Mutually unbiased bases via complex projective trigonometry

We give a synthetic construction of a complete system of mutually unbiased bases in $\mathbb{C}^3$.


Synthetic description
In quantum mechanics, the concept of mutually unbiased bases [1] is of fundamental importance.Recall that orthonormal bases (e i ) and (f j ) for C n are called mutually unbiased if the absolute value of the Hermitian inner product e i , f j for all i, j is independent of i and j (and therefore necessarily equals 1 √ n ).No knowledge of quantum theory is required for reading this article.
A complete system of mutually unbiased bases in C n is a system of n+1 such bases.It is still unknown whether a complete system of bases exists for C 6 i.e., in CP 5 .A complete system of mutually unbiased bases is known to exist when the vector space dimension is a prime power.We propose a new approach to constructing mutually unbiased bases, and implement it in the case of the complex projective plane.
Fix a complex projective line Recall that ℓ is isometric to a 2-sphere of constant Gaussian curvature.Let A, B, C, D ∈ ℓ be the vertices of a regular inscribed tetrahedron, i.e., an equidistant 4-tuple of points on the 2-sphere.There is a synthetic construction of a complete system of mutually unbiased bases (MUBs), where each basis includes one of the 4 points A, B, C, D. Let be the complex projective line consisting of points at maximal distance from A ∈ CP 2 .On the projective line ℓ of (1.1), let E ∈ ℓ be the antipodal point of A, so that E = ℓ ∩m, where m is the complex projective line of (1.2).Then {A, E} is a pair of points at maximal geodesic distance in CP 2 .In homogeneous coordinates, this corresponds to the fact that their representing vectors in C 3 are orthonormal with respect to the standard Hermitian inner product.Choose a great circle (i.e., closed geodesic) passing through E. We view S 1 as the equator of m and denote by the corresponding north and south poles.Then S 1 ⊆ m is the set of equidistant points from A ′ and A ′′ .Let γ ABE be the closed geodesic (great circle) passing through the three points A, B, E ∈ ℓ.Consider the real projective plane RP 2 B ⊆ CP 2 that includes the great circles γ ABE and S 1 .Consider also the real projective plane RP 2 C ⊆ CP 2 that includes the great circles γ ACE and S 1 .Theorem 1.1.A complete system of MUBs is constructed as follows.The first basis is {A, A ′ , A ′′ } ⊆ CP 2 .On the complex projective line passing through A ′ and A ′′ , let S 1 be the equidistant circle from A ′ and A ′′ .Choose an equilateral triangle EE ′ E ′′ in S 1 .Let ℓ be the projective line through A and E. We complete A to an equidistant 4-tuple A, B, C, D on ℓ.Next, we complete B to a basis B , and similarly for B ′′ ∈ γ AE ′′ .Next, we complete C to a basis {C, C ′ , C ′′ } as we did for B; the same for D.
The remainder of the paper presents a proof of Theorem 1.1.

Complex projective trigonometry
Consider a pair of unit vectors v, w in C n endowed with its Hermitian inner product , .Associated with the pair v, w, there is a pair of angles: (1) α = arccos Re v, w is the usual angle between the vectors when C n is identified with R 2n , and (2) θ = arccos | v, w | is the least angle between vectors in the complex lines spanned by v and w.Letting P w be the orthogonal projection of w to the complex line spanned by v, we denote by ψ (2.1) the angle between v and P w, so that by the spherical law of cosines (see [2, p. 17]) we have Given a pair of geodesics issuing from a point A ∈ CP n , the corresponding angle α (resp.θ) is defined similarly using their unit tangent vectors at A. Consider a geodesic triangle in CP n with sides of length a, b, c and angles α and θ at the point opposite the side c.We normalize the metric on CP n so that the sectional curvature satisfies 1  4 ≤ K ≤ 1.Then each complex projective line CP 1 ⊆ CP n is isometric to a unit sphere.The following formula, generalizing the law of cosines of spherical trigonometry, goes back to Shirokov [3] and was exploited in [4] in 1984 as well as in 1991 in [5, p. 176]: Let A ∈ CP 2 and let m ⊆ CP 2 be the complex projective line consisting of points at maximal distance π from A (the notation is consistent with that introduced in Section 1).Let E, X ∈ m and consider a geodesic triangle AEX contained in a copy of a real projective plane (of curvature 1  4 according our normalisation).As before, α and θ are the angles between the tangent vectors at A to the chosen geodesics γ AE and γ AX .
Consider the tangent vector of the geodesic γ AX at the point X.If we parallel translate this vector along the geodesic γ XE to a vector u at the point E, then (1) the angle at E between u and the tangent vector to the geodesic γ AC is the angle ψ of (2.1) by [5, p. 177]; (2) the distance d(E, X) between E and X is d(E, X) = 2θ (the factor of 2 is due to our normalisation of the metric).
We complete A to a basis {A, A ′ , A ′′ } by choosing a pair of antipodal points A ′ , A ′′ ∈ m.
The following identity plays a key role in the construction.Relative to the above normalisation of curvature, a totally geodesic CP 1 has Gaussian curvature 1 whereas a totally geodesic RP 2 has Gaussian curvature Consider the equator S 1 ⊆ m equidistant from A ′ and A ′′ , and choose an equilateral triangle EE ′ E ′′ on S 1 , so that d(E, E ′ ) = 2π 3 and similarly for the other two pairs.The complex projective line through A and E is denoted ℓ as in Section 1.
We choose a geodesic arc γ AE ⊆ ℓ ⊆ CP 2 and a point B ∈ γ AE so that d(A, B) = d.Next, we consider the complex projective line through A and E ′ .On this line, we choose a geodesic γ AE ′ ⊆ CP 2 so that the angle ψ between γ AE and γ AE ′ is ψ = π (here α = 2π 3 and θ = π 3 ).Then the geodesics γ AE and γ AE ′ lie in a common real projective plane RP 2 which also includes the equator S 1 ⊆ m.We choose the point B ′ ∈ γ AE ′ so that d(A, B ′ ) = d.We choose the point B ′′ above E ′′ similarly.Note that the angle ψ between γ AE ′ and γ AE ′′ is also π, because parallel transport around S 1 in RP 2 of a vector u orthogonal to S 1 gives −u.

1 4 . 2 = 3 . 1 √ 3 ( 2 3
Let d be the spherical sidelength of a regular inscribed tetrahedron in CP 1 .Then geodesic arcs of length d, π − d, and π 2 form a right-angle triangle in RP 2 with hypotenuse d, so that cos d Mutually unbiased bases By definition, another basis {B, B ′ , B ′′ } is unbiased with respect to {A, A ′ , A ′′ } if and only if the distance d between A and B is d = 2 arccos again the factor of 2 is due to our normalisation), and similarly for all the other distances between a member of the first basis and a member of the second basis.Note that cos d = 2 cos 2 d 2 − 1 = −1 = − 1 3 , i.e., d = arccos − 1 3 is the spherical sidelength of a regular inscribed tetrahedron.