Almost commuting unitaries with spectral gap are near commuting unitaries

Osborne, Tobias J.

doi:10.1090/s0002-9939-09-10026-6

Cited by 4 publications

(7 citation statements)

References 18 publications

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“…Note that we then obtain a stronger form of Osborne's result in [11] only requiring one matrix to have a spectral gap. We only need to transform one unitary into a Hermitian matrix, but we use a modification of a fractional linear transformation instead.…”

Section: Almost Commuting Hermitian and Normal Matricesmentioning

confidence: 71%

“…The geometry of a square, which is Some other examples (which we will discuss below) are that of almost commuting Hermitian and unitary matrices (which is related to the geometry of a cylinder/anulus) and two almost commuting unitaries (the geometry of the torus). The latter does not always have nearby commuting matrices, but [11] showed that if both unitaries have a spectral gap then one obtains nearby commuting unitaries. The proof given there involves using a matrix logarithm to reduce the unitary matrices with a spectral gap into Hermitian matrices.…”

Section: Almost Commuting Hermitian and Normal Matricesmentioning

confidence: 99%

“…Proof. Just as in [11], we can multiply V by a phase so that the spectral gap is centered at 1. Set f θ (x) = x−i x+i and g θ (z) = i 1+z 1−z .…”

Section: Almost Commuting Hermitian and Normal Matricesmentioning

confidence: 99%

“…(For example, proving (Q ) as in Remark 7.5. ) First, one can use these estimates to get results for other types of matrices, like two unitary matrices with a spectral gap in [11] or other situations like in [9]. We discuss a few of these in Section 18.…”

Section: Introductionmentioning

confidence: 99%

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Constructive Approaches to Lin's Theorem for Almost Commuting Matrices

Herrera¹

2020

Preprint

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Lin's theorem states that for all > 0, there is a δ > 0 such that for allWe present full details of the approach in [1], which seemingly is the closest result to a general constructive proof of Lin's theorem. Constructive results for some special cases are presented along with applications to the problem of almost commuting matrices where B is assumed to be normal and also to macroscopic observables.

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Section: Almost Commuting Hermitian and Normal Matricesmentioning

confidence: 71%

Section: Almost Commuting Hermitian and Normal Matricesmentioning

confidence: 99%

“…Proof. Just as in [11], we can multiply V by a phase so that the spectral gap is centered at 1. Set f θ (x) = x−i x+i and g θ (z) = i 1+z 1−z .…”

Section: Almost Commuting Hermitian and Normal Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Constructive Approaches to Lin's Theorem for Almost Commuting Matrices

Herrera¹

2020

Preprint

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“…For unitary matrices the above is however known to be generally false [11]. This is due to a K-theoretic obstruction [12][13][14], though it is true if this obstruction vanishes [7,8,15], or under the assumption of a spectral gap [16]. Imposing a form of self-duality analogous to time-reversal symmetry the relevant K-theoretic obstruction reduces to the spin Chern number of a fermionic system [17], highlighting a link between the fields of topologically ordered quantum systems [18] and approximately commuting matrices.…”

Section: Introductionmentioning

confidence: 99%

Approximate symmetries of Hamiltonians

Chubb

Flammia

2017

Journal of Mathematical Physics

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We explore the relationship between approximate symmetries of a gapped Hamiltonian and the structure of its ground space. We start by considering approximate symmetry operators, defined as unitary operators whose commutators with the Hamiltonian have norms that are sufficiently small. We show that when approximate symmetry operators can be restricted to the ground space while approximately preserving certain mutual commutation relations. We generalize the Stone-von Neumann theorem to matrices that approximately satisfy the canonical (Heisenberg-Weyl-type) commutation relations, and use this to show that approximate symmetry operators can certify the degeneracy of the ground space even though they only approximately form a group. Importantly, the notions of "approximate" and "small" are all independent of the dimension of the ambient Hilbert space, and depend only on the degeneracy in the ground space. Our analysis additionally holds for any gapped band of sufficiently small width in the excited spectrum of the Hamiltonian, and we discuss applications of these ideas to topological quantum phases of matter and topological quantum error correcting codes. Finally, in our analysis we also provide an exponential improvement upon bounds concerning the existence of shared approximate eigenvectors of approximately commuting operators under an added normality constraint, which may be of independent interest.

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Almost commuting matrices, localized Wannier functions, and the quantum Hall effect

Hastings¹,

Loring²

2010

Journal of Mathematical Physics

104

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On the oscillator realization of conformal U(2, 2) quantum particles and their particle-hole coherent states On a classification of irreducible almost commutative geometries, a second helping For models of noninteracting fermions moving within sites arranged on a surface in three-dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are K-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward and does not require translational invariance or introduce a flux torus. We further show that there is a Z 2 index obstruction to approximating almost commuting self-dual matrices by exactly commuting self-dual matrices and present additional conjectures regarding the approximation of almost commuting real and self-dual matrices by exactly commuting real and self-dual matrices. The motivation for considering this problem is the case of physical systems with additional antiunitary symmetries such as time-reversal or particle-hole conjugation. Finally, in the case of the sphere-mathematically speaking, three almost commuting Hermitians whose sum of square is near the identity-we give the first quantitative result, showing that this index is the only obstruction to finding commuting approximations. We review the known nonquantitative results for the torus.Given a list of bounded operators on infinite dimensional Hilbert space, it is often natural to seek a finite-rank projection P that almost commutes with that set. The C ‫ء‬ -algebraist would do so in the study of quasidiagonality. 1-3 In physics, we are interested in a projection onto a band of energy states separated from the rest of the spectrum by an energy gap; assuming the underlying Hamiltonian is local, this projection will itself be local due to the gap, and hence will approximately commute with a list of observables.Whatever exact relations might be known to hold for the original operators ͑H 1 , ... ,H r ͒ will generally hold only approximately for the compressions ͑PX 1 P , ... , PX r P͒. In a lattice model, the projection might be from a finite dimensional space to a space whose dimension is much lower, but still the outcome is finite-rank operators that approximately satisfy some relations. Can these be approximated by finite-rank operators that exactly satisfy those relations?For example, if X 1 and X 2 in B͑H͒ satisfy −I Յ X j Յ I and ͓X 1 , X 2 ͔ = 0, then P almost commuting with the X j implies − I Յ PX j P Յ I, a͒ Electronic We especially want to know if these almost commuting Hermitian operators are close to commuting Hermitian operators in the corner PB͑H͒P Х M k ͑C͒. It is sufficient to answer this question: can two almost commuting Hermitian matrices be approximated by commuting Hermitian matrices? The answer is yes. This is Lin's theorem. 4 The si...

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Almost commuting unitaries with spectral gap are near commuting unitaries

Cited by 4 publications

References 18 publications

Constructive Approaches to Lin's Theorem for Almost Commuting Matrices

Constructive Approaches to Lin's Theorem for Almost Commuting Matrices

Approximate symmetries of Hamiltonians

Almost commuting matrices, localized Wannier functions, and the quantum Hall effect

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