Reachable cardinalities of the Butson mixed Schur construction, and a quantum Latin square of order seven with cardinality 45

Abstract

A quantum Latin square of order n is an n × n array of unit vectors of ℂⁿ every row and every column of which is an orthonormal basis; its cardinality is the number of entries once vectors differing by a global phase are identified. Zhang, Lv and Cao tabulate, at the orders their determination of the spectrum leaves open, which cardinalities are known to occur and which are uncertain; the order-seven row of that table prints 45 as uncertain. We exhibit a quantum Latin square of order seven whose cardinality is exactly 45. The witness is a mixed Schur product Φ(H,K)ᵢⱼ=7^(-1/2)(hᵢodot kⱼ) of two Butson Hadamard matrices H,K ∈ BH(7,6), presented by explicit exponent matrices over ℤ/6, and the certificate is exact and integral: for a Butson pair the cardinality is the size of a sumset |S+T| of two n-element subsets of (ℤ/m)ⁿ modulo constants, and at m=6 the Butson condition is equivalent to two integer equations on each column difference profile. The sufficient shift criterion previously used to assemble such witnesses certifies none of the 84 ordered pairs of distinct columns of ours, and it is the equivalence that replaces it. We also record, general in n and m, the ceiling cardΦ(H,H) ≤ n(n+1)/2 for the symmetric construction — sharp at n=7, where it gives the value 28 that the same table lists as attainable and isolated — and the bracket n ≤ cardΦ(H,K) ≤ n². In particular the dichotomy |S+T| ∈ {p,p²}, which holds at a prime order p for matrices monomially equivalent to the Fourier matrix Fₚ, fails once the modulus is allowed to differ from the order. Every theorem below is machine-checked in Lean 4; the exhaustive enumerations reported alongside them are stated as measurements and are marked as such.

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