Non-Commutative Martingale Transforms

Randrianantoanina, Narcisse

doi:10.1006/jfan.2002.3952

Cited by 34 publications

(4 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [52], the continuum limit of discrete toy models for holography was studied finding that, generically, this limit is extremely discontinuous. appearing (31) (recall that X i = M k 2 ,k k , X ii = S k ,n ⊗ (ε,π )1 /2 S k ,n 1 ). Recall also that ∂ i j (ε) = (ε 11 ,...,ε i ,...,ε nn )− (ε 11 ,...,−ε i j ,...,ε nn ) 2…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Geometry of Banach Spaces: A New Route Towards Position Based Cryptography

Junge

Kubicki

Palazuelos

et al. 2022

Commun. Math. Phys.

View full text Add to dashboard Cite

In this work we initiate the study of position based quantum cryptography (PBQC) from the perspective of geometric functional analysis and its connections with quantum games. The main question we are interested in asks for the optimal amount of entanglement that a coalition of attackers have to share in order to compromise the security of any PBQC protocol. Known upper bounds for that quantity are exponential in the size of the quantum systems manipulated in the honest implementation of the protocol. However, known lower bounds are only linear. In order to deepen the understanding of this question, here we propose a position verification (PV) protocol and find lower bounds on the resources needed to break it. The main idea behind the proof of these bounds is the understanding of cheating strategies as vector valued assignments on the Boolean hypercube. Then, the bounds follow from the understanding of some geometric properties of particular Banach spaces, their type constants. Under some regularity assumptions on the former assignment, these bounds lead to exponential lower bounds on the quantum resources employed, clarifying the question in this restricted case. Known attacks indeed satisfy the assumption we make, although we do not know how universal this feature is. Furthermore, we show that the understanding of the type properties of some more involved Banach spaces would allow to drop out the assumptions and lead to unconditional lower bounds on the resources used to attack our protocol. Unfortunately, we were not able to estimate the relevant type constant. Despite that, we conjecture an upper bound for this quantity and show some evidence supporting it. A positive solution of the conjecture would lead to stronger security guarantees for the proposed PV protocol providing a better understanding of the question asked above.

show abstract

Section: Discussionmentioning

confidence: 99%

“…• we estimate UMD(M n,m ) from known bounds for the UMD constant of the p-Schatten class S p , for 1 < p < ∞. It is known that these spaces are UMD and the following estimate for UMD(S p ) is available [31]:…”

Section: Some Key Estimates Of Type Constantsmentioning

confidence: 99%

Geometry of Banach Spaces: A New Route Towards Position Based Cryptography

Junge

Kubicki

Palazuelos

et al. 2022

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…A lot of classical martingale inequalities have been generalised to the noncommutative setting (see e.g. [23,24,25,36,37,38]). Among these articles, the work due to Junge and Xu [25] is of special importance.…”

Section: Introductionmentioning

confidence: 99%

“…(ST) for 1 < p ≤ 2 follows from (ST) for p ≥ 2 by duality. For (MT), Junge and Xu refer to [36], where the estimate is established for semifinite von Neumann algebras.…”

Section: Introductionmentioning

confidence: 99%

Johnson–Schechtman inequalities for noncommutative martingales

Jiao

Sukochev²,

Zanin³

et al. 2017

Journal of Functional Analysis

View full text Add to dashboard Cite

We establish distributional estimates for noncommutative martingales, in the sense of decreasing rearrangements of the spectra of unbounded operators, which generalises the study of distributions of random variables. Our results include distributional versions of the noncommutative Stein, dual Doob, martingale transform and Burkholder-Gundy inequalities. Our proof relies upon new and powerful extrapolation theorems. As an application, we obtain some new martingale inequalities in symmetric quasi-Banach operator spaces and some interesting endpoint estimates. Our main approach demonstrates a method to build the noncommutative and classical probabilistic inequalities in an entirely operator theoretic way.

show abstract

Weak Type Estimates for the Noncommutative Vilenkin–Fourier Series

Scheckter

Sukochev

2018

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

Non-Commutative Martingale Transforms

Cited by 34 publications

References 28 publications

Geometry of Banach Spaces: A New Route Towards Position Based Cryptography

Geometry of Banach Spaces: A New Route Towards Position Based Cryptography

Johnson–Schechtman inequalities for noncommutative martingales

Weak Type Estimates for the Noncommutative Vilenkin–Fourier Series

Contact Info

Product

Resources

About