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.Comment: 41 page
The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the Hausdorff-Young inequality, Young's inequality, the Hirschman-Beckner uncertainty principle, the Donoho-Stark uncertainty principle. We characterize the minimizers of the uncertainty principles. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's λ-lattices, modular tensor categories etc.
We introduce a pictorial approach to quantum information, called holographic software. Our software captures both algebraic and topological aspects of quantum networks. It yields a bi-directional dictionary to translate between a topological approach and an algebraic approach. Using our software, we give a topological simulation for quantum networks. The string Fourier transform (SFT) is our basic tool to transform product states into states with maximal entanglement entropy. We obtain a pictorial interpretation of Fourier transformation, of measurements, and of local transformations, including the n-qudit Pauli matrices and their representation by Jordan-Wigner transformations. We use our software to discover interesting new protocols for multipartite communication. In summary, we build a bridge linking the theory of planar para algebras with quantum information.
We introduce a new notion of an angle between intermediate subfactors and prove various interesting properties of the angle and relate it to the Jones index. We prove a uniform 60 60 to 90 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 of intermediate subfactors of an irreducible subfactor has at most exponential growth with respect to the Jones index. This answers a question of Longo’s published in 2003.
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 3D 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 multiquons and their transformations in a natural way. We obtain a 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, which are widely used in tensor networks. We obtain a 3D representation of the controlled NOT (CNOT) gate that is considerably simpler than earlier work, and a 3D topological protocol for teleportation.quon language | picture-language | quantum information | joint relation | topological algebra T opological quantum information was formulated by Kitaev (1) and Freedman et al. (2). Here, we formulate a 3D topological picture-language that we call the "quon language"-suggesting quantum particles. It leads to strikingly elementary mathematical proofs and insights into quantum information protocols. In our previous work, we represented qudits, the basic unit of quantum information, using charged strings in 2D. This fits naturally into the framework of planar para algebras (3-6). We call this our "two-string model."We also found a "four-string model" in 2D, in which we represent a 1-qudit vector as a neutral pair of particle-antiparticle charged strings (3, 4). These charged strings have the properties of parafermions. The presence of charges leads to para isotopy relations, which reflect the parafermion multiplication laws. Neutral pairs satisfy isotopy, a very appealing property. However, braiding two strings from different qudits destroys individual qudit neutrality, and this problem seemed unsurmountable for multiqudit states. So can one isolate those transformations that map the neutral pairs into themselves?Here, we solve this problem by defining "quons." We embed the neutral pairs of charged strings representing qudits into the interior of a 3-manifold. The quon language has the flavor of a topological field theory with strings. The resulting composites of 3-manifolds and strings give us quon states, transformations of quons, and quon measurements. However, the composites contain a further aspect: There are topological relations that involve both the strings and the manifolds. We call them "joint relations." These joint relations provide basic grammatical structure as well as insight into our language.In String-Genus Joint Relation, we see that, if a neutral string surrounds a genus in the manifold, then one can remove them both. In Topological Relations for C*-Hopf Algebras, we use this joint relation to obtain an elementary understanding of Frobenius and C * -Hopf algebra relations stated in Bi-Frobenius Algebras. These relations are key in tensor net...
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