Semidefinite Representation for Convex Hulls of Real Algebraic Curves

Scheiderer, Claus

doi:10.1137/17m1115113

Cited by 27 publications

(23 citation statements)

References 28 publications

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“…The proof of the Helton-Nie conjecture in dimension two can be found in [35]. 4 The main observation of this section is a consequence of the previous theorem and the following simple Lemma: Lemma 9.…”

Section: Sdps For General Povmsmentioning

confidence: 87%

Universality and Optimality in the Information–Disturbance Tradeoff

Hashagen

Wolf

2018

Ann. Henri Poincaré

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We investigate the tradeoff between the quality of an approximate version of a given measurement and the disturbance it induces in the measured quantum system. We prove that if the target measurement is a non-degenerate von Neumann measurement, then the optimal tradeoff can always be achieved within a two-parameter family of quantum devices that is independent of the chosen distance measures. This form of almost universal optimality holds under mild assumptions on the distance measures such as convexity and basis-independence, which are satisfied for all the usual cases that are based on norms, transport cost functions, relative entropies, fidelities, etc. for both worst-case and average-case analysis. We analyze the case of the cb-norm (or diamond norm) more generally for which we show dimension-independence of the derived optimal tradeoff for general von Neumann measurements. A SDP solution is provided for general POVMs and shown to exist for arbitrary convex semialgebraic distance measures.

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Section: Sdps For General Povmsmentioning

confidence: 87%

Universality and Optimality in the Information–Disturbance Tradeoff

Hashagen

Wolf

2018

Ann. Henri Poincaré

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“…This would provide examples of real algebraic varieties whose convex hull is a spectrahedral shadow. This topic has been studied for instance in [Sch18a,RS10] and is related to the Helton-Nie conjecture [HN10]. Such questions draw connections between discotopes and the world of convex algebraic geometry, optimization and semidefinite programming.…”

Section: Discussionmentioning

confidence: 99%

The Geometry of Discotopes

Gesmundo¹,

Meroni²

2021

Preprint

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We study a class of semialgebraic convex bodies called discotopes. These are instances of zonoids, objects of interest in real algebraic geometry and random geometry. We focus on the face structure and on the boundary hypersurface of discotopes, highlighting interesting birational properties which may be investigated using tools from algebraic geometry. When a discotope is the Minkowski sum of two-dimensional discs, the Zariski closure of its set of extreme points is an irreducible hypersurface. In this case, we provide an upper bound for the degree of the hypersurface, drawing connections to the theory of classical determinantal varieties.

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“…In fact, for every semi-algebraic set S ⊆ R n of dimension at least two we prove that there exist polynomial maps ϕ : R n → R m for which the closed convex hull of ϕ(S) in R m has no semidefinite representation. This is in marked contrast to the case where S has dimension one, when it is known that the closed convex hull of S is always a spectrahedral shadow [36].…”

Section: Introductionmentioning

confidence: 88%