Joint numerical range and its generating hypersurface

Chien, Mao-Ting; Nakazato, Hiroshi

doi:10.1016/j.laa.2009.07.034

Cited by 18 publications

(25 citation statements)

References 12 publications

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“…The dual variety S * F ⊂ P n * is the complex projective variety which is the closure of the set of tangent hyperplanes of S F at smooth points [22,28,24]. The boundary generating hypersurface [15] of F is the real affine part of the dual variety, S * F (R) := {(x 1 , . .…”

Section: Remark 13 (Real Varieties)mentioning

confidence: 99%

“…Despite the long history of the problem [8,37,14,15,12], a classification of the joint numerical range of triples of matrices (n = 3) is unknown even in the case d = n = 3. Our motivation to tackle this problem is quantum mechanics, as we explain in Section 2.…”

Section: Introductionmentioning

confidence: 99%

“…Ovals, where (e, s) = (0, 0), are studied in [37]. An example of (e, s) = (4, 0) is in [32], one of (e, s) = (1, 1) is in [15], and one of (e, s) = (2, 1) is in [9].…”

Section: Introductionmentioning

confidence: 99%

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Classification of joint numerical ranges of three hermitian matrices of size three

Szymański

Weis

Życzkowski

2018

Linear Algebra and its Applications

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Abstract. The joint numerical range W (F ) of three hermitian 3-by-3 matrices F = (F 1 , F 2 , F 3 ) is a convex and compact subset in R 3 . We show that W (F ) is generically a three-dimensional oval. Assuming dim(W (F )) = 3, every one-or two-dimensional face of W (F ) is a segment or a filled ellipse. We prove that only ten configurations of these segments and ellipses are possible. We identify a triple F for each class and illustrate W (F ) using random matrices and dual varieties.

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Section: Remark 13 (Real Varieties)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classification of joint numerical ranges of three hermitian matrices of size three

Szymański

Weis

Życzkowski

2018

Linear Algebra and its Applications

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“…Numerical ranges. We are looking forward to exploring how finite-order differentiability of ∂W connects to algebraic curves [40,20] and critical value curves [38,36,37] of W . We hope that our two-dimensional results will be useful to understand higher-dimensional projections of state spaces, as they appear in the context of entanglement [57] and state representation problems [55].…”

Section: Introductionmentioning

confidence: 99%

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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“…, (A m x, x)). These concepts are useful in studying the joint behaviors of several operators, and have been studied extensively; see for example [1,4,6,9,11] and their references.…”

Section: Introductionmentioning

confidence: 99%

Maps preserving the joint numerical radius distance of operators

Poon

2012

Linear Algebra and its Applications

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Dedicated to Professor David Lutzer on the occasion of his retirement. AbstractDenote the joint numerical radius of an m-tuple of bounded operators A = (A 1 , . . . , A m ) by w(A). We give a complete description of maps f satisfying w(A − B) = w(f (A) − f (B)) for any two m-tuples of operators A = (A 1 , . . . , A m ) and B = (B 1 , . . . , B m ). We also characterize linear isometries for the joint numerical radius, and maps preserving the joint numerical range of A.

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Joint numerical range and its generating hypersurface

Cited by 18 publications

References 12 publications

Classification of joint numerical ranges of three hermitian matrices of size three

Classification of joint numerical ranges of three hermitian matrices of size three

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

Maps preserving the joint numerical radius distance of operators

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