Wigner-Yanase skew information vs. quantum Fisher information

Among concepts describing the information contents of quantum mechanical density operators, both the Wigner-Yanase skew information and the quantum Fisher information defined via symmetric logarithmic derivatives are natural generalizations of the classical Fisher information. We will establish a relationship between these two fundamental quantities and show that they are comparable.

     

Uncertainty principle and quantum Fisher information

A family of inequalities, related to the uncertainty principle, has been recently proved by S. Luo, Z. Zhang, Q. Zhang, H. Kosaki, K. Yanagi, S. Furuichi and K. Kuriyama. We show that the inequalities have a geometric interpretation in terms of quantum Fisher information. Using this formulation one may naturally ask if this family of inequalities can be further extendend, for example to the RLD quantum Fisher information. We show that this is impossible by producing a family of counterexamples.     

On a refinement of Heisenberg uncertainty relation by means of quantum Fisher information

We show that an uncertainty relation proved by Luo and Yanagi and related to Wigner-Yanase-Dyson information cannot hold for an arbitrary quantum Fisher information. We show that, by changing a costant, a similar trace inequality holds in full generality.     

Affine Fisher information inequalities

We show that the Lp Petty projection inequality is a special case of the integral affine (p,λ)-Fisher information inequality for 1?p<n.     

Fisher information under restriction of Shannon information in multi-terminal situations

Fisher information generally decreases by summarizing observed data into encoded messages. The present paper studies the amount of Fisher information included in independently summarized messages from correlated information sources; that is, the amount of Fisher information when sequences x ^N and y ^N of N independent observations of random variables x and y are encoded (summarized) independently of each other into meassages m _X and m _Y . The problem is to obtain the maximal amount of Fisher information when the size of the summarized data or Shannon message information is limited. The problem is solved in the case of completely compressed symmetric data summarization. An achievable bound is given in the general case. Information geometry, which is a powerful new differential geometrical method applicable to statistics and systems theory, is applied to this problem, proving its usefulness in information theory as well.The present work is supported in part by Grant-in-Aid for Scientific Research #61030014, Ministry of Education, Science and Culture of Japan.     

Operator-valued free Fisher information and modular frames

We introduce the operator-valued free Fisher information for a random variable in an operator-valued noncommutative probability space and point out its relations to the amalgamated freeness. Using M. Frank and D. Larson's modular frame notion we can construct the conjugate variable for an operator-valued semicircle variable with conditional expectation covariance. Then we obtain its free Fisher information and show it is equal to the index of the conditional expectation. At last the conjugate variable with respect to a modular frame operator for a semicircle variable is also constructed.

     

Skew information decreases under quantum measurements

We prove that the skew information decreases under nonselective quantum measurements that do not disturb the conserved observable. We further conjecture that the skew information also decreases under selective quantum measurements, and we prove this for two-dimensional quantum systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 109–119, April, 2007.     

Fisher information inequalities and the central limit theorem

We give conditions for an O(1/n) rate of convergence of Fisher information and relative entropy in the Central Limit Theorem. We use the theory of projections in L² spaces and Poincaré inequalities, to provide a better understanding of the decrease in Fisher information implied by results of Barron and Brown. We show that if the standardized Fisher information ever becomes finite then it converges to zero.OTJ is a Fellow of Christs College, Cambridge, who helped support two trips to Yale University during which this paper was written.Mathematics Subject Classification (2000):Primary: 62B10 Secondary: 60F05, 94A17     

An application of the convolution inequality for the Fisher information

A characterization of the normal distribution by a statistical independence on a linear transformation of two mutually independent random variables is proved by using the convolution inequality for the Fisher information.     

A state-space approach to quantum permutations

In this exposition of quantum permutation groups, an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and interprets the states of an algebra of continuous functions on a quantum permutation group as quantum permutations. This interpretation allows talk of an element of a quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated by various quantum permutation group phenomena.     

On the essential spectrum of quantum waveguides

The main aim of the paper is the study of essential spectra of electromagnetic Schrödinger operators with variable potentials in cylindric domains , where is a bounded domain with a smooth boundary provided by admissible boundary conditions. Applying the limit operators method, we obtain explicit estimates of the essential spectrum for a wide class of quantum waveguides. We also consider a numerical example of calculations of the discrete spectrum of horizontally stratified quantum waveguides applying a method of the decomposition of solutions of spectral problems for one‐dimensional Schrödinger operators as a power series with respect to the spectral parameter. Copyright © 2012 John Wiley & Sons, Ltd.     

Applying the Linblad equation to quantum dissipative systems

For quantum systems with linear dissipation, we obtain the representation of the Linblad equation in the canonical form via Hermitian operators. Based on this representation, we derive equations for the entropy density and for the statistical projection operator. We consider the quantum harmonic oscillator with linear dissipation as an example. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 288–294, August, 2006. An erratum to this article is available at .     

A geometric approach to the canonical reformulation of quantum mechanics

The measure of distinguishability between two neighboring preparations of a physical system by a measurement device naturally defines a line element on the preparation space of the system. We show that quantum mechanics can be derived from the invariance of this line element in the canonical formulation. We also discuss the canonical formulation of quantum statistical mechanics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 479–486, June, 2006.     

THE ESTIMATION OF PRIOR FROM FISHER INFORMATION

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Digital quantum batteries: Energy and information storage in nanovacuum tube arrays

Dielectric material between capacitor electrodes increases the capacitance. However, when the electric field exceeds a threshold, electric breakdown in the dielectric discharges the capacitor suddenly and the stored energy is lost. We show that nanovacuum tubes do not have this problem because (i) electric breakdown can be suppressed with quantization phenomena, and (ii) the capacitance is large at small gap sizes. We find that the energy density and power density in nanovacuum tubes are large compared to lithium batteries and electrochemical capacitors. The electric field in a nanovacuum tube can be sensed with MOSFETs in the insulating walls. Random access arrays of nanovacuum tubes with an energy gate, to charge the tube, and an information gate attached to the MOSFET, to sense the electric field in the tube, can be used to store both energy and information. © 2010 Wiley Periodicals, Inc. Complexity 2010     

How to distinguish quantum covariances using uncertainty relations

To each operator monotone function it is possible to associate a quantum version of the classical covariance. We show that: i) only for regular quantum covariances one can prove non-trivial uncertainty relations; ii) the usual quantum covariance gives the best inequalities in this setting.