Maximum-Entropy Inference and Inverse Continuity of the Numerical Range

Weis, Stephan

doi:10.1016/s0034-4877(16)30022-2

Cited by 7 publications

(5 citation statements)

References 52 publications

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“…We prove that points of discontinuity of ρ * A correspond to crossings of class C 1 between the ground state energy λ and a higher energy level. This was proved earlier [76] using functional analysis and a result [45] about lower semi-continuity of the (set-valued) inverse of the numerical range map |x → x|Ax . Here we give a direct proof using extensions x W,± of the reverse Gauss map x W , which parametrize homeomorphically all sufficiently small one-sided neighborhoods in the set of smooth extreme points of W , which contains all discontinuities of ρ * A .…”

Section: Introductionmentioning

confidence: 55%

“…The main results of this section were proved earlier [76]. To prove Theorem 5.3, the following Theorem 6.1 on the inverse numerical range map f −1 A was translated to ρ * A by exploiting that the state space M d is stable [56,64], which means that the mid-point map (ρ, σ) → 1 2 (ρ + σ) is open.…”

Section: It Follows Immediately From Theorem 53 and (53) That ρ *mentioning

confidence: 91%

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Signatures of quantum phase transitions from the boundary of the numerical range

Spitkovsky

Weis

2018

Journal of Mathematical Physics

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The ground state energy of a finite-dimensional one-parameter Hamiltonian and the continuity of a maximum-entropy inference map are discussed in the context of quantum critical phenomena. The domain of the inference map is a convex compact set in the plane, called the numerical range. We study the differential geometry of its boundary in relation to the ground state energy. We prove that discontinuities of the inference map correspond to C 1 -smooth crossings of the ground state energy with a higher energy level. Discontinuities may appear only at C 1 -smooth points of the boundary of the numerical range considered as a manifold. Discontinuities exist at all C 2 -smooth non-analytic boundary points and are essentially stronger than at analytic points or at points which are merely C 1 -smooth (non-exposed points).2010 Mathematics Subject Classification. 82B26, 52A10, 47A12, 15A18, 32C25, 62F30, 94A17, 54C08, 46T20.

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Section: Introductionmentioning

confidence: 55%

Section: It Follows Immediately From Theorem 53 and (53) That ρ *mentioning

confidence: 91%

Signatures of quantum phase transitions from the boundary of the numerical range

Spitkovsky

Weis

2018

Journal of Mathematical Physics

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“…6, this is closely connected to the failure of Kippenhahn's conjecture. On the other hand, one of us has recently shown [47] that for all d ∈ N the maximumentropy inference ρ * : W (u 1 , u 2 ) → M d has at most finitely many discontinuities.…”

Section: Multiply Generated Round Boundary Pointsmentioning

confidence: 99%

Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach

Rodman

Spitkovsky

Szkola

et al. 2015

Journal of Mathematical Physics

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We study the continuity of an abstract generalization of the maximum-entropy inference -a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we use numerical range methods in the case of quantum inference which refers to two observables. One result is a complete characterization of points of discontinuity for 3 × 3 matrices.

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“…As it happens, the described continuity problems are equivalent. The map ρ * is continuous at α ∈ W (A) if and only if f −1 A is strongly continuous at α [36], and ρ * is indeed discontinuous at all isolated multiply generated round boundary points of W (A), see Sec. 6 of [29].…”

Section: Introductionmentioning

confidence: 99%

Pre-images of extreme points of the numerical range, and applications

Spitkovsky¹,

Weis

2016

Operators and Matrices

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Abstract. We extend the pre-image representation of exposed points of the numerical range of a matrix to all extreme points. With that we characterize extreme points which are multiply generated, having at least two linearly independent pre-images, as the extreme points which are Hausdorff limits of flat boundary portions on numerical ranges of a sequence converging to the given matrix. These studies address the inverse numerical range map and the maximum-entropy inference map which are continuous functions on the numerical range except possibly at certain multiply generated extreme points. This work also allows us to describe closures of subsets of 3-by-3 matrices having the same shape of the numerical range.Mathematics subject classification (2010): 47A12, 54C10, 62F30, 94A17.

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Maximum-Entropy Inference and Inverse Continuity of the Numerical Range

Cited by 7 publications

References 52 publications

Signatures of quantum phase transitions from the boundary of the numerical range

Signatures of quantum phase transitions from the boundary of the numerical range

Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach

Pre-images of extreme points of the numerical range, and applications

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