# Pauli exclusion principle and Entangled pairs

It is true for fermions in the same potential that the total wavefunction of two particles must be antisymmetric with respect to exchange of electrons. Which means the spin wavefunction is given by

$\chi=\frac{1}{\sqrt{2}}[\chi_+ (1)\chi_- (2)-\chi_+ (2)\chi_- (1)]$

which looks very much like the bell state,

$\beta_{11}=\frac{1}{\sqrt{2}}[ |01\rangle - |10 \rangle]$.

So, can we call those fermions, entangled states, as long as they are within the potential or there is something fundamentally special about entangled states (e.g. difference in measurement statistics) which makes them more unique?

Apologies if the question is too simple for the level of this website. However, apparently it has made a lot of confusion for many people!

• It is not necessarily true that this is the form of the spin wavefunction. The total wavefunction must be antisymmetric, yes. But you could have a state, for example, where the spatial wavefunction is asymmetric and the spin wavefunction is symmetric. – Greg P Jan 6 '11 at 23:10
• It's a good question, subject to what Greg just pointed out. – David Z Jan 6 '11 at 23:12
• Entangled pairs are usually spatially separated, right? I thought this is where the name comes from: you entangle (i.e. correlate) two different systems. While here you have just a superposition of states of one system. I don't think people call any old superposition entanglement. – Marek Jan 6 '11 at 23:24
• @Greg is right. To make his point complete, the shape of the spatial component often depends on angular momentum. See e.g. this discussion about approximate deuterium's eigenstates. – Marek Jan 6 '11 at 23:28
• @ Greg: I understand your point however, I just mentioned an instance in which they are very similar. – iii Jan 6 '11 at 23:33

## 3 Answers

if you replace "0" and "1" by "up" and "down", you get a similar state for two spins - which is referred to as the singlet. All these states are mathematically analogous except that the states "0" or "1", or "up" and "down", or "plus" and "minus" (as indices of your $\chi$) may mean physically different things - i.e. these states may influence the interactions of the system with other degrees of freedom differently.

For example, the spin "up" and "down" likes to add some $-\mu.B$ energy in a magnetic field that depends on the direction of the spin. Other degrees of freedom interact differently - and must be prepared by different apparata, depending on the context. At the level of "information", you always have two subsystems whose 1 qubit of information is correlated with the other in the same way; from the viewpoint of all physics, they can be very different things (just think about all the ways how qubits may be realized in quantum computers).

However, the state of the form $|01-10\rangle$ is always entangled: the quantum numbers of the two fermions (or subsystems) $1,2$ in the state are nontrivially correlated. This doesn't prove any interaction - it just proves that they were prepared to have correlated properties.

To see that the state is entangled, regardless of the symbols, note that it cannot be written as a tensor product of a state for the fermion or subsystem 1, multiplied by another state of the subsystem or fermion 2. Equivalently, you may trace over the 2 degrees of freedom, to get a density matrix for the subsystem 1. And you will get $\mbox{diag}(0.5,0.5)$ which has a nonzero entropy $\rho \ln(\rho)$, proving that the state isn't pure. Because the induced 1-particle state isn't pure, it proves that the original state of the two particles was entangled.

Almost all states in the multi-particle Hilbert space are entangled, of course. However, there are often reasons to assume that two systems are not entangled - because they didn't influence each other in the past (or at least not much).

Let's start from the definition of entangled state.

Briefly -- if the state of your system can be described by separately defining the states of its components, then we call the state of this system a separable state.

If such a description is impossible -- then the state is an entangled state.

Now, for both your examples it is impossible to factorize the states of individual particles in the description of the state of the total system. Therefore both of these states are entangled states.

Every state can be written in the way you mention, in terms of two states 1 and 2, for an appropriate choice of states 1 and 2. By itself it does not indicate any entanglement. What makes a state entangled is a specific property of the two states 1 and 2, namely that they are physical states belonging to two subsystems which do not interact with each other (for example systems spatially separated from each other). Only then it is interesting to talk about entanglement, which is roughly speaking the degree of correlation between the two states, which cannot be undone by operating on either one of the two subsystems separately.

• I'd be careful about saying it's only interesting to talk about entanglement when the participants are non-interacting; in fact, much interesting work is being done at the interface of condensed matter and quantum information in understanding the entanglement structure of strongly interacting many body systems. – wsc Jan 11 '11 at 3:11
• Good point, I’d say it is the degree of correlation between subsystems that is not an expression of their interaction. At the level of this discussion, I’d leave this subtlety to the footnotes. – user566 Jan 11 '11 at 3:15