# Mechanism of quantum entanglement; proof of quantumness

A system is said to be entangled if its state cannot be expressed as a product of states. (i.e $\psi_{AB}\neq\psi_A \psi_B$) Physically it is a direct result of conservation of energy/momentum, but what (quantum) mechanical procedure(s) take a particle or particles and create an entangled state, and how would this procedure confirm that a force is quantum in nature?

This question comes from an experiment designed to test whether gravity is in fact a quantum force; see https://www.quantamagazine.org/physicists-find-a-way-to-see-the-grin-of-quantum-gravity-20180306/. The article makes me think that the particle(s) to be entangled interact through some fundamental force, and if this force is quantum then the interaction may cause a superposition of states that may be entangled, otherwise the interaction will be deterministic and no superposition may occur; however, I could not find some analysis of that statement and it thus seems non sequitur.

UPDATE:

Thanks to all that have answered. Each answer (so far) has given me a more clear understanding of how states are entangled; however, they are missing the second part to the question, perhaps because I was not clear enough:

From the answers I see that an entangled state is created via some operation/evolution (such as CNOT) but if the force in question (i.e. gravity) is not quantum in nature does it imply that such an operation does not exist, and if not, may a mathematical proof be provided?

• Your claim that "Physically [entanglement] is a direct result of conservation of energy/momentum" is incorrect. There's many types of entanglement, and only a few can be traced to any sort of conservation law. This is an understandable misconception which can be directly traced to many misleading popular expositions of entanglement, but it's still wrong. – Emilio Pisanty Jul 9 '18 at 20:37
• @EmilioPisanty, could you elaborate? My current understanding of entanglement is strictly mathematical in the two-particle case; I don't know of any type of entanglement. – user195162 Jul 9 '18 at 22:49
• Is there a clear-cut definition what a "classical force" is? Otherwise, I'd say that if a force creates entanglement, it cannot be quantum, pretty much as a definition. Thus, if it creates entanglement, it must necessarily be quantum. – Norbert Schuch Jul 10 '18 at 0:50
• @EmilioPisanty, please point out some types of entanglement that cannot be traced to conservation laws. – S. McGrew Jul 13 '18 at 17:14
• @S.McGrew That probably deserves its own thread. – Emilio Pisanty Jul 13 '18 at 17:28

It's very hard to see what exactly you're asking so I want to provide a bit of a broader context: Entanglement is not a second-class citizen of the country that is quantum mechanics. She does not need permission to appear, or "special forces" to create her, or any such thing. In fact I will go one step further to say, we wish she did! If we could limit her scope of appearance, so that she only appeared when we performed some rare sorcery, we would have a quantum computer by now that could do more than prove that with high probability, fifteen is three times five.

She is the twin sister of coherence, the “waviness” that we started QM in order to explain in the first place. In fact they never appear in the same location at the same time, so much so that many physics students expect us teachers to dramatically reveal that they were the same all along, that coherence is really “self-entanglement” or some such, and this is why when you entangle two systems their internal coherence disappears. (I don’t know how to define this well enough to satisfy them with a theorem.)

Coherence says that a quantum bit can be in a superposition of states, $a|0\rangle + b |1\rangle$. If you have two bits, one of which is in the above state, the other is just known to be $0$, then the joint state of the whole system is$$\big(a|0\rangle + b |1\rangle\big)\otimes\big(|0\rangle\big)=a|00\rangle + b |10\rangle.$$ Now almost any interaction which couples these two bits will entangle them, even if the interaction itself has a purely classical description. For example the interaction $(u,v)\mapsto(u, u\text{ xor } v)$ is a typical classical computer operation that can be realized I quantum mechanics as the unitary transform\begin{align} |00\rangle&\mapsto|00\rangle\\ |01\rangle&\mapsto|01\rangle\\ |10\rangle&\mapsto|11\rangle\\ |11\rangle&\mapsto|10\rangle\\ \end{align} And in QM we call this the “CNOT” gate. It has a completely classical description but when you happen to apply it to the above state which has a coherent bit for $u$, you will find the entangled state, $$\operatorname{CNOT}_{1\to 2}\Big(\big(a|0\rangle + b |1\rangle\big)\otimes\big(|0\rangle\big)\Big)=a|00\rangle + b |11\rangle.$$ That state is entangled. Notice how entanglement in this notation “looks like” the coherence looked, that $a|x\rangle+b|y\rangle$ pattern.

What has happened is a sort of failure of transitivity, or something: we have three things playing together: classical reversible logic gates (in this case xor), composing two systems and decomposing them later (the role of $\otimes$), and coherence. By itself, logic gates play well with composition: this is how computers have scaled to terabytes. By itself coherence plays nice with composition. Logic gates by themselves play nice with coherence. But you put all three together and you discover that they don’t all play perfectly well together: you may start with a composition and then be unable to cleanly decompose it into $|x\rangle\otimes|y\rangle$ after.

So the problem in general with our quantum computers, is actually exactly of this form. Our 0 and 1 bits, have some sort of different physical interaction with the atoms around them. In other words, the bits are not fully isolated from the atoms around them, and each bit’s interaction with its environment is slightly different depending on whether the bit is in the zero state or the one state.

Quantum mechanics says that that's a very bad thing. Like, QM doesn't make value judgments directly: but our goal was to make the bits coherent, and any entanglement with any outside environment--including any atoms that we don't immediately have under control--reduces the coherence of the system, and transfers it to the entanglement of the system plus environment. So QM translates any differential interaction, no matter how classically describable, to entanglement and then it tells us that this will make it progressively harder to achieve the goal we set for ourselves as these little entanglements sap away more and more of our coherence.

The article makes me think that the particle(s) to be entangled interact through some fundamental force, and if this force is quantum then the interaction may cause a superposition of states that may be entangled.

Here's a good answer explaining what entanglement is.

Here's a question I asked, asking what types of Hamiltonians generate entanglement. Putting it in your language, an interaction Hamiltonian would be the "quantum force" you are looking for. (technically this entangling-interaction in the form of energy. If you want it to be called an entangling "force" instead of entangling energy, you'd have to go from "potential energy" to force - which is basically just finding a derivative.)

What is an "entangling force"? When you have two quantum systems and you they can now interact, this will create entanglement between them. As explained here, the basically any type of interaction will create entanglement.

Is there something "quantum" about certain "forces"? While in classical physics forces are usually the starting point, typically in quantum we use "generating functions" which create unitary operators. The specialness of these unitary operators that their quantum state can't "leak out" in some unknown or undetectable channel (like heat lost in friction). Interestingly, these "not-quantum" systems that DO leak information out actually also create entanglement. The quantum information by leaking out to the environment is interacting with the system, this (as said above) creates entanglement between the two subsystems (the state we're interested in, and the environment subsystem it's interacting with). Most physicists believe that all complete systems are these non-leaky unitary ones, and that we only get things that don't appear to preserve quantum features like superposition when they lose information to the environment.

Entanglement generation can be very simple. The simplest example of entanglement I can think of is a beamsplitter. If you put a photon through a beamsplitter, you can generate the state $\frac{1}{\sqrt{2}}(|1\rangle |0\rangle + |0\rangle |1 \rangle)$ This cannot be written as a product of two individual states, and is therefore an entangled state. So the process of creating entanglement can actually be very simple.

Let me illustrate this with a simple process that can generate entangement. Consider two spin 1/2 particles all aligned in the $x$ direction, i.e. they are in the state $|\rightarrow\rightarrow\rangle$. Now, starting at time $t=0$, apply to the system with the Ising Hamiltonian (if you like a more vivid picture, just think of suddenly put the two particles close enough to each other, this process is called quenches) $$H= J\sigma^1_z\sigma^2_z.$$ The eigenstates of the Hamiltonian are $|\uparrow \uparrow\rangle$, $|\downarrow \downarrow\rangle$ (with eigenvalue $J$) and $|\uparrow \downarrow\rangle$, $|\downarrow \uparrow\rangle$ (with eigenvalue $-J$). However, the original state is $|\rightarrow \rightarrow\rangle=\frac{1}{2}(|\uparrow \uparrow\rangle+|\downarrow \downarrow\rangle+|\uparrow \downarrow\rangle+|\downarrow \uparrow\rangle)$. Therefore, after some time $t\neq 0$, the state will evolve to $$\frac{1}{2}\exp(-itJ/\hbar)(|\uparrow \uparrow\rangle+|\downarrow \downarrow\rangle)+\frac{1}{2}\exp(itJ/\hbar)(|\uparrow \downarrow\rangle+|\downarrow \uparrow\rangle)$$ which is an entangled state.

I do not know if you are familiar with the math or not. But if you are not, the short answer is that almost any "interaction" can introduce entanglement. And humans believe that any fundamental force is quantum/interacting in nature. Therefore, they can all generate entanglement in a process that is more or less similar to the one stated above

In answer to the last part of your question: gravity, whether fundamentally quantum mechanical or not, can cause entanglement. Almost anything that can be done to a beam of photons by refraction or diffraction can be done gravitationally on a cosmic scale. So, an entanglement experiment using lasers and refractive and diffractive optical elements will provide the same results if recreated using gravitating masses and cosmic distances.

• If this is true, then wouldn't the aforementioned experiment give no information whatsoever into the nature of gravity? Here is the link to one of the papers (I don't have access) : journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.240401 – user195162 Jul 14 '18 at 20:31
• You can download the paper for free at [arxiv.org/pdf/1707.06050.pdf]. I think you're right that the cosmic-scale entanglement experiment does not give any information about the quantum nature of gravity. It merely indicates that gravity does not need to be fundamentally quantum mechanical in order to cause entanglement between photons. The experiment proposed in the paper is a very different experiment and might possibly, as the authors suggest, reveal something about the quantum nature of gravity. – S. McGrew Jul 14 '18 at 21:10