Noncommutativity does not imply complete incompatibility

Abstract

In every complex dimension d ≥ 4, two orthonormal bases can have noncommuting coordinate projectors for every pair of nonempty proper index sets while failing complete incompatibility.

Definition 1.1 (Coordinate orthogonal projectors).

$$\forall d\in \mathbb{N},\forall b\in \mathrm{OrthonormalBasis}\left(\mathrm{Fin}\left(d\right),\mathbb{C},\mathrm{EuclideanSpace}\left(\mathbb{C},\mathrm{Fin}\left(d\right)\right)\right),\forall S\in \mathrm{Finset}\left(\mathrm{Fin}\left(d\right)\right),\forall i\in \mathrm{Fin}\left(d\right),\forall j\in \mathrm{Fin}\left(d\right),\mathrm{projector}\left(b,S,i,j\right)=\sum _{k\in S}b\left(k\right)\left(i\right)\cdot \mathrm{star}\left(b\left(k\right)\left(j\right)\right)$$

Formalization. D5/S3/Quantum/Measurements/NoncommutingNotCompletelyIncompatible.projector (✓ std3).

Citation. Stephan De Bièvre (2023). Relating incompatibility, noncommutativity, uncertainty and Kirkwood-Dirac nonclassicality. DOI: 10.1063/5.0110267. URL: https://arxiv.org/abs/2207.07451v1.

Commentary.

Section 3, p. 5: “Given bases 𝒜 and ℬ, we introduce a family of orthogonal projectors as follows. For every S,T ⊂ ⟦1,d⟧ := {1,2,…,d},” followed by Π_𝒜(S) = ∑{i∈S} |a_i⟩⟨a_i| and Π_ℬ(T) = ∑{j∈T} |b_j⟩⟨b_j|. The displayed matrix entry is exactly this sum of rank-one projectors. star is complex conjugation; b(k)(i) is the i-th coordinate of the k-th basis vector.

Definition 1.2 (Condition (ii)).

$$\forall d\in \mathbb{N},\forall a\in \mathrm{OrthonormalBasis}\left(\mathrm{Fin}\left(d\right),\mathbb{C},\mathrm{EuclideanSpace}\left(\mathbb{C},\mathrm{Fin}\left(d\right)\right)\right),\forall b\in \mathrm{OrthonormalBasis}\left(\mathrm{Fin}\left(d\right),\mathbb{C},\mathrm{EuclideanSpace}\left(\mathbb{C},\mathrm{Fin}\left(d\right)\right)\right),(\mathrm{NonCommutingProjectors}\left(a,b\right))\iff (\forall S\in \mathrm{Finset}\left(\mathrm{Fin}\left(d\right)\right),\forall T\in \mathrm{Finset}\left(\mathrm{Fin}\left(d\right)\right),(\mathrm{Nonempty}\left(S\right))\Rightarrow ((S\ne \mathrm{univ}\left(\mathrm{Fin}\left(d\right)\right))\Rightarrow ((\mathrm{Nonempty}\left(T\right))\Rightarrow ((T\ne \mathrm{univ}\left(\mathrm{Fin}\left(d\right)\right))\Rightarrow (\mathrm{projector}\left(a,S\right)\cdot \mathrm{projector}\left(b,T\right)\ne \mathrm{projector}\left(b,T\right)\cdot \mathrm{projector}\left(a,S\right))))))$$

Formalization. D5/S3/Quantum/Measurements/NoncommutingNotCompletelyIncompatible.NonCommutingProjectors (✓ std3).

Citation. Stephan De Bièvre (2023). Relating incompatibility, noncommutativity, uncertainty and Kirkwood-Dirac nonclassicality. DOI: 10.1063/5.0110267. URL: https://arxiv.org/abs/2207.07451v1.

Commentary.

Proposition 7(ii), p. 16: “For all S,T ⊂ ⟦1,d⟧, with 1 ≤ |S|, |T| < d, [Π_𝒜(S),Π_ℬ(T)] ≠ 0.” Nonempty proper finite subsets express precisely the two cardinality bounds. The two products are complex matrix products, so their inequality expresses a nonzero operator commutator.

Definition 1.3 (Condition (i): complete incompatibility).

$$\forall d\in \mathbb{N},\forall a\in \mathrm{OrthonormalBasis}\left(\mathrm{Fin}\left(d\right),\mathbb{C},\mathrm{EuclideanSpace}\left(\mathbb{C},\mathrm{Fin}\left(d\right)\right)\right),\forall b\in \mathrm{OrthonormalBasis}\left(\mathrm{Fin}\left(d\right),\mathbb{C},\mathrm{EuclideanSpace}\left(\mathbb{C},\mathrm{Fin}\left(d\right)\right)\right),(\mathrm{CompletelyIncompatible}\left(a,b\right))\iff (\forall S\in \mathrm{Finset}\left(\mathrm{Fin}\left(d\right)\right),\forall T\in \mathrm{Finset}\left(\mathrm{Fin}\left(d\right)\right),(\mathrm{card}\left(S\right)+\mathrm{card}\left(T\right)\le d)\Rightarrow (\inf \left(\mathrm{span}\left(\mathbb{C},\mathrm{image}\left(a,\mathrm{coe}\left(S\right)\right)\right),\mathrm{span}\left(\mathbb{C},\mathrm{image}\left(b,\mathrm{coe}\left(T\right)\right)\right)\right)=\mathrm{bot}\left(\mathbb{C},\mathrm{EuclideanSpace}\left(\mathbb{C},\mathrm{Fin}\left(d\right)\right)\right)))$$

Formalization. D5/S3/Quantum/Measurements/NoncommutingNotCompletelyIncompatible.CompletelyIncompatible (✓ std3).

Citation. Stephan De Bièvre (2023). Relating incompatibility, noncommutativity, uncertainty and Kirkwood-Dirac nonclassicality. DOI: 10.1063/5.0110267. URL: https://arxiv.org/abs/2207.07451v1.

Commentary.

Definition 4, p. 14: “We say that two bases 𝒜 and ℬ are completely incompatible (COINC) if and only if all index sets S,T in ⟦1,d⟧ for which |S|+|T|≤d have the property that Π_𝒜(S)ℋ∩Π_ℬ(T)ℋ={0}.” The image of each coordinate projector is the complex span of the indicated basis vectors. coe denotes SetLike.coe from Finset (Fin d) to Set (Fin d); image is set image; inf is intersection of complex submodules; bot is the zero submodule. Empty index sets remain in scope.

Definition 1.4 (De Bièvre’s conjecture).

$$(claim)\iff (\forall d\in \mathbb{N},(4\le d)\Rightarrow (\exists a\in \mathrm{OrthonormalBasis}\left(\mathrm{Fin}\left(d\right),\mathbb{C},\mathrm{EuclideanSpace}\left(\mathbb{C},\mathrm{Fin}\left(d\right)\right)\right),\exists b\in \mathrm{OrthonormalBasis}\left(\mathrm{Fin}\left(d\right),\mathbb{C},\mathrm{EuclideanSpace}\left(\mathbb{C},\mathrm{Fin}\left(d\right)\right)\right),(\mathrm{NonCommutingProjectors}\left(a,b\right))\wedge (\neg (\mathrm{CompletelyIncompatible}\left(a,b\right)))))$$

Formalization. D5/S3/Quantum/Measurements/NoncommutingNotCompletelyIncompatible.claim (✓ std3).

Citation. Stephan De Bièvre (2023). Relating incompatibility, noncommutativity, uncertainty and Kirkwood-Dirac nonclassicality. DOI: 10.1063/5.0110267. URL: https://arxiv.org/abs/2207.07451v1.

Commentary.

Section 5, p. 16: “We conjecture it is true that (ii) does not imply (i) in all dimensions d ≥ 4, but we have not produced such examples in other dimensions than 4 and 6.” Thus for each natural dimension at least four there exist two arbitrary orthonormal bases satisfying condition (ii) and failing condition (i).

Theorem 1.5 (The conjecture holds in every dimension at least four).

$$claim$$

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

Resolves. Problems/de-bievre-2022-noncommuting-not-coinc-all-dimensions (proved) by D5/S3/Quantum/Measurements/NoncommutingNotCompletelyIncompatible.result.

Source. Repository-derived.

Acknowledgement. Stephan De Bièvre (2023). Relating incompatibility, noncommutativity, uncertainty and Kirkwood-Dirac nonclassicality. DOI: 10.1063/5.0110267. URL: https://arxiv.org/abs/2207.07451v1.

Commentary.

Take the standard basis and the columns of H = I − (2/q)wwᵀ, where w = (1,2,…,2) and q = 4d−3. This real Householder reflection is unitary over the complex space. For a nonempty proper set T let m = ∑_{k∈T} w_k², so 0 < m < q. Every off-diagonal entry of its coordinate projector equals (2w_iw_j/q²)(2m−q(1_T(i)+1_T(j))). The last factor is nonzero: indicator sums zero and two use the strict mass bounds, and indicator sum one uses the oddness of q. A coordinate inside S and one outside S then give a nonzero commutator entry for every nonempty proper S. Finally 2e₀−e₁ is nonzero and equals 2He₀−He₁, so it lies in both two-coordinate spans for S = T = {0,1}. Their total cardinality is four, which is at most d.

References

• Truth anchor: D5/S3/Quantum/Measurements/NoncommutingNotCompletelyIncompatible.CompletelyIncompatible
• Truth anchor: D5/S3/Quantum/Measurements/NoncommutingNotCompletelyIncompatible.NonCommutingProjectors
• Truth anchor: D5/S3/Quantum/Measurements/NoncommutingNotCompletelyIncompatible.claim
• Truth anchor: D5/S3/Quantum/Measurements/NoncommutingNotCompletelyIncompatible.projector
• Truth anchor: D5/S3/Quantum/Measurements/NoncommutingNotCompletelyIncompatible.result