Metric spaces and positive definite functions

Schoenberg, I. J.

doi:10.1090/s0002-9947-1938-1501980-0

Cited by 617 publications

(331 citation statements)

References 9 publications

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“…A proof of this fact can be found in reference [11]. Alternatively, this can be proved starting from some classical results of harmonic analysis due to Schoenberg [16], [17]. The basic property of the JSD that makes Schoenberg theorem applicable is that s JS X is a definite negative kernel, that is, for all finite collection of real number (ζ i ) i≤N and for all corresponding finite sets (x i ) i≤N of points in X + N , the implication…”

Section: Jensen-shannon Divergencementioning

confidence: 93%

Wootters? distance revisited: a new distinguishability criterium

et al. 2005

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The notion of distinguishability between quantum states has shown to be fundamental in the frame of quantum information theory. In this paper we present a new distinguishability criterium by using a information theoretic quantity: the Jensen-Shannon divergence (JSD). This quantity has several interesting properties, both from a conceptual and a formal point of view. Previous to define this distinguishability criterium, we review some of the most frequently used distances defined over quantum mechanics' Hilbert space. In this point our main claim is that the JSD can be taken as a unifying distance between quantum states. 89.70.+c

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Section: Jensen-shannon Divergencementioning

confidence: 93%

Wootters? distance revisited: a new distinguishability criterium

et al. 2005

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“…A proof of this last fact can be found in references [17,18]. Alternatively, this can be proved by using some results of harmonic analysis due to I. Schoenberg [19]. The basic fact that makes Schoenberg's theorem applicable to the classical JSD resides in that it is a definite negative kernel, that is, for all finite collection of real numbers (c i ) i≤N , and for all corresponding probability distributions…”

Section: Classical Jensen-shannon Divergence and Its Quantum Extementioning

confidence: 99%

Metric character of the quantum Jensen-Shannon divergence

et al. 2008

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In a recent paper, the generalization of the Jensen Shannon divergence (JSD) in the context of quantum theory has been studied (Phys. Rev. A 72, 052310 (2005)). This distance between quantum states has shown to verify several of the properties required for a good distinguishability measure. Here we investigate the metric character of this distance. More precisely we show, formally for pure states and by means of a numerical procedure for mixed states, that its square root verifies the triangle inequality.

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“…Отметим простые факты, связанные с изложенным. Описание функций с указанным выше свойством восходит к работе Шенберга [4], этот круг вопросов рассмотрен также в работе [5]. Из результатов этих работ выте-кает, что указанные функции допускают представление…”

Section: случайный процессunclassified

Об асимптотике спектра операторов, связанных с $p-$ адическим полями

Ismagilov¹

2014

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Metric spaces and positive definite functions

Cited by 617 publications

References 9 publications

Wootters? distance revisited: a new distinguishability criterium

Wootters? distance revisited: a new distinguishability criterium

Metric character of the quantum Jensen-Shannon divergence

Об асимптотике спектра операторов, связанных с $p-$ адическим полями

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