arXiv subjects

Annalisa Marzuoli

Publications and source records attributed to Annalisa Marzuoli.

At least 19 recordsLinked to original sources

arXiv2024

Knotted optical fields: Seifert fibrations, braided open books and topological constraints

This paper presents a novel framework for studying knotted and braided configurations of optical fields, moving beyond the conventional Hopfion solution based on the Hopf fibration. By employing the Seifert fibration, a preferred framing is introduced to characterize the ``knottiness" of tubular neighbourhoods of knots embedded in the 3-sphere. This approach yields a specific presentation of Seifert surfaces and facilitates the description of knotted optical fields, drawing on an enriched version of Rañada's formulation. The relation between knots and braids enables one to explore the connections between configurations generated experimentally through the controlled over- and under-crossings of light beams, and concepts from geometric group theory and algebraic topology. This framework, emphasizing the significance of ``braided open books", sheds light on topological constraints inherent in classes of braids suitable for representing interlaced optical fields. These constraints primarily pertain to braids with $n\leq3$ strands -- aligning with existing experimental implementations -- and homogeneous braids with any number of strands. This paper establishes a theoretical foundation for understanding and manipulating knotted optical fields, suggesting potential applications and avenues for further research in this domain.

math-ph↗

arXiv2018

Projective Ponzano-Regge spin networks and their symmetries

We present a novel hierarchical construction of projective spin networks of the Ponzano-Regge type from an assembling of five quadrangles up to the combinatorial 4-simplex compatible with a geometrical realization in Euclidean 4-space. The key ingrendients are the projective Desargues configuration and the incidence structure given by its space-dual, on the one hand, and the Biedenharn--Elliott identity for the 6j symbol of SU(2), on the other. The interplay between projective-combinatorial and algebraic features relies on the recoupling theory of angular momenta, an approach to discrete quantum gravity models carried out successfully over the last few decades. The role of Regge symmetry --an intriguing discrete symmetry of the $6j$ which goes beyond the standard tetrahedral symmetry of this symbol-- will be also discussed in brief to highlight its role in providing a natural regularization of projective spin networks that somehow mimics the standard regularization through a q-deformation of SU(2).

math-ph↗

arXiv2016

Conformal variations and quantum fluctuations in discrete gravity

After an overview of variational principles for discrete gravity, and on the basis of the approach to conformal transformations in a simplicial PL setting proposed by Luo and Glickenstein, we present at a heuristic level an improved scheme for addressing the gravitational (Euclidean) path integral and geometrodynamics.

gr-qc↗

arXiv2014

Symmetric angular momentum coupling, the quantum volume operator and the 7-spin network: a computational perspective

A unified vision of the symmetric coupling of angular momenta and of the quantum mechanical volume operator is illustrated. The focus is on the quantum mechanical angular momentum theory of Wigner's 6j symbols and on the volume operator of the symmetric coupling in spin network approaches: here, crucial to our presentation are an appreciation of the role of the Racah sum rule and the simplification arising from the use of Regge symmetry. The projective geometry approach permits the introduction of a symmetric representation of a network of seven spins or angular momenta. Results of extensive computational investigations are summarized, presented and briefly discussed.

quant-ph↗

arXiv2013

Topological effective field theories for Dirac fermions from index theorem

Dirac fermions have a central role in high energy physics but it is well known that they emerge also as quasiparticles in several condensed matter systems supporting topological order. We present a general method for deriving the topological effective actions of (3+1) massless Dirac fermions living on general backgrounds and coupled with vector and axial-vector gauge fields. The first step of our strategy is standard (in the Hermitian case) and consists in connecting the determinants of Dirac operators with the corresponding analytical indices through the zeta-function regularization. Then, we introduce a suitable splitting of the heat kernel that naturally selects the purely topological part of the determinant (i.e. the topological effective action). This topological effective action is expressed in terms of gauge fields using the Atiyah-Singer index theorem which computes the analytical index in topological terms. The main new result of this paper is to provide a consistent extension of this method to the non Hermitian case where a well-defined determinant does not exist. Quantum systems supporting relativistic fermions can thus be topologically classified on the basis of their response to the presence of (external or emergent) gauge fields through the corresponding topological effective field theories.

cond-mat.mes-hall↗

arXiv2013

Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials

The action of the quantum mechanical volume operator, introduced in connection with a symmetric representation of the three-body problem and recently recognized to play a fundamental role in discretized quantum gravity models, can be given as a second order difference equation which, by a complex phase change, we turn into a discrete Schrödinger-like equation. The introduction of discrete potential-like functions reveals the surprising crucial role here of hidden symmetries, first discovered by Regge for the quantum mechanical 6j symbols; insight is provided into the underlying geometric features. The spectrum and wavefunctions of the volume operator are discussed from the viewpoint of the Hamiltonian evolution of an elementary "quantum of space", and a transparent asymptotic picture emerges of the semiclassical and classical regimes. The definition of coordinates adapted to Regge symmetry is exploited for the construction of a novel set of discrete orthogonal polynomials, characterizing the oscillatory components of torsion-like modes.

quant-ph↗

arXiv2012

Fault-tolerant mosaic encoding in knot-based cryptography

The cryptographic protocol based on topological knot theory,recently proposed by the authors, is improved for what concerns the efficiency of the encoding of knot diagrams and its error robustness. The standard Dowker-Thistlethwaite code, based on the ordered assignment of two numbers to each crossing of a knot diagram and not unique for some classes of knots, is replaced by a system of eight prototiles (knot mosaics) which, once assembled according to a set of combinatorial rules, reproduces unambiguously any unoriented knot diagram. A Reed-Muller scheme is used to encode with redundancy the eight prototiles into blocks and, once the blank tile is added and suitably encoded, the knot diagram is turned into an N X N mosaic, uniquely associated with a string of length 4 N^2 bits. The complexity of the knot, measured topologically by the number of crossings, is in turn polynomially related to the number of tiles of the associated mosaic, and for knot diagrams of higher complexity the mosaic encoding provides a design of the knot-based protocol which is fault-tolerant under random 1-bit flips. It is also argued that the knot mosaic alphabet might be used in other applications which require high-capacity data transmission.

math-ph↗

arXiv2012

BF-theory in graphene: a route toward topological quantum computing?

Besides the plenty of applications of graphene allotropes in condensed matter and nanotechnology, we argue that graphene sheets might be engineered to support room-temperature topological quantum processing of information. The argument is based on the possibility of modeling the monolayer graphene effective action by means of a 3d Topological Quantum Field Theory of BF-type able to sustain non-Abelian anyon dynamics. This feature is the basic requirement of recently proposed theoretical frameworks for fault-tolerant and decoherence protected quantum computation.

quant-ph↗

arXiv2010

Post Quantum Cryptography from Mutant Prime Knots

By resorting to basic features of topological knot theory we propose a (classical) cryptographic protocol based on the `difficulty' of decomposing complex knots generated as connected sums of prime knots and their mutants. The scheme combines an asymmetric public key protocol with symmetric private ones and is intrinsecally secure against quantum eavesdropper attacks.

math-ph↗

arXiv2010

3nj Morphogenesis and Semiclassical Disentangling

Recoupling coefficients (3nj symbols) are unitary transformations between binary coupled eigenstates of N=(n+1) mutually commuting SU(2) angular momentum operators. They have been used in a variety of applications in spectroscopy, quantum chemistry and nuclear physics and quite recently also in quantum gravity and quantum computing. These coefficients, naturally associated to cubic Yutsis graphs, share a number of intriguing combinatorial, algebraic, and analytical features that make them fashinating objects to be studied on their own. In this paper we develop a bottom--up, systematic procedure for the generation of 3nj from 3(n-1)j diagrams by resorting to diagrammatical and algebraic methods. We provide also a novel approach to the problem of classifying various regimes of semiclassical expansions of 3nj coefficients (asymptotic disentangling of 3nj diagrams) for n > 2 by means of combinatorial, analytical and numerical tools.

quant-ph↗

arXiv2010

Quantum Tetrahedra

We discuss in details the role of Wigner 6j symbol as the basic building block unifying such different fields as state sum models for quantum geometry, topological quantum field theory, statistical lattice models and quantum computing. The apparent twofold nature of the 6j symbol displayed in quantum field theory and quantum computing -a quantum tetrahedron and a computational gate- is shown to merge together in a unified quantum-computational SU(2)-state sum framework.

math-ph↗

arXiv2009

Microscopic description of 2d topological phases, duality and 3d state sums

Doubled topological phases introduced by Kitaev, Levin and Wen supported on two dimensional lattices are Hamiltonian versions of three dimensional topological quantum field theories described by the Turaev-Viro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state are shown in case of the honeycomb lattice and the gauge group being a finite group by means of the well-known duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three dimensional geometrical interpretation are given.

quant-ph↗

arXiv2009

Efficient quantum processing of 3-manifold topological invariants

A quantum algorithm for approximating efficiently 3--manifold topological invariants in the framework of SU(2) Chern-Simons-Witten (CSW) topological quantum field theory at finite values of the coupling constant k is provided. The model of computation adopted is the q-deformed spin network model viewed as a quantum recognizer in the sense of Wiesner and Crutchfield, where each basic unitary transition function can be efficiently processed by a standard quantum circuit. This achievement is an extension of the algorithm for approximating polynomial invariants of colored oriented links found in Refs 2,3. Thus all the significant quantities - partition functions and observables - of quantum CSW theory can be processed efficiently on a quantum computer, reflecting the intrinsic, field-theoretic solvability of such theory at finite k. The paper is supplemented by a critical overview of the basic conceptual tools underlying the construction of quantum invariants of links and 3-manifolds and connections with algorithmic questions that arise in geometry and quantum gravity models are discussed.

quant-ph↗

arXiv2008

Braiding and entanglement in spin networks: a combinatorial approach to topological phases

The spin network quantum simulator relies on the su(2) representation ring (or its q-deformed counterpart at q= root of unity) and its basic features naturally include (multipartite) entanglement and braiding. In particular, q-deformed spin network automata are able to perform efficiently approximate calculations of topological invarians of knots and 3-manifolds. The same algebraic background is shared by 2D lattice models supporting topological phases of matter that have recently gained much interest in condensed matter physics. These developments are motivated by the possibility to store quantum information fault-tolerantly in a physical system supporting fractional statistics since a part of the associated Hilbert space is insensitive to local perturbations. Most of currently addressed approaches are framed within a 'double' quantum Chern-Simons field theory, whose quantum amplitudes represent evolution histories of local lattice degrees of freedom. We propose here a novel combinatorial approach based on `state sum' models of the Turaev-Viro type associated with SU(2)_q-colored triangulations of the ambient 3-manifolds. We argue that boundary 2D lattice models (as well as observables in the form of colored graphs satisfying braiding relations) could be consistently addressed. This is supported by the proof that the Hamiltonian of the Levin-Wen condensed string net model in a surface Sigma coincides with the corresponding Turaev-Viro amplitude on Sigma x [0,1] presented in the last section.

math-ph↗

arXiv2006

Coupling of quantum angular momenta: an insight into analogic/discrete and local/global models of computation

In the past few years there has been a tumultuous activity aimed at introducing novel conceptual schemes for quantum computing. The approach proposed in (Marzuoli A and Rasetti M 2002, 2005a) relies on the (re)coupling theory of SU(2) angular momenta and can be viewed as a generalization to arbitrary values of the spin variables of the usual quantum-circuit model based on `qubits' and Boolean gates. Computational states belong to finite-dimensional Hilbert spaces labelled by both discrete and continuous parameters, and unitary gates may depend on quantum numbers ranging over finite sets of values as well as continuous (angular) variables. Such a framework is an ideal playground to discuss discrete (digital) and analogic computational processes, together with their relationships occuring when a consistent semiclassical limit takes place on discrete quantum gates. When working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum finite states--machines which in turn represent discrete versions of topological quantum computation models. We argue that our model embodies a sort of unifying paradigm for computing inspired by Nature and, even more ambitiously, a universal setting in which suitably encoded quantum symbolic manipulations of combinatorial, topological and algebraic problems might find their `natural' computational reference model.

cs.CC↗

arXiv2006

Quantum automata, braid group and link polynomials

The spin--network quantum simulator model, which essentially encodes the (quantum deformed) SU(2) Racah--Wigner tensor algebra, is particularly suitable to address problems arising in low dimensional topology and group theory. In this combinatorial framework we implement families of finite--states and discrete--time quantum automata capable of accepting the language generated by the braid group, and whose transition amplitudes are colored Jones polynomials. The automaton calculation of the polynomial of (the plat closure of) a link L on 2N strands at any fixed root of unity is shown to be bounded from above by a linear function of the number of crossings of the link, on the one hand, and polynomially bounded in terms of the braid index 2N, on the other. The growth rate of the time complexity function in terms of the integer k appearing in the root of unity q can be estimated to be (polynomially) bounded by resorting to the field theoretical background given by the Chern-Simons theory.

quant-ph↗

arXiv2005

Spin networks, quantum automata and link invariants

The spin network simulator model represents a bridge between (generalized) circuit schemes for standard quantum computation and approaches based on notions from Topological Quantum Field Theories (TQFT). More precisely, when working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum automata which in turn represent discrete versions of topological quantum computation models. Such a quantum combinatorial scheme, which essentially encodes SU(2) Racah--Wigner algebra and its braided counterpart, is particularly suitable to address problems in topology and group theory and we discuss here a finite states--quantum automaton able to accept the language of braid group in view of applications to the problem of estimating link polynomials in Chern--Simons field theory.

gr-qc↗

arXiv2004

Computing Spin Networks

We expand a set of notions recently introduced providing the general setting for a universal representation of the quantum structure on which quantum information stands. The dynamical evolution process associated with generic quantum information manipulation is based on the (re)coupling theory of SU(2) angular momenta. Such scheme automatically incorporates all the essential features that make quantum information encoding much more efficient than classical: it is fully discrete; it deals with inherently entangled states, naturally endowed with a tensor product structure; it allows for generic encoding patterns. The model proposed can be thought of as the non-Boolean generalization of the quantum circuit model, with unitary gates expressed in terms of 3nj coefficients connecting inequivalent binary coupling schemes of n+1 angular momentum variables, as well as Wigner rotations in the eigenspace of the total angular momentum. A crucial role is played by elementary j-gates (6j symbols) which satisfy algebraic identities that make the structure of the model similar to "state sum models", employed in discretizing Topological Quantum Field Theories and quantum gravity. The spin network simulator can thus be viewed also as a Combinatorial QFT model for computation. The semiclassical limit (large j's) is discussed.

quant-ph↗