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On Certainty and Generalized Information Measures

Pradeep Kumar Bhatia

Published Jan 1, 2010• 6 citations
OpenAlex

Abstract

By considering the relation between certainty and information, three generalized classes of information measures have been derived, which include the information measures studied by Van der Lubbe et al. [12] as their special cases.

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Abstract Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete positive, operator-valued measure (POVMs) that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical two-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.

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Errors in Nobel Prize for Physics (6) ──Improper Heisenberg Uncertainty Principle

Fu Yuhua

One of the reasons for 1932 Nobel Prize for physics is “for the creation of quantum mechanics”. As well-known, the most famous contribution of Heisenberg is therefore one of the most importment reasons for Heisenberg was awarded the Nobel Prize is the creation of principle. While, this paper points out that the original principle is improper. Considering all the possible situations (including the case that people can create laws), the author presents certainty-uncertainty with general form and variable dimension fractal form. According to the classification of Neutrosophy, certainty-uncertainty can be divided into three principles in different conditions: certainty principle, namely a particle’s position and momentum can be known simultaneously; uncertainty principle, namely a particle’s position and momentum cannot be known simultaneously; and neutral (fuzzy) indeterminacy principle, namely whether or not a particle’s position and momentum can be known simultaneously is undetermined. The special cases of certainty-uncertainty include the original principle and Ozawa inequality. In addition, in accordance with the original discussing high-speed particle’s speed and track with Newton mechanics is unreasonable; but according to certainty-uncertainty Newton mechanics can be used to discuss the problem of gravitational defection of a photon orbit around the Sun (it gives the same result of deflection angle as given by general relativity). Finally, for the reason that in physics the principles, laws and the like that are regardless of the principle (law) of conservation of energy may be invalid; therefore certainty-uncertainty should be restricted (or constrained) by principle (law) of conservation of energy, and thus it can satisfy the principle (law) of conservation of energy.

•May 1, 2015•0 citations

State-dependent and state-independent uncertainty relations for skew information and standard deviation

Sahil

Abstract In this work, we derive state-dependent uncertainty relations (uncertainty equalities) in which commutators of incompatible operators (not necessarily Hermitian) are explicitly present and state-independent uncertainty relations based on the Wigner-Yanase (-Dyson) skew information. We derive uncertainty equality based on standard deviation for incompatible operators with mixed states, a generalization of previous works in which only pure states were considered. We show that for pure states, the Wigner-Yanase skew information based state-independent uncertainty relations become standard deviation based state-independent uncertainty relations which turn out to be tighter uncertainty relations for some cases than the ones given in previous works, and we generalize the previous works for arbitrary operators. As the Wigner-Yanase skew information of a quantum channel can be considered as a measure of quantum coherence of a density operator with respect to that channel, we show that there exists a state-independent uncertainty relation for the coherence measures of the density operator with respect to a collection of different channels. We show that state-dependent and state-independent uncertainty relations based on a more general version of skew information called generalized skew information which includes the Wigner-Yanase (-Dyson) skew information and the Fisher information as special cases hold. In qubits, we derive tighter state-independent uncertainty inequalities for different form of generalized skew informations and standard deviations, and state-independent uncertainty equalities involving generalized skew informations and standard deviations of spin operators along three orthogonal directions. Finally, we provide a scheme to determine the Wigner-Yanase (-Dyson) skew information of an unknown observable which can be performed in experiment using the notion of weak values.

arXiv:2402.03159•Physica Scripta•Oct 2, 2024•1 citation

On the q-parameter spectrum of generalized information-entropy measures with no cut-off prescriptions

Marco Masi

arXiv:cond-mat/0605529•Physics Letters A•May 27, 2006•3 citations
Abstract Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete positive, operator-valued measure (POVMs) that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical two-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.
One of the reasons for 1932 Nobel Prize for physics is “for the creation of quantum mechanics”. As well-known, the most famous contribution of Heisenberg is therefore one of the most importment reasons for Heisenberg was awarded the Nobel Prize is the creation of principle. While, this paper points out that the original principle is improper. Considering all the possible situations (including the case that people can create laws), the author presents certainty-uncertainty with general form and variable dimension fractal form. According to the classification of Neutrosophy, certainty-uncertainty can be divided into three principles in different conditions: certainty principle, namely a particle’s position and momentum can be known simultaneously; uncertainty principle, namely a particle’s position and momentum cannot be known simultaneously; and neutral (fuzzy) indeterminacy principle, namely whether or not a particle’s position and momentum can be known simultaneously is undetermined. The special cases of certainty-uncertainty include the original principle and Ozawa inequality. In addition, in accordance with the original discussing high-speed particle’s speed and track with Newton mechanics is unreasonable; but according to certainty-uncertainty Newton mechanics can be used to discuss the problem of gravitational defection of a photon orbit around the Sun (it gives the same result of deflection angle as given by general relativity). Finally, for the reason that in physics the principles, laws and the like that are regardless of the principle (law) of conservation of energy may be invalid; therefore certainty-uncertainty should be restricted (or constrained) by principle (law) of conservation of energy, and thus it can satisfy the principle (law) of conservation of energy.
Abstract In this work, we derive state-dependent uncertainty relations (uncertainty equalities) in which commutators of incompatible operators (not necessarily Hermitian) are explicitly present and state-independent uncertainty relations based on the Wigner-Yanase (-Dyson) skew information. We derive uncertainty equality based on standard deviation for incompatible operators with mixed states, a generalization of previous works in which only pure states were considered. We show that for pure states, the Wigner-Yanase skew information based state-independent uncertainty relations become standard deviation based state-independent uncertainty relations which turn out to be tighter uncertainty relations for some cases than the ones given in previous works, and we generalize the previous works for arbitrary operators. As the Wigner-Yanase skew information of a quantum channel can be considered as a measure of quantum coherence of a density operator with respect to that channel, we show that there exists a state-independent uncertainty relation for the coherence measures of the density operator with respect to a collection of different channels. We show that state-dependent and state-independent uncertainty relations based on a more general version of skew information called generalized skew information which includes the Wigner-Yanase (-Dyson) skew information and the Fisher information as special cases hold. In qubits, we derive tighter state-independent uncertainty inequalities for different form of generalized skew informations and standard deviations, and state-independent uncertainty equalities involving generalized skew informations and standard deviations of spin operators along three orthogonal directions. Finally, we provide a scheme to determine the Wigner-Yanase (-Dyson) skew information of an unknown observable which can be performed in experiment using the notion of weak values.