VOLUME 77, NUMBER 5 PHYSICAL REVIEW LETTERS 29 JULY 1996
Quantum Chaos in Terms of Entropy for a Periodically Kicked Top
Robert Alicki, Danuta Makowiec, and Wieslaw Miklaszewski
Institute of Theoretical Physics and Astrophysics, University of Gdan ´sk, Wita Stwosza 57, PL 80-952 Gdan ´sk, Poland
(Received 1 December 1995)
A recently developed approach to the quantum dynamical entropy for classical and infinite quantum
dynamical systems involving von Neumann entropy for multitime correlation matrices is adapted
to finite quantum systems and implemented numerically for a periodically kicked top. Significant
quantitative differences between chaotic and regular regime are observed. [S0031-9007(96)00816-2]
PACS numbers: 05.45.+b, 03.65.Sq, 65.50.+m
“While the idea of an entropy is of great help in
understanding classical mechanical systems, nobody has
been able to find its analog in quantum mechanics; therein
lies the great unresolved mystery of quantum chaos”
(Gutzwiller [1], p. 143).
In this Letter we argue that the recently developed
approach to the dynamical entropy for infinite quantum
systems (e.g., systems in the thermodynamical limit) and
classical systems [2] can be adapted to describe significant
differences between regular and chaotic regime in finite
quantum systems [8].
We recall briefly the construction of the Kolmogorov-
Sinai (KS) entropy for classical dynamical systems. The
reversible dynamical map T acts on the phase space
M, leaving invariant the certain probability measure m.
Any initial finite partition E E
1
, E
2
,..., E
l
of M into
disjoint sets evolves in (discrete) time, producing finer
and finer partitions of the form E
j
1
> T E
j
2
> ··· >
T
n21
E
j
n
; j
1
,..., j
n
1, 2, . . . , l . The KS entropy (dy-
namical entropy) is now defined as
h
KS
T sup
E
lim
n!`
μ
2
1
n
X
j
1
,...,j
n
mE
j
1
> T E
j
2
3 > ··· > T
n21
E
j
n
ln mE
j
1
> T E
j
2
3 > ··· > T
n21
E
j
n
∂
, (1)
where the supremum is taken over all finite initial par-
titions. Dynamical systems with h
KS
. 0 are called
chaotic, and for M being a finite dimensional manifold
(and under some conditions) the following relation be-
tween dynamical entropy and Lyapunov exponents l
a
(which measure the exponential divergence of trajectories)
holds:
h
KS
T
X
l
a
.0
l
a
. (2)
It is obvious that the above definition so crucially depen-
dent on the notion of phase space and its transformations
cannot be directly translated into the quantum domain.
One needs a new one which involves such general mathe-
matical structures such as algebra of observables, states
as positive normalized linear functionals, and dynamics as
an automorphism of observables, which are common for
both quantum and classical theory.
We begin with the basic mathematical framework treat-
ing classical and quantum systems on the same footing.
Let X X
1
, X
2
,..., X
l
be a set of (generally complex)
observables satisfying the following normalization condi-
tion:
l
X
i 1
X
i
X
i
1 . (3)
We call such a family of observables a partition of
unity. In the classical case, X
i
’s are complex-valued
functions and the asterisk denotes complex conjugation.
In the quantum case, X
j
’s are operators acting on the
Hilbert space of the system with the asterisk being
the Hermitian conjugation. The discrete time dynamics
in the Heisenberg picture is given by a linear map
Q acting on observables: X ! QX . For classical
systems, Q is generated by the dynamics T on the phase
space: QX j X T
21
j, while for quantum ones it
is defined in terms of the unitary operator U such that
QX U
XU. We also need a fixed time-invariant
reference state given in the classical case by a T -invariant
probability measure m on the phase space. For a quantum
system we take as a reference state the density matrix r
commuting with U. The mean value of the observable
X at this state will be denoted by X . For a classical
case X
R
X jmdj, while for a quantum system
X tr rX .
For every evolution step n 1, 2, . . . , we construct a
multitime correlation matrix
V
n
i
1
,...,i
n
;j
1
,...,j
n
X
j
1
QX
j
2
··· Q
n21
X
j
n
3Q
n21
X
i
n
··· QX
i
2
X
i
1
,
i
k
, j
k
1, 2, . . . , l , k 1,..., n . (4)
Due to the condition (3) V
n
can be treated as a
positively defined, l
n
3 l
n
complex-valued matrix with
a trace equal to one. Having the eigenvalues of V
n
p
i
1
,...,i
n
$ 0;
P
p
i
1
,...,i
n
1, we can calculate the von
838 0031-9007 96 77(5) 838(4)$10.00 © 1996 The American Physical Society