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 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. Thus it is impossible to project it by measurement to a separable state, as discussions about entanglement always assume.

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 with a few local degrees of freedom each (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.

Realistic discussions about entanglement in quantum information theory usually consider the entanglement of spin/polarization degrees of freedom of photons transversally localized in different rays, electrons localized in quantum dots, etc.. For example, two electrons in two quantum dots dot1 and dot2 are in the antisymmetrized state
|psi1,psi2> = |psi1,dot1> tensor |psi2,dot2> - |psi2,dot2> tensor |psi1,dot1>
in 2-particle space (depending on 6 coordinates apart from the spins, the latter represented by psi1 and psi2), but as the positions are ignored (being just used to identify which electron is where), they are treated as effective electrons in the 2-particle state psi1 tensor psi2 residing in the tiny 4-dimensional tensor product C^2 tensor C^2. If spin up is measured at dot 1, the spin part psi1 collapses to up, etc.. Note that there is no position measurement involved in observing the spin, as the quantum dots already (and permanently) have measured the presence of a particle in each dot.

Thus in this small effective space, talking about electron entanglement is meaningful, as disregarding position (responsible for the antisymmetrizing) reduces the state space to a tensor product. But in the full 2-particle space, the attempt to reason with entanglement fails on the formal level and is at best paradoxical if just considered informally.

In the above, I took as only dof left the spins, as this is the situation that prevails in quantum information theory. However, the reduced Hilbert space can take other degrees of freedom into account. In place of the quantum dot we may have a bigger region that confines each electron. Then the electron has also spatial degrees of freedom. As long as these additional degrees of freedom are local to each dot (corresponding to a confining double well potential with infinitely high walls), one still has the tensor product structure, and hence a setting where entanglement makes sense, and in which one can consider to prepare an entangled 2-electron state. (If the walls are not infinitely high, such a representation is still approximately valid, as the tunneling probability is very small for sufficxiently high walls.)

To summarize: A discussion of entanglement of indistinguishable particles is possible for a pair of particles in a double well potential, but only in a reduced description that accounts of the fact that one particle is in each well and has to stay there because of the walls of the well. Thus only a small subset of all possible wave functions has physical relevance. Having a way to distinguish the particles means that one can restrict attention to a small tensor product Hilbert space inside the big Hilbert space of antisymmetric wave functions to (approximately or exactly, depending on the form of the well) describe the degrees of freedom left.

Arnold Neumaier (
A theoretical physics FAQ