What is actually being entangled?

Question

I read that even observation can result in entanglement so I assume it is true for any interaction but I like to know what physical properties can be entangled beside spin states? Can we entangle charge and mass too? If I were to entangled the spin of particle A and particle B then can I also entangle other physical property of particle A and a new particle C? Last but not least is it possible entangle their position states?

Answer 1

I read that even observation can result in entanglement so I assume it is true for any interaction but I like to know what physical properties can be entangled beside spin states?

The discipline of physics has many branches and theories, still based on the quantum mechanics we all learn in universities, have developed different models to illuminate different issues. In the process words are brought in which have an everyday meaning to describe strict quantum mechanical phenomena. I have found on this board that "entanglement" in quantum computing means different mathematical definitions than it means for me using it as a particle physicist.

I often answer with the every day use definition from webster :

a. to wrap or twist together : INTERWEAVE

In this definition every state of matter that can be described by a given quantum mechanical wavefunction is entangled by definition, since it is a probabilistic system. With this definition any conserved quantum number in an interaction of two particles is "entangled", for example, since if you know the overall quantum numbers , and you measure the quantum numbers of one of them, you must know the others, from energy, momentum, angular momentum, lepton numbers etc. conservation.

But I have found out that in the quantum computing world only spin is implied when using the term "entanglement".

So it depends on the context of the discussion.

The "observation can result in entanglement" statement seems to come from the wider definition.

Can we entangle charge and mass too?

If you can use the conservation laws. A nucleus is entangling mass and charge to first order, since the mass is a close multiple of the individual masses and the charge too.

See what I mean by context?

If I were to entangled the spin of particle A and particle B then can I also entangle other physical property of particle A and a new particle C?

If you can write a wavefunction for the three particles, and use quantum numbers why not?

Last but not least is it possible entangle their position states?

There is no conservation law for position states, but maybe in a quantum lattice? This I do not have an opinion on.

Answer 2

Any pair of degrees of freedom of any (quantum) system can, in principle, be entangled. Standard examples that come to mind are:

1. Entanglement between polarisation (or other internal degree of freedom, such as spin) of different particles. Think two photons in different spatial locations whose polarisations are entangled. You can get this e.g. via SPDC.

2. Entanglement between position of different particles. E.g. two photons which are in a superposition of being both "left" and both "right". Think of a pair of photons produced via SPDC and evolved each through a polarising beamsplitter.

3. Entanglement in the Fock states of different modes. This is a bit trickier to realise, but you can think of a single photon in a superposition of two modes as a case of entanglement between the occupation states of the two modes: $|1\rangle+|2\rangle\simeq|10\rangle+|01\rangle\equiv (a_1^\dagger+a_2^\dagger)|\text{vac}\rangle$ (this would only be usable if one is able to perform nonlinear measurements mixing vacuum $|\text{vac}\rangle$ and occupied states $a_i^\dagger|\text{vac}\rangle$).

4. Entanglement between different degrees of freedom of a single particle. E.g. a single photon whose polarisation state is entangled with its position, or OAM, or frequency.

5. Entanglement between degrees of freedom of different types of particles. What you can have e.g. in optomechanical systems.

Some of these cases are not what people often referred to as "entanglement", as they don't involve spatially separated particles. They nonetheless fit the definition of entanglement, in the sense of a state $\rho\in\mathcal H_A\otimes\mathcal H_B$ that cannot be written as convex mixture of products states $\rho_A\otimes\rho_B$.

If I were to entangled the spin of particle A and particle B then can I also entangle other physical property of particle A and a new particle C?

I'm not sure if this what you mean by this, but if $A$ is maximally entangled with $B$, it cannot be entangled with something else. Look up monogamy of entanglement.