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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
Square-Root Measurement for Quantum Symmetric Mixed
State Signals
Kentaro Kato, Associate Member, IEEE, and
Osamu Hirota, Senior Member, IEEE
Abstract—A certain class of mixed states artificially constructed that is
used in quantum cryptography is crucial for representation of signals. This
correspondence shows that the square-root measurement gives the optimal
measurement for such a class of mixed states.
Index Terms—Error probability, quantum cryptography, quantum detection theory, quantum mixed state signals, square-root measurement.
I. INTRODUCTION
The problem of optimization of quantum measurement is one of the
interesting and essential topics in quantum information theory [1]–[3].
This problem is to design the receiver for quantum state signals in order
to realize highly reliable communication systems. A quantum measurement that minimizes the average probability of detection error is referred to as an optimal receiver. To design the optimal receiver, one can
relay on quantum detection theory, which gives some strategies such
as Bayes strategy, Neyman–Pearson strategy, and mini-max strategy
[1]–[5].
The mathematical structures and error performances of the optimal
receivers for various signals have been discussed by several authors
[7]–[17] since Helstrom’s pioneering work [6]. For example, the structures and performances of the optimal receivers for the coherent orthogonal signals and the simplex signals were clarified by Yuen et
al. [7]. In 1982, Helstrom developed the iterative procedures of the
Bayes-cost reduction, in which the minimum error probabilities of the
quadrature amplitude-shift keyed (ASK) coherent state signals and the
ternary ASK signals of thermal coherent states were numerically computed [8]. The case of the multi-ary symmetric pure state signals was
first discussed by Helstrom [1]. Charbit et al. discussed on the optimal
receiver for the multi-ary phase-shift keyed (PSK) signals in the study
of the cutoff rate of individual quantum measurement [9] and Osaki et
al.analytically clarified the structures of the optimal receivers for the
binary PSK (BPSK), 3PSK, quaternary PSK (QPSK), and 3ASK coherent state signals [10]. Furthermore, Ban et al. discussed the relation
between the optimal measurement and the square-root measurement
in the case of the symmetric pure state signals [11]. However, except
few cases such as the binary mixed state signals [6] and the signals
consisting of the mixture of equiprobable eigenstates [13], there are no
analytical solutions of the optimal receiver for mixed state signals.
On the other hand, it is well known that the square-root measurement
(SRM) plays an important role in the proof of the quantum channel
coding theorem [18]–[20]. From this we can see the SRM is asymptotically optimum when we decode classical codes consisting of quantum
state signals. Originally, the SRM was referred in Helstrom’s book as
Manuscript received May 16, 2002; revised July 25, 2003. This work was
supported in part by the Core Research for Evolutional Science and Technology,
Japan Science and Technology Corporation.
K. Kato is with the Faculty of Science and Engineering, Chuo University, Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan (e-mail: [email protected]).
O. Hirota is with the Tamagawa University Research Institute and the
Department of Information and Communication Engineering, Tamagawa
University, Tamagawagakuen 6-1-1, Machida, Tokyo 194-8610, Japan (e-mail:
[email protected]).
Communicated by P. W. Shor, Associate Editor for Quantum Information
Theory.
Digital Object Identifier 10.1109/TIT.2003.820050
the suboptimum quantum receiver [1]. In the Helstrom’s iterative procedure of the Bayes-cost reduction [8] and the other numerical calculation methods [21]–[23], the SRM is used to give the initial measurement process. After the paper of Helstrom’s iterative algorithm, the
SRM has been investigated in mutual information criterion rather than
error probability criterion. In 1994, Hausladen and Wootters proposed
that the SRM is a “pretty good” measurement to attain the accessible
information when the signals are equiprobable and almost orthogonal
[24]. This result came to fruition in the quantum channel coding theorem. Furthermore, Sasaki et al. employed the SRM to show the first
examples of the superadditibity of the maximum mutual informations
of quantum channels [25], [26].
In general, the SRM is not the optimal measurement in error probability criterion. It was, however, pointed out that the SRM is the optimal measurement for the symmetric pure state signals by Ban et al.
in terms of Bayes strategy [11]. This motivates to consider new signal
classes that the SRM can be optimum. Indeed, it is expanded to the case
of the binary pure state linear codes by Usuda et al. [14], the case of
the multiply symmetric pure state signals by Barnett [15], the case of
the geometrically uniform pure state signals by Eldar and Forney [16],
and the case of the receiver-side symmetric pure state signals in entanglement-assisted communication systems [17]. Following these results,
we will try to expand Ban’s result to the case of mixed state signals.
Furthermore, it has been emphasized by Yuen [28] that the quantum
detection theory plays the most important role in security analysis
for some quantum cryptosystems. Especially, in Yuen–Kim type of
quantum cryptosystem [29], the security of the system depends on the
ability of signal detection of the eavesdropper. The legitimate users
in the above quantum cryptosystem have to design the transmission
signal such that the density operators for the signal look like mixed
states to the eavesdropper. In such cases, the mixed states may
be artificial. A role of the quantum detection theory in the above
mentioned scheme is to clarify the ultimate limitation of detectability
for those states. A main purpose of this correspondence is to provide
an analytical solution in detectability analysis for a specific class of
the artificially constructed mixed states. In Section II, we survey the
framework of quantum detection theory. In Section III, we introduce
the mixed state signals to be considered. We derive the SRM for the
mixed state signals defined in the previous section in Section IV,
and the optimality of the SRM for the mixed state signals under
consideration is shown in Section V. Finally, we show a simple
example and summarize our results.
II. AVERAGE PROBABILITY OF ERROR AND BAYES STRATEGY
We now consider a multi-ary quantum communication system. According to quantum detection theory, each signal is described by a density operator on a signal Hilbert space Hsig , which is nonnegative definite and has unit trace
^i
0; Tr^i = 1;
i = 1; 2; . . . ; M:
When we perform quantum measurement to detect quantum state signals, a signal detector is mathematically described by a positive operator-valued measure (POVM). For discrete decision cases, the POVM
is denoted as follows:
5 = f5^ 1 ; 5^ 2 ; . . . ; 5^ M g
where
0018-9448/03$17.00 © 2003 IEEE
5^ j 0;
M
5^ j = ^1
j =1
j
= 1; 2; . . . ; M 3
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
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and where M 3 is the number of decision, ^
1 is the identity of the signal
^ j of the POVM is called a detecHilbert space, and each element 5
tion operator. Using the density operator ^i of quantum signal and the
^ j of quantum measurement, the conditional probdetection operator 5
ability detecting the letter “j ” when the letter “i” is true is given by
TABLE I
14) AND POSSIBLE
(
,
j = Tr5^ ^ :
P (j i)
j
i
Hence the average probability of error is defined as
Pe
=10
M
j
i P (i i)
i=1
where i is a priori probability of the quantum state signal ^i , that is,
i > 0;
M
i
= 1; 2; . . . ; M
^i
= 1:
i
i=1
In quantum detection theory, some of the strategies that minimize
the average probability of error have been developed in various situations [1]–[5]. Bayes strategy is one of the strategies, which is employed
when a priori probabilities of signals are given. When Bayes strategy
is employed, the optimal detection operators satisfy the next two conditions [2], [4]
5^ (opt) ( ^ 0 ^ )5^ (opt) = 0;
^ (opt) 0 ^ 0;
^ 5
i
i
k
i
j
k
j
8 (i; j )
8 i:
j
i
k
i
s
1
i;
i
N
N
j 0
(n
n=1
y = V^ y V^ = V^ M = ^1:
1)M +i
ih 0
(n
1)M +i
5^ srm = G^ 01 2 ^ G^ 01
=
i
(4)
j;
^
G
= 1; 2; . . . ; M
(5)
;
i
= 1; 2; . . . ; M
M
^i :
i=1
Applying to the doubly symmetric mixed state signals, it becomes
N
5^ srm =
j 0
(n
i
n=1
1)M +i
ih 0
j
(n
1)M +i
M
j ih j
(7)
where
j i = 0^ 0 j i;
1=2
0^ s
s
s=1
s
and where the set fjs i : s = 1; 2; . . . ; M 0 g is an orthonormal basis
of the signal Hilbert space. In this case, the detection operators of the
SRM satisfy
5^ srm 5^ srm = 5^ srm ;
i
i;j
j
i
8 (i; j )
where i;j is Kronecker’s delta (see Appendix I).
By using the orthonormal basis fjs ig, each of the (k; l)-elements
of the represent matrix of ^i is given by
h j^ j i = N1
k
i
=2
i
where
s
According to their result, the optimal measurement process for the symmetric pure state signals is given by the square-root measurement. We
would like to expand this result to mixed state signal cases, hence we
introduce a new signal class by using the symmetric pure states of (3).
We now restrict the M 0 symmetric pure states of (3) to be linearly independent. By using the M 0 symmetric pure states, we define new signals
such that
1
^ =;
where k is an arbitrary integer. This property indicates a symmetry of
the mixed state signals. From the properties of (3) and (6), we call the
signals defined in (5) the doubly symmetric mixed state signals. Note
that the terminology “doubly symmetric” used here is different from
that used in [15]. In our case, “doubly” means the symmetries of the
signal states (6) and the component states (3) by the unitary operator
of (4).
(3)
where the unitary operator V^ satisfies
V^ V^
(6)
As mentioned earlier, a signal detector is described by a POVM. It
is, however, difficult to obtain a closed-form analytical expression of
the optimal POVM directly from the necessary and sufficient conditions of Bayes strategy in general. Fortunately, we can take an alternative approach, that is, we will employ a special measurement called
the square-root measurement (SRM). For a given signal set f^i : i =
1; 2; . . . ; M g, the SRM is defined as
As mentioned in the Introduction, we sometimes encounter mixed
states which are constructed artificially in a quantum cryptosystem.
At the present time, we cannot predict what kind of artificial mixed
states are employed in quantum cryptosystems. However, if we can
show some analysis of quantum detection problem for such states, it
will be useful. Although we cannot solve the optimum quantum detection scheme for any kind of such mixed states, we can show that there
exists an analytical solution if the density operators have certain symmetry.
In 1996, Ban et al. [11] showed the optimal measurement process
for the symmetric pure state signals defined by
0
s = 1; 2; . . . ; M
= 1; 2; . . . ; M
i
IV. SQUARE-ROOT MEASUREMENT
III. DOUBLY SYMMETRIC MIXED STATE SIGNALS
s
01 ^1 V^ ykM +i01 ;
kM +i
(2)
This is the necessary and sufficient condition of Bayes strategy. Therefore, our task is to find the optimal POVM for a given signal set by
using this condition.
i = V^ 01 j
= V^
(1)
k
j
where the integer N satisfies M 0 = M N (see Table I). The dimension
of the signal Hilbert space spanned by fj s ig is given as dim Hsig =
M 0 . Each of the mixed state signals has the property
i
N
(01 2 )
=
l
n=1
01)M +i (01=2 )(n01)M +i;l ;
0
0
k = 1; 2; . . . ; M ; l = 1; 2; . . . ; M
k;(n
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
where (01=2 )k;l is the (k; l)-element of the square root of the Gram
matrix given by
(01 2 ) = 1
=
k;l
p v 0 0
M
(k
s
M0
l)s
(8)
a) k + l = even.
^ V^ M . Because of U^ N = ^1, we may change
Here we put U
the start point of the index “n” in (9). By replacing n with (k +
l)=2 + n in the first sum of (9), it becomes
s=1
N
^ given by
is the eigenvalue of 0
and where s
M
=
1
(t
t
Similarly, each of the (k; l)-elements of the representation matrix of
the detection operator of the SRM for the doubly symmetric mixed
state signals is given by
N
h j5^ j i =
k
srm
l
j
n=1
k;(n01)M +j (n01)M +j;l :
N
1=2
1
V. OPTIMAL DETECTION OF THE DOUBLY SYMMETRIC
MIXED STATE SIGNALS
where
N
p(0) =
h jV^ j i
nM
1
n=1
2
i
i
0 ^ ) 5^ = 0;
j
i
i
j
j
N
N
j 0
(k
N
k=1 l=1
1)M +i
k;l
h 0
(l
1)M +j
j
(i;j )
N
=
1=2
k;l
=
N
h j0^ 0 V^ 0
1=2
1
n=1
N
1=2
n=1
1
=1
(l
n)M +j
N
0
1=2
(n
k)M +j
2 h j0^ 0 V^ 0
1
1=2
(l
n
1
0i j
)M
i
1
j i
1
k
1
i
:
=1
N
j
1=2
1=2
l
0n ) j
(
+n
h j0^ 0 U^
0n V^ j 0i j
1=2
1=2
1
2 h j0^ 0 U^
1
1=2
1
i
1
i
i
1
0n V^ j 0i j
=1
1
j i
2 h j0^ 0 U^
N
i
0n )0k V^ j 0i j
2 h j0^ 0 U^ 0
1
1
) V^ 0 j
1=2
h j0^ 0 U^
n=1
j i
+n
l
h j0^ 0 U^ (
1
=
)0
+n
1=2
1
n
0i j
1
1
2 h j0^ 0 U^ 0(
1
0
j i
h j0^ 0 V^ 0
1
n
k)M
2 h j0^ 0 V^ 0
1
0
(n
i
j i
+n
1=2
1
1
(i;j )
:
1
0n V^ j 0i j
1=2
h j0^ 0 U^ (
where
F
j
n
2 h j0^ 0 U^
n
(i;j )
i
(i;j )
k;l
iF
k
l
h j0^ 0 U^
=1
Substituting the detection operators of the SRM for the doubly symmetric mixed state signals, the left-hand side of the condition becomes
5^ srm (^ 0 ^ ) 5^ srm = 1
1
Therefore, we obtain Fk;l = 0.
b) k + l = odd.
Replacing n with (k + l + 1)=2+ n in the first sum of (9) and
n0 with (k + l 0 1)=2 0 n0 in the second sum, we have
8 (i; j ):
j
n
1=2
1
Let us prove this proposition. From the assumption that a priori probabilities of the signals are equal, the first condition of Bayes strategy is
reduced as follows:
5^ (^
0n V^ j 0i j
1=2
1
n
F
:
1
i
1
2 h j0^ 0 U^ 0( 0 ) j i
N
=
= 1 0 p(0)
i
h j0^ 0 U^ ( 0 )0 V^ 0 j i
=1
n
1
Pe
j
1
1=2
1
We have thus obtained the analytical expression of the SRM for the
doubly symmetric mixed state signals.
srm
) V^ 0 j
Similarly, by replacing n0 with (k + l)=2 0 n0 in the second sum
of (9), it becomes
N
Here we show that the SRM is the optimal receiver for the doubly
symmetric mixed state signals with equal a priori probabilities. Furthermore, we will see that the minimum value of the average probability of error is given by
i
j i
+n
2 h j0^ 0 U^
1
1
+n
l
h j0^ 0 U^
n=1
s
s
1=2
1
=
+n
2 h j0^ 0 U^ 0(
1)s
t=1
0k j
1=2
1
h j iv0 0
p
where v = exp[2i=M 0 ] and i = 01, and where the eigenvalues of
0^ are strictly positive, > 0, because j i are linearly independent.
s
h j0^ 0 U^
n=1
+n
1
i
1
i
j i
1
= 0:
i
(9)
^ 01=2 ] = 0 [11]. Hence, the
and where we have used the relation [V^ k ; 0
(i;j )
question is whether Fk;l is zero for arbitrary fixed parameters (i; j )
and (k; l). In order to examine it, we consider next the following two
cases: a) the sum of k and l is even, and b) the sum of k and l is odd.
We have thus proved the first half of the proposition.
Next we show that the SRM for the doubly symmetric mixed state
signals satisfies the second condition of Bayes strategy. From the assumption that the signals are equiprobable, i = 1=M , the second condition of Bayes strategy is reduced in the next form
M
i=1
^i
^i 5
0 ^ 0;
j
j
= 1; 2; . . . ; M:
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
Substituting the detection operators of the SRM for the doubly symmetric mixed state signals to the left-hand side of the condition, it becomes
M
^i
^i 5
srm
i=1
where Z^
0 ^ = N1 V^ 0
M +j
j
where the permutation is defined by
01 Z^ V^ y0M +j 01
= Y^ 0 X^ and where we have defined each term as
Y^
^
X
N
1=2
1
n=1
N
1
1
n
1
1=2
n
=2
n
:
=
t
s
t
N
s
=
M +1
; n
n=1
0
N
(01 2 )
=
Y 0 X
s;t
s;t
..
.
!
=
n
M +s;t
01)M +1 (01=2 )(n01)M +1;t
0
0
s = 1; 2; . . . ; M ; t = 1; 2; . . . ; M :
= 1; 2; . . . ; r
k
where 2A2y (1; . . . ; k; 1; . . . ; k) is the k th principle minor [27]. In
order to apply this lemma to our case, we now set Z 0 001=2 Z 001=2 :
Reversely, we may rewrite this relation as Z = 01=2 Z 0 01=2 : Because
of the positive definiteness of the Gram matrix 0, the matrix Z is nonnegative definite if the matrix Z 0 is nonnegative definite. The (s; t)-elements of Z 0 are calculated as
(01 2 )1 ( 01)
=
; n
n=1
0
Y 0
N
n=1
s;t
s;t
M +1
(001 2 )( 01)
=
n
M +s;t
!
MN;
! (N 0 1)(M 0 1) + 1;
..
.
!
2Y 0 2y = 2001
ij
1;
0;
NM
0 1):
N
det 2Y 0 2y (1; . . . ; k; 1; . . . ; k)
= det 2Z 0 2y (1; . . . ; k; 1; . . . ; k)
k
> 0;
= 1; 2; . . . ; N (M 0 1):
Hence, our question turns into whether the matrix Y is positive definite
or not. From (8), the elements of the matrix Y are given by
Ys;t
=
1
2
M0
M
M
N
p
k
l v
01)(k0l)M +(t0s)l
(n
n=1 k=1 l=1
and the eigenvalues of Y are calculated as
= 1
"s
M
M
p 0
(m
m=1
1)N +s s :
Since the eigenvalues s of the Gram matrix 0 are strictly positive,
the eigenvalues "s of the matrix Y are also strictly positive. Hence, Y
is a positive definite matrix. Therefore, the SRM satisfies the second
condition of Bayes strategy.
From (5) and (7), the conditional probability P (j ji) is given as follows:
j
^j
P (j i) = Tr5
srm
^i
N
h jU^ V^ 0 j i
p(i 0 j ):
=
for i = (j )
for i 6= (j )
01=2 2y > 0
Y0
and
:
On the matrix X 0 , it is easy to see that the ((n 0 1)M +1; (n 0 1)M +
1)-elements of X 0 , 1 8 n N , are one, and the others are zero. Here
we introduce a permutation matrix 2 defined by
2 =
0 1);
Using the permutation matrix 2 previously defined, we have
s;(n01)M +1 (n01)M +1;t
0 X0
n(M
..
.
so that if the matrix Y is strictly positive definite, then
(01 2 )( 01)
det 2A2y (1; . . . ; k; 1; . . . ; k) > 0;
N
0 1;
! N (M 0 1) + n;
! (n 0 1)(M 0 1) + 1;
N M
Each of the elements of the ((n01)M +1)th rows and ((n01)M +1)th
columns, 1 8 n N , is zero. Therefore, the rank of the matrix, rank
Z , is N (M 0 1) at most (see Appendix II). In general, an nth square
matrix A, which is a Hermitian and have rank A = r (n r), is
nonnegative definite if and only if there exists the permutation matrix
2 such that
Zs;t
M
2X 0 2y = diag(0; 0; . . . ; 0; 1; 1; . . . ; 1)
( 01)
s;(n
;
1;
..
.
=2
n=1
0 =
!
NM
t
(01 2 )1 ( 01)
M
0 1) + 1;
N (M
..
.
(N 0 1)M + 1
(N 0 1)M + 2
= h jZ^j i
= h jY^ j i 0 h jX^ j i
s
!
!
nM
0^ 1 2 U^ 01 j1 ih1 jU^ y 010^ 1
=
n=1
n
1
2
(n 0 1)M + 1
(n 0 1)M + 2
:
h j0^ U^ 0 j iU^ y 0 0^
Hence, our task is to check whether the operator Z^ is nonnegative definite or not. Indeed, by using the representation matrix of Z^ given by
the orthonormal basis fjs ig, we can check the nonnegativeness of Z^ .
The (s; t)-element of the representation matrix Z is given by
Zs;t
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1
n
i
j
2
1
n=1
The function p(i) defined above satisfies the following relations:
0
p( i) = p(i);
p(i + kM )
= p(i);
M
i=1
p(i) =
1
(10)
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where k is an arbitrary integer (Appendix III). With the function p(i),
the minimum value of the average probability of error for the doubly
symmetric mixed state signals is given by
P srm = 1 0 p(0):
e
We have thus proved the proposition.
VII. CONCLUSION
We have shown that the optimum measurement for the doubly symmetric mixed states with equal a priori probabilities is the SRM constructed by signal states themselves. These types of artificial mixed
states appear in signal design for quantum cryptosystem, especially in
Yuen–Kim protocol for quantum key distribution. So our result may be
applicable to calculate a security problem for a quantum cryptosystem.
VI. A SIMPLE EXAMPLE
APPENDIX I
ORTHOGONALITY OF THE SRM
In this section, we consider the binary signals consisting of coherent
states as a simple example. The binary signals are given as follows:
^1 = 1 (j
2
1
^2 = (j
2
The detection operators of the SRM for the doubly symmetric
mixed state signals are orthogonal although it is not orthogonal
in general. Furthermore, the detection operators of the SRM is
orthogonal for a larger large-signal class. Here, we will state this
property. To start with, we consider M 0 linearly independent pure
states j s i; s = 1; 2; . . . ; M 0 , whose states need not be symmetric.
Using these pure states, we define
ih j + j ih j)
ih j + j ih j)
1
2
1
3
3
2
4
4
where
j
j
j
j
1
3
2
4
i = ji
i = j0i
i = jii
i =j0ii
1=2
s
s
j
= 1; 8 i
and where the integer N satisfies the relation M 0 = MN
(Table I),
(i;n)
0^ =
4
i;n)
s
a2 b c2 b3
b3 a1 b c1
2 =
c2 b3 a2 b
b c1 b3 a1
s
c
s
c
( )
=
M
=1
i=1 n
j~
and the representation matrix of the detection operators of the SRM are
given by
0
0
0
0
0 0
0 0 0 0
0 0
0 1 0 0
;
5srm
=
:
2
1 0
0 0 0 0
0 0
0 0 0 1
srm
srm
By using the matrices of 1 , 2 and 51 , 52 , one can directly check
that the SRM is a optimal measurement in this case. The minimum
value of the average probability of error is given by
P
e
c
s
N
j~
(i;n)
n=1
(i;n)
i
ih ~
(i;n)
j
^ of f^i g becomes
and then the Gram operator G
G^ =
M
^
i
i=1
s
= 1 0 1 (1 + f + f ):
2
s
(i;n)
i
=M
5^ srm
= j1 ih1 j + j3 ih3 j
1
= j2 ih2 j + j4 ih4 j
5^ srm
2
srm
s
i M j
^ = M
The detection operators of the SRM for the binary signals are given by
1
0
5srm
=
1
0
0
M
the signal ^i defined by (11) is represented as
s
c
= 1:
M
=1
N
by defining
s
s
i;n)
i;n
s
c
(
n=1
has a one-to-one correspondence to the set fs : s = 1; 2; . . . ; M 0 g,
that is, s = f(i; n) and (i; n) = f 01 (s) with a suitable function f .
Since a signal quantum state has a unit trace
M
s
c
N
8 (i; n);
> 0;
f(i; n) : i = 1; 2; . . . ; M; n = 1; 2; . . . ; N g
j ih j:
s
s=1
(
and where we have assumed that the index set
a1 = 1 (1 + f + f ) ;
a2 = 1 (1 0 f 0 f )
4
4
1
2
c1 = k + f 0 f ;
c2 = 1 k2 0 f + f
4
4
b = 1 (K + iK )
4
f = 1 (1 + k2 )2 0 4K 2; f = 1 (1 0 k2 )2 0 4K 2
2
2
2
and where k = exp[0jj ], K = k cos[jj2 ], and K = k sin[jj2 ].
c
(i;n)
where
(i;n)
c
j;
(11)
where
c
ih
i = 1; 2; . . . ; M
i;n)
n=1
By using this basis, the representation matrices of the signals are given
by
a1 b c1 b3
b3 a2 b c2
1 =
;
c1 b3 a1 b
3
b c2 b a2
(
i
and where ji is the coherent state of light with complex amplitude .
In this case, an orthonormal basis of the signal Hilbert space is given
by
j i = 0^ 0 j i;
N
^ =
N
M
(i;n)
i=1 n=1
=M
j~
M
(i;n)
j
j ~ ih ~ j
s
s=1
ih ~
s
M ^:
Hence, the detection operators of the SRM for the mixed state signals
of (11) is given as follows:
5^ srm = G^ 01 2 ^ G^ 01 2
= 1 ^01 2 ^ ^01 2
M
=
^01 2 j ~( ) ih ~(
=
i
=
i
=
=
i
N
=
i;n)
i;n
n=1
=
N
j~
(i;n)
n=1
ih~
(i;n)
j
j^0
1=2
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003
where
3317
ACKNOWLEDGMENT
j~(i;n) i ^01=2 j ~(i;n) i:
From [26], the set
fj~
(i;n)
The authors would like to thank T. S. Usuda, M. Ban, M. Sohma,
and M. Osaki for helpful discussions and encouraging comments.
i: i = 1; 2; . . . ; M ; n = 1; 2; . . . ; N g
REFERENCES
is an orthonormal basis of the signal Hilbert space spanned by fj
s = 1; 2; . . . ; M 0 g, that is,
h~
(i;n)
j~
(i ;n )
i =
s
i:
(i;n);(i ;n )
(i; n) = (i0 ; n0 )
(i; n) 6= (i0 ; n0 ):
= 1;
0;
Therefore, the detection operators of the SRM for the mixed state signals defined by (11) are orthogonal.
APPENDIX II
RANK OF THE MATRIX Z
N , and for each t, 1 t M 0
For arbitrary fixed n, 1 n
Z(n01)M +1;t
N
=
(01 2 )1 ( 01)
=
M +1
; n
=1
n
2 (0 ) 0
0
0 (0 ) 0
1=2
(n
N
n
1)M +(n
1=2
=1
1=2
(n
1)M +1;t
1)M +1;(n
01)M +1
2 (0 ) 0
N
=
(n
1)M +1;t
(01 2 )1 ( 01)
=
M +1
; n
=1
n
2 (0 )
1=2
N
0
01)01)M +1;t
((n +n
(01 2 )1 (( 0
=
n
=1
1=2
;
01)M +1
n+1)
n
2 (0 ) 0
N
=
(n
(01 2 )1 ( 01)
=
n
1)M +1;t
M +1
; n
=1
2 (0 )
1=2
N
0
01)01)M +1;t
((n +n
(01 2 )1 ( 01)
=
n
=1
1=2
2 (0 )
= 0:
M +1
; n
01)01)M +1;t
((n +n
Therefore, the rank of the matrix Z is N (M
0 1) at most.
APPENDIX III
PROOF OF (10)
On the first and the second relation of (10):
(straightforward calculation).
On the third relation of (10): From the relation
we have
M
j =1
j
P (j M )
=
M
p(M
j =1
=
M
j =1
=
M
j =1
= 1:
p(j
0 j)
0 M)
p(j )
M
j =1
j
P (j M )
= 1,
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