arXiv subjects

Olivier Brunet

Publications and source records attributed to Olivier Brunet.

15 recordsLinked to original sources

arXiv2026

A Formal Theory for Finite-Dimensional Possibilistic Quantum Mechanics

In this work, we present a logical formalism for reasoning about quantum systems in finite dimension. Contrary to the usual approach in quantum logic, our formalism is based classical first-order logic, which allows us to use the tools of model theory in our study. In particular, we show that our formal theory is complete, meaning that it entirely determines the behaviour of quantum systems. Moreover, we provide a characterization of the models of our formal theory, thus providing new insights in the study of hidden variable models of quantum theory.

quant-ph↗

arXiv2019

A Purely Algebraic Summation Method

It is mathematical folklore that 1 + 2 + 3 + 4 + ... = --1/12. This result is usually achieved using elaborate analytical methods, such as zeta function regularization or Ramanujan summation. However, in its notebooks, Ramanujan has also provided a very simple derivation which relied instead on algebraic manipulations. Recently, a video from Numberphile has presented a similar derivation of the result (provoking lots of discussions and debates about the meaning of such an equality). But this derivation, simple as it is, is usually considered as less rigorous than those using more elaborate analytical methods. However, this derivation is indeed perfectly rigourous, and in this article, we will define a general algebraic construction which we will use as a framework for expressing this derivation and, more generally, for providing a new summation method.

math.CA↗

arXiv2017

Geometric Properties of Paths in Relativistic Lagrangian Mechanics

Considering an extension of the principle of covarience to the action along a path in relativistic Lagrangian mechanics, we motivate the use of geometric -- i.e. covariant and parameter invariant -- Lagrangian functions. We then study some properties of geometric Lagrangians, and introduce the notion of deviation of a path, which is a covariant measure of how much a path departs from a geodesic. Finally, we apply this notion of the twin paradox, and provide a rigorous resolution of it.

math-ph↗

arXiv2016

Geometric Time and Causal Time in Relativistic Lagrangian Mechanics

In this article, we argue that two distinct types of time should be taken into account in relativistic physics: a geometric time, which emanates from the structure of spacetime and its metrics, and a causal time, indicating the flow from the past to the future. A particularity of causal times is that its values have no intrinsic meaning, as their evolution alone is meaningful. In the context of relativistic Lagrangian mechanics, causal times corresponds to admissible parameterizations of paths, and we show that in order for a langragian to not depend on any particular causal time (as its values have no intrinsic meaning), it has to be homogeneous in its velocity argument. We illustrate this property with the example of a free particle in a potential. Then, using a geometric Lagrangian (i.e. a parameterization independent Lagrangian which is also manifestly covariant), we introduce the notion of ageodesicity of a path which measures to what extent a path is far from being a geodesic, and show how the notion can be used in the twin paradox to differentiate the paths followed by the two twins.

gr-qc↗

arXiv2016

On the Possibility of Quantum Circuits Part I: the Epistemic Level

We present a formulation of quantum circuits where the focus is set on whether a given circuit (made of unitary operators and projective measurements with definite outcomes) does reflect an actually realizable physical experiment. In order to do this, we introduce verifications statements which are purely epistemic assertions indicating whether a outcome is possible at some point and develop our formalism which, in the end, consists in a set of logical rules about verification statements. Finally, we argue that our formalism provides a Lorentz-invariant realistic formulation of quantum circuits and illustrate this by considering a circuit corresponding to Hardy's paradox and showing how our formalism prevents making contradictory assertions regarding our knowledge about the circuit.

quant-ph↗

arXiv2015

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 characteristics that any model (and hence, any formulation of quantum mechanics compatible with its predictions 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, using a strengthening of the Kochen-Specker theorem, we show that 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, so that quantum mechanics can be considered complete as long as the possibility of outcomes is considered.

quant-ph↗

arXiv2013

Orthogonality and Dimensionality

In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as the cardinality of a maximal collection of mutually orthogonal elements (which, for instance, can be seen as spatial directions). Following this idea, we develop a formalism based on two basic ingredients, namely an orthogonality relation and matroids which are a very generic algebraic structure permitting to define a notion of dimension. Having obtained what we call orthomatroids, we then show that, in high enough dimension, the basic ingredients of orthomatroids (more precisely the simple and irreducible ones) are isomorphic to generalized Hilbert lattices, so that the latter are a direct consequence of an orthogonality-based characterization of dimension.

quant-ph↗

arXiv2013

Weakening the Born Rule - Towards a Stateless Formulation of Quantum Mechanics

The notion of state vector is, in quantum mechanics, as central as it is problematic, as illustrates the wealth of publications about the sub- jects, 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.

quant-ph↗

arXiv2012

Dynamics and Hidden Variables

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 pre-existing outcomes of some measurements (those corresponding to the three Pauli matrices), we analyse the evolution of our representation when some unitary gates (namely, the Hadamard gate, the π/2 phase shifter and the controlled-not) 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.

quant-ph↗

arXiv2008

Partial Description of Quantum States

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.

quant-ph↗

arXiv2007

An Intrisic Topology for Orthomodular Lattices

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.

quant-ph↗

arXiv2006

A priori Knowledge and the Kochen-Specker Theorem

We introduce and formalize a notion of "a priori knowledge" about a quantum system, and show some properties about this form of knowledge. Finally, we show that the Kochen-Specker theorem follows directly from this study. This version is a draft version, the bibliography in particular is extremely scarce. Comments welcome.

quant-ph↗

arXiv2006

Representation Systems and Quantum Structures

Two important classes of quantum structures, namely orthomodular posets and orthomodular lattices, can be characterized in a classical context, using notions like partial information and points of view. Using the formalism of representation systems, we show that these quantum structures can be obtained by expressing conditions on the existence of particular points of view, of particular ways to observe a system.

quant-ph↗

arXiv2005

A Rule-Based Logic for Quantum Information

In the present article, we explore a new approach for the study of orthomodular lattices, where we replace the problematic conjunction by a binary operator, called the Sasaki projection. We present a characterization of orthomodular lattices based on the use of an algebraic version of the Sasaki projection operator (together with orthocomplementation) rather than on the conjunction. We then define of a new logic, which we call Sasaki Orthologic, which is closely related to quantum logic, and provide a rule-based definition of this logic.

cs.LO↗

arXiv2003

Dynamic Quantum Logic for Quantum Programs

We present a way to apply quantum logic to the study of quantum programs. This is made possible by using an extension of the usual propositional language in order to make transformations performed on the system appear explicitly. This way, the evolution of the system becomes part of the logical study. We show how both unitary operations and two-valued measurements can be included in this formalism and can thus be handled logically.

quant-ph↗