Are the particles that are bounded by chemical bonds or nuclei joined by nuclear forces entangled or are they pure states? In addition to that, are the subatomic particles in the atoms and nuclei, excitations of a single electron field, quark field, entangled or are just pure states in general when measured? That is, if we don't measure anything, usual postulates of quantum mechanics say that states are in linear superposition, but are they mixed, pure or entangled states in general?

  • Entanglement is a property of interacting identical particles, so yes all of your examples are entangled – Lewis Miller 2 days ago

First, here's some clarification about the vocabulary: The overall state of a system can be pure even if its parts are entangled with each other. The state of part of a system is called mixed (rather than pure) if that part is entangled with the rest of the system.

In general, when two or more particles are in a bound state, their properties are entangled with each other. Here are a few examples:

  • In the lowest-energy state of a hydrogen atom, the spins of the electron and proton are entangled with each other. To be specific, they are in the superposition $$ |\psi\rangle\sim \big|\uparrow\,\downarrow\big\rangle - \big|\downarrow\,\uparrow\big\rangle \tag{1} $$ where the first arrow indicates the spin-direction of the electron and the second arrow indicates the spin-direction of the proton. (Reference: Griffiths, Introduction to Quantum Mechanics, section 6.5, "Hyperfine splitting".) For simplicity, I'm only showing the spin degrees of freedom here.

  • Positronium is a short-lived bound state of an electron and a positron (anti-electron). In positronium the two particles form a two-particle "orbital" around their center of mass. Schematically, the wavefunction for this orbital is a spherically-symmetric superposition of configurations in which the electron and positron are on opposite sides of the center of mass, like this: $$ |\psi\rangle\sim \sum_r f(r)\sum_\mathbf{u}\ \big|r\mathbf{u},-r\mathbf{u}\big\rangle \tag{2} $$ where the "sum" (actually an integral) is over all unit vectors $\mathbf{u}$ from the center of mass, and the first and second factors represent the electron and positron coordinates, respectively. The function $f(r)$ is a radial wavefunction. (I'm being very schematic here, to highlight the entanglement.) Because of the sum over $\mathbf{u}$, the locations of the electron and positron are entangled with each other.

  • Pions are short-lived spin-0 particles made of (roughly speaking) a quark and an anti-quark. Pions can be electrically neutral or electrically charged. The electrically neutral version is a superposition of $|u\bar u\rangle$ and $|d\bar d\rangle$, where $u$ is an up quark, $d$ is a down quark, and an overhead bar indicates an antiparticle. In words, the neutral pion is entangled with respect to quark flavor.

These are just a few examples, and they are typical. Notice that in each of the preceding examples, I was careful to say what properties are entangled with each other.

When talking about a system with multiple identical particles, like a helium atom with two electrons, or a hydrogen molecule ion with two protons bound together by an electron, we need to be specific about what we mean by "entanglement". The wavefunction for two identical spin-0 bosons must be symmetric with respect to their locations: $\psi(x_1,x_2)=\psi(x_2,x_1)$. For example, the wavefunction may have the form $$ \psi(x_1,x_2)=f(x_1)g(x_2)+f(x_2)g(x_1). $$ In this sense, we could say that their locations are always entangled with each other, just because they are bosons. A more interesting case is a wavefunction of the form $$ \psi(x_1,x_2)=f(x_1)f(x_2)+g(x_1)g(x_2), $$ where $f$ and $g$ are localized in two widely separated locations. In this case, we can say that the locations of the particles are entangled whether or not they are bosons. This is a superposition of "two particles here and no particles there" with "no particles here and two particles there".

In the quantum computing / quantum information literature, people often talk about the overall entanglement between "subsystems" rather than talking about entanglement beteween specific sets of properties of those subsystems. The latter concept is more general, and this more-general version is what I had in mind when writing this answer.

(An example of how to quantify entanglement with respect to a given set of observables is described in another post.)

Quantum systems become entangled through interaction with each other. Entanglement is broken when the entangled particles decohere through interaction with the environment; for example, when a measurement is made.

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