De Finetti Theorems for Braided Parafermions

Abstract

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).

arXiv comment25 pages

References in corpus (23)

• Quantum entanglement
• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Unknown Quantum States: The Quantum de Finetti Representation
• Exotic non-Abelian anyons from conventional fractional quantum Hall states
• Parafermionic edge zero modes in Z_n-invariant spin chains
• Symmetry implies independence
• One-and-a-half quantum de Finetti theorems
• Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
• A de Finetti representation for finite symmetric quantum states
• A Generalization of Quantum Stein's Lemma
• Parafermions in Interacting Nanowire Bundle
• A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation
• De Finetti theorems for easy quantum groups
• Quantum de Finetti theorem under fully-one-way adaptive measurements
• Planar Para Algebras, Reflection Positivity
• A quasipolynomial-time algorithm for the quantum separability problem
• De Finetti theorem on the CAR algebra
• Quons: A 3D Language for Quantum Information
• Finite de Finetti theorem for conditional probability distributions describing physical theories
• Noncommutative independence from the braid group $B_{\infty }$
• Holographic Software for Quantum Networks
• Constructive Simulation and Topological Design of Protocols