Conditional Entropy and the Rokhlin Metric on an Orthomodular Lattice with Bayessian State

Khare, Mona; Roy, Shraddha

doi:10.1007/s10773-007-9581-1

Cited by 16 publications

(8 citation statements)

References 16 publications

(9 reference statements)

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“…α βα ββ α =+ between the partitions α and β of the quantum logic Λ , which preserves the metric properties (Khare & Roy, 2008), as it holds for the probabilistic scheme.…”

Section: Logic Of Quantum Mechanics and Informational Distancementioning

confidence: 87%

Navigation of Quantum-Controlled Mobile Robots

Kagan¹,

Ben‐Gal²

2011

Recent Advances in Mobile Robotics

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Recent Advances in Mobile Robotics 312 quantum mechanics (Ballentine, 2006); for a complete review of quantum computation and information theory, see, e.g., (Nielsen & Chuang, 2000). Basic notation and properties of quantum-mechanical systemsIn general, in the considerations of the finite quantum-mechanical systems, it is postulated (Ballentine, 2006) with the unit sum of the diagonal elements, ( ) tr 1 σ = ; and that the observed value of the quantum-mechanical system is specified by the eigenvalues of the matrix σ . Since matrix σ is Hermitial, its eigenvalues are real numbers. If matrix σ is diagonal, then such a representation of the state is equivalent to the representation of the state of stochastic classical system, in which diagonal elements jj σ form a probability vector (Holevo, 2001 From such a property of observation, it follows that in contrast to the classical systems, the actual state of the quantum-mechanical system obtains a value, which was measured by the observer, and further evolution of the system starts from this value. In the other words, the evolution the quantum-mechanical system depends on the fact of its observation. www.intechopen.com Navigation of Quantum-Controlled Mobile Robots 313An actual evolution of the quantum-mechanical system is governed by the evolution operators, which are applied to the state matrix σ or state vector s . Below, we consider the states and operators, which are used in quantum information theory. Concepts of the quantum information theoryThe elementary state, which is considered in quantum information theory (Nielsen & Chuang, 2000), is called qubit (quantum bit) and is represented by a two-element complex vector , which correspond to the bit values "0" and "1" known in classical information theory (Cover & Thomas, 1991). In general, vectors 0 and 1 determine the states "spin up" and "spin down" of electron, i.e. 0" " ≡ ↑ and 1" " ≡ ↓ . 12" " =⋅ → −⋅ ← , the pairs of the vectors 0 a n d 1, a n d "" → and "" ← can be used interchangeably.In general, the evolution of the qubits is governed by the use of the following operators: The Pauli operators are the most known qubits operators that are in use in general quantum mechanics, while the other three operators are more specific for quantum information theory. According to the Kitaev-Solovey theorem (Nielsen & Chuang, 2000), an algebra, which consists of the qubits 0 and 1 , and CNOT, Hadamard and phase shift operators, forms a universal algebra that models any operation of the Boolean algebra :The other types of the qubit gates, e.g. phase shift operator S and its derivatives, cannot be represented by the classical operators and require quantum computing devices. In such computing, it is assumed that each matrix operation is conducted in one computation step providing a power of quantum computing. The indicated dependence of quantum states on the observation process allows an implementation of such operations by the use of adaptive computation schemes. Below, we demonstrate a relation between quantum operati...

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“…α βα ββ α =+ between the partitions α and β of the quantum logic Λ , which preserves the metric properties (Khare & Roy, 2008), as it holds for the probabilistic scheme.…”

Section: Logic Of Quantum Mechanics and Informational Distancementioning

confidence: 87%

Navigation of Quantum-Controlled Mobile Robots

Kagan¹,

Ben‐Gal²

2011

Recent Advances in Mobile Robotics

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“…It should be noted that other fuzzy analogies of entropy are presented in [12][13][14][15][16][17][18][19][20][21][22][23][24][25]. It is known that there are many possibilities for defining operations with fuzzy sets; an overview can be found in [26].…”

Section: Introductionmentioning

confidence: 99%

Kullback-Leibler Divergence and Mutual Information of Experiments in the Fuzzy Case

Markechová

2017

Axioms

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Abstract:The main aim of this contribution is to define the notions of Kullback-Leibler divergence and conditional mutual information in fuzzy probability spaces and to derive the basic properties of the suggested measures. In particular, chain rules for mutual information of fuzzy partitions and for Kullback-Leibler divergence with respect to fuzzy P-measures are established. In addition, a convexity of Kullback-Leibler divergence and mutual information with respect to fuzzy P-measures is studied.

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“…The final section presents conclusions and some suggestions for further research. It is noted that certain basic studies on entropy of fuzzy partitions and related notions were done in [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Entropy of Fuzzy Partitions and Entropy of Fuzzy Dynamical Systems

Markechová

Riečan

2016

Entropy

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Abstract:In the paper we define three kinds of entropy of a fuzzy dynamical system using different entropies of fuzzy partitions. It is shown that different definitions of the entropy of fuzzy partitions lead to different notions of entropies of fuzzy dynamical systems. The relationships between these entropies are studied and connections with the classical case are mentioned as well. Finally, an analogy of the Kolmogorov-Sinai Theorem on generators is proved for fuzzy dynamical systems.

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Conditional Entropy and the Rokhlin Metric on an Orthomodular Lattice with Bayessian State

Cited by 16 publications

References 16 publications

Navigation of Quantum-Controlled Mobile Robots

Navigation of Quantum-Controlled Mobile Robots

Kullback-Leibler Divergence and Mutual Information of Experiments in the Fuzzy Case

Entropy of Fuzzy Partitions and Entropy of Fuzzy Dynamical Systems

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