The inverse Gram matrix of an orthocross MIC is half-integral

Abstract

For every dimension d and every orthonormal basis of C^d, the Gram matrix G of the orthocross MIC is invertible and every entry of G^(-1) is an integer or a half-integer. This proves the conjecture of J. B. DeBrota, C. A. Fuchs and B. C. Stacey (arXiv:1812.08762), who observed it numerically.

Definition 1.1 (Basis and cross vectors).

$$\mathrm{vec}\left(X_j\right)=e_j,\mathrm{vec}\left(T_{jk}\right)=e_j+e_k,\mathrm{vec}\left(V_{jk}\right)=e_j+i\cdot e_k$$

Formalization. D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.vec (✓ std3).

Citation. John B. DeBrota; Christopher A. Fuchs; Blake C. Stacey (2020). The Varieties of Minimal Tomographically Complete Measurements. DOI: 10.1142/S0219749920400055. URL: https://arxiv.org/abs/1812.08762v5.

Commentary.

The index set consists of the d basis indices X_j and, for every pair j < k, a real cross index T_jk and an imaginary cross index V_jk, d^2 indices in all. The vectors are e_j, e_j + e_k and e_j + i e_k.

Definition 1.2 (Normalising weights).

$$\mathrm{weight}\left(X_j\right)=1,\mathrm{weight}\left(T_{jk}\right)=\mathrm{weight}\left(V_{jk}\right)=\frac{1}{2}$$

Formalization. D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.weight (✓ std3).

Citation. John B. DeBrota; Christopher A. Fuchs; Blake C. Stacey (2020). The Varieties of Minimal Tomographically Complete Measurements. DOI: 10.1142/S0219749920400055. URL: https://arxiv.org/abs/1812.08762v5.

Commentary.

The weight is 1 for a basis vector and 1/2 for a cross vector, whose squared norm is 2, so that w(alpha) v_alpha v_alpha^* is a rank-one projector.

Definition 1.3 (The projectors Pi_alpha).

$$\mathrm{proj}\left(U,\alpha \right)=\mathrm{weight}\left(\alpha \right)\cdot (U\mathrm{vec}\left(\alpha \right))(U\mathrm{vec}\left(\alpha \right){)}^{\ast }$$

Formalization. D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.proj (✓ std3).

Citation. John B. DeBrota; Christopher A. Fuchs; Blake C. Stacey (2020). The Varieties of Minimal Tomographically Complete Measurements. DOI: 10.1142/S0219749920400055. URL: https://arxiv.org/abs/1812.08762v5.

Commentary.

For a unitary U, the columns U e_j form the chosen orthonormal basis, and Pi_alpha(U) is the projector onto U v_alpha: the paper’s Gamma_jj, (1/2)(|j> + |k>)(<j| + <k|) and (1/2)(|j> + i|k>)(<j| - i<k|) for j < k.

Definition 1.4 (The frame operator Omega).

$$\mathrm{frame}\left(U\right)=\sum _{\alpha }\mathrm{proj}\left(U,\alpha \right)$$

Formalization. D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.frame (✓ std3).

Citation. John B. DeBrota; Christopher A. Fuchs; Blake C. Stacey (2020). The Varieties of Minimal Tomographically Complete Measurements. DOI: 10.1142/S0219749920400055. URL: https://arxiv.org/abs/1812.08762v5.

Commentary.

Omega is the sum of the d^2 projectors.

Definition 1.5 (The orthocross MIC elements).

$$\mathrm{mic}\left(U,\alpha \right)=\mathrm{frame}{\left(U\right)}^{-\frac{1}{2}}\mathrm{proj}\left(U,\alpha \right)\mathrm{frame}{\left(U\right)}^{-\frac{1}{2}}$$

Formalization. D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.mic (✓ std3).

Citation. John B. DeBrota; Christopher A. Fuchs; Blake C. Stacey (2020). The Varieties of Minimal Tomographically Complete Measurements. DOI: 10.1142/S0219749920400055. URL: https://arxiv.org/abs/1812.08762v5.

Commentary.

E_alpha = Omega^(-1/2) Pi_alpha Omega^(-1/2), where Omega^(-1/2) is the positive square root of the inverse of the positive definite matrix Omega.

Definition 1.6 (The Gram matrix).

$$\mathrm{gram}\left(U,\alpha ,\beta \right)=\mathrm{tr}\left(\mathrm{mic}\left(U,\alpha \right)\mathrm{mic}\left(U,\beta \right)\right)$$

Formalization. D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.gram (✓ std3).

Citation. John B. DeBrota; Christopher A. Fuchs; Blake C. Stacey (2020). The Varieties of Minimal Tomographically Complete Measurements. DOI: 10.1142/S0219749920400055. URL: https://arxiv.org/abs/1812.08762v5.

Commentary.

The Gram matrix of the MIC, G_alpha beta = tr(E_alpha E_beta).

Definition 1.7 (The conjecture).

$$claim\iff (\forall d\in \mathbb{N},\forall U\in \mathrm{UnitaryGroup}\left(d\right),(\mathrm{IsUnit}\left(\det \left(\mathrm{gram}\left(U\right)\right)\right))\wedge (\forall a\in \mathrm{Idx}\left(d\right),\forall b\in \mathrm{Idx}\left(d\right),\exists z\in \mathbb{Z},2\cdot (\mathrm{gram}{\left(U\right)}^{-1})(a,b)=z))$$

Formalization. D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.claim (✓ std3).

Citation. John B. DeBrota; Christopher A. Fuchs; Blake C. Stacey (2020). The Varieties of Minimal Tomographically Complete Measurements. DOI: 10.1142/S0219749920400055. URL: https://arxiv.org/abs/1812.08762v5.

Commentary.

The conjecture of the paper, for every dimension d and every orthonormal basis (every unitary U): G is invertible and 2 (G^(-1))_ab is an integer for all indices a and b of Idx(d), the d^2 indices X_j, T_jk and V_jk.

Theorem 1.8 (Half-integrality of the inverse Gram matrix).

$$claim$$

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

Resolves. Problems/debrota-2020-orthocross-gram-inverse-half-integer (proved) by D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.result.

Source. Repository-derived.

Acknowledgement. John B. DeBrota; Christopher A. Fuchs; Blake C. Stacey (2020). The Varieties of Minimal Tomographically Complete Measurements. DOI: 10.1142/S0219749920400055. URL: https://arxiv.org/abs/1812.08762v5.

Commentary.

Since the square of Omega^(-1/2) is Omega^(-1), G_alpha beta = tr(Pi_alpha Omega^(-1) Pi_beta Omega^(-1)), and a unitary change of basis does not change it, so U = 1 suffices. In the standard basis Omega has d on the diagonal, (1 - i)/2 above it and (1 + i)/2 below it. The matrices D_T = E_jk + E_kj, D_V = i(E_kj - E_jk) and D_X = E_jj - E_jj C - C E_jj, where C is the off-diagonal part of Omega, satisfy tr(D_alpha Pi_beta) = delta_alpha beta; there are d^2 of them and they are linearly independent, so every matrix Y equals the sum over beta of tr(Y Pi_beta) D_beta. It follows that G M = 1 for M_beta gamma = tr(D_beta Omega D_gamma Omega), so G is invertible and G^(-1) = M. The matrices (1 - i) Omega and (1 - i) D_alpha have Gaussian-integer entries, and 4 = (1 + i) i (1 - i)^3, so every entry of 4 Z with Z = Omega D_gamma Omega is (1 + i) times a Gaussian integer a + b i. Z is Hermitian, so M_T gamma = Z_kj + Z_jk, M_V gamma = i Z_jk - i Z_kj and M_X gamma = Z_jj minus the sum over q of C_jq Z_qj + C_qj Z_jq are read off from these entries, and in each case 2M is an integer: a - b, -(a + b), and a sum of the integers a or -b with 2 Z_jj.

References

• Truth anchor: D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.claim
• Truth anchor: D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.frame
• Truth anchor: D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.gram
• Truth anchor: D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.mic
• Truth anchor: D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.proj
• Truth anchor: D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.result
• Truth anchor: D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.vec
• Truth anchor: D5/S3/Quantum/Measurement/OrthocrossGramHalfInteger.weight