  Quantum mechanics of composite systems, gives rise to certain special states
called entangled states. A physical system, that is in an entangled state
displays an intricate correlation between its subsystems. There are also some
composite quantum states (classically correlated states or separable states)
that are not entangled. It is generally claimed, often without a rigorous proof
to support, that these intricate correlations of an entangled state cannot
occur in a classical system. This expository article, provides an elementary
proof that entangled states cannot arise in the setting of classical mechanics.
In addition, a detailed description of the origin of entanglement in quantum
systems is included. The mathematical concepts that are necessary for this
purpose are presented. The absence of entanglement in the classical setting is
due to the fact that every pure classical state of a composite system is a
product state, that is, a tensor product of two pure states of the subsystems.
In contrast, there are pure composite quantum states that cannot be expressed
in the form of a product state or even by a convex sum of product states.
Roughly speaking, this is because classical states are positive valued
functions on the phase-space while quantum states are positive linear
operators. The structure of the tensor product between two commutative spaces
of scalar valued functions is drastically different from that of the tensor
product between two non-commutative spaces of linear operators. In other words,
entanglement is a non-commutative phenomenon.
