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Partial Description of Quantum States
, 2008
"... One of the most central and controversial element of quantum mechanics is the use of non zero vectors of a Hilbert space (or, more generally, of one dimension subspaces) for representing the state of a quantum system. In particular, the question whether such a representation is complete has been deb ..."
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One of the most central and controversial element of quantum mechanics is the use of non zero vectors of a Hilbert space (or, more generally, of one dimension subspaces) for representing the state of a quantum system. In particular, the question whether such a representation is complete has been debated since almost the early days of quantum mechanics. In this article, we develop an alternate way to formalize knowledge about the state of quantum systems, based solely on experimentally accessible elements, namely on outcomes of finite measurements. We introduce what we call partial description which, given a feasible measurement, indicates some outcomes which are known to be impossible (i.e. known to have a probability equal to 0 to occur) and hence have to be discarded. Then, we introduce partial states (which are partial descriptions providing as much information as possible) and compare this way to describe quantum states to the orthodox one, using vector rays. Finally, we show that partial states allow to describe quantum states in a strictly more expressive way that the orthodox description does. 1
arXiv:1210.4262. Dynamics and Hidden Variables
, 2013
"... Abstract We study the way the unitary evolution of spin 1/2 particules can be represented in a counterfactual definiteness setting. More precisely, by representing the state of such a particule by a triplet of values corresponding to the supposedly preexisting outcomes of some measurements (those c ..."
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Abstract We study the way the unitary evolution of spin 1/2 particules can be represented in a counterfactual definiteness setting. More precisely, by representing the state of such a particule by a triplet of values corresponding to the supposedly preexisting outcomes of some measurements (those corresponding tothethreePaulimatrices), weanalysetheevolutionofourrepresentationwhen some unitary gates (namely, the Hadamard gate, the π/2 phase shifter and the controllednot) are applied. Then, we describe in terms of triplets the creation of an EPR pair and discuss the possibility of having this representation comply with the predictions of quantum mechanics. Finally, we show that this is not possible unless one of the assumptions used to build our model is dropped. 1
Constructible Models of Orthomodular Quantum Logics
"... Abstract: We continue in this article the abstract algebraic treatment of quantum sentential logics [39]. The Notions borrowed from the field of Model Theory and Abstract Algebraic Logic AAL (i.e., consequence relation, variety, logical matrix, deductive filter, reduced product, ultraproduct, ultra ..."
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Abstract: We continue in this article the abstract algebraic treatment of quantum sentential logics [39]. The Notions borrowed from the field of Model Theory and Abstract Algebraic Logic AAL (i.e., consequence relation, variety, logical matrix, deductive filter, reduced product, ultraproduct, ultrapower, Frege relation, Leibniz congruence, Suszko congruence, Leibniz operator) are applied to quantum logics. We also proved several equivalences between state property systems (JauchPironAerts line of investigations) and AAL treatment of quantum logics (corollary 18 and 19). We show that there exist the uniquely defined correspondence between state property system and consequence relation defined on quantum logics. We also signalize that a metalogical property Lindenbaum property does not hold for the set of quantum logics.
An Intrisic Topology for Orthomodular Lattices
, 2008
"... We present a general way to define a topology on orthomodular lattices. We show that in the case of a Hilbert lattice, this topology is equivalent to that induced by the metrics of the corresponding Hilbert space. Moreover, we show that in the case of a boolean algebra, the obtained topology is the ..."
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We present a general way to define a topology on orthomodular lattices. We show that in the case of a Hilbert lattice, this topology is equivalent to that induced by the metrics of the corresponding Hilbert space. Moreover, we show that in the case of a boolean algebra, the obtained topology is the discrete one. Thus, our construction provides a general tool for studying orthomodular lattices but also a way to distinguish classical and quantum logics. 1
Weakening the Born Rule – Towards a Stateless Formulation of Quantum Mechanics
, 2013
"... The notion of state vector is, in quantum mechanics, as central as it is problematic, as illustrates the wealth of publications about the subjects, including in particular the many attempts to obtain an acceptable interpretation of quantum mechanics. In this article, we propose a different approach, ..."
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The notion of state vector is, in quantum mechanics, as central as it is problematic, as illustrates the wealth of publications about the subjects, including in particular the many attempts to obtain an acceptable interpretation of quantum mechanics. In this article, we propose a different approach, and initiate the study of a formulation of quantum mechanics, where the notion of state is entirely replaced by assertions about measurement outcomes. We define a notion of “verification ” which represents the knowledge that one may have about the possible outcomes of the measurements performed on a quantum system, and express a set of logical rules which allow to reason about quantum systems using verification assertions only, and thus making no reference to the problematic notion of state. 1
c © Olivier Brunet This work is licensed under the Creative Commons Attribution License. Quantum Measurements from a Logical Point of View
"... We introduce a logic modelling some aspects of the behaviour of the measurement process, in such a way that no direct mention of quantum states is made, thus avoiding the problems associated to this rather evasive notion. We then study some properties of the models of this logic, and deduce some cha ..."
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We introduce a logic modelling some aspects of the behaviour of the measurement process, in such a way that no direct mention of quantum states is made, thus avoiding the problems associated to this rather evasive notion. We then study some properties of the models of this logic, and deduce some characteristics that any model (and hence, ideally, any formulation of quantum mechanics compatible with its prediction and relying on a notion of measurement) should verify. The main results we obtain are that in the case of a Hilbert space of dimension at least 3, no model can lead to the certain prediction of more than one atomic outcome. Moreover, if the Hilbert space is finite dimensional, then we are able to precisely describe the structure of the predictions of any model of our logic. In particular, we show that all the models of our logic do exactly make the same predictions regarding whether a given sequence of outcomes is possible or not. As Jaynes puts it so vividly, “our present [quantum mechanical] formalism is not purely epistemological; it is a peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature – all scrambled up by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble ” [10]. One origin for theses difficulties is, in our opinion, the excessive reliance of the quantum mechanical
Article Classical Probability and Quantum Outcomes
, 2014
"... www.mdpi.com/journal/axioms/ ..."
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