arXiv subjects

Sachin J. Valera

Publications and source records attributed to Sachin J. Valera.

4 recordsLinked to original sources

arXiv2024

Topological Quantum Teleportation and Superdense Coding -- Without Braiding

We present the teleportation and superdense coding protocols for a family of anyon theories coming from Tambara-Yamagami categories, of which the lowest rank theories describe Ising anyons. In contrast to the usual approach to anyonic computation, we relax the requirement that we should be able to braid anyons. This is motivated by the goal of designing basic protocols that require less control over quasiparticles, and which may therefore be amenable to realisation in near-term systems. Since these implementations are braid-free, they are also compatible with Majorana modes on a 1d quantum wire.

quant-ph↗

arXiv2021

Fusion Structure from Exchange Symmetry in (2+1)-Dimensions

Until recently, a careful derivation of the fusion structure of anyons from some underlying physical principles has been lacking. In [Shi et al., Ann. Phys., 418 (2020)], the authors achieved this goal by starting from a conjectured form of entanglement area law for 2D gapped systems. In this work, we instead start with the principle of exchange symmetry, and determine the minimal prescription of additional postulates needed to make contact with unitary ribbon fusion categories as the appropriate algebraic framework for modelling anyons. Assuming that 2D quasiparticles are spatially localised, we build a functor from the coloured braid groupoid to the category of finite-dimensional Hilbert spaces. Using this functor, we construct a precise notion of exchange symmetry, allowing us to recover the core fusion properties of anyons. In particular, given a system of $n$ quasiparticles, we show that the action of a certain $n$-braid $β_{n}$ uniquely specifies its superselection sectors. We then provide an overview of the braiding and fusion structure of anyons in the usual setting of braided $6j$ fusion systems. By positing the duality axiom of [A. Kitaev, Ann. Phys., 321(1) (2006)] and assuming that there are finitely many distinct topological charges, we arrive at the framework of ribbon categories.

quant-ph↗

arXiv2021

Skein-Theoretic Methods for Unitary Fusion Categories

Unitary fusion categories (UFCs) have gained increased attention due to emerging connections with quantum physics. We consider a fusion rule of the form $q\otimes q \cong \mathbf{1}\oplus\bigoplus^k_{i=1}x_{i}$ in a UFC $\mathcal{C}$, and extract information using the graphical calculus. For instance, we classify all associated skein relations when $k=1,2$ and $\mathcal{C}$ is ribbon. In particular, we also consider the instances where $q$ is antisymmetrically self-dual. Our main results follow from considering the action of a rotation operator on a "canonical basis". Assuming self-duality of the summands $x_{i}$, some general observations are made e.g. the real-symmetricity of the $F$-matrix $F^{qqq}_q$. We then find explicit formulae for $F^{qqq}_q$ when $k=2$ and $\mathcal{C}$ is ribbon, and see that the spectrum of the rotation operator distinguishes between the Kauffman and Dubrovnik polynomials.

math.QA↗

arXiv2020

Anyons and the HOMFLY Skein Algebra

We give an exposition of how the Kauffman bracket arises for certain systems of anyons, and do so outside the usual arena of Temperley-Lieb-Jones categories. This is further elucidated through the discussion of the Iwahori-Hecke algebra and its relation to modular tensor categories. We then proceed to classify the framed link-invariants associated to a system of self-dual anyons $q$ with $\sum_{x}N_{qq}^{x}\leq2$. In particular, we construct a trace on the HOMFLY skein algebra which can be expanded via gauge-invariant quantities, thereby generalising the case of the Kauffman bracket. Various examples are provided, and we deduce some interesting properties of these anyons along the way.

math-ph↗