First of all, protons and anti-protons carry half-integer spin, so the first statement is wrong. However the value of the spin doesn't have anything to do with whether they can be entangled or not. Whatever the value of the spin, particles of different spin components can be defined in entangled states, because it's possible to define operators that flip a component of the spin.

So it is possible to define a space of states, where evolution starts in a state where a spin component taking the value +1/2 (in units of Planck's constant) and ends in a state where it takes the value -1/2 and the space of intermediate states is described by linear combinations of these states. The same reasoning applies to states of integer spin.

The reason it's not possible to entangle a proton and an anti-proton is that a ``charge flipping'' operator doesn't exist, because electric charge is always conserved during evolution. It's not possible to start with a state of a proton and evolve it into a state of an anti-proton, along with the linear combinations that would describe the intermediate states, which would describe a proton entangled with an anti-proton. (Incidentally, one other reason is that a proton carries baryon number, opposite that of an anti-proton and baryon number is, also, globally conserved. Though it might not be by certain processes, electric charge is always conserved.)

We need to go beyond Standard model and look for violations of certain symmetries. In the case of beta decay, parity is violated under Standard model. Let us collaborate to identify symmetry elements which are violated when we go beyond Standard Model.

Whatever the physics beyond the Standard Model is, it will not change the fact that electric charge is a Lorentz invariant quantity, so it is, simply, meaningless to discuss transitions between a proton and an anti-proton. They're not possible, which implies that any linear combination of a state of a proton and a state of anti-proton can't be entangled. The evolution operator is block-diagonal in that basis.

Principles of Nonlinear Dynamics could be helpful in certain situations.

The Law of Charge conservation is violated under Majorana Dynamics.

Optical simulation of charge conservation violation and Majorana dynamics ROBERT KEIL, 1,2 CHANGSUK NOH, 3 AMIT RAI, 3,4 SIMON STÜT and others

Optica, Vol. 2, No. 5 / May 2015

I am working on setting up experiments which would fulfil goal to prepare entangled states in which both proton and antiproton participate.

yes that's real in CEREN

Hi,

In fact, there are some studies for both entanglement and polarization in Baryon-Antibaryon.