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Quantum logic. A brief outline
, 2005
"... A more complete introduction of the author can be found in the book ..."
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A more complete introduction of the author can be found in the book
Statistical structures underlying quantum mechanics and social science, eprint arXiv:quantph/0307234
, 2003
"... Common observations of the unpredictability of human behavior and the influence of one question on the answer to another suggest social science experiments are probabilistic and may be mutually incompatible with one another, characteristics attributed to quantum mechanics (as distinguished from clas ..."
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Common observations of the unpredictability of human behavior and the influence of one question on the answer to another suggest social science experiments are probabilistic and may be mutually incompatible with one another, characteristics attributed to quantum mechanics (as distinguished from classical mechanics). This paper examines this superficial similarity in depth using the FoulisRandall Operational Statistics language. In contradistinction to physics, social science deals with complex, open systems for which the set of possible experiments is unknowable and outcome interference is a graded phenomenon resulting from the ways the human brain processes information. It is concluded that social science is, in some ways, “less classical ” than quantum mechanics, but that generalized “quantum ” structures may provide appropriate descriptions of social science experiments. Specific challenges to extending “quantum” structures to social science are identified. 1
Quantum Diagonalization of Hermitean Matrices
, 2000
"... To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource. It allows one, in principle, to diagonalize hermitean (N × N) matrices by quantum mechanical ..."
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To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource. It allows one, in principle, to diagonalize hermitean (N × N) matrices by quantum mechanical measurements only. To do so, one considers the given matrix as an observable of a single spin with appropriate length s which can be measured using a generalized SternGerlach apparatus. Then, each run provides one eigenvalue of the observable. As it is based on the ‘collapse of the wave function ’ associated with a measurement, the procedure is neither a digital nor an analog calculation—it defines thus a new quantum mechanical method of computation. Nonclassical features of quantum mechanics such as Heisenberg’s uncertainty relation and entanglement have intrigued physicists for several decades. From a classical point of view, quantum mechanics imposes constraints on the ways to talk about nature. An electron does not “have ” position and momentum as does a billiard ball. Similarly, if
Constructive Mathematics and Quantum Physics
, 1999
"... This paper is dedicated to the memory of Prof. Gottfried T. Ru ttimann ..."
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This paper is dedicated to the memory of Prof. Gottfried T. Ru ttimann
The quantum way to diagonalize hermitean matrices
, 2003
"... An entirely quantum mechanical approach to diagonalize hermitean matrices has been presented recently. Here, the genuinely quantum mechanical approach is considered in detail for (2 × 2) matrices. The method is based on the measurement of quantum mechanical observables which provides the computation ..."
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An entirely quantum mechanical approach to diagonalize hermitean matrices has been presented recently. Here, the genuinely quantum mechanical approach is considered in detail for (2 × 2) matrices. The method is based on the measurement of quantum mechanical observables which provides the computational resource. In brief, quantum mechanics is able to directly address and output eigenvalues of hermitean matrices. The simple lowdimensional case allows one to illustrate the conceptual features of the general method which applies to (N × N) hermitean matrices. (Fortschr. Phys. 51, 248 (2003))
QUANTUM LOGIC AND QUANTUM COMPUTATION
, 812
"... This quantum logic of qubits (also called quantum computational logic [8, 18]) is a formalism of finite tensor products of twodimensional Hilbert spaces and will be ..."
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This quantum logic of qubits (also called quantum computational logic [8, 18]) is a formalism of finite tensor products of twodimensional Hilbert spaces and will be
Quantum logic. A brief outline
, 2005
"... Quantum logic has been introduced by Birkhoff and von Neumann as an attempt to base the logical primitives, the propositions and the relations and operations among them, on quantum theoretical entities, and thus on the related empirical evidence of the quantum world. We give a brief outline of quant ..."
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Quantum logic has been introduced by Birkhoff and von Neumann as an attempt to base the logical primitives, the propositions and the relations and operations among them, on quantum theoretical entities, and thus on the related empirical evidence of the quantum world. We give a brief outline of quantum logic, and some of its algebraic properties, such as nondistributivity, whereby emphasis is given to concrete experimental setups related to quantum logical entities. A probability theory based on quantum logic is fundamentally and sometimes even spectacularly different from probabilities based on classical Boolean logic. We give a brief outline of its nonclassical aspects; in particular violations of BooleBell type consistency constraints on joint probabilities, as well as the KochenSpecker theorem, demonstrating in a constructive, finite way the scarcity and even nonexistence of twovalued states interpretable as classical truth assignments. A more complete introduction of the author can be found in the book Quantum Logic (Springer, 1998)
PROBABILISTIC FORCING IN QUANTUM LOGICS
"... Summary. It is shown that orthomodular lattice can be axiomatized as an ortholattice with a unique operation of identity (bi–implication) instead of the operation of implication and a corresponding algebraic unified quantum logic is formulated. A statistical YES–NO physical interpretation of the qua ..."
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Summary. It is shown that orthomodular lattice can be axiomatized as an ortholattice with a unique operation of identity (bi–implication) instead of the operation of implication and a corresponding algebraic unified quantum logic is formulated. A statistical YES–NO physical interpretation of the quantum logical propositions is then provided to establish a support for a novel YES–NO representation of quantum logic which is also given together with a conjecture about possible completion of quantum logic by means of probabilistic forcing. 1.
A Bibliography of Publications in the Journal of Mathematical Physics: 19801984
, 1
"... E 6 [720, 2098, 2059]. E 7 [720]. E 8 [720]. exp[i(!=c) (r 0 ct)]=r [1856]. F (4) [1178]. F 4 [720, 1798]. G [829]. G(3) [1178]. g 0 2 [604]. G 2 [1073, 847, 1187, 1614, 296, 788]. GL(4; R) [2195]. GL(n; R) [851]. Gln [52]. Gsl(2) [1872]. G oe [SU(2)] [1869]. H [977, 79, 250, 1199, 502, 1252, ..."
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E 6 [720, 2098, 2059]. E 7 [720]. E 8 [720]. exp[i(!=c) (r 0 ct)]=r [1856]. F (4) [1178]. F 4 [720, 1798]. G [829]. G(3) [1178]. g 0 2 [604]. G 2 [1073, 847, 1187, 1614, 296, 788]. GL(4; R) [2195]. GL(n; R) [851]. Gln [52]. Gsl(2) [1872]. G oe [SU(2)] [1869]. H [977, 79, 250, 1199, 502, 1252, 1148, 913]. H [1439]. He [637]. I [2089]. i [2073]. II [299, 1220, 375]. III\Omega III [163]. II [161]. III [162]. rv(r)dr + v(r)dr ! 1 [209]. I [158]. II [159]. III [160]. ISO(n) [16, 680]. IU(n) [577]. IX [528]. 1340, 1864, 2171, 2059, 106, 240]. J fffi [586]. K [418]. K 3 [493]. l [478, 1668, 266, 1790, 1459, 1460]. L 1=2 W (R l ) [593]. L [262]. [905]. P (') [336]. LCRG [1228]. M 4 [374, 167, 529]. j x j 0(n+1)=2 [1952]. N [1165, 1826, 1950, 1302, 2009, 834, 1257, 955, 1844, 390, 313, 1584, 1522, 42, 536, 1930, 604, 1478, 1484, 2101, 888, 1075]. n + 1 [1994]. N = 2 [1091]. n t = (n n x ) x [1034]. N 1 [796]. O(3) [1654, 794, 795, 1420]. O(4