International Summer School on Quantum Information

Even though the effect of a quantum measurement cannot be described within classical physics, relevant features of an entangled quantum state are accessible to semiclassical analysis. The description of quantum states in Hilbert space corresponds to the description of classical states as surfaces, or more generally, manifolds in phase space. Subsystems are related to specific subspaces of the Hilbert space, which in their turn correspond to lower dimensional phase spaces. An entangled state corresponds semiclassically to a surface that cannot be decomposed into a product of lower dimensional surfaces lying in these specific subspaces. A full description of entangled states can only be achieved in the language of density operators. The natural semiclassical theory then involves Wigner functions and their Fourier transform the chord function, or the quantum characteristic function. This extension is similar to that involved in proceeding from subspaces to product spaces. Indeed, we need the product of the {\it bra} space and the {\it ket} space, in the Dirac notation. The various representations of the density operators then result from alternative choices of allowed (Lagrangian) coordinate planes in double phase space. The partial trace of a pure state density operator over a subsystem generates the reduced density operator in the remaining subspace. This is a mixed state unless the original state is a product. The corresponding reduced Wigner function is obtained by projecting the original Wigner function, just as one would generate a marginal distribution from a higher dimensional probability distribution. The reduced chord function is a section of the original chord function. The purity of the reduced density operator, i.e. the square of its trace, is a good measure of entanglement, obtained by integrating the square of either the reduced Wigner function, or the square-modulus of the reduced chord function. The Wigner function can be associated to measurements of the eigenvalues, $\pm 1$, of the operators corresponding to reflections in phase space, i.e. generalized parity operators. These are wavelike properties which appear unintuitive if the quantum system is pictured as corresponding to a classical particle. It is thus shown that a deceptively {\it classical looking} state, prepared without violating full classical correspondence, is entangled: Its parity eigenvalues are as fully correlated as those in the classic example of Einstein, Podolsky and Rosen. The exact classical propagation of Wigner and chord functions is employed to follow explicitly the decoherence of a simple open system with an arbitrarily large environment, interacting via a quadratic periodic Hamiltonian. As well as entangling the subsystems, the initial state of the environment is averaged over. In the first iteration, the evolution of the open system agrees with a Markovian approximation, but, already for the second iteration, this only holds as a weak coupling limit.