Fusion of anyons
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Anyon fusion is the process by which multiple anyons behave as one larger composite anyon. Anyon fusion is essential to understanding the physics of non-abelian anyons and how they can be used in quantum information.
If identical abelian anyons each with individual statistics (that is, the system picks up a phase when two individual anyons undergo adiabatic counterclockwise exchange) all fuse together, they together have statistics . This can be seen by noting that upon counterclockwise rotation of two composite anyons about each other, there are pairs of individual anyons (one in the first composite anyon, one in the second composite anyon) that each contribute a phase . An analogous analysis applies to the fusion of non-identical abelian anyons. The statistics of the composite anyon is uniquely determined by the statistics of its components.
Non-abelian anyon fusion rulesEdit
Non-abelian anyons have more complicated fusion relations. As a rule, in a system with non-abelian anyons, there is a composite particle whose statistics label is not uniquely determined by the statistics labels of its components, but rather exists as a quantum superposition (this is completely analogous to how two fermions known to each have spin 1/2 and 3/2 are together in quantum superposition of total spin 1 and 2). If the overall statistics of the fusion of all of several anyons is known, there is still ambiguity in the fusion of some subsets of those anyons, and each possibility is a unique quantum state. These multiple states provide a Hilbert space on which quantum computation can be done.
Specifically, two non-abelian anyons labeled and have a fusion rule given by , where the formal sum over goes over all labels of possible anyon types in the system (as well as the trivial label denoting no particles), and each is a nonnegative integer which denotes how many distinct quantum states there are in which and fuse into (This is true in the abelian case as well, except in that case, for each and , there is one type of anyon for which and for all other , .) Each anyon type should also have a conjugate antiparticle among the list of possible anyon types, such that , i.e. it can annihilate with its antiparticle. The anyon type label does not specify all of the information about the anyon, but the information that it does indicate is topologically invariant under local perturbations.
For example, the Fibonacci anyon system, one of the simplest, consists of labels and ( denotes a Fibonacci anyon), which satisfy fusion rule (corresponding to ) as well as the trivial rules and (corresponding to ).
The Ising anyon system consists of labels , and , which satisfy fusion rules , , and the trivial rules.
The operation is commutative and associative, as it must be to physically make sense with fused anyons. Furthermore, it is possible to view the coefficients as matrix entries of a matrix with row and column indices and ; then the largest eigenvalue of this matrix is known as the quantum dimension of anyon type .
Fusion rules can also be generalized to consider in how many ways a collection can be fused to a final anyon type .
Hilbert spaces of fusion processesEdit
The fusion process where and fuse into corresponds to a dimensional complex vector space , consisting of all the distinct orthonormal quantum states in which and fuse into . This forms a Hilbert space. When , such as in the Ising and Fibonacci examples, is at most just a one dimensional space with one state. The direct sum is a decomposition of the tensor product of the Hilbert space of individual anyon and the Hilbert space of individual anyon . In topological quantum field theory, is the vector space associated with the pair of pants with waist labeled and legs and .
More complicated Hilbert spaces can be constructed corresponding to the fusion of three or more particles, i.e. for the quantum systems where it is known that the fuse into final anyon type . This Hilbert space would describe, for example, the quantum system formed by starting with a quasiparticle and, via some local physical procedure, splitting up that quasiparticle into quasiparticles (because in such a system all the anyons must necessarily fuse back into by topological invariance). There is an isomorphism between and for any . As mentioned in the previous section, the permutations of the labels are also isomorphic.
One can understand the structure of by considering fusion processes one pair of anyons at a time. There are many arbitrary ways one can do this, each of which can be used to derive a different decomposition of into pairs of pants. One possible choice is to first fuse and into , then fuse and into , and so on. This approach shows us that , and correspondingly where is the matrix defined in the previous section.
This decomposition manifestly indicates a choice of basis for the Hilbert space. Different arbitrary choices of the order in which to fuse anyons will correspond to different choices of basis.