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 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...

An ideal characterization of the Clifford operators J M Farinholt -

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