On SIC-POVMs and MUBs in Dimension 6

• For prime-power dimensions, finite affine planes and maximal MUB sets can be constructed, whereas corresponding non-prime-power constructions are unknown.Source: §1 · p. 1
• Numerical evidence indicates that SIC-POVMs may exist in all dimensions, but the relation between SIC-POVMs and affine planes is more abstract than for some MUB constructions.Source: §1 · p. 1
• Dimension six admits an explicit algebraic SIC-POVM, despite the absence of a finite affine plane of order six.Source: §1 · p. 1
• Several sets of three MUBs in dimension six are maximal because they cannot be extended to four MUBs.Source: §1 · p. 1

• The Heisenberg group H_d is generated by the cyclic shift X and phase operator Z, with operators X^aZ^b forming a basis of all d × d matrices.Source: §2 · pp. 1–2
• Modulo its center, H_d is isomorphic to Z_d × Z_d, while its generators are orthogonal under the trace inner product.Source: §2 · p. 2
• The Fourier matrix and P_d act on H_d by conjugation, with the Fourier matrix interchanging X and Z up to phase.Source: §2 · p. 2
• The Jacobi group is generated by X, Z, DFT_d, and P_d, and the group generated by DFT_d and P_d acts transitively on H_d modulo its center.Source: §2 · p. 2
• The Jacobi group is a semidirect product involving H_d and SL(2, Z_d), with H_d normal and the corresponding quotient isomorphic to SL(2, Z_d).Source: §2 · p. 2

• A SIC-POVM contains d^2 rank-one operators whose pairwise inner products are constant, making the measurement informationally complete and symmetric.Source: §3 · pp. 2–3
• A normalized fiducial state generates a Heisenberg-invariant SIC-POVM, and Jacobi-group basis changes produce equivalent SIC-POVMs with the same Heisenberg representation.Source: §3 · p. 3
• The construction searches one representative from each nontrivial conjugacy class of J_6 for an eigenvector whose Heisenberg orbit satisfies the SIC equations.Source: §3 · p. 3
• The resulting polynomial system is solved algebraically with MAGMA over a number field extending the rationals by degree 96.Source: §3 · pp. 3–4
• The construction yields 48 SIC-POVMs from one fiducial orbit and 96 in total after including the complex-conjugated state; this matches [6, Table 1].Source: §3 · p. 4

General construction

• Three mutually unbiased bases exist in every dimension through eigenbases of X, Z, and XZ^k when gcd(k, d) = 1.Fourier transformation gives the analogous construction using X^kZ.Source: §4 · pp. 5–6

Construction

• 16 orthonormal bases can be constructed from 48 vectors, each unbiased with respect to the eigenbases BX and BZ of X and Z.The vectors arise as real solutions of polynomial equations obtained from unbiasedness conditions.Source: §4 · p. 6

Maximal triples

• Each triple {BX, BZ, Bi} is maximal because no vector is unbiased with respect to all three bases.The 12-element sets of vectors unbiased with respect to 12 others contain none of the bases Bi.Source: §4 · pp. 6–7

Symmetry extension

• Theorem 2 extends under Jacobi-group transformations to eigenbases of XaZb and Xa′Zb′ whenever ab′ − a′b = 1 mod 6.The transformed bases are obtained from BX and BZ by a global change of basis.Source: §4 · p. 7

Maximal triples

• The specific three-base constructions from Lemma 2 are maximal in dimension six, so no vector is unbiased with respect to all three.This applies to the triples involving BX, BXZ^k, and BZ for k = 1, 5.Source: §4 · pp. 6–7

Scope

• Any two Heisenberg-related bases satisfying the stated subgroup condition cannot belong to a set of four MUBs.Consequently, any set of four or more MUBs in dimension six, if it exists, cannot be related to the Heisenberg group.Source: §4 · pp. 7–8

• Because dimension four is a prime power, a maximal set of five mutually unbiased bases exists.Source: §5 · p. 8
• Starting from the eigenbases BZ and BX of Z and X, direct computation gives only three mutually unbiased bases.Source: §5 · p. 8
• The third unbiased basis is parameterized by two parameters a and b in [0, π).Source: §5 · p. 8
• For all choices of parameters a and b, the three MUBs are maximal because no further vector is unbiased with respect to all three bases.Source: §5 · p. 8

• The authors explicitly construct a SIC-POVM in dimension six, the smallest dimension that is not a prime power.Source: §6 · p. 8
• The results support the conjecture that dimension six has at most three MUBs for constructions based on Heisenberg-group operators.Source: §6 · p. 8
• The maximal number of mutually unbiased bases in non-prime-power dimensions remains open, including the smallest case d = 6.Source: §6 · p. 8
• The paper notes that MUBs appear more strongly related to finite geometries than SIC-POVMs.Source: §6 · p. 8

• The appendix lists 48 vectors unbiased with respect to the eigenbases BX and BZ of X and Z.Source: Appendix · p. 9
• The listed vectors are not normalized, and their first coordinate is fixed to one.Source: Appendix · p. 9
• The vector expressions use ω = exp(2πi/12), a primitive 12-th root of unity, together with θ.Source: Appendix · p. 9
• The cited appendix passages give explicit coordinate formulas for constructed vectors indexed 3, 7, 8, and 11, with coordinates expressed algebraically using ω and θ.Source: Appendix · pp. 9–10