How can one understand entanglement when there is only a single particle?

Question

As per my knowledge, at least 2 particles are needed to show entanglement in the context of some property. But nowadays, mode entanglement is a well known phenomena which says that a single particle (say photon) can be entangled within its degrees of freedom. How can one understand this fact?

Answer 1

You do not need two particles to have entanglement. Entanglement is a property that any bipartite system can have. A (pure) bipartite system is entangled if it cannot be written as tensor product of other states. Bipartite here means that the state $\lvert\psi\rangle$ lives in a space of the form $\mathcal H_A \otimes \mathcal H_B$. What $\mathcal H_A$ and $\mathcal H_B$ represent, physically, strongly depends on the context.

In other words, one should always say that "this and this modes are entangled", instead of saying that "this and this particle are entangled". Indeed, even when there are two particles, the entanglement is between some of their properties (often their spins), not between the particles themselves.

This means that a state can be entangled if it has any kind of nontrivial tensor product structure. A few examples are:

1. A single photon (or any other particle with a spin) in a state like $\lvert0,H\rangle + \lvert1,V\rangle$, meaning that the photon is at some position (denoted here with $\lvert0\rangle$) with polarization $\lvert H\rangle$ and at some other position (denoted here with $\lvert1\rangle$) with polarization $\lvert V\rangle$. In this case $\mathcal H_A$ is the space of all possible spatial modes (positions) that the photon can occupy, while $\mathcal H_B$ is the Hilbert space of the polarization modes of the same photon.
2. A single photon in a superposition of two spatial modes (or any other kind of modes really), ignoring its polarization state. Even such a state can be considered as entangled if one considers the associated Fock space. In this case $\mathcal H_A$ is the space of possible occupation numbers of the first spatial mode, while $\mathcal H_B$ is the space of possible occupation numbers of the second spatial mode. Whether this state is to be considered as entangled is also directly addressed in (S. J. van Enk 2005).
3. The spin degrees of freedom of a pair of (say) electrons, when in a state like $\lvert00\rangle+\lvert11\rangle$.

An important point is however that certain kinds of entanglement are harder to exploit than others. In fact, from a fundamental perspective, basically any state is entangled. Take for example any photon. Its wavefunction will always be delocalised over a continuum of positions, therefore the corresponding state is always to be considered entangled. One does not however generally speak of "entanglement" in this context, simply because it is hard to do anything useful with this form of entanglement. If you cannot use it, there is no much use of talking about it.

On the other hand, the entanglement structure of a pair of entangled spins is (relatively speaking) very easy to manipulate and use, hence why it is more commonly used.