How Are Entanglement and the Pauli Exclusion Principle Related?

Entanglement and the Pauli exclusion principle often appear together when discussing electrons, but they answer different questions. Pauli exclusion limits which states identical fermions may occupy. Entanglement describes whether a joint quantum state can be separated into independent states for its parts.

The connection comes from the exchange symmetry required for identical particles. That symmetry can link the particles’ spin and spatial states, and some familiar allowed states—especially the two-electron spin singlet—are entangled.

What each principle says

For identical fermions, exchanging the particle labels changes the sign of the complete wavefunction. For two particles with coordinates and spin labels represented together by $x_1$ and $x_2$, this means

$$\Psi(x_1,x_2)=-\Psi(x_2,x_1).$$

Electrons are fermions, so their joint states must obey this antisymmetry. If two identical fermions are put in the same one-particle state, antisymmetry makes the proposed two-particle state vanish. This is the mathematical basis of the exclusion principle.

Entanglement is a different property. For a pure state of two subsystems, the state is entangled if it cannot be written as a product of a state for one subsystem and a state for the other. For a mixed state, entanglement means that it cannot be expressed as a convex mixture of product states; a mixed state can fail to be a product while still being separable. The antisymmetry requirement may constrain the allowed states, but it does not by itself mean that every state is entangled in every useful sense.

Why spin and position must be considered together

For two electrons, the total state combines spatial and spin parts. If the spatial part is symmetric under exchange, the spin part must be antisymmetric, so that the total state is antisymmetric. If the spatial part is antisymmetric, the spin part must instead be symmetric.

The two-electron spin singlet is

$$|S=0,m=0\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow\rangle_1|\downarrow\rangle_2-|\downarrow\rangle_1|\uparrow\rangle_2\right).$$

It is antisymmetric under exchange. The three spin-triplet states are symmetric under exchange, so they must be paired with an antisymmetric spatial state. This requirement explains, for example, why two electrons with the same spin cannot both occupy the same spatial orbital.

Worked example: two electrons in one orbital

Suppose two electrons occupy the same normalized spatial orbital $\phi(\mathbf r)$. The spatial part of their state is

$$\phi(\mathbf r_1)\phi(\mathbf r_2),$$

which is symmetric under swapping the electron coordinates. To make the full state antisymmetric, the spin part must be the singlet. Thus the complete state is

$$\Psi=\phi(\mathbf r_1)\phi(\mathbf r_2)\frac{|\uparrow\rangle_1|\downarrow\rangle_2-|\downarrow\rangle_1|\uparrow\rangle_2}{\sqrt{2}}.$$

This is allowed: the electrons share a spatial orbital, but they do not occupy the same complete one-particle state. One has spin up and the other spin down in each term of the state.

The spin state is entangled. If you measure the spin of either electron, its reduced state is an equal mixture of up and down, while measurements of both spins are perfectly anticorrelated along the same axis. The joint state cannot be factored into an independent state for each spin.

A caveat about identical particles

The labels 1 and 2 in equations are usually bookkeeping labels, not tags that let you identify one electron permanently. As a result, statements about “entanglement between particles” require care for indistinguishable particles. In some settings it is more useful to discuss entanglement between spatial modes or other experimentally addressable subsystems. Exchange antisymmetry alone is not generally counted as useful entanglement; the answer depends on what subsystems and operations are physically available.

This caveat does not change the exclusion principle or the singlet’s spin correlations. It clarifies that entanglement depends on how the system is divided into subsystems, whereas the fermionic exchange rule applies to the complete state.

Common mistakes

• Treating Pauli exclusion as a ban on sharing a location. It forbids identical fermions from occupying the same complete one-particle state, not from being at the same position.
• Calling every antisymmetric state entangled. Antisymmetry is a particle-exchange requirement. Whether there is entanglement of interest depends on the chosen subsystems and the available measurements.
• Looking at spin or position in isolation. The total state must have the correct exchange symmetry; the spin part alone need not be antisymmetric in every allowed state.
• Assuming exclusion applies to all particles. The Pauli principle applies to identical fermions, such as electrons, not to bosons such as photons.