# Uniformly distributed state in the Weyl basis

The Weyl basis (also known as Weyl-Heisenberg) is an orthonormal, unitary, and non-Hermitian basis for the Hilbert space of dimension $$d$$. The basis elements are given by $$U_{ab} = \sqrt{\omega^{ab}}X^aZ^b,$$ where $$X,Z$$ are the shift and clock matrices and $$\omega = e^{i2\pi/d}$$. I'm wondering whether for all $$d$$ there exists a pure state $$|\psi\rangle$$ such that $$|\langle\psi| U_{ab}|\psi\rangle|$$ is constant (except of course for $$U_{00}$$).

Expanding the hypothetical state in this basis, $$|\psi\rangle\langle\psi| = \frac1d \sum_{ab} \alpha_{a,b} U_{ab},$$ we see that demanding it to be Hermitian translates to $$\alpha_{a,b}\alpha_{-a,-b} = 1$$ (note that these coefficients are in general complex), have trace one results in $$\alpha_{0,0} = 1$$, and have trace squared equal one results in $$|\alpha_{a,b}| = 1/\sqrt{d+1}$$ for all other $$a,b$$.

Simply setting $$\alpha_{a,b} = 1/\sqrt{d+1}$$ does give me the desired state for $$d=2$$ and $$d=3$$, but doesn't work for higher dimensions, negative eigenvalues show up. It's easy enough to find the state numerically for higher $$d$$, so I guess it always exists, but I can't find a good method to calculate the coefficients analytically. Also, it seems a natural question to ask, but I couldn't find anything about it in the literature.

• out of curiosity, did this question arise from thinking about MUBs?
– glS
Aug 20, 2021 at 22:27
• No, I was just wondering about different bases for quantum states. The analogous problem for the standard basis $|i\rangle\langle j|$ is absolutely trivial, so I expected this one to be easy as well. Apparently it's not. I don't see any connection to MUBs. I did compute the state numerically for the cursed dimension 6, so it's not a matter of primes versus composite numbers. Aug 21, 2021 at 9:47

If $$|\langle\psi| U_{ab}|\psi\rangle|$$ is constant then $$\{\frac{1}{d} U_{ab}|\psi \rangle \langle\psi| U_{ab}^\dagger\}$$ is a SIC-POVM of Weyl-Heisenberg type.
It's conjectured that such a structure exists in every dimension $$d$$ (it's also known as Zauner's conjecture). It's still an open question, though numerically it was verified at least for all $$d \le 150$$.