# Is conservation of charge valid in QM?

In electromagnetism, we have the continuity equation

$$\frac{\partial}{\partial t} \rho(\vec{r},t) = - \vec{\nabla} \cdot \vec{j}$$

which, upon integration over a finite volume $$V$$ bound by surface $$S$$ yields

$$\frac{d}{dt} \int_V \rho(\vec{r},t) = -\int_S \vec{j} \cdot d\vec{S}$$

The interpretation usually given of this equation is that the charge lost in any finite volume $$V$$ is accounted for in the flow of charge leaving the surface. Thus, the global conservation of charge law can be made stronger, we require any physical process to satisfy a local conservation charge as well. An implication of this is that charge cannot disappear and reappear someplace else, i.e. teleport. However, I know that one can teleport quantum information if one shares entanglement. I wonder if there is a way to teleport charge as well. One reason I suspect this may be possible is that there is nothing in the continuity equation explicitly forbidding this, since the equation assumes the charge $$\rho$$ is continuous so it doesn't actually apply on a quantum scale. On the other hand, I suspect it may not be possible since any such protocol would have to destroy a particle on one end a quantum network and create it at the other, so some mass would have to be transferred as well (maybe?), which is distinct from a usual teleportation where there is no mass transfer.

I think my question can be made more clear with an example. In the CHSH or magic square or whatever quantum game you like, the use of entanglement allows the players Alice and Bob to share information. In particular, this information is encoded in, say, the spin of the entangled electrons they both hold. Thus in a sense we can say that they send each other (with high probability) the spin of their particle and use that to glean information about what measurement they made on their system. In a sense, Alice sent Bob the spin of electron after a measurement. So fundamental properties of elementary particles can be transmitted. Is there a way to send information about charge in a similar fashion that would seem to violate local conservation of charge?

• Comments are not for extended discussion; this conversation has been moved to chat. Jul 24, 2020 at 19:39

Contrary to what science fiction may tell us, quantum teleportation does not involve the physical teleportation of matter. It only teleports the state of matter. In that sense, it teleports the information that is associated with a particle and not the particle itself. So, electric charge remains where it is.

In response to the comment: the spin is a degree of freedom that can change and therefore it is part of the state of a particle. Charge, on the other hand is a fixed property of a particle and does not change. Therefore, one cannot teleport it.

• I understand that science fiction is not real. Read my edit for more clarification on the question. Jul 23, 2020 at 3:35

The entanglement is the coupling of different quantum systems. The experimenters bring entangled systems to their own journies. And when a quantum system is measured, the other quantum system is collapsed at the same time because they are entangled. While this process, no material is transferred.

It is the non-locality of the world which makes this possible and it doesn't involve any material transfer. So we don't need to worry about the charge conservation during this experiment.

And contrary to your belief, no information is transferred during this process. If it does, it will violate the relativistic causality. And if the causality can be violated, you can send information to your own past which doesn't make any sense.

In the quantum teleportation experiment, the quantum state of Alice is sent to Bob. The goal is to copy Alice's qubit state at Bob's site. But the information is transferred through the classical channel. The quantum channel is merely to share the entangled state and when the whole system is represented with Bell state, the entangled state in Bob's side simulates Alice's qubit state by a phase difference. After Bob gets the information from Alice through the classical channel, Bob does a unitary operation for the phase difference to his quantum state to copy the quantum information sent by Alice. (The copy is not perfect though because it violates no-cloning theorem.)

In summary, the information is sent by the classical channel and the quantum channel is only a tool to make the copy of the quantum state easier. And this doesn't violate the charge conservation.

• I am not wondering if entanglement sends charge. I know it does not. I am wondering if there exists or could exist a (not necessarily analogous) protocol to the sending of spin that could send charge. Jul 23, 2020 at 3:37
• @Eulerian I added the argument explaining no information transfer. Jul 23, 2020 at 3:40
• @kevin012 There are protocols that make use of entanglement + transfer of classical information to perform the transfer of quantum information. This, does not violate causality and I believe is what Eulerian refers to as "information teleportation" Jul 23, 2020 at 3:47
• @LucasBaldo give me some link. Let me have a look. Jul 23, 2020 at 3:51
• Jul 23, 2020 at 3:56

Elementary particle physics view:

The underlying level of nature is quantum mechanical and its behavior is encapsulated in the standard model of particle physics..

Charge is a conserved quantized quantity always carried by the charged particles in the table during interactions, and the probability of interaction is calculated using quantum field theory and Feynman diagrams to define the integrals to be calculated. The value of the charge is in multiples of 1/3 ( quark charges) or it is an integer number, the unit being the electron charge

This is the state of matter from which all macroscopic observations arise.

I wonder if there is a way to teleport charge as well

As all the other conserved quantum numbers it might be used in quantum entanglement problems, for example if a gamma ray creates an $$e^+e^-$$ pair and you manage to measure the $$e^+$$ you know that the one that escaped is the $$e^-$$, and depending on the experiment what its energy is. That is all.