A remark on quasi-isometries

It d−pends. Wigner's symmetry theorem implies that transformations that preserve transition probabilities of pure quantum states are linear maps on the level of density operators. We investigate the stability of this implication. On the one hand, we show that any transformation that preserves transition probabilities up to an additive ε in a separable Hilbert space admits a weak linear approximation, i.e. one relative to any fixed observable. This implies the existence of a linear approximation that is 4 √ εd-close in Hilbert-Schmidt norm, with d the Hilbert space dimension. On the other hand, we prove that a linear approximation that is close in norm and independent of d does not exist in general. To this end, we provide a lower bound that depends logarithmically on d. CONTENTS

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“…In order to close this gap, more sophisticated tools from Banach space theory might be useful (see end remark in Ref. [13]).…”

Section: Discussionmentioning

confidence: 99%

Are almost-symmetries almost linear?

Cuesta

Wolf

2019

Journal of Mathematical Physics

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“…In the work of N. J. Kalton [1][2][3], we can find novel ideas and methods for the stability of functional equations that depart from the classical methods of Hyers, Ulam and Rassias [4]. In Ref.…”

Section: Introductionmentioning

confidence: 99%

“…In Ref. [3] (see Theorem 2.2), Kalton provides a sharp bound on the stability of the additive map in R n for the so-called singular case. His proof makes use of probabilistic and geometric methods in Banach space theory.…”

Section: Introductionmentioning

confidence: 99%

Type and Cotype Constants and the Linear Stability of Wigner’s Symmetry Theorem

Cuesta

2019

Symmetry

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show abstract

“…In the work of N. J. Kalton [1][2][3] we can find novel ideas and methods for the stability of functional equations which depart from the classical methods of Hyers, Ulam and Rassias [4]. In Ref.…”

Section: Introductionmentioning

confidence: 99%

“…In Ref. [1] (see Theorem 2.2) Kalton provides a sharp bound on the stability of the additive map in R n for the so-called singular case. His proof makes use of probabilistic and geometric methods in Banach space theory.…”

Section: Introductionmentioning

confidence: 99%

(Co-)type and the linear stability of Wigner's symmetry theorem

Cuesta¹

2019

Preprint

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We study the relation between the linear stability of almost-symmetries and the geometry of the Banach spaces on which these transformations are defined. We show that any transformation between finite dimensional Banach spaces that preserves transition probabilities up to an additive error admits an approximation by a linear map, and the quality of the approximation depends on the type and cotype constants of the involved spaces.

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A remark on quasi-isometries

Abstract: Abstract. We show that if f : Bn → R n is an −quasi-isometry, with < 1, defined on the unit ball Bn of R n , then there is an affine isometry h : Bn → R n with f (x)−h(x) ≤ C (1+log n) where C is a universal constant. This result is sharp.

Cited by 19 publications

References 13 publications

Are almost-symmetries almost linear?

Are almost-symmetries almost linear?

Type and Cotype Constants and the Linear Stability of Wigner’s Symmetry Theorem

(Co-)type and the linear stability of Wigner's symmetry theorem

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