h . For s = 1, Eq. (51) has the unique solution p(1, E) = 1 / 2 h g + l / E - O ( E / h g ) / E < E -1, where g = (3./h)gl/2(L/h) and 1 + ~¢/E-f+ 1
O(E) =
2
The important quantity is the cut-off rate given by the formula 6' = / x 0 r ' I ' p ( I ' E ) ) = 2(2-~g + 1 - O ( E / h g ) ) + log O ( E / h g ) .
(52)
We now pass to the expurgated bound in the case of a Gaussian state without squeezing. By putting ot = LI and I] = diag[E, El, (48) is simplified as ~z(zr, s, p) = s 2 p E + log
1 + p2E2 - 2 p E +
hgs
]j
(53)
The optimization of the expurgated bound can be performed analytically. Taking the partial derivative with respect to p we obtain the equation p2 _ p
(~gs
+
1)
1
+ 2h gs E = O.
The solution of this equation, satisfying p < E - 1 , is p(s, E) --
1 1 + 2hgs E
O(E/hgs) E
Substituting this in (15), we obtain the following expression, which is to be maximized with respect to s _> I,
ERROR EXPONENTS FOR QUANTUM CHANNEI.~ WITH CONSTRAINED INPUTS
~t(rr, s , p ( s , E ) ) - s R = 2 ( 2 - ~ g + S - s O ( E / h g s )
355
) +slogO(E/hgs)-sR.
This function is readily shown to be concave in s. Taking the derivative with respect to s, we obtain the equation
O(E/hgs) = e R, the solution of which is s =
E
2hg~/e 2R - e R"
(54)
If this is larger than 1, which is equivalent to
R < logO(E/hgs) = 0-~/2(7r, 1, p(1, E)), then the maximum is achieved for the value of s given by (54) and is equal to E ~g(1 - v/1 - e -R) = Eex(R) > Er(R),
(55)
(which up to a factor coincides with the expurgated bound for the classical Gaussian channel). In particular /z(Tr, 1, p(1, E)) is equal to /2(~r, 1, p(1, E)), that is the cut-off rate C. In the range O-~/2(zr, 1, p(1, E)) _< R _< ~s/~(zr, 1, p(1, E)),
(56)
where the optimizing s is equal to 1, we have the linear bound
Eex(R) = Er(R) =/z(zr, 1, p(1, E)) - R,
(57)
with the quantities ~/z(~r, 1, p(1, E)), defined by (52). Finally, provided /x(~r, s, p) is indeed concave, in the range O ~s/z(~r, 1, p(1, E)) < R < C we have Eex(R) < Er(R) with Er(R) given implicitly by (14).
3. Reliability function at the zero rate. Finally, let us calculate the bounds (16) for the quantum Gaussian channels; we can calculate them for the general Gaussian state in one mode, unlike the random coding bound or the expurgated bound for E(R). There exists a unitary operator such that diag[y:, y-2] = O~Ot/)., where V is a parameter related to squeezing (y = 1 corresponds to no squeezing). We further
356
A.S. HOLEVO, M. SOHMA and O. HIROTA
assume that diag[crl, o.2] = 0 ZOt. Substituting (44) to the fight-hand side of (16) and transforming the variables of the integral by the unitary operator O, we obtain 1
E(+0) < - - max [y--2o. 1 -It- y2o.2] = F2E/X,
(58)
- 2~. ~reT'l
where we suppose y _> 1 and then the optimum distribution is given by [o'1, o.2] = [0, 2E]. On the other hand, substituting (47) to the left-hand side of (16), we obtain
yEE/hg <_E(+0).
(59)
Thus we have obtained the bounds for E(+0). In particular, for a pure (squeezed) state we get the exact value, given by simple formula E ( + 0 ) = 2yEE/h. Appendix
1. Characteristic function of the product of two Gaussian states. The following multiplication formula
TrS1S2V(z) - (2~) t
TrS1V(v)TrS2V(z - v)exp -~v Az dr,
(60)
where ~ denotes the number of the modes, was established in [10]. Substituting
TrSjV(z) = exp im}z - ~z otjz
(j = 1, 2)
(61)
into (60), and performing the integration, we obtain
TrSIS2 V (z) = [det A - l ( a l + a 2 ) ] - ~ exp --~(ml + m2)t(~l + ~2)-l(ml + m2)
[1
]
x exp -~((ot2 + (i/2)A)z -- 2im2)t(otl + ot2)-l((al -- (i/2)A)z --2iml) . (62)
2. Proof of the relation (44)for the fidelity between two Gaussian states. Using (24), (28) and (62), we have
Tr Sx/~m~Sv~mV(z)
[1
= exp -~(ml
-
m2)t(2~l/2(a)ot)-l(ml
1
-
m2)
]
× exp [--lzt (~ot~l/2(ot) -- 8A~l/2(ot)-lot-l A) z + irltz + (tz ] ,
(63)
ERROR EXPONENTS FOR QUANTUM CHANNELS WITH CONSTRAINED INPUTS
357
where
ml + m2 n --
=
2
,
(64)
4AOt-I#I/2(Ot) -1 (ml -- m2).
(65)
Let us consider a trace-class operator T with the characteristic function I t TrT V (z) = exp [imt z + ~t z - -~ z otz] .
Then,
(66)
using the formula (60), we have Tr T* T V (z) = exp[~tot -1 ~ ]TrS 2 V (z),
(67)
where S is the Gaussian density operator with the characteristic function
TrSV(z) = exp [i(m + ~Aot1 t otz]. 1 1~) t z _ ~z L
z
(68)
.I
This yields T r I T [ = T r T ~ / T - ~ V ( 0 ) = e x p [ l ( t o t -1 ( ] .
(69)
Combining (63) and (69) we obtain (44), which also holds for the multimode case. Acknowledgment. A. S. Holevo is grateful to Prof. O. Hirota for his hospitality at the Research Institute of Tamagawa University where this paper was initiated. REFERENCES [1] S. L. Braunstein: Squeezing as an irreducible resource, LANL Report no. quant-ph/9904002. [2] M. V. Burnashev and A. S. Holevo: On reliability function of quantum communication channel, Probl. Peredachi Inforr~, 34 (1988), 1-13 (in Russian); LANL Report no. quant-ph/9?03013, 1997. [3] C. M. Caves and P. B. Drummond: Rev. Mod. Phys. 66 (1994), 481-538. [4] A. Furusawa, J. S0rensen, S. L. Braunstein, C. Fuchs, H. J. Kimble and E. S. Polzik: Science 282 (1998), 706. [5] R. G. Gallager: Information Theory and Reliable Communications, J. Wiley, New York 1968; Section 7.3. [6] J. P. Gordon: Quantum effects in communication systems, Proc. IRE, 50, N9 (1962), 1898-1908. [7] J. P. Gordon: Noise at Optical Frequencies; Information Theory, in Quantum Electronics and Coherent Light, Proc. Int. School Phys. "Enrico Fermi", Course XXXI, ed. P. A. Miles, Academic Press, New York 1964, pp. 156-181. [8] C. W. Helstrom: Quantum Detection and Estimation Theory, Academic Press, New York 1976. [9] A. S. Holevo: On quasieqnivalence of quasffTee states, Teor. Mat. Fi~, vol. 13, pp. 184-199, 1972 (in Russian). English translation by Consultants Bureau, Plenum Publ. Corp., 1071-1082, 1973.
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A. S. HOLEVO, M. $OHMA and O. HIROTA
[10] A. S. Holevo: On the optimal measurement of the mean value of a Gaussian state, Techn. Cybern., NS, (1974), 152-158 (in Russian). [11] A. S. Holevo: Coding theorems for quantum channels, Tamagawa University Research Review, N4, 1998. LANL Report no. quant-ph/9809023. [12] A. S. Holevo: Sending quantum information with G-aussian states, Quantum Communication, Computing, and Measurement 2, pp. 75-82, Kluwer Academic/Plenum Publishers, New York 2000. [13] A. S. Holevo, M. Sohma and O. Hirota, "The capacity of quantum Ganssian channels ," Phys. Rev. A, 59, N3 (1999), pp. 1820-1828. [14] A. S. Holevo: Reliability function of general classical-quantum channel, LANL Report no. quantph/9907087. To appear in IEEE Trans. Inform. Theory, 46, N6, 2000. [15] R. S. Ingarden: Rep. Math. Phys. 10, (1976), 43-72.
REPORTS ON MATHEMATICAL PHYSICS
No. 3
ERROR EXPONENTS FOR QUANTUM CHANNELS WITH CONSTRAINED INPUTS A. S. HOLEVO Steldov Mathematical Institute, Gubkina 8, Moscow 117966, Russia (e-mall: [email protected]) M. SOHMA Matsushita Research Institute Tokyo Inc., 3-10-1, Higashi-mita, Tama-ku, Kawasaki, Kanagawa 214-8501, Japan (e-mail: [email protected])
and O. HIROTA Research Institute for Quantum Communication, Tamagawa University, Machida, Tokyo 194, Japan (e-mal: [email protected])
(Received August 28, 2000)
Dedicated to Professor R. S. lngarden on the occasion of his eightieth birthday The purpose of this work is the study of the error probability for quantum channels with infinite (continuous) alphabets and additive constraints onto the input signal. We derive the lower bounds for the reliability function of such channels in terms of quantum Gallager's functions and compute them for quantmn Gaussian channels. Keywords: quantum communication channel, reliability function, Ganssian state.
1.
Introduction The importance of considering channels with infinite (continuous) alphabets and with constrained inputs was clear from the beginning of quantum communications [6, 7, 15] and was confirmed by a recent outburst of interest in the so called "continuous variables" systems in quantum optics, see e.g. [1, 4]. Recent results concern the classical capacity of such channels, see e.g. [11, 13]. However, as is well known in information theory, a much more detailed and practically applicable description of the asymptotical channel performance is given by the reliability function, which is essentially the speed of exponential decay of the error probabilities at different rates. In the paper [2], quantum analogs of the random coding and the expurgation [3431
344
A. S. HOLEVO, M. SOI'iMA and O. HIROTA
lower bounds for the reliability function were given for the case of pure signal states represented by rank-one density operators, and in [14] the expurgation bound was generalized to arbitrary signal states. These papers dealt with the case of finite input alphabet. In the present paper we use the tools from the classical information theory [5] to generalize these bounds to quantum channels with infinite alphabets and constrained inputs. As an application, we compute them for particularly important case of quantum Gaussian channels, by using analytical tools developed in [9]. For the readers convenience we start with repeating some basic notions, see [11]. We take as the input alphabet A an arbitrary Borel subset in a finite-dimensional Euclidean space E. We assume that the channel is given by weakly continuous mapping x --* Sx from the input alphabet A to the set of density operators in a Hilbert space 7-/. (The weak continuity means continuity of all matrix elements (~lSx~b); ~ , ~ ~ ~). We assume that the states Sx have a finite von Neumann entropy H (Sx) = - Tr Sx log Sx 1, moreover sup H(Sx) < oo. xEA
The input is described by an a priori probability distribution zr = Jr(dx) on A. We assume that a continuous positive function f on E is fixed and consider the set Pl of probability distributions rr on A satisfying L f ( x ) T r ( d x ) < E.
(1)
Note that the alphabet A may be discrete; in that case the integrals should be simply replaced by sums. Let us consider also the product channel in the Hilbert space H®n = 9g ® . . . ® with the input alphabet A n consisting of words w = (xl . . . . . xn) of length n, and the density operator Sw = Sx, ® . . . ® corresponding to the word w. We call by code of size M a sequence (wl, X l ) . . . . . (wM, XM), where w k are words of length n, satisfying the additive constraint f (xl) + . . . + f (xn) < nE,
(2)
and {Xk} is a family of positive operators in ~®n, satisfying ~ 1 Xj < I. Defining Xo = I - ~_,j Xj, we have a resolution of identity in ~®" [11]. An output k > 1 means the decision that the word w t was transmitted, while the output 0 is interpreted as the evasion of any decision. The conditional probability to make a decision in favour of a message w k provided that the codeword w j was transmitted is TrSwiXk, in particular, the probability 1 Throughout the paper we use the natural logarithms.
ERROR EXPONENTS FOR QUANTUM CHANNELS WITH CONSTRAINED INPUTS
345
to make a wrong decision is equal to
Pj (•, X) = 1 - TrSwj Xj. The quality of the code (~V, X) is measured by the error probabilities Pmax(W, X) = max Pj (W, X) I<_j<_M
or 1
M
x) =
Pj(W, x),
j=l
which are known to be essentially equivalent from the point of view of information theory [5]. We shall be interested in the speed of the exponential decay of the error probabilities when n ~ oo and M = e nR for different rates R below the classical capacity of the channel C given by the expression [11]
:u ?(fa Sx ' XOfa 'Sx' x'1 The computation of the quantity C is itself an interesting and nontrivial problem for a general quantum Gaussian channel (see [13] for the case of squeezed-state channel). In Section 2 we derive the two upper bounds for the error probabilities in terms of the quantum Gallager functions /z(zr, s, p),/2(a', s, p), which are the most important and detailed analytical characteristics of the channel performance. In Section 3 we give some new formulae for quantum Gaussian states, which are then applied in Section 4 for computation of the Gallager functions for an arbitrary quantum Gaussian channel. The optimization of the resulting bounds involves Legendre transforms of the Gallager functions, which in general cannot be performed analytically. We consider it only for the case of a channel without squeezing in one mode. Finally, we were able to evaluate the reliability function at the zero rate, which is another important quantity characterizing the channel. 2.
The upper bounds for the error probability
Let us consider first the channel with the pure signal states Sx, let Jr be an a priori distribution satisfying the restriction (1), and let f be a function satisfying the condition for the central limit theorem for random variables f(xi), namely
A f (x)27r(dx) < oo. Following the basic Shannon idea of random coding, we consider the random ensemble of M codewords of the length n, in which the codewords are independent,
346
A. S. HOLEVO, M. SOHMA and O. HIROTA
identically distributed (i.i.d.), and each codeword has the probability distribution P(w
=
(X1 . . . . .
Xn) ~ B) = fB rr(dxO.., rr(dxn).
To account for the constraint (2), one introduces the modified ensemble of i.i.d. codewords with the probability distribution
n,~
Xs,n
f(ik)
ar(dxl)... ~:(dxn),
(3)
where
Xs,~(Y) =
nE-8<_y<_nE,
1
if
0,
otherwise,
(4)
and v~,~ = P ( E n - ~5 <_ ~-,~k=lf(Xk) < nE). It is easy to see that the corresponding expectations of any positive random variable ¢ depending on one word w = (Xl . . . . . xn) satisfy a Markov-type inequality Ivl~~ <
\ vn2 /
M exp{p ~ [ f ( x D k=l
- E]} ~,
(5)
where p > 0 [5]. THEOREM 1. Let Sx be pure states, then the following random coding bound holds for the codes of size M = e nR with R < C:
(eP'~ 2 Ivl~mxf P(W, X) _< 2 - -
where /z(zr, s, p) = - logTr
exp{-n[/z(n, s, p) - sR]},
(fa)1, ePtffx)-E1SxTr(dx )
,
(6)
(7)
and 0 < s < 1, 0 < p, 0 < 8 are arbitrary parameters. By the central limit theorem, the quantity vn,8 satisfies limn~oo ~/rffvn,s > 0 [5], thus adding only o(n) to the exponential in (6).
Proof: The bound (6) is obtained in the same way as Proposition 1 in [2], that is by evaluating the expectation of the average error probability using random, independently chosen codewords, but with the modified codeword distribution. To prove this, we use the estimate obtained in Theorem 2 of [11]. It was shown that for an ensemble of independent identically distributed codewords M inf ab(W, X) < 2(M - 1)STr(MSw) TM, X
0 < s < 1.
(8)
ERROR EXPONENTS FOR QUANTUM CHANNELS WITH CONSTRAINED INPUTS
347
Using this inequality with the modified probability distribution 15~, and then evaluating the right-hand side of the resulting inequality with the help of (5), we obtain
(ep,~l+s Msinf P(W, X) < 2 ( M - 1) s - \v,,8,/
n Tr(M exp{p E [ f ( x k ) -- E]}Sw) l+s.
(9)
k=l
Taking into account that M exp{p E l f ( x \ )
- E]}Sw =
exp(p[f(x) - E])Sx~r(dx)
,
k=l
the bound (6) follows. [] There is a conjecture [2] that the random coding bound for arbitrary density operators is obtained by replacing (7) with 1 \ l+s , /z(rr, x, p) = - logTr fA ePtf(x)-elSxr-~Jr(dx))
(10)
which is a natural noncommutative generalization of the classical bound [5]. THEOREM 2. Let Sx be arbitrary density operators. Then the following expurgated
bound holds rninPmax(W'X)
R + 2 1 ° g 2et'85] Vn., } / / '/
(11)
where /2(zr, s, p) = - s log
fafa epty~x)+/~y)-zel (
~ rr(dx)rc(dy).
Tr
(12)
Proof: We obtain this by modifying the proof of Theorem in Section 3 of [14]. Indeed, according to the relation (28) from [14] there exists a code (W, X) of the size M for which Pj(W,X)<
4(M-1)M~
Tr
,
j=l
. . . . . M;
l#j.
(13)
Using the analog of the inequality (5) for two words and the fact that the words are i.i.d., we find
< ( eAe , ~ -2 [ f a f \ vn,~j whence (11) follows.
e "tf
348
A. S. HOLEVO, M. S O H M
and O. HIROTA
We shall denote by Pc(M, n) any of the minimal error probabilities
rain Pr x(W, X),
rnini'(W, X).
W,X
W,X
Introducing the reliability function 1
1
E(R) = lim sup - log n~oo n Pe(e nR, n)' which characterizes the exponential rate of convergence of the error probability, we get the lower bound for E(R) in the case of pure state channel,
E(R) > max{Er(R), Eex(R)}, where
Er(R) ---- max (max max Iz(Tr, s, p) - sR),
(14)
E~.( R) = max(max max ~z(Tr, s, p) - s R).
(15)
O
0
For arbitrary states the bound Er(R) is still unavailable, and we content ourselves with Eex(R). We call #(Jr, s, p), ~z(rr, s, p) the quantum Gallager functions. The quantity = max max/2(a', 1, p) 0
is the cut-off rate of the channel, giving an idea of its performance at intermediate rates. Another interesting characteristic of the channel is the value E ( + 0 ) of the reliability function at the zero rate, which we can evaluate both from below and above. THEOREM 3.
-min
rt ~7~1
trEIDI J J
Proof: The first inequality follows from the lower bound for E ( + 0 ) given by Eex(0) = maxmax max/2(zr, s, p), l
0_
(17)
by letting s --~ cx~, p ~ 0. As a first step for obtaining an upper bound for E(+0), we recall the following inequality in the proof of Proposition 2 from [14],
Pmax(W, X) > max -
w~w' "4
Tr
(18)
ERROR EXPONENTS FOR QUANTUM CHANNELS w r r H CONSTRAINED INPUTS
349
where w, w' are codewords from W = (w I . . . . . toM). Generalizing the proof of Proposition 3 from [2], we can estimate the logarithm of the right-hand side of (18) as follows, maxlogTr
>- M ( M
W~W !
--
1)
n
-
- 1)
M(M
logTr V/~wV/~w,
E W,WtE'V~
rain E l°gTr S x / ~ w t k ) ~ l
t
n
-M(M_
l) E k i k j l o g T r
(19)
v/-ffiiv~j
LJ
> - M-
min ~
1 {~i}E7~d1
>
rain
- M-
2riIrj log Tr
(20)
J,j
1 ~'1
log Tr
~ (dx) ~r(dy).
Here w(k) is the k-th component of the codebloek w and ki = ~riM is the number of codeblocks w with Sw(l~,t) = Si, where kopt is the value of k which minimizes Eww, logTr I In addition, ~,¢ is the set of probability distributions with the power constraint f(xi)Tri < E.
(21)
i
Thus the upper bound for E ( + 0 ) is given by (16). 3.
The quantum
Gaussian
[]
channel
We are going to apply the results of the previous section to the quantum Gaussian channel in one mode, which is described below. However, we do this in the way easy to generalize to the multimode case (see [13]). To define the quantum Gaussian state, we consider the unitary operator in 7-[ for a vector z = [Zq, zp] t, V(z) = exp i (Zqq + Zpp),
(22)
where q and p are operators satisfying the Heisenberg CCR [q, p] = ifiI.
(23)
DEFINITION. The density operator S is called Gaussian, if its quantum characteristic function has the form T r S V (z) = exp [imt z - l zt otz] , where m is a column vector and a is a real symmetric matrix.
(24)
350
A. S. HOLEVO, M. SOHMA and O. HIROTA
For simplicity we introduce the column vector R = [q, p]t, by which (22) can be rewritten as V ( z ) = exp(iRtz). Then we can show that m = TrSR,
i t~ - ~ A = TrRSR t.
(25)
The mean m = [mq, mp] t can be an arbitrary vector; the necessary and sufficient condition on the correlation matrix ot is the generalized Robertson uncertainty relation c~ - (i/2)A > 0, where
A =
[0 :]
(26)
-h is the commutation matrix. Let So be the Gaussian density operator with the mean m = 0 and the correlation matrix oc Consider the family of operators S m = Z~(m)SoZ~(m) t, where 7~(m) = e x p ~ ( m p q - mqp) is the unitary displacement operator in 7-t. Then the density operator Sm has the characteristic function (24). In an optical communication system, So describes background noise, comprising quantum noise, and m is the classical signal. Thus the mapping m --~ Sm gives a quantum Gaussian channel. The natural input power constraint is given by (1) with
f(m)
=
1 2 -1-m 2) = ~m 1 t m. ~(mq
(27)
In preparation for obtaining lower bounds for the reliability function, we shall calculate the characteristic function of the s-th power of the Gaussian density operator. LEMMA. For any s > 0, TrSSmV (z) = .Ms(or) exp [imtz - ½ztt~ff, (or)z],
(28)
where the functions .Ms, ~s are defined below by the relations (39). Proof: Consider the density operator Sm with the correlation matrix ot and the mean vector m. Then there exists the linear symplectic transformation £ such that £ A £ t = A and ~t = £ot£. t = )d, where I is the identity matrix [13]. For this transformation we can find the unitary operator U satisfying
v(r.'z) = u t v ( z ) U .
(29)
Applying the unitary operator U and the displacement operator D ( - m ) to S, we obtain the Gaussian density operator S0 with the correlation matrix t~ = ~.I and the mean vector m = 0, 7So = U D ( - m ) S m 2 9 ( - m ) t U t. (30)
ERROR EXPONENTS FOR QUANTUM CHANNELS WITH CONSTRAINED INPUTS
351
Then the density operator So has the spectral decomposition
~n~(~./~'l--1/2~n
1
50 -- )./h + 1/2 _ \ X / h + 1/2,1 In)(nl,
(31)
which allows us to calculate the s-th power of $0 as follows, -
S~) =
(
1
)' ~
A./h + 1/2
(~./h - 1/2~ ~n
,,--o \A.Ih '+ 1/2,1
+ 1/2 = \~./h + 1 / 2 ]
(2./h + 1/2)'
(32)
In)(nl In)(nl],
(33)
where
with
~. = ~.gs ( ~ ) ,
(34)
(d + 1/2)' + ( d - 1/2) ~ gs(d) = - ~ . (d + 1/2), - (d - 1/2) ~"
(35)
1
This implies that up to a constant factor, 5g is a Gaussian density operator with the correlation matrix ~I. We shall use the matrix A - l t ~ which is diagonalizable and has purely imaginary eigenvalues +i ~. It is important that under the linear transformations it changes as the matrix of an operator. We shall also need the matrix function abs(.), which is defined as follows: for a diagonalizable matrix M = Tdiag(mj)T -1, we put absM = Tdiag(Imjl)T -1, see [12] for details. We can rewrite (34) as ~ [ .-~ otgs ( a b s ( A - l o t ) )
(36)
.
Going back to the initial basis, we obtain
1
I
it
]
TrSSV(z) = fs(X/h--------~exp imtz - ~z Otgs (abs(A-lo0) z ,
(37)
where
L(d)=(d+l/2)S-(d-1/2)
s.
(38)
Then we set 1
Afs(t~) = [detfs(abs(A-la))] - : , and (28) follows.
~s(u) = gs(abs(A-lo0),
(39) []
352
A.S. HOLEVO,M. SOHMAand O. HIROTA
This result can be easily extended to the mulfirnode Gaussian state, by using the fact that the operator A-lot is diagonalizable as explained in [12, 13]. If we restrict our attention to one mode, then detot = dett~ = ~2,
abs(A_lot) _ x / ~ o t t,
(4o)
and
JV's(ot)= fs ( dVrd-~/h)-1 ,
(41)
~s(ot)=gs ( d~/'d-~et~/h)I.
(42)
The case s = 1/2 considered in [9] has special features. The functions 1
(d + 1/2) 1/2 + (d - 1/2) 1/2
gl/2(d) = 2---d" (d + 1/2)1/2 - (d - 1/2)1/2, fl/2(d) = (d + 1/2) 1/2 - (d - 1/2) 1/2 are then related by the equation
2dgl/2(d)fl/2(d)2= 1.
(43)
In Appendix we give the proof of the following formula for the the two Gaussian states Sm~, S,n2: Tr
= exp - ~ ( m l - m 2 ) t o t - l ( m l
-
fidelity between
m2) ,
(44)
which will be used in the next section to evaluate the zero-rate value of the reliability function. This formula also has an independent interest, because of the importance of the fidelity in many questions of quantum information theory.
4.
Evaluating reliability function of a quantum Gaussian channel
1. Computation of the Gallager functions. By an appropriate generalization of can take the a priori distribution Gaussian
the argument in the classical case, we
2~r dex/-d-~-~exp where the trace of E satisfies the relation power constraint.
mtE-lm
din,
(45)
l(~qq .~_ ~pp): E following from the
ERROR EXPONENTS FOR QUANTUM CHANNELS WITH CONSTRAINED INPUTS
353
The Gallager function /z(~r, s, p) is given by [
/
"
1
\ l+s
Iz(zr, s,p) =-logTr~JaeP[½mtm-E]Sm~rc(dm) )
•
Using (28),-we obtain /z(Tr, s, p) = (1 + s ) p E - ( 1
+ s) log [det(I - pI2)-1/2A/'r~7+(u) ]
_ log [)V-l+s (G.~+ (or)or + ( ~ - 1 - p / ) - l ) ] .
(46)
On the other hand, using (28) and the formula for the product of two Gaussian states (see Appendix), we have
[1:,
Tr S v / ~ S v ~ 2 = V r S q ~ I Sv~2V(0) =exp
-
m2)t(2~½(oOoO-l(ml
-
m2)
]
,
(47)
and hence
1, × exp [p(lm'm + in n-
S
= 2spE + ~ log det [(I -
p~g)(/-
p ~ + 2 ( 2 s ~ (~)ot) -1E)].
(48)
2. Lower bounds for the reliability function. Now let us turn to the optimization problems (14) and (15), which involve the Legendre transform, and for this the concavity in s of the functions lz(zr, s, p(s, E)), ~z(zr, s, p(s, E)), where p(s, E) is the maximizing value of p for fixed s, is essential. Although it has not been shown that /z(zr, s, p) provides the random coding bounds for mixed state signals, we would assume it in this section. If we consider only pure state signals, the following statements are valid strictly. We shall restrict ourselves to the case where So is a Ganssian state without squeezing, which is a simple example of the mixed Gaussian state. Then ot = ~.I O. > hi2), and the optimum a priori distribution is given by the Gaussian state with the correlation matrix ~ = diag[E, E]. Eq. (46) is simplified as /z(zr, s, p) ---- (1 + s)log f__~ (X/h) + p(1 + s)E + log [A(s, p)l+s _ B(s, p)l+s], (49) where
A(s, p) = ((~.lh)gr~+ (klh) + 1/2)(1 - pE) + EIh, B(s, p) = ((~.lh)g~+~ (X/h) - 1/2)(1 - pE) + Ell~.
(50)
354
A.S. HOLEVO, M. SOHMA and O. HIROTA
Trying to maximize /~(Jr, s, p) with respect to p we obtain the equation 2(1/h - (p)~/h)gr~+~ (~./h))(a(s, p)S _ B(s, p)S) = p ( a ( s , p)S + B(s, p)S).
(51)
This equation can be solved explicitly only for s = 0, 1. Thus, contrary to the classical case [5], the maximum in (14) in general can be found only numerically. The concavity of the function /z0r, s, p(s, E)) can be shown analytically in the case of pure states; we conjecture it holds in general, and we postpone the study of this question to a subsequent publication. For s = 0 we have p = 0 and C = ~--~Iz(zr, O, O) = [(Ns + N + 1) log(Ns + N + 1) - (Ns + N ) log(Ns + N)]
- [ ( N + 1) log(N + 1) - N logN], where N = L / h - 1/2 and Ns = E / h . For s = 1, Eq. (51) has the unique solution p(1, E) = 1 / 2 h g + l / E - O ( E / h g ) / E < E -1, where g = (3./h)gl/2(L/h) and 1 + ~¢/E-f+ 1
O(E) =
2
The important quantity is the cut-off rate given by the formula 6' = / x 0 r ' I ' p ( I ' E ) ) = 2(2-~g + 1 - O ( E / h g ) ) + log O ( E / h g ) .
(52)
We now pass to the expurgated bound in the case of a Gaussian state without squeezing. By putting ot = LI and I] = diag[E, El, (48) is simplified as ~z(zr, s, p) = s 2 p E + log
1 + p2E2 - 2 p E +
hgs
]j
(53)
The optimization of the expurgated bound can be performed analytically. Taking the partial derivative with respect to p we obtain the equation p2 _ p
(~gs
+
1)
1
+ 2h gs E = O.
The solution of this equation, satisfying p < E - 1 , is p(s, E) --
1 1 + 2hgs E
O(E/hgs) E
Substituting this in (15), we obtain the following expression, which is to be maximized with respect to s _> I,
ERROR EXPONENTS FOR QUANTUM CHANNEI.~ WITH CONSTRAINED INPUTS
~t(rr, s , p ( s , E ) ) - s R = 2 ( 2 - ~ g + S - s O ( E / h g s )
355
) +slogO(E/hgs)-sR.
This function is readily shown to be concave in s. Taking the derivative with respect to s, we obtain the equation
O(E/hgs) = e R, the solution of which is s =
E
2hg~/e 2R - e R"
(54)
If this is larger than 1, which is equivalent to
R < logO(E/hgs) = 0-~/2(7r, 1, p(1, E)), then the maximum is achieved for the value of s given by (54) and is equal to E ~g(1 - v/1 - e -R) = Eex(R) > Er(R),
(55)
(which up to a factor coincides with the expurgated bound for the classical Gaussian channel). In particular /z(Tr, 1, p(1, E)) is equal to /2(~r, 1, p(1, E)), that is the cut-off rate C. In the range O-~/2(zr, 1, p(1, E)) _< R _< ~s/~(zr, 1, p(1, E)),
(56)
where the optimizing s is equal to 1, we have the linear bound
Eex(R) = Er(R) =/z(zr, 1, p(1, E)) - R,
(57)
with the quantities ~/z(~r, 1, p(1, E)), defined by (52). Finally, provided /x(~r, s, p) is indeed concave, in the range O ~s/z(~r, 1, p(1, E)) < R < C we have Eex(R) < Er(R) with Er(R) given implicitly by (14).
3. Reliability function at the zero rate. Finally, let us calculate the bounds (16) for the quantum Gaussian channels; we can calculate them for the general Gaussian state in one mode, unlike the random coding bound or the expurgated bound for E(R). There exists a unitary operator such that diag[y:, y-2] = O~Ot/)., where V is a parameter related to squeezing (y = 1 corresponds to no squeezing). We further
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assume that diag[crl, o.2] = 0 ZOt. Substituting (44) to the fight-hand side of (16) and transforming the variables of the integral by the unitary operator O, we obtain 1
E(+0) < - - max [y--2o. 1 -It- y2o.2] = F2E/X,
(58)
- 2~. ~reT'l
where we suppose y _> 1 and then the optimum distribution is given by [o'1, o.2] = [0, 2E]. On the other hand, substituting (47) to the left-hand side of (16), we obtain
yEE/hg <_E(+0).
(59)
Thus we have obtained the bounds for E(+0). In particular, for a pure (squeezed) state we get the exact value, given by simple formula E ( + 0 ) = 2yEE/h. Appendix
1. Characteristic function of the product of two Gaussian states. The following multiplication formula
TrS1S2V(z) - (2~) t
TrS1V(v)TrS2V(z - v)exp -~v Az dr,
(60)
where ~ denotes the number of the modes, was established in [10]. Substituting
TrSjV(z) = exp im}z - ~z otjz
(j = 1, 2)
(61)
into (60), and performing the integration, we obtain
TrSIS2 V (z) = [det A - l ( a l + a 2 ) ] - ~ exp --~(ml + m2)t(~l + ~2)-l(ml + m2)
[1
]
x exp -~((ot2 + (i/2)A)z -- 2im2)t(otl + ot2)-l((al -- (i/2)A)z --2iml) . (62)
2. Proof of the relation (44)for the fidelity between two Gaussian states. Using (24), (28) and (62), we have
Tr Sx/~m~Sv~mV(z)
[1
= exp -~(ml
-
m2)t(2~l/2(a)ot)-l(ml
1
-
m2)
]
× exp [--lzt (~ot~l/2(ot) -- 8A~l/2(ot)-lot-l A) z + irltz + (tz ] ,
(63)
ERROR EXPONENTS FOR QUANTUM CHANNELS WITH CONSTRAINED INPUTS
357
where
ml + m2 n --
=
2
,
(64)
4AOt-I#I/2(Ot) -1 (ml -- m2).
(65)
Let us consider a trace-class operator T with the characteristic function I t TrT V (z) = exp [imt z + ~t z - -~ z otz] .
Then,
(66)
using the formula (60), we have Tr T* T V (z) = exp[~tot -1 ~ ]TrS 2 V (z),
(67)
where S is the Gaussian density operator with the characteristic function
TrSV(z) = exp [i(m + ~Aot1 t otz]. 1 1~) t z _ ~z L
z
(68)
.I
This yields T r I T [ = T r T ~ / T - ~ V ( 0 ) = e x p [ l ( t o t -1 ( ] .
(69)
Combining (63) and (69) we obtain (44), which also holds for the multimode case. Acknowledgment. A. S. Holevo is grateful to Prof. O. Hirota for his hospitality at the Research Institute of Tamagawa University where this paper was initiated. REFERENCES [1] S. L. Braunstein: Squeezing as an irreducible resource, LANL Report no. quant-ph/9904002. [2] M. V. Burnashev and A. S. Holevo: On reliability function of quantum communication channel, Probl. Peredachi Inforr~, 34 (1988), 1-13 (in Russian); LANL Report no. quant-ph/9?03013, 1997. [3] C. M. Caves and P. B. Drummond: Rev. Mod. Phys. 66 (1994), 481-538. [4] A. Furusawa, J. S0rensen, S. L. Braunstein, C. Fuchs, H. J. Kimble and E. S. Polzik: Science 282 (1998), 706. [5] R. G. Gallager: Information Theory and Reliable Communications, J. Wiley, New York 1968; Section 7.3. [6] J. P. Gordon: Quantum effects in communication systems, Proc. IRE, 50, N9 (1962), 1898-1908. [7] J. P. Gordon: Noise at Optical Frequencies; Information Theory, in Quantum Electronics and Coherent Light, Proc. Int. School Phys. "Enrico Fermi", Course XXXI, ed. P. A. Miles, Academic Press, New York 1964, pp. 156-181. [8] C. W. Helstrom: Quantum Detection and Estimation Theory, Academic Press, New York 1976. [9] A. S. Holevo: On quasieqnivalence of quasffTee states, Teor. Mat. Fi~, vol. 13, pp. 184-199, 1972 (in Russian). English translation by Consultants Bureau, Plenum Publ. Corp., 1071-1082, 1973.
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[10] A. S. Holevo: On the optimal measurement of the mean value of a Gaussian state, Techn. Cybern., NS, (1974), 152-158 (in Russian). [11] A. S. Holevo: Coding theorems for quantum channels, Tamagawa University Research Review, N4, 1998. LANL Report no. quant-ph/9809023. [12] A. S. Holevo: Sending quantum information with G-aussian states, Quantum Communication, Computing, and Measurement 2, pp. 75-82, Kluwer Academic/Plenum Publishers, New York 2000. [13] A. S. Holevo, M. Sohma and O. Hirota, "The capacity of quantum Ganssian channels ," Phys. Rev. A, 59, N3 (1999), pp. 1820-1828. [14] A. S. Holevo: Reliability function of general classical-quantum channel, LANL Report no. quantph/9907087. To appear in IEEE Trans. Inform. Theory, 46, N6, 2000. [15] R. S. Ingarden: Rep. Math. Phys. 10, (1976), 43-72.