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Arthur Jaffe

Publications and source records attributed to Arthur Jaffe.

At least 37 records · Page 2Linked to original sources
arXiv2022

Quantifying scrambling in quantum neural networks

We characterize a quantum neural network's error in terms of the network's scrambling properties via the out-of-time-ordered correlator. A network can be trained by optimizing either a loss function or a cost function. We show that, with some probability, both functions can be bounded by out-of-time-ordered correlators. The gradients of these functions can be bounded by the gradient of the out-of-time-ordered correlator, demonstrating that the network's scrambling ability governs its trainability. Our results pave the way for the exploration of quantum chaos in quantum neural networks.

quant-ph↗
arXiv2021

Quantum scrambling with classical shadows

Quantum dynamics is of fundamental interest and has implications in quantum information processing. The four-point out-of-time-ordered correlator (OTOC) is traditionally used to quantify quantum information scrambling under many-body dynamics. Due to the OTOC's unusual time ordering, its measurement is challenging. We propose higher-point OTOCs to reveal early-time scrambling behavior, and present protocols to measure any higher-point OTOC using the shadow estimation method. The protocols circumvent the need for time-reversal evolution and ancillary control. They can be implemented in near-term quantum devices with single-qubit readout.

quant-ph↗
arXiv2020

A Mathematical Picture Language Project

The mathematical picture language project that we began in 2016 has already yielded interesting results. We also point out areas of mathematics and physics where we hope that it will prove useful in the future.

math-ph↗
arXiv2020

Quantum Fourier Analysis

{\em Quantum Fourier analysis} is a new subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform $\FS$, as a map between suitably defined $L^{p}$ spaces, leading to a new uncertainty principle for relative entropy. We cite several applications of the quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a new topological inequality, and we outline several open problems.

math.OA↗
arXiv2019

Duality of Graph Invariants

We study a new set of duality relations between weighted, combinatoric invariants of a graph $G$. The dualities arise from a non-linear transform $\mathfrak{B}$, acting on the weight function $p$. We define $\mathfrak{B}$ on a space of real-valued functions $\mathcal{O}$ and investigate its properties. We show that three invariants (weighted independence number, weighted Lovász number, and weighted fractional packing number) are fixed points of $\mathfrak{B}^{2}$, but the weighted Shannon capacity is not. We interpret these invariants in the study of quantum non-locality.

math.FA↗
arXiv2019

De Finetti Theorems for Braided Parafermions

The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states on the parafermion algebra, including finding a distinction that depends on whether the order of the parafermion algebra is square free. We characterize the extremal nature of product states (an inverse de Finetti theorem).

math-ph↗
arXiv2019

Reflection Positivity and Levin-Wen Models

The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin-Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.

math-ph↗
arXiv2018

Reflection Positivity Then and Now

We review the discovery of reflection positivity. We also explain a new geometric approach and proof of the reflection positivity property.

math.HO↗
arXiv2017

Holographic Software for Quantum Networks

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.

quant-ph↗
arXiv2017

A Mathematical Picture Language Program

We give an overview of our philosophy of pictures in mathematics. We emphasize a bi-directional process between picture language and mathematical concepts: abstraction and simulation. This motivates a program to understand different subjects, using virtual and real mathematical concepts simulated by pictures.

math.HO↗
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

Reflection Positive Doubles

Here we introduce reflection positive doubles, a general framework for reflection positivity, covering a wide variety of systems in statistical physics and quantum field theory. These systems may be bosonic, fermionic, or parafermionic in nature. Within the framework of reflection positive doubles, we give necessary and sufficient conditions for reflection positivity. We use a reflection-invariant cone to implement our construction. Our characterization allows for a direct interpretation in terms of coupling constants, making it easy to check in concrete situations. We illustrate our methods with numerous examples.

math-ph↗
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

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↗