Mutually Unbiased Bases are Complex Projective 2-Designs

I. INTRODUCTION

• Complementary observables have mutually unbiased eigenbases, and the converse association also holds for a pair of mutually unbiased bases.Measuring one observable after preparing an eigenstate of the other gives probability 1/d for each outcome.Source: §I · p. 1
• d + 1 complete measurements are necessary and sufficient to attain the lower bound for reconstructing a density matrix when the observables are pairwise complementary.Complementary observables have mutually unbiased eigenbases.Source: §I · p. 1
• Dimension 6 remains an open case for this tomography approach because a set of seven mutually unbiased bases has not been constructed.The text notes that the process is known in dimensions 3, 4, and 5.Source: §I · p. 1

II. MUTUALLY UNBIASED BASES

• Two orthonormal bases are mutually unbiased when every pair of basis vectors has squared inner product 1/d..Source: §II · p. 1
• Maximal sets of d + 1 mutually unbiased bases are known in prime power dimensions, but it is unknown whether this bound is attainable in other dimensions.The cited Latin-square construction in square dimensions yields w + 2 bases and never gives a maximal set.Source: §II · pp. 1–2
• Wootters and Fields' construction gives q + 1 mutually unbiased bases in dimension q for odd prime powers.Source: §II · p. 1
• The Galois-ring construction gives 2n + 1 mutually unbiased bases in dimension 2^n [13].Source: §II · p. 2
• The Latin-square construction gives w + 2 bases in dimension d^2 from w mutually orthogonal Latin squares and a complex Hadamard matrix [23].Source: §II · p. 2
• The examples exhibit maximal sets of four bases in dimension 3 and five bases in dimension 4.Source: §II · p. 2
• M(d) denotes the maximum number of mutually unbiased bases in C^d.Source: §II · p. 2

III. WELCH’S LOWER BOUNDS

• Welch derived the bounds to lower-bound the maximal cross-correlation of spreading sequences in synchronous code-division multiple-access systems.Blichfeld and Sidelnikov derived similar bounds for real unit vectors.Source: §III · p. 2
• A finite unit-vector set is a t-design exactly when it attains the Welch bounds for every k ≤ t.The section presents this as an equivalence between the design condition and equality in the bounds.Source: §III · p. 2
• Vectors differing only by a complex phase are treated as equivalent, and the resulting quotient of the unit sphere is identified with complex projective space.The section favors the quotient-sphere notation because unit-vector normalization is common in quantum computing.Source: §III · p. 2

IV. COMPLEX PROJECTIVE t-DESIGNS

• A finite set is a t-design when its average matches the projective-space integral for every function in Hom(t,t)°.Hom(k,k)° consists of restrictions of polynomials homogeneous of degree k in each variable family.Source: §IV · p. 3
• A set is a t-design exactly when its moment identities hold for every x and 0 ≤ k ≤ t, equivalently when it attains the Welch bounds for those k.Theorem 1 gives these three conditions as equivalent.Source: §IV · p. 3
• The proof derives the moment identity by applying cubature to overlap polynomials, then uses tensor powers and a nonnegative squared norm to recover exact cubature.The latter direction establishes cubature for the stated polynomial space by linearity.Source: §IV · p. 3

V. UNIFORM TIGHT FRAMES

• A frame spans the space while allowing linear dependence; it is tight when its bounds agree and uniform when its vectors have unit norm.The frame bounds are required to be nonzero real constants.Source: §V · p. 3
• A finite vector set is a uniform tight frame exactly when it is a WBE-sequence set and a projective 1-design.The proof applies the frame constants |F|/d and Theorem 1.Source: §V · p. 3
• Any projective 1-design can be obtained by projecting an orthogonal basis from a higher-dimensional space whose basis vectors have equal norm..Source: §V · p. 3

• A union of d+1 mutually unbiased bases forms a 2-design with angle set {0, 1/d} and d(d + 1) elements.Source: §VI · p. 4
• The forward proof verifies equality in the Welch bounds for k = 0, 1, and 2, then invokes the design criterion.The union has d(d + 1) elements.Source: §VI · p. 4
• Conversely, every 2-design with angle set {0, 1/d} and d(d + 1) elements is a union of d+1 mutually unbiased bases.Source: §VI · p. 4
• For the converse, each design point has d − 1 orthogonal partners, and the resulting orthogonal neighborhoods form orthonormal bases that are mutually unbiased when distinct.The proof establishes the neighborhood size by counting shared orthogonal vectors.Source: §VI · p. 4
• Zauner conjectures that non-prime-power dimensions admit no 2-design with angle set {0, 1/d} and d(d + 1) elements; the converse theorem was not previously known, while Theorem 3 was obtained earlier by Zauner.Source: §VI · p. 4

• A SIC-POVM in dimension d is a 2-design with angle set {1/(d + 1)} and d^2 elements.Source: §VII · p. 4
• The proof verifies Welch-bound equality for k = 0, 1, and 2 to establish the design property.The argument uses the SIC-POVM's d^2 vectors and presents a new, simple proof of the result previously shown in [16].Source: §VII · p. 4

• Maximal MUB unions and complex projective 2-designs with d^2 + d elements and angle set {0, 1/d} are equivalent, and these sets attain the Welch bounds for k = 0, 1, and 2.Source: §VIII · p. 5
• The result generalizes Barnum's finding for a particular MUB system and the corresponding statement over the real numbers in.Source: §VIII · p. 5
• Zauner conjectures that affine 2-designs do not exist in dimensions with two distinct prime factors.Source: §VIII · p. 5