Almost Commuting Unitary Matrices

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the finite dimensional unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius normOur strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain non-residually finite central extensions of lattices in some simple p-adic Lie groups. These groups act on high rank Bruhat-Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

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“…Z 2 is not Frobenius-stable. In [51], Voiculescu proved that the matrices [21,51]). By the inequalities…”

Section: Some Examples Of Non Frobenius-stable Groupsmentioning

confidence: 99%

Stability, Cohomology Vanishing, and Nonapproximable Groups

Chiffre

Glebsky

Lubotzky

et al. 2020

Forum of Mathematics, Sigma

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“…The proof draws on a connection to homology, in particular using a homotopy invariant to establish the lower bound on the distance to commuting approximants. A succinct and elementary proof of the result is given by Exel and Loring [EL89].…”

Section: Limits For Commuting Approximantsmentioning

confidence: 99%

Interactive Proofs with Approximately Commuting Provers

Coudron

Vidick

2015

Automata, Languages, and Programming

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The class MIP * of promise problems that can be decided through an interactive proof system with multiple entangled provers provides a complexity-theoretic framework for the exploration of the nonlocal properties of entanglement. Very little is known in terms of the power of this class. The only proposed approach for establishing upper bounds is based on a hierarchy of semidefinite programs introduced independently by Pironio et al. and Doherty et al. in 2006. This hierarchy converges to a value, the fieldtheoretic value, that is only known to coincide with the provers' maximum success probability in a given proof system under a plausible but difficult mathematical conjecture, Connes' embedding conjecture.No bounds on the rate of convergence are known.We introduce a rounding scheme for the hierarchy, establishing that any solution to its N-th level can be mapped to a strategy for the provers in which measurement operators associated with distinct provers have pairwise commutator bounded by O(ℓ 2 / √ N) in operator norm, where ℓ is the number of possible answers per prover.Our rounding scheme motivates the introduction of a variant of quantum multiprover interactive proof systems, called MIP * δ , in which the soundness property is required to hold against provers allowed to operate on the same Hilbert space as long as the commutator of operations performed by distinct provers has norm at most δ. Our rounding scheme implies the upper bound MIP * δ ⊆ DTIME(exp(exp(poly)/δ 2 )).In terms of lower bounds we establish that MIP * 2 − poly , with completeness 1 and soundness 1 − 2 − poly , contains NEXP. The relationship of MIP * δ to MIP * has connections with the mathematical literature on approximate commutation. Our rounding scheme gives an elementary proof that the Strong Kirchberg Conjecture implies that MIP * is computable. We also discuss applications to device-independent cryptography.

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“…The proof of this will be broken into lemmas and propositions. Theorem 2.3 is a variation on the main result in [5]. That result had a smaller lower bound, but the bound applied to the distance to any pair of commuting matrices, not just commuting unitary matrices.…”

Section: The Winding Number Invariantmentioning

confidence: 99%

Quantitative K-Theory Related to Spin Chern Numbers

Loring¹

2014

SIGMA

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Abstract. We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine which values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to be correct. Specifically, the Pfaffian-Bott index can be computed by the "log method" for commutator norms up to a specific constant.

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Almost Commuting Unitary Matrices

Abstract: Abstract. A pair of square matrices is said to be almost commuting if their commutator is small in norm. We give an elementary proof of a theorem of Voiculescu showing that not all almost commuting pairs can be perturbed to a commuting pair.

Cited by 11 publications

References 2 publications

Stability, Cohomology Vanishing, and Nonapproximable Groups

Stability, Cohomology Vanishing, and Nonapproximable Groups

Interactive Proofs with Approximately Commuting Provers

Quantitative K-Theory Related to Spin Chern Numbers

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