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).
Highlights
Parafermion Algebras and the parafermion planar para algebras (PAPPA) ModelThe parafermion algebra P Fm of degree d is the Zd -graded *-algebra generated by { c j }mj=1, with m possibly infinite
Størmer [Sto69] proposed a non-commutative version of the de Finetti theorem, and he demonstrated that extremal, symmetric states on infinite, tensor-product C∗ algebras can be expressed in terms of product states
We have proposed and proved a new type of de Finetti theorem for the parafermion algebra P F∞ with respect to the action of braid group B∞ that braids qudits
Summary
The parafermion algebra P Fm of degree d is the Zd -graded *-algebra generated by { c j }mj=1, with m possibly infinite. Each two-string braid can be expressed in terms of the generators of the parafermion algebra, as shown in formula (8.1) of [JL17], bk(2) =. The left-most single braid b2(2j)−1 and the right-most single braid b2(2j)+1 illustrated in (10) commute, so their relative vertical order does not matter This decomposition shows that b j is in the algebra generated by the four parafermions c2 j−1, c2 j , c2 j+1, c2 j+2. We are especially concerned with shifts of pairs of generators of the parafermion algebra, as they correspond to the action of the four-string braids. For any b ∈ Let (P F φ∞B∞)B.∞LdetenSoBt∞e denote the set the fixed point of B∞-invariant algebra states under on P F∞. Let B = b1 · · · b denote the unitary transformation implementing this element of the braid group on the GNS Hilbert space H.
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