Why do Bell states have all real coefficients?

Question

Canonically, the four Bell or EPR states for 2-qubit systems are given by:

$|\Phi^{\pm}\rangle = \frac{1}{\sqrt{2}} \left( |00\rangle \pm |11\rangle\right)$

$|\psi^{\pm}\rangle = \frac{1}{\sqrt{2}} \left( |01\rangle \pm |10\rangle\right)$.

I'm wondering why is it that we never mention states such as $|\psi\rangle = \frac{1}{\sqrt{2}} \left( |00\rangle + i|11\rangle\right)$? That state is also entangled, and it is maximally entangled (if you take either partial trace, you get the identity matrix, thus maximally mixed). Is there a reason why we don't call those "Bell states", or is it simply the case that there is no unique way of defining them?

Answer 1

I would call $|\psi\rangle = \frac{1}{\sqrt 2} (|00\rangle + i|11\rangle)$ also a Bell state due to the properties that you have already listed. There is nothing really special of $|\Phi^+\rangle$ over $|\psi\rangle$.

Though, it is easier to calculate with real coefficients and the four standard Bell states do form a orthonormal basis of the 2-qubit Hilbert space. (But you can also construct an ONB with maximally entangled states with non-real coefficients, of course.)

Answer 2

It's because we can get away with it. It's always easier to deal with only real coefficients.

For dimension $d$ there is a very simple and convenient construction of generalized Bell states: $$|\Psi_{ij}\rangle = (I \otimes X^iZ^j)|\phi^+\rangle,$$ where $X,Z$ are the $d$-dimensional shift and clock operators, and $|\phi^+\rangle = \frac1{\sqrt d}\sum_{i=0}^{d-1}|ii\rangle $. Note that they always form an orthonormal basis and are maximally entangled, but for $d \ge 3$ they have complex coefficients.

First I thought this was necessarily so, but as Craig Gidney pointed out one can build a real Bell basis for power of 2 dimensions simply by taking tensor products (and permuting subsystems) of the $d=2$ case. It is unclear, however, whether it exists in all dimensions.

Any maximally entangled state can be written in the form $(I \otimes U) |\phi^+\rangle$ for some unitary $U$, and the inner product of two states in this form is given by $\operatorname{tr}(V^\dagger U)/d$ (if they are built with unitaries $U,V$). Therefore finding a real Bell basis in dimension $d$ reduces to finding a set of $d^2$ real unitaries $\{U_i\}$ such that $\operatorname{tr}(U_i^\dagger U_j) = d \delta_{ij}$.