--------------------------------------------
Indistinguishable particles and entanglement
--------------------------------------------

Commonly, outside of quantum field theory, there is a lot of 
imprecision in talking about indistinguishable particles, even in 
serious work.

In particular, many people regard indistinguishable particles as being 
intrinsically entangled. However, to talk usefully about entanglement, 
the systems that are considered to be entangled must be distinguishable 
(usually, by their position or momentum). 

Indeed, the common assumption in a formal definition of of entangled 
systems, here quoted from 
    http://en.wikipedia.org/wiki/Entangled_state  
is: ''The Hilbert space of the composite system is the tensor product''.
This is violated for indistinguishable bosons or fermions, where the 
Hilbert space of the composite system is the symmetrized and 
antisymmetrized tensor product, respectively.

The state space of indistinguishable particles is very different from 
that of distinguishable particles. Almost nothing particle-like 
survives this change of basic setting. In particular, the prerequisites 
for Alice and Bob doing the usual kinds of experiments on entanglement 
cannot be performed in principle if the particles involved are 
indistinguishable.

For example, two electrons are on the fundamental level 
indistinguishable; their joint wave function is proportional to 
|12>-|21>; no other form of superposition is allowed. Similarly, 
two photons are on the fundamental level indistinguishable; their 
joint wave function is proportional to |12>+|21>; no other form of
superposition is allowed. On the other hand, a photon and an electron 
are intrinsically distinguishable (e.g. by their spin), and arbitrary 
superposition are possible.
 

But if, on a more practical level, one knows already that (by 
preparation) there are two electrons or two photons, one moving to the 
left and one moving to the right, one can use this information to 
distinguish the electrons or photons: The particles are now described 
by states in the tensor product of two independent, much smaller 
effective Hilbert spaces (rather than a single antisymmetrized 
2-particle Hilbert space with degrees of freedom for every pair of 
positions), which makes them distinguishable effective objects. 
Therefore, one can assign separate state information to each of them,
and construct arbitrary superpositions of the resulting tensor product 
states. These effective, distinguishable particles can be (and 
typically are) entangled.