Geometric construction of biangular grassmannian frames

Introduction

Finite frame theory provides redundant expansions of vectors in finite-dimensional Hilbert spaces, offering resilience against noise, erasures, and channel errors in signal processing and quantum communications [14,7] . In quantum information theory, quantum state tomography, coding theory, and spherical packings, significant attention is devoted to frames that minimise maximal pairwise cross-correlations, known as Grassmannian frames. While Equiangular Tight Frames (ETFs) represent optimal line packings achieving the Welch bound, their existence is severely restricted by dimensional and structural constraints [15] . When ETFs do not exist, one naturally turns to 𝐾-angular frames with small angle set cardinalities (𝐾 = 2,3). In this paper, we develop a rigorous computational algebraic geometry framework to analyse unit-norm tight biangular (𝐾 = 2) Grassmannian frames over the complex field ℂ. By embedding complex projective lines into spaces of rank-one Hermitian operators via the second Veronese map, we reformulate frame constraints as polynomial ideals and solve the existence question using computational commutative algebra. The remainder of this paper is structured as follows. In Section 2, we collect foundational definitions from finite frame theory, spherical embeddings, and algebraic geometry over ℂ. In Section 3, we formulate frame vectors as projective points of the Veronese variety and define system polynomial ideals. In Section 4, we provide concrete worked algebraic examples (𝑀 = 2,3,5). In Section 5, we prove the main non-existence theorem for tight biangular frames in ℂ 2 .

Preliminaries and Notations

Throughout this paper, we work exclusively over the complex field ℂ. Let ℋ = ℂ 𝑚 denote the 𝑚-dimensional complex Hilbert space equipped with the standard inner product ⟨𝑥, 𝑦⟩ = 𝑦 * 𝑥 = ∑ 𝑥 𝑖 𝑚 𝑖=1 𝑦 𝑖 and norm ∥ 𝑥 ∥= √⟨𝑥, 𝑥⟩. Let {𝑒 𝑗 } 𝑗=1 𝑚 denote the canonical orthonormal basis for ℂ 𝑚 , and 𝐼 𝑚 denote the 𝑚 × 𝑚 identity matrix. -tight frame, every frame vector 𝑣 𝑗 ∈ 𝛷 satisfies the Parseval-type energy identity [15] ∑|⟨𝑣 𝑗 , 𝑣 𝑙 ⟩| b. The Grassmannian constant for the pair (𝑀, 𝑚) over ℂ is [7] 𝜇 𝑀,𝑚 (ℂ): = inf = 𝑀 -1. Following [15] , we recall some relevant results for completeness.

Finite Frames in Complex

Proposition 2.7 (Equidistribution and Parity of Biangular Tight Frames). Let 𝛷 = {𝑣 𝑗 } 𝑗=1

𝑀 ∈ 𝛺 𝑀,𝑚 (ℂ) be a biangular tight frame with angles 𝑐 1 , 𝑐 2 . a. 𝛷 is automatically equidistributed with multiplicities m 1 , m 2 satisfying m 1 + m 2 = M -1 and

b. If 𝑀 is odd, the multiplicities m 1 and m 2 must both be even positive integers.

Conway-Hardin-Sloane Spherical Embedding. The spherical embedding technique of Conway, Hardin, and

Sloane is used to map complex frame vectors to higherdimensional real spheres.

Theorem 2.8 (Spherical Embedding of Tight Frames). Let 𝛷 = {𝑣 𝑗 } 𝑗=1 𝑀 ∈ 𝛺 𝑀,𝑚 (ℂ) be a unit-norm frame in ℂ 𝑚 , and set 𝐷 = 𝑚 2 -1. a. The frame vectors map isometrically to the sphere of

b. If 𝛷 is a tight frame, the embedded vectors sum to zero in

Gröbner Bases and

defined by a homogeneous ideal 𝐼 ℎ . c. Hilbert's Weak Nullstellensatz [19]: The algebraic variety 𝕍(𝐼) is empty (𝕍(I) = ∅) if and only if 1 ∈ 𝐼, which occurs if and only if the reduced Gröbner basis of 𝐼 under any monomial ordering is {1}. d. In terms of Krull dimension, 𝕍(I) = ∅ ⇔ dim(R/I) = -1.

Frame Vectors As Projective Points of a Veronese Variety

Let ℋ = ℂ 𝑚+1 and consider a frame 𝛷 = {𝑣 𝑗 } 𝑗=1 𝑀 ⊂ ℋ. Each frame vector 𝑣 𝑗 = (𝑧 𝑗0 , … , 𝑧 𝑗𝑚 ) ∈ ℂ 𝑚+1 defines a rank-one Hermitian projection operator 𝑃 𝑣 𝑗 = 𝑣 𝑗 𝑣 𝑗 *

Since scaling 𝑣 𝑗 by a complex scalar 𝜆 ∈ 𝑈(1) leaves 𝑃 𝜆𝑣 𝑗 = |𝜆| 2 𝑃 𝑣 𝑗 = 𝑃 𝑣 𝑗 invariant, each operator 𝑃 𝑣 𝑗 (denoted simply by 𝑃 𝑗 ) represents a unique projective point in ℙ(Herm(𝑚 + 1, ℂ)) ≅ ℂℙ (𝑚+1) 2 -1 . Definition 3.1 (Veronese Embedding). The second Veronese embedding 𝜈 2: ℂℙ 𝑚 → ℂℙ (𝑚+1) 2 -1 is defined by [20] 𝜈 2: [𝑧 𝑗0: …: 𝑧 𝑗𝑚 ] ↦ [𝑧 𝑗𝑖 𝑧 𝑗𝑘 ] 0≤𝑖,𝑘≤𝑚 .

The image 𝜈 2 (ℂℙ 𝑚 ) corresponds to rank-one Hermitian operators and is realised as the algebraic variety defined by the vanishing of all 2 × 2 minors of

3. Gauge-fixing ideal (𝐼 fix ): By unitarily rotating the frame, we may fix the first frame vector to 𝑣 1 = (1,0) ⊤ 𝐼 fix = (𝑧 10 -1, 𝑧 11 ).

The full ideal defining the configuration variety of (𝑀, 2) biangular frames is given by the sum of these ideals

Concrete Worked Algebraic Examples

To build geometric intuition and demonstrate the algebraic framework, we present explicit worked examples for small vector configurations in ℂ 2 .

Example 4.1 (Orthonormal Basis: 𝑀 = 2 in ℂ 2 ). Consider 𝑀 = 2 vectors in ℂ 2 given by the canonical basis

The rank-one Hermitian projection operators on 𝜈 2 (ℂℙ 1 ) ⊂ ℂℙ 3 are

Checking frame properties a. Tightness: ). Consider the 3-vector frame in ℂ 2 (the Mercedes-Benz ETF)

The corresponding Hermitian rank-one projection matrices are given by [15] 𝑣

The five vectors are distinct unit vectors. Computing the pairwise inner-product magnitudes for distinct vectors gives Hence, the set of distinct off-diagonal inner-product magnitudes is

Thus, 𝛷 non is biangular. On the other hand, its frame operator is

Therefore, 𝛷 non is a biangular (𝐾 = 2), unit-norm frame in ℂ 2 that is not tight. This example demonstrates that biangularity alone does not imply tightness, and hence the tightness condition is the essential additional constraint in the nonexistence of tight biangular frames consisting of five vectors in ℂ 2 .

Example 4.4 (Tight Triangular Frame: 𝑀 = 5 in ℂ 2 ). Consider the orthoplectic 5-vector frame 𝛷 tri = {𝑣 1 , … , 𝑣 5 } ⊂ ℂ 2 from [15] 𝑣

This frame satisfies 𝑆 = ∑ 𝑣 𝑗 . Decomposing

) -20 = 5 2 + 15 -20 = -5 2 .

The contradiction 0 = -5/2 proves that no set of five rankone operators {𝑃 1 , … , 𝑃 5 } ⊂ 𝜈 2 (ℂℙ 1 ) can simultaneously satisfy biangular trace metrics and the tightness linear operator identity. (Equivalently, in terms of polynomial ideals, 1 ∈ 𝐼 5 ⊂ ℂ[𝑧 10 , … , 𝑧 51 ], so 𝕍(𝐼 5 ) = ∅ and dim𝐼 5 = -1).

Classification and Non-Existence Theorem in ℂ 𝟐

We now present the general non-existence theorem for biangular tight Grassmannian frames in ℂ 2 .

Theorem = 1. The direct geometric contradiction 3/4 ≠ 1 proves that no complex vector configuration in ℂ 2 can simultaneously satisfy the frame tightness and biangularity polynomial constraints. Consequently, the configuration variety is empty (𝕍(𝐼 4 ) = ∅). By Hilbert's Weak Nullstellensatz, 𝕍(𝐼 4 ) = ∅ if and only if 1 ∈ 𝐼 4 , which forces the coordinate ring quotient to be the zero ring (𝑅 4 /𝐼 4 = {0}), yielding Krull dimension dim𝐼 4 = -1.

Induction for 𝑀 ≥ 4:

Assume by induction that 𝕍(𝐼 𝑘 ) = ∅ for some 𝑘 ≥ 4. For 𝑀 = 𝑘 + 1, the ideal of constraints 𝐼 𝑘+1 contains all constraint generators of 𝐼 𝑘 plus additional polynomial constraints imposed by the (𝑘 + 1)-th frame vector. That is,

Since the zero-locus operator 𝕍(⋅) is inclusion-reversing, we obtain ∅ = 𝕍(𝐼 𝑘 ) ⊇ 𝕍(𝐼 𝑘+1 ),

Which forces 𝕍(𝐼 𝑘+1 ) = ∅. Hence, no (𝑀, 2) biangular tight Grassmannian frames exist in ℂ 2 for any 𝑀 ≥ 4.

Reconstruction of Frame Vectors From Veronese Varieties

When the coordinate ring quotient R/I has finite non-zero Krull dimension, frame vector configurations can be reconstructed algebraically from the zero-dimensional solution variety 𝕍(𝐼) or geometrically via spectral factorisation of points on the Veronese variety.

Algebraic reconstruction via variety solving:

Let 𝐼 ⊂ 𝑅 = ℂ[𝑧 10 , … , 𝑧 𝑀1 ] be an ideal defining a valid frame configuration variety 𝕍(𝐼). If dim(𝑅/𝐼) = 0, the variety consists of a finite set of discrete points corresponding to isolated frame configurations up to global unitary transformations. Solving the system of polynomial equations 𝑓 𝑖 (𝑧) = 0 for generators 𝑓 𝑖 ∈ 𝐼 yields the exact algebraic coordinates (𝑧 𝑗0 , 𝑧 𝑗1 ) ∈ ℂ 2 for each frame vector 𝑣 𝑗 ∈ 𝛷.

Geometric reconstruction via spectral factorization:

Alternatively, given a collection of rank-one projection operators {𝑃 1 , … , 𝑃 𝑀 } ⊂ 𝜈 2 (ℂℙ 𝑚-1 ) lying on the Veronese variety, the pairwise trace inner products form the positive semi-definite Gramian matrix 𝐺 ∈ ℂ 𝑀×𝑀 with entries 𝐺 𝑗𝑙 = Tr(𝑃 𝑗 𝑃 𝑙 ). By spectral decomposition, 𝐺 factorises as

Where 𝑈 ∈ 𝕌(𝑀) is unitary and 𝛬 = diag(𝜆 1 , … , 𝜆 𝑚 , 0, … ,0) ∈ ℂ 𝑀×𝑀 is the diagonal matrix containing the non-zero eigenvalues of 𝐺. The synthesis matrix 𝑇 * ∈ ℂ 𝑚×𝑀 containing the reconstructed frame vectors as its columns is uniquely recovered via 𝑇 * = 𝛬 1/2 𝑈 * ∈ ℂ 𝑚×𝑀 , Where 𝛬 1/2 = diag(√𝜆 1 , … , √𝜆 𝑚 , 0, … ,0) ∈ ℂ 𝑚×𝑀 . The columns of 𝑇 * yield the exact unit-norm frame vectors {𝑣 1 , … , 𝑣 𝑀 } ⊂ ℂ 𝑚 satisfying the prescribed geometric angularity and spectral tightness conditions. This confirms that points on the Veronese variety allow complete, explicit geometric reconstruction of the unit-norm frame vectors.

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