Quantum field logic

Algebraic quantum field theory, or AQFT for short, is a rigorous analysis of the structure of relativistic quantum mechanics. It is formulated in terms of a net of operator algebras indexed by regions of a Lorentzian manifold. In several cases the mentioned net is represented by a family of von Neumann algebras, concretely, type III factors. Local quantum field logic arises as a logical system that captures the propositional structure encoded in the algebras of the net. In this framework, this work contributes to the solution of a family of open problems, emerged since the 30s, about the characterization of those logical systems which can be identified with the lattice of projectors arising from the Murray-von Neumann classification of factors. More precisely, based on physical requirements formally described in AQFT, an equational theory able to characterizethe type III condition in a factor is provided. This equational system motivates the study of a variety of algebras having an underlying orthomodular lattice structure. A Hilbert style calculus, algebraizable in the mentioned variety, is also introduced and a corresponding completeness theorem is established

Approximate transformations of bipartite pure-state entanglement from the majorization lattice

We study the problem of deterministic transformations of an \textit{initial} pure entangled quantum state, $|\psi\rangle$, into a \textit{target} pure entangled quantum state, $|\phi\rangle$, by using \textit{local operations and classical communication} (LOCC). A celebrated result of Nielsen [Phys. Rev. Lett. \textbf{83}, 436 (1999)] gives the necessary and sufficient condition that makes this entanglement transformation process possible. Indeed, this process can be achieved if and only if the majorization relation $\psi \prec \phi$ holds, where $\psi$ and $\phi$ are probability vectors obtained by taking the squares of the Schmidt coefficients of the initial and target states, respectively. In general, this condition is not fulfilled. However, one can look for an \textit{approximate} entanglement transformation. Vidal \textit{et. al} [Phys. Rev. A \textbf{62}, 012304 (2000)] have proposed a deterministic transformation using LOCC in order to obtain a target state $|\chi^\mathrm{opt}\rangle$ most approximate to $|\phi\rangle$ in terms of maximal fidelity between them. Here, we show a strategy to deal with approximate entanglement transformations based on the properties of the \textit{majorization lattice}. More precisely, we propose as approximate target state one whose Schmidt coefficients are given by the supremum between $\psi$ and $\phi$. Our proposal is inspired on the observation that fidelity does not respect the majorization relation in general. Remarkably enough, we find that for some particular interesting cases, like two-qubit pure states or the entanglement concentration protocol, both proposals are coincident.Comment: Revised manuscript close to the accepted version in Physica A (10 pages, 1 figure

Representing continuous t-norms in quantum computation with mixed states

A model of quantum computation is discussed in (Aharanov et al 1997 Proc. 13th Annual ACM Symp. on Theory of Computation, STOC pp 20–30) and (Tarasov 2002 J. Phys. A: Math. Gen. 35 5207–35) in which quantum gates are represented by quantum operations acting on mixed states. It allows one to use a quantum computational model in which connectives of a four-valued logic can be realized as quantum gates. In this model, we give a representation of certain functions, known as t-norms (Menger 1942 Proc. Natl Acad. Sci. USA 37 57–60), that generalize the triangle inequality for the probability distributionvalued metrics. As a consequence an interpretation of the standard operations associated with the basic fuzzy logic (H´ajek 1998 Metamathematics of Fuzzy Logic (Trends in Logic vol 4) (Dordrecht: Kluwer)) is provided in the frame of quantum computatio

Lógicas multivaluadas y la axiomatización de la lógica computacional cuántica

El significado de una sentencia elemental en la lógica asociada a la computación cuántica está representado por la cantidad de información cuántica codificada en una colección de qbits, el equivalente cuántico de los bits clásicos (0, 1 o F, V), o de qmixe

Multi-class classification based on quantum state discrimination

We present a general framework for the problem of multi-class classification using classification functions that can be interpreted as fuzzy sets. We specialize these functions in the domain of Quantum-inspired classifiers, which are based on quantum state discrimination techniques. In particular, we use unsharp observables (Positive Operator-Valued Measures) that are determined by the training set of a given dataset to construct these classification functions. We show that such classifiers can be tested on near-term quantum computers once these classification functions are “distilled” (on a classical platform) from the quantum encoding of a training dataset. We compare these experimental results with their theoretical counterparts and we pose some questions for future research

Generalized coherence vector applied to coherence transformations and quantifiers

One of the main problems in any quantum resource theory is the characterization of the conversions between resources by means of the free operations of the theory. In this work, we advance on this characterization within the quantum coherence resource theory by introducing the generalized coherence vector of an arbitrary quantum state. The generalized coherence vector is a probability vector that can be interpreted as a concave roof extension of the pure states coherence vector. We show that it completely characterizes the notions of being incoherent, as well as being maximally coherent. Moreover, using this notion and the majorization relation, we obtain a necessary condition for the conversion of general quantum states by means of incoherent operations. These results generalize the necessary conditions of conversions for pure states given in the literature, and show that the tools of the majorization lattice are useful also in the general case. Finally, we introduce a family of coherence quantifiers by considering concave and symmetric functions applied to the generalized coherence vector. We compare this proposal with the convex roof measure of coherence and others quantifiers given in the literature.Comment: 21 pages, 2 figures (close to the published version

Many worlds and modality in the interpretation of quantum mechanics: an algebraic approach

Many worlds interpretations (MWI) of quantum mechanics avoid the measurement problem by considering every term in the quantum superposition as actual. A seemingly opposed solution is proposed by modal interpretations (MI) which state that quantum mechanics does not provide an account of what `actually is the case', but rather deals with what `might be the case', i.e. with possibilities. In this paper we provide an algebraic framework which allows us to analyze in depth the modal aspects of MWI. Within our general formal scheme we also provide a formal comparison between MWI and MI, in particular, we provide a formal understanding of why --even though both interpretations share the same formal structure-- MI fall pray of Kochen-Specker (KS) type contradictions while MWI escape them.Comment: submitted to the Journal of Mathematical Physic

Optimal common resource in majorization-based resource theories

We address the problem of finding the optimal common resource for an arbitrary family of target states in quantum resource theories based on majorization, that is, theories whose conversion law between resources is determined by a majorization relationship, such as it happens with entanglement, coherence or purity. We provide a conclusive answer to this problem by appealing to the completeness property of the majorization lattice. We give a proof of this property that relies heavily on the more geometric construction provided by the Lorenz curves, which allows to explicitly obtain the corresponding infimum and supremum. Our framework includes the case of possibly non-denumerable sets of target states (i.e. targets sets described by continuous parameters). In addition, we show that a notion of approximate majorization, which has recently found application in quantum thermodynamics, is in close relation with the completeness of this lattice. Finally, we provide some examples of optimal common resources within the resource theory of quantum coherence.Fil: Bosyk, Gustavo Martin. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física La Plata; ArgentinaFil: Bellomo, Guido. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; ArgentinaFil: Holik, Federico Hernán. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física La Plata; ArgentinaFil: Freytes, H.. Università Degli Studi Di Cagliari.; ItaliaFil: Sergioli, Giuseppe. Università Degli Studi Di Cagliari.; Itali

Modality, Potentiality and Contradiction in Quantum Mechanics

In [11], Newton da Costa together with the author of this paper argued in favor of the possibility to consider quantum superpositions in terms of a paraconsistent approach. We claimed that, even though most interpretations of quantum mechanics (QM) attempt to escape contradictions, there are many hints that indicate it could be worth while to engage in a research of this kind. Recently, Arenhart and Krause [1, 2, 3] have raised several arguments against this approach and claimed that, taking into account the square of opposition, quantum superpositions are better understood in terms of contrariety propositions rather than contradictory propositions. In [17] we defended the Paraconsistent Approach to Quantum Superpositions (PAQS) and provided arguments in favor of its development. In the present paper we attempt to analyze the meanings of modality, potentiality and contradiction in QM, and provide further arguments of why the PAQS is better suited, than the Contrariety Approach to Quantum Superpositions (CAQS) proposed by Arenhart and Krause, to face the interpretational questions that quantum technology is forcing us to consider.Comment: Published in: New Directions in Paraconsistent Logic, J-Y B\'eziau M. Chakraborty & S. Dutta (Eds.), Springer, in press. arXiv admin note: text overlap with arXiv:1404.518

Fuzzy Propositional Logic Associated with Quantum Computational Gates

We apply residuated structures associated with fuzzy logic to develop certain aspects of information processing in quantum computing from a logical perspective. For this purpose, we introduce an axiomatic system whose natural interpretation is the irreversible quantum Poincare structure.Comment: 41 pages, no figure