scholarly journals Implementation of comparative probability by normal states. Infinite dimensional case

1990 ◽  
Vol 132 (3) ◽  
pp. 581-592 ◽  
Author(s):  
Simba A. Mutangadura
Keyword(s):  
Dimensional Case ◽  
Infinite Dimensional ◽  
Comparative Probability ◽  
Normal States

CHAOTIC BEHAVIOR OF ORBITS CLOSE TO A HETEROCLINIC CONTOUR

1996 ◽  
Vol 06 (01) ◽  
pp. 69-79 ◽  
Author(s):  
M. BLÁZQUEZ ◽  
E. TUMA
Keyword(s):  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Chua’S Circuit ◽  
Closed Contour ◽  
Chua's Circuit ◽  
Infinite Dimensional ◽  
Focus Type

We study the behavior of the solutions in a neighborhood of a closed contour formed by two heteroclinic connections to two equilibrium points of saddle-focus type. We consider both the three-dimensional case, as in the well-known Chua's circuit, as well as the infinite-dimensional case.

scholarly journals Directed-Completeness of Quantum Statistical Experiments in the Randomization Order

2020 ◽  
Vol 27 (01) ◽  
pp. 2050003
Author(s):  
Yui Kuramochi
Keyword(s):  
Equivalence Relations ◽  
Statistical Experiment ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Statistical Experiments ◽  
Classical Statistics ◽  
Outcome Space ◽  
Linear Amplifier ◽  
Normal States ◽  
Ideal Quantum

A parametrized family of normal states on a von Neumann algebra is called a statistical experiment, which generalizes the corresponding concepts in classical statistics and finite-dimensional quantum systems. We introduce randomization preorder and equivalence relations for statistical experiments with a fixed parameter set and for normal channels with a fixed input space by post-processing completely positive channels. In this paper, we prove that the set of equivalence classes of statistical experiments or those of normal channels is an upper and lower directed-complete partially ordered set with respect to the randomization order, i.e. any increasing or decreasing net of statistical experiments or channels has its supremum or infimum in the randomization order. We also show that if the outcome space of each statistical experiment or channel of a randomization-monotone net is commutative, the outcome space of the supremum or infimum can also be taken to be commutative. We consider two examples of homogeneous Markov processes of channels on infinite-dimensional separable Hilbert spaces, namely block-diagonalization with irrational translation and ideal quantum linear amplifier channels, and explicitly derive their infima. Throughout the paper, the concept of channel conjugation is used to obtain results for decreasing channels from those for increasing channels.

scholarly journals A new proof of Donoghue's interpolation theorem

2004 ◽  
Vol 2 (3) ◽  
pp. 253-265 ◽  
Author(s):  
Yacin Ameur
Keyword(s):  
Hilbert Space ◽  
Radon Measure ◽  
Interpolation Theorem ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Positive Radon Measure ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
New Interpretation ◽  
Necessary And Sufficient

We give a new proof and new interpretation of Donoghue's interpolation theorem; for an intermediate Hilbert spaceH∗to be exact interpolation with respect to a regular Hilbert coupleH¯it is necessary and sufficient that the norm inH∗be representable in the form‖f‖∗=(∫[0,∞](1+t−1)K2(t,f;H¯)2dρ(t))1/2with some positive Radon measureρon the compactified half-line[0,∞]. The result was re-proved in [1] in the finite-dimensional case. The purpose of this note is to extend the proof given in [1] to cover the infinite-dimensional case. Moreover, the presentation of the aforementioned proof in [1] was slightly flawed, because we forgot to include a reference to ‘Donoghue's Lemma’, which is implicitly used in the proof. Hence we take this opportunity to correct that flaw.

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