Mutually unbiased bases via complex projective trigonometry

1 Synthetic description

In quantum mechanics, the concept of mutually unbiased bases (MUBs) [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 MUBs 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 C P 5 . A complete system of MUBs is known to exist when the vector space dimension is a prime power. We propose a new approach to construct MUBs 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

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 MUBs, where each basis includes one of the four 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

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

be the closed geodesic (great circle) passing through the three points A , B , E ∈ ℓ . Consider the real projective plane

that includes the great circles γ A B E and S 1 . Consider also the real projective plane RP C 2 ⊆ CP 2 that includes the great circles γ A C E and S 1 .

The remainder of this article presents a proof of Theorem 1.1.

2 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:

• α = arccos Re ⟨ v , w ⟩ is the usual angle between the vectors when C n is identified with R 2 n , and

• θ = arccos ∣ ⟨ v , w ⟩ ∣ is the least angle between vectors in the complex lines spanned by v and w .

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 to 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 A E X contained in a copy of a real projective plane (of curvature 1 4 according to our normalization). As before, α and θ are the angles between the tangent vectors at A to the chosen geodesics γ A E and γ A X .

Consider the tangent vector of the geodesic γ A X at the point X . If we parallelly translate this vector along the geodesic γ X E to a vector u at the point E , then

• the angle at E between u and the tangent vector to the geodesic γ A C is the angle ψ of (2.1) by [5, p. 177];

• the distance d ( E , X ) between E and X is d ( E , X ) = 2 θ

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 normalization of curvature, a totally geodesic CP 1 has Gaussian curvature 1, whereas a totally geodesic RP 2 has Gaussian curvature 1 4 . Let d be the spherical side length 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

3 MUBs

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 1 3 (again the factor of 2 is due to our normalization), 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 = 2 3 − 1 = − 1 3 , i.e., d = arccos − 1 3 is the spherical side length of a regular inscribed tetrahedron.

Consider the equator S 1 ⊆ m equidistant from A ′ and A ″ and choose an equilateral triangle E E ′ E ″ on S 1 , so that d ( E , E ′ ) = 2 π 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 A E ⊆ ℓ ⊆ CP 2 and a point B ∈ γ A E 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 A E ′ ⊆ CP 2 so that the angle ψ between γ A E and γ A E ′ is ψ = π (here α = 2 π 3 and θ = π 3 ). Then the geodesics γ A E and γ A E ′ lie in a common real projective plane RP 2 which also includes the equator S 1 ⊆ m . We choose the point B ′ ∈ γ A E ′ so that d ( A , B ′ ) = d . We choose the point B ″ above E ″ similarly. Note that the angle ψ between γ A E ′ and γ A E ″ is also π , because parallel transport around S 1 in RP 2 of a vector u orthogonal to S 1 gives − u .

By construction, d ( A , B ) = d . Let us calculate d ( A ′ , B ) . Consider the right-angle triangle A ′ E B in RP 2 . As cos d 2 = 1 3 by the unbiased condition, we have sin d 2 = 2 3 . Hence, by the spherical theorem of cosines,

as required. The equality A B = A ′ B is equivalent to (2.4). A similar calculation shows that cos d ( A ″ , B ) = − 1 3 , as well. By symmetry, each of the points B ′ and B ″ is also at geodesic distance d from A , A ′ and A ″ . It remains to check that the triple { B , B ′ , B ″ } corresponds to an orthonormal basis. Indeed, ψ = π implies cos α = cos ψ cos θ = − cos θ by (2.2). We apply (2.3) at the point A we obtain

and therefore d ( B , B ′ ) = π as required.

4 Completing the MUBs

To specify a third basis, we choose C ∈ ℓ so as to complete A and B to an equilateral triangle A B C with side d (and angle 2 π 3 ). Then the corresponding angle θ vanishes, and hence α = ψ = 2 π 3 . We complete C to a basis { C , C ′ , C ″ } as in Section 3, using a real projective plane which includes the geodesic γ A C E and the equator S 1 ⊆ m (the same equator as before, equidistant between A ′ and A ″ ). To check that the C -basis is unbiased relative to the B -basis, note that ψ ( B , B ′ ) = π and ψ ( B , C ) = 2 π 3 hence ψ ( B ′ , C ) = π − 2 π 3 = π 3 , while θ = π 3 as before. Hence, cos d ( B , C ′ ) = 1 9 + 8 9 cos ψ cos θ − 2 ⋅ 4 9 sin 2 θ = − 1 3 , as required. The other pairs are checked similarly.

To specify a fourth basis, we choose D ∈ ℓ to complete the equilateral triangle A B C to a regular inscribed tetrahedron, and proceed as before. This completes the proof of Theorem 1.1.

Such a synthetic construction of MUBs involves less redundancy than complex Hadamard matrix presentations, and may therefore be useful in shedding light on MUBs in higher-dimensional complex projective spaces, such as CP 5 where the maximal number of MUBs is unknown.