CENTRAL LIMITS OF SQUEEZING OPERATORS
L. Accardi
Dipartimento di Matematica
Centro Matematico V. Volterra
Universit`a di Roma II, Roma, Italy
A. Bach
Institut fur Theoretische Physik I
Universit¨at M¨unster, M¨unster, FRG
1
Contents
1 Introduction 3
2 The De Moivre-Laplace theorem in the 2-dimensional case 3
3 d-level systems: 1 copy 5
4 d-level systems: N copies 8
5 Inclusion of the number operator 10
6 Squeezing states 14
2
1 Introduction
In [1] we have proved a quantum De Moivre-Laplace theorem based on a
modification of the Giri-von Waldenfels quantum central limit theorem. In
[2] P.A. Meyer outlined a method based on direct calculations which, taking
advantage of the explicit structure of the algebra of 2 × 2 matrices, allows
a drastic simplification of the proof of the main result of the first part of
our paper and relates it with a similar result obtained, independently and
simultaneusly, by Parthasarathy [5]. In the first part of the present note we
simplify the Parthasarathy-Meyer method and extend it to deal with arbi-
trary d-dimensional Bernoulli processes, where d is a natural integer (cfr.
Sections (3),(4). We also prove another statement in Meyer’s note (cf. The-
orem (5.1)). Finally (Section (6) ) we show that the method of proof used
in [1] allows, with minor modifications, to solve the problem of the central
limit approximation of the squeezing states - a problem left open in [1] and to
which , due to the nonlinearity of the coupling, Parthasarathy-Meyer direct
computational method cannot be applied.
Acknowledgments.
Part of this paper was written while L.A. was visiting the University of
Strasbourg, continued while L.A. was visiting the Center for Mathematical
system theory in Gainesville and completed while A.B. was visiting the Uni-
versity of Rome II. The authors are grateful to P.A.Meyer for several usefull
comments. L.A. also acknowledges support from Grant AFOSR 870249 and
ONR N00014-86-K-0538 through the Center for Mathematical System The-
ory, University of Florida.
2 The De Moivre-Laplace theorem in the 2-
dimensional case
Throughout this paper we adopt the notations of [1] with the only excep-
tion that, in the first four secions, we use Meyer’ s normalization for the
N-coherent vectors and the N-Weyl opertors . Thus, in particular, ϕ
N
(0)
denotes the vacuum state in ⊗
n
C
2
; W
0
(z) = exp(zs
+
− zs) the Weyl op-
erator on C
2
; W
N
(z) = ⊗
N
W
0
(z/
√
N) the N − Weyl operator on ⊗
N
C
2
;
while Φ(0), W (z) = exp(za
+
−za) denotes the corresponding objects for the
3
harmonic oscillator. As usual
s
+
=
0 1
0 0
; s
−
=
0 0
1 0
; n
+
=
1 0
0 0
; (1)
ϕ
N
(0) = ⊗
N
e
1
; e
1
=
0
1
The main result of the first part of [1] is:
Theorem 1 For every natural integer k and for every k ∈, z
1
. . . , z
R
∈ C
lim
N→∞
< ϕ
N
(0), w
N
(z
1
) . . . w
N
(z
R
)ϕ
N
(0) >=< Φ(0), W (z
1
) . . . W (z
R
)Φ(0) >
(2)
uniformly for z
1
, . . . , z
R
in a bounded set.
Proof. For z
j
∈ C one has
< ϕ
N
(0), W
N
(z
1
) . . . W
N
(z
k
)ϕ
N
(0) >= (3)
=< ⊗
N
e
i
, ⊗
N
W
o
z
1
√
N
. . .⊗
N
W
o
z
k
√
N
⊗
N
e
i
>=< e
i
, W
o
z
i
√
N
e . . . W
o
z
k
√
N
e
1
>
N
=
X
(j
1
,...j
k
)
∈N
k
1
j
1
! . . . j
k
!
< e
i
,
→
Y
i=1,...k
z
i
√
N
s
+
−
z
i
√
N
s
−
j
i
e
1
>
N
=
n
∞
X
n=0
N
−n/2
X
(j
1
...j
k
)∈N
k
P
k
i=k
j
i
=n
1
j
1
! . . . j
k
!
< e
1
,
→
Y
i=1,...,k
(z
i
s
+
− z
i
s
−
)
j
i
e
1
>
o
N
Denote R
N
the right hand side of (3) with the sum in n starting from 3 rather
than 0. Then
|R
N
| ≤
∞
X
n=3
N
−n/2
P
j
1
,...,j
k
P
i
j
i
= n
1
j
1
! . . . j
k
!
k
→
Y
i=1,...,k
(z
i
s
+
− z
i
s
−
)
j
i
k≤ (4)
≤
∞
X
n=3
N
−n/2
X
1
j
i
! . . . j
k
!
| z
1
|
j
1
. . . | z
k
|
j
k
≤
2e
|z
1
|
. . . e
|z
k
|
N
3/2
= o
N
−1/2
4
For n = 0 there is only one term equal to 1; for n = 1 the contribution is
zero, since < e
1
, S
±
e
i
>= 0. The term n = 2 is equal to
1
N
P
(j
1
,...,j
k
)∈N
k
P
i
j
i
= 2
1
j
1
! . . . j
k
!
< e
1
,
→
Y
i=1
...k
(z
i
s
+
− z
i
s
−
)
j
i
e
1
>=
=
1
N
n
k
X
i=1
1
2
< e
1
,
z
i
s
+
− z
i
s
−
2
e
1
> +
X
i<j
< e
1
,
z
i
s
+
− z
i
s
−
z
j
s
+
− z
j
s
−
e
1
>
o
=
=
1
N
1
2
k
X
i=1
−z
i
z
i
+
X
i<j
−z
j
z
i
!
In conclusion we obtain
< ϕ
N
(0), W
N
(z
1
), . . . W
n
(z
k
)ϕ
N
(0) >
=
(
1 +
1
N
"
−
1
2
k
X
i=1
|z
i
|
2
−
X
i<j
z
i
z
j
#
+ o(N
−1/2
)
)
N
(5)
and the algebraic identity
−
1
2
X
i
|z
i
|
2
−
X
i<j
z
i
z
j
= −
1
2
X
i
z
i
2
− i Im
X
i<j
z
i
z
j
shows that the right hand (side of (5) converges to the right hand side of (2).
The uniformity of the convergence is obvious in view of (4).
3 d-level systems: 1 copy
Our strategy to deal with d-level systems is to exploit the isomorphism be-
tween the symmetric tensor product (⊗
d−1
C)
+
of d-1 copies of C
2
and C
d
,
which intertwines the Lie algebra representation of SO(3; R) given by
S
±
=
d−1
X
k=1
j
k
(s
±
) ,
d−1
X
k=1
j
k
(s
3
) = S
3
5