Quon 3D language for quantum information

Abstract

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...

Quon Language: Surface Algebras and Fourier Duality

“…It is shown in [LWJ17] that |GHZ and |M ax are graphic quons, and the corresponding surface tangles are given by Inspired by this observation, we generalize |GHZ and |M ax to n-quons on genus-g surfaces for the MTC C .…”

“…In tensor networks, the |GHZ and |M ax are represented as two trivalent vertices: and . They have been considered as two fundamental tensors in [Laf03], see also [LWJ17,Bia17,CK17].…”

“…Quon language is a 3D picture language that we can apply to simulate mathematical concepts [LWJ17]. It was designed to answer a question in quantum information, where the underling symmetry is the group Z 2 for qubits and Z d for qudits.…”

Block maps and Fourier analysis

“…If we apply the sum set estimate to group subfactors and take p, q to be central projections of the group algebra acting on the regular representation, then Equation (5) is equivalent to the following result about representations of a finite group: The SFT has been applied to quantum information in [8,20]. 1 It is shown that the SFT on quons for a unitary modular tensor category is the modular S matrix in [19,20]. Therefore we can apply our results on Fourier analysis to quons and obtain new results about the modular tensor category and the S matrix.…”

Charged string tensor networks

“…Building on a series of results (4-6) Liu, Wozniakowski and Jaffe have recently developed a topological variant of tensor networks which among other results, lead to their discovery of an elegant charged string braiding for the controlled-NOT gate (a.k.a. the Feynman gate) (7).…”