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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.

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  • $\begingroup$ out of curiosity, did this question arise from thinking about MUBs? $\endgroup$
    – glS
    Aug 20 at 22:27
  • $\begingroup$ 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. $\endgroup$ Aug 21 at 9:47
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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 $d \le 150$.

It's also quite a known problem. The paper that popularized this hypothesis already have 800 citations according to Google Scholar

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    $\begingroup$ I'm familiar with SIC-POVMs, I just had fail to see the connection. Thanks a lot! So it turns out the problem is actually hard. $\endgroup$ Aug 21 at 18:46

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