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Zhengwei Liu

Publications and source records attributed to Zhengwei Liu.

At least 91 records · Page 5Linked to original sources
arXiv2017

An angle between intermediate subfactors and its rigidity

We introduce a new notion of angle between intermediate subfactors and prove various interesting properties of the angle and relate it with the Jones' index. We prove a uniform 60 to 90 degree bound for the angle between minimal intermediate subfactors of a finite index irreducible subfactor. From this rigidity we can bound the number of minimal (or maximal) intermediate subfactors by the kissing number in geometry. As a consequence, the number intermediate subfactors of an irreducible subfactor has at most exponential growth with respect to the Jones index. This answers a question of Longo published in 2003.

math.OA↗
arXiv2017

Lifting shadings on symmetrically self-dual subfactor planar algebras

In this note, we discuss the notion of symmetric self-duality of shaded planar algebras, which allows us to lift shadings on subfactor planar algebras to obtain Z/2Z-graded unitary fusion categories. This finishes the proof that there are unitary fusion categories with fusion graphs 4442 and 3333.

math.OA↗
arXiv2017

Quon language: surface algebras and Fourier duality

Quon language is a 3D picture language that we can apply to simulate mathematical concepts. We introduce the surface algebras as an extension of the notion of planar algebras to higher genus surface. We prove that there is a unique one-parameter extension. The 2D defects on the surfaces are quons, and surface tangles are transformations. We use quon language to simulate graphic states that appear in quantum information, and to simulate interesting quantities in modular tensor categories. This simulation relates the pictorial Fourier duality of surface tangles and the algebraic Fourier duality induced by the S matrix of the modular tensor category. The pictorial Fourier duality also coincides with the graphic duality on the sphere. For each pair of dual graphs, we obtain an algebraic identity related to the $S$ matrix. These identities include well-known ones, such as the Verlinde formula; partially known ones, such as the 6j-symbol self-duality; and completely new ones.

math.QA↗
arXiv2017

Block maps and Fourier analysis

We introduce block maps for subfactors and study their dynamic systems. We prove that the limit points of the dynamic system are positive multiples of biprojections and zero. For the Z2 case, the asymptotic phenomenon of the block map coincides with that of that 2D Ising model. The study of block maps requires a further development of the recent work of the authors on the Fourier analysis of subfactors. We generalize the notion of sum set estimates in additive combinatorics for subfactors and prove the exact inverse sum set theorem. Using this new method, we characterize the extremal pairs of Young's inequality for subfactors, as well as the extremal operators of the Hausdorff-Young inequality.

math.OA↗
arXiv2017

Jones-Wassermann subfactors for modular tensor categories

The representation theory of a conformal net is a unitary modular tensor category. It is captured by the bimodule category of the Jones-Wassermann subfactor. In this paper, we construct multi-interval Jones-Wassermann subfactors for unitary modular tensor categories. We prove that these subfactors are self-dual. It generalizes and categorifies the self-duality of finite abelian groups and we call it modular self-duality.

math.OA↗
arXiv2017

Quons: A 3D Language for Quantum Information

We present a 3D, topological picture-language for quantum information. Our approach combines charged excitations carried by strings, with topological properties that arise from embedding the strings in the interior of a three-dimensional manifold with boundary. A quon is a composite that acts as a particle. Specifically a quon is a hemisphere containing a neutral pair of open strings with opposite charge. We interpret multi-quons and their transformations in a natural way. We obtain a new type of relation, a string-genus "joint relation," involving both a string and the 3D manifold. We use the joint relation to obtain a topological interpretation of the $C^{*}$ Hopf algebra relations, that are widely used in tensor networks. We obtain a 3D representation of the Controlled NOT or CNOT gate (that is considerably simpler than earlier work) and a 3D topological protocol for teleportation.

quant-ph↗
arXiv2017

Constructive Simulation and Topological Design of Protocols

We give a topological simulation for tensor networks that we call the two-string model. In this approach we give a new way to design protocols, and we discover a new multipartite quantum communication protocol. We introduce the notion of topologically-compressed transformations. Our new protocol can implement multiple, non-local compressed transformations among multi-parties using one multipartite resource state.

quant-ph↗
arXiv2016

Classification of Thurston-relation subfactor planar algebras

Bisch and Jones suggested the skein theoretic classification of planar algebras and investigated the ones generated by 2-boxes with the second author. In this paper, we consider 3-box generators and classify subfactor planar algebras generated by a non-trivial 3-box satisfying a relation proposed by Thurston. The subfactor planar algebras in the classification are either $E^6$ or the ones from representations of quantum $SU(N)$. We introduce a new method to determine positivity of planar algebras and new techniques to reduce the complexity of computations.

math.OA↗
arXiv2016

Uncertainty Principles for Kac Algebras

In this paper, we introduce the notation of bi-shift of biprojections in subfactor theory to unimodular Kac algebras. We characterize the minimizers of Hirschman-Beckner uncertainty principle and Donoho-Stark uncertainty principle for unimodular Kac algebras with biprojections and prove Hardy's uncertainty principle in terms of minimizers.

math.OA↗
arXiv2016

Compressed Teleportation

In a previous paper we introduced holographic software for quantum networks, inspired by work on planar para algebras. This software suggests the definition of a compressed transformation. Here we utilize the software to find a CT protocol to teleport compressed transformations. This protocol serves multiple parties with multiple persons.

quant-ph↗
arXiv2016

Planar Para Algebras, Reflection Positivity

We define a planar para algebra, which arises naturally from combining planar algebras with the idea of $\mathbb{Z}_{N}$ para symmetry in physics. A subfactor planar para algebra is a Hilbert space representation of planar tangles with parafermionic defects, that are invariant under para isotopy. For each $\mathbb{Z}_{N}$, we construct a family of subfactor planar para algebras which play the role of Temperley-Lieb-Jones planar algebras. The first example in this family is the parafermion planar para algebra (PAPPA). Based on this example, we introduce parafermion Pauli matrices, quaternion relations, and braided relations for parafermion algebras which one can use in the study of quantum information. An important ingredient in planar para algebra theory is the string Fourier transform (SFT), that we use on the matrix algebra generated by the Pauli matrices. Two different reflections play an important role in the theory of planar para algebras. One is the adjoint operator; the other is the modular conjugation in Tomita-Takesaki theory. We use the latter one to define the double algebra and to introduce reflection positivity. We give a new and geometric proof of reflection positivity, by relating the two reflections through the string Fourier transform.

math.QA↗
arXiv2016

Yang-Baxter relation planar algebras

We give a new type of Schur-Weyl duality for the representations of a family of quantum subgroups and their centralizer algebra. We define and classify singly-generated, Yang-Baxter relation planar algebras. We present the skein theoretic construction of a new parameterized planar algebra. We construct infinitely many new subfactors and unitary fusion categories, and compute their trace formula as a closed-form expression, in terms of Young diagrams.

math.OA↗
arXiv2016

Qudit Isotopy

We explore a general diagrammatic framework to understand qudits and their braiding, especially in its relation to entanglement. This involves understanding the role of isotopy in interpreting diagrams that implement entangling gates as well as some standard quantum information protocols. We give qudit Pauli operators $X,Y,Z$ and comment on their structure, both from an algebraic and from a diagrammatic point of view. We explain alternative models for diagrammatic interpretations of qudits and their transformations. We use our diagrammatic approach to define an entanglement-relay protocol for long-distance entanglement. Our approach rests on algebraic and topological relations discovered in the study of planar para algebras. In summary, this work provides bridges between the new theory of planar para algebras and quantum information, especially in questions involving entanglement.

quant-ph↗
arXiv2015

The generator conjecture for $3^G$ subfactor planar algebras

We state a conjecture for the formulas of the depth 4 low-weight rotational eigenvectors and their corresponding eigenvalues for the $3^G$ subfactor planar algebras. We prove the conjecture in the case when $|G|$ is odd. To do so, we find an action of $G$ on the reduced subfactor planar algebra at $f^{(2)}$, which is obtained from shading the planar algebra of the even half. We also show that this reduced subfactor planar algebra is a Yang-Baxter planar algebra.

math.OA↗
arXiv2015

A super-Eddington wind scenario for the progenitors of type Ia supernovae: binary population synthesis calculations

The super-Eddington wind scenario has been proposed as an alternative way for producing type Ia supernovae (SNe Ia). The super-Eddington wind can naturally prevent the carbon--oxygen white dwarfs (CO WDs) with high mass-accretion rates from becoming red-giant-like stars. Furthermore, it works in low-metallicity environments, which may explain SNe Ia observed at high redshifts. In this article, we systematically investigated the most prominent single-degenerate WD+MS channel based on the super-Eddington wind scenario. We combined the Eggleton stellar evolution code with a rapid binary population synthesis (BPS) approach to predict SN Ia birthrates for the WD+MS channel by adopting the super-Eddington wind scenario and detailed mass-accumulation efficiencies of H-shell flashes on the WDs. Our BPS calculations found that the estimated SN Ia birthrates for the WD+MS channel are ~0.009-0.315*10^{-3}{yr}^{-1} if we adopt the Eddington accretion rate as the critical accretion rate, which are much lower than that of the observations (<10% of the observed SN Ia birthrates). This indicates that the WD+MS channel only contributes a small proportion of all SNe Ia. The birthrates in this simulation are lower than previous studies, the main reason of which is that new mass-accumulation efficiencies of H-shell flashes are adopted. We also found that the critical mass-accretion rate has a significant influence on the birthrates of SNe Ia. Meanwhile, the results of our BPS calculations are sensitive to the values of the common-envelope ejection efficiency.

astro-ph.SR↗
arXiv2014

Singly generated planar algebras of small dimension, Part III

The first two authors classified subfactor planar algebra generated by a non-trivial 2-box subject to the condition that the dimension of 3-boxes is at most 12 in Part I; 13 in Part II of this series. They are the group planar algebra for $\mathbb{Z}_3$, the Fuss-Catalan planar algebra ; and the group/subgroup planar algebra for $\mathbb{Z}_2\subset \mathbb{Z}_5\rtimes \mathbb{Z}_2$. In the present paper, we extend the classification to 14 dimensional 3-boxes. They are all BMW. Precisely it contains a depth 3 one from quantum $SO(3)$, and a one-parameter family from quantum $Sp(4)$.

math.OA↗
arXiv2014

The hybrid CONe WD + He star scenario for the progenitors of type Ia supernovae

The hybrid CONe white dwarfs (WDs) have been suggested to be possible progenitors of type Ia supernovae (SNe Ia). In this article, we systematically studied the hybrid CONe WD + He star scenario for the progenitors of SNe Ia, in which a hybrid CONe WD increases its mass to the Chandrasekhar mass limit by accreting He-rich material from a non-degenerate He star. According to a series of detailed binary population synthesis simulations, we obtained the SN Ia birthrates and delay times for this scenario. The SN Ia birthrates for this scenario are ~0.033-0.539*10^(-3)yr^(-1), which roughly accounts for 1-18% of all SNe Ia. The estimated delay times are ~28Myr-178Myr, which are the youngest SNe Ia predicted by any progenitor model so far. We suggest that SNe Ia from this scenario may provide an alternative explanation of type Iax SNe. We also presented some properties of the donors at the point when the WDs reach the Chandrasekhar mass. These properties may be a good starting point for investigating the surviving companions of SNe Ia, and for constraining the progenitor scenario studied in this work.

astro-ph.SR↗