  Classical surfaces in phase space correspond to quantum states in Hilbert
space. Subsystems specify factor spaces of the Hilbert space. An entangled
state corresponds semiclassically to a surface that cannot be decomposed into a
product of lower dimensional surfaces. Such a classical factorization never
exists for ergodic eigenstates of a chaotic Hamiltonian. The space of quantum
operators corresponds to a double phase space. The various representations of
the density operator then result from alternative choices of allowed coordinate
planes. In the case of the Wigner function and its Fourier transform, the chord
function, or the quantum characteristic function, this is a phase space on its
own. The reduced Wigner function, representing the partial trace of a density
operator over a subsystem is the projection of the original Wigner function;
the reduced chord function is obtained as a section. The purity of the reduced
density operator, the square of its trace, is a measure of entanglement,
obtained by integrating either the square of the reduced Wigner function, or
the square-modulus of the reduced chord function. Bell inequalities for general
parity measurements can be violated even for classical looking states with
positive Wigner functions that have evolved classically from product states.
These include the original EPR state. Entanglement with an unknown environment
results in decoherence. An example is that of the centre of mass of a large
number of independent particles, entangled with internal variables. In this
case, the Central Limit Theorem for Wigner functions leads to some aspects of
Markovian evolution for the reduced system.
