is quantum motion?
The question of how complex
quantum motion is of fundamental importance with deep
entanglement and decoherence.
In classical mechanics, complex systems are characterized by
positive Lyapunov exponent i.e. by local exponential
have positive algorithmic complexity and, in terms of the
dynamical description, almost all orbits are random and
In spite of many efforts, the problem of characterizing the
complexity of a quantum system is still open. Indeed the above
of complexity cannot be transferred, sic
et simpliciter, to quantum mechanics, where there is no
of trajectories. Still, a comparison between classical and
dynamics can be made by studying the evolution in time of the
and quantum phase space distributions, both ruled by linear
We have proposed [1,2] the number of harmonics of the Wigner
function as a suitable measure of the complexity of a quantum
recall that in classical mechanics the number of harmonics of
classical distribution function in phase space grows linearly
integrable systems and exponentially for chaotic systems, with
growth rate related to the rate of local exponential instability
classical motion. Thus the growth rate of the number of
harmonics is a
measure of classical complexity. Since the phase space approach
equally used for both classical and quantum
number of harmonics of the Wigner function appears as the
quantity to measure the complexity of a quantum state.
We have shown  that this measure, which reduces to the
well-known measure of complexity in classical systems and which
is valid for both pure and mixed states in single-particle and
many-body systems, takes into account the combined role of chaos
and entanglement in the realm of quantum mechanics. The
effectiveness of the measure has been illustrated in the example
of the Ising chain in a homogeneous tilted magnetic field. We
have provided numerical evidence that the multipartite
entanglement generation leads to an exponential increase with
time of the number of harmonics of the Wigner function until
saturation in both integrable and chaotic regimes. The growth
rate of the associated entropy can be used to detect quantum
phase transitions. The proposed entropy measure can also
distinguish between integrable and chaotic many-body dynamics by
means of the size of long-term fluctuations which become smaller
when quantum chaos sets in.
In this context, we have proposed  the Wigner separability entropy as a measure of complexity of a quantum state. This quantity measures the number of terms that effectively contribute to the Schmidt decomposition of the Wigner function with respect to a chosen phase space decomposition.
We have proved that the Wigner separability entropy is equal to the operator space entanglement entropy, measuring entanglement in the space of operators, and, for pure states, to twice the entropy of entanglement. The quantum to classical correspondence between the Wigner separability entropy and the separability entropy of the classical phase space Liouville density is illustrated by means of numerical simulations of chaotic maps. In this way, the separability entropy emerges as an extremely broad complexity quantifier in both the classical and quantum realms.
Growth of the number of harmonics in a chaotic system, both in
mechanics at different values of the effective Planck constant
 G. Benenti and G. Casati, How
is quantum motion?, Phys. Rev. E 79, 025201(R) (2009).
 V.V. Sokolov, O.V. Zhirov, G. Benenti and G. Casati, Complexity of quantum states and
reversibility of quantum motion, Phys. Rev. E 78, 046212 (2008).
 V. Balachandran, G. Benenti, G. Casati and J. Gong, Phase-space characterization of
complexity in quantum many-body dynamics, Phis. Rev. E
82, 046216 (2010).
 G. Benenti, G.G. Carlo and T. Prosen, Wigner separability entropy
and complexity of quantum dynamics, Phis. Rev. E
85, 051129 (2012).