No Pauli-orbit mutually unbiased bases for four qubits or for an odd number of qubits greater than one

The stabilizer Rényi entropy of order $\alpha$ measures how far a pure state of $n$ qubits is from the stabilizer states, the pure states in which $2^n$ Pauli operators have expectation value $\pm 1$. It is computed from the expectation values of the $4^n$ Pauli operators in the state. For two qubits, a numerical search by Liu, Low and Yin finds that the pure states with the largest stabilizer Rényi entropy of order 2 are the fiducials of mutually unbiased bases (MUBs) for the Pauli group, the states whose orbit under the Pauli group splits into MUBs. A conjecture of Liu, Low and Yin includes an $n$-qubit statement, which asserts that for every number $n$ of qubits other than 1 and 3, the largest stabilizer Rényi entropy of every order $\alpha \ge 2$ is attained exactly by these fiducials. In a fiducial, every squared expectation value of a non-identity Pauli operator equals 0 or $2^{-n}$, because the state and its image under the operator lie in one basis or in two mutually unbiased bases. We prove that for four qubits and for every odd number of qubits greater than one, no state has all these squares in $\{0, 2^{-n}\}$, so that no fiducial exists. For an odd number $n \ge 3$ of qubits, the Pauli components of the equation $\rho^2 = \rho$ for the density matrix $\rho$ of such a state would make its nonzero expectation values rational, whereas the modulus $2^{-n/2}$ of these values is irrational. For four qubits, a Clifford unitary and the arithmetic of Gaussian integers reduce such a state to one whose entries in the computational basis are fourth roots of unity times $1/4$, where a Clifford unitary is a unitary that maps every Pauli operator by conjugation to a Pauli operator up to sign. An exact enumeration of these states up to diagonal Clifford unitaries then finds no fiducial. The $n$-qubit statement therefore fails for four qubits and for every odd number of qubits from five on.