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Star matrices force large orthostochastic dimension

Abstract

Star matrices contradict an eventual upper bound of three on Gutkin’s internal dimension.

Rows and columns are indexed by Fin n, with values 0 through n − 1; the first index is the row. This corresponds to the upper index in Gutkin’s notation, whose lower index denotes the column. EuclideanSpace ℝ (Fin d) is the real d-dimensional Euclidean space. The set Matrix.doublyStochastic ℝ (Fin n) consists of real n by n matrices with nonnegative entries and every row and column summing to one.

Definition 1.1 (Norms and two orthogonality systems).

Formalization. D5/S3/Quantum/Matrix/GutkinOrthostochasticDimensionRefutation.IsGutkinLift (✓ std3).

Citation. Eugene Gutkin (2013). On a multi-dimensional generalization of the notions of orthostochastic and unistochastic matrices. DOI: 10.1016/j.geomphys.2013.06.009. URL: https://arxiv.org/abs/1301.2537v1.

Commentary.

Lemma 1, equations (1)–(2), and the definition of Iso and ν prescribe the two orthogonality systems and the entrywise squared norms. The sum over r is column orthogonality, and the sum over c is row orthogonality. Both systems are required here; their equivalence is not assumed. The right-hand side is one for equal indices and zero otherwise.

Definition 1.2 (Real d-orthostochastic matrices).

Formalization. D5/S3/Quantum/Matrix/GutkinOrthostochasticDimensionRefutation.IsOrthostochasticOfDim (✓ std3).

Citation. Eugene Gutkin (2013). On a multi-dimensional generalization of the notions of orthostochastic and unistochastic matrices. DOI: 10.1016/j.geomphys.2013.06.009. URL: https://arxiv.org/abs/1301.2537v1.

Commentary.

Definition 1 calls a bistochastic matrix P real d-orthostochastic when P = ν(V) for a vector-valued matrix V in Iso(ℝ,n,d). Thus bistochasticity is required together with a lift satisfying the norm equations and both orthogonality systems.

Definition 1.3 (The infimum of the universal dimensions).

Formalization. D5/S3/Quantum/Matrix/GutkinOrthostochasticDimensionRefutation.dMin (✓ std3).

Citation. Eugene Gutkin (2013). On a multi-dimensional generalization of the notions of orthostochastic and unistochastic matrices. DOI: 10.1016/j.geomphys.2013.06.009. URL: https://arxiv.org/abs/1301.2537v1.

Commentary.

Conjecture 1 defines d_min(n) as the smallest internal dimension for which every bistochastic n by n matrix is d-orthostochastic. The formula uses the infimum in ℕ of exactly that set of dimensions. A nonempty subset of ℕ contains its infimum; the infimum of the empty set in ℕ is zero.

Definition 1.4 (Gutkin’s eventual dimension bound).

Formalization. D5/S3/Quantum/Matrix/GutkinOrthostochasticDimensionRefutation.claim (✓ std3).

Citation. Eugene Gutkin (2013). On a multi-dimensional generalization of the notions of orthostochastic and unistochastic matrices. DOI: 10.1016/j.geomphys.2013.06.009. URL: https://arxiv.org/abs/1301.2537v1.

Commentary.

Conjecture 1 asserts that there exists a threshold n₀ such that 2 ≤ d_min(n) ≤ 3 for every n ≥ n₀, over the real field.

Definition 1.5 (A symmetric star matrix).

Formalization. D5/S3/Quantum/Matrix/GutkinOrthostochasticDimensionRefutation.starMatrix (✓ std3).

Source. Repository-derived.

Acknowledgement. Eugene Gutkin (2013). On a multi-dimensional generalization of the notions of orthostochastic and unistochastic matrices. DOI: 10.1016/j.geomphys.2013.06.009. URL: https://arxiv.org/abs/1301.2537v1.

Commentary.

For n ≥ 2, put a = 1/(n − 1). The central diagonal entry is zero, every edge from the central index 0 has weight a, and every other diagonal entry has weight 1 − a. All remaining entries vanish. This matrix tests the universal dimension bound in Gutkin’s conjecture. The formula defines it for every natural n: the subtraction n − 1 is performed in ℕ before coercion to ℝ, and real division by zero is zero.

Theorem 1.6 (The eventual dimension bound is false).

Proof. Machine-checked in Lean as D5/S3/Quantum/Matrix/GutkinOrthostochasticDimensionRefutation.result (✓ std3). ∎

Resolves. Problems/gutkin-2013-orthostochastic-dimension-refutation (refuted) by D5/S3/Quantum/Matrix/GutkinOrthostochasticDimensionRefutation.result.

Source. Repository-derived.

Acknowledgement. Eugene Gutkin (2013). On a multi-dimensional generalization of the notions of orthostochastic and unistochastic matrices. DOI: 10.1016/j.geomphys.2013.06.009. URL: https://arxiv.org/abs/1301.2537v1.

Commentary.

For n ≥ 2 let S = starMatrix(n) and a = 1/(n − 1). Its entries are S₀₀ = 0, S₀ⱼ = Sⱼ₀ = a for j ≠ 0, Sⱼⱼ = 1 − a for j ≠ 0, and zero elsewhere. They are nonnegative. Row 0 sums to (n − 1)a = 1; every other row sums to a + (1 − a) = 1. Symmetry gives the same column sums, so S is bistochastic.

Suppose V lifts S in ℝᵈ. Fix distinct nonzero columns j and k. For a row r ≠ 0, if r = j then Sᵣₖ = 0, while if r ≠ j then Sᵣⱼ = 0. The norm equations make the corresponding vector zero, so every cross term ⟨Vᵣⱼ,Vᵣₖ⟩ vanishes except possibly the term in row 0. Column orthogonality therefore gives ⟨V₀ⱼ,V₀ₖ⟩ = 0.

For each j ≠ 0 the norm equation gives ‖V₀ⱼ‖² = a > 0. The n − 1 vectors V₀ⱼ are thus nonzero and pairwise orthogonal. They are linearly independent in the d-dimensional real Euclidean space, which forces n − 1 ≤ d.

Assume the conjectured threshold n₀ exists and take n = max(n₀,5). Then 2 ≤ dMin(n) ≤ 3. The set defining dMin(n) cannot be empty, since its infimum would then be zero. Its natural-number infimum belongs to it, so S has a lift in dimension dMin(n). The dimension bound gives 4 ≤ n − 1 ≤ dMin(n), contradicting dMin(n) ≤ 3.

References