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Six-anyon triple-encoded subspace and leakage complement

Definition

Assume the Fibonacci fusion rules $1 \otimes x = x \otimes 1 = x$ and $τ \otimes τ = 1 \oplus τ$ , with each listed summand of multiplicity one. For six ordered $τ$ charges, let $W_{a}$ be the fixed-input, fixed-total fusion space of total charge $a \in {1, τ}$ in the tree that first fuses leaves $1, 2, 3$ and $4, 5, 6$ separately, then fuses those triple outputs. Let $V_{a}$ be the same fixed-charge space in the left-associated tree.

To identify these spaces, write $A = ((τ_{1} \otimes τ_{2}) \otimes τ_{3})$ . Use the object associator composite $g = (α_{A \otimes τ_{4}, τ_{5}, τ_{6}}^{- 1}) \circ α_{A, τ_{4}, τ_{5} \otimes τ_{6}}^{- 1} \circ ({id}_{A} \otimes α_{τ_{4}, τ_{5}, τ_{6}})$ from the triple-grouped input tree to the left-associated tree. On a fusion morphism $f$ into $a$ , define $G_{a} (f) = f \circ g^{- 1} : V_{a}$ ; this is a unitary change of fusion-tree coordinates.

For one triple of total charge $τ$ , its two orthonormal path vectors are indexed by the charge $x \in {1, τ}$ after the first two anyons: both $x \otimes τ \to τ$ channels have multiplicity one. Write these vectors as $∣ x ⟩$ . For $x, y \in {1, τ}$ and $a \in {1, τ}$ , let $∣ x, y; a ⟩$ be the grouped-tree vector with first triple path $x$ , first triple total $τ$ , second triple path $y$ , second triple total $τ$ , and final vertex channel $τ \otimes τ \to a$ . The last channel has multiplicity one. Define $C_{a} : = G_{a} (span {∣ x, y; a ⟩ : x, y \in {1, τ}}) \subseteq V_{a} .$ In the ordered basis $(G_{a} ∣ 1, 1; a ⟩, G_{a} ∣ 1, τ; a ⟩, G_{a} ∣ τ, 1; a ⟩, G_{a} ∣ τ, τ; a ⟩)$ , the inclusion $J_{a} : C^{4} \to V_{a}$ is an isometry; its code projector is $P_{a} = J_{a} J_{a}^{*}$ . The same ordered local triple paths are used for $a = 1$ and $a = τ$ ; only the final pair-charge coordinate differs. The orthogonal complements are called leakage spaces.

Facts & Assumptions

Given: The stated Fibonacci fusion rules, the multiplicity-aware fusion-tree basis, the unitary associator, and the fixed-charge encoding convention.

[A1]

Assume the Fibonacci fusion rules stated in the Definition. The actual Fibonacci data carrier states $1 \otimes x = x \otimes 1 = x$ and $τ \otimes τ = 1 \oplus τ$ with multiplicity one in each channel. Fibonacci fusion data and chiral convention

[F1]

The fusion-tree definition constructs each path vector by composing its vertex multiplicity-basis maps. Multiplicity-aware fusion-tree basis

[F2]

The fusion-tree lemma proves that these path vectors form a complete orthonormal basis and counts them by the products of the vertex multiplicities. Fusion-tree basis and dimension recursion

[F3]

An isometric inclusion $J$ has code projector $J J^{*}$ onto its image. Fixed-charge encoding and code projector

[F4]

Each associator is unitary, so the stated finite associator composite induces a unitary map between the two fusion-tree Hom spaces. Unitary fusion category F-symbol unitarity

Proof

1.1A1F2F4algebra

The fusion rule gives $\dim V (τ^{2} \to 1) = \dim V (τ^{2} \to τ) = 1$ and therefore $\dim V (τ^{3} \to 1) = 1$ , $\dim V (τ^{3} \to τ) = 2$ . Applying the fusion-tree count [F2] to the grouped tree, total charge $1$ receives one basis vector from the $(1, 1)$ triple-output pair and four from $(τ, τ)$ , so $\dim W_{1} = 1 + 2 \cdot 2 = 5$ . Total charge $τ$ receives two vectors from $(1, τ)$ , two from $(τ, 1)$ , and four from $(τ, τ)$ , so $\dim W_{τ} = 2 + 2 + 4 = 8$ . The map $G_{a}$ in [F4] is unitary, hence $\dim V_{1} = 5$ and $\dim V_{τ} = 8$ .

1.2F1F2F3F4givenalgebra

Each selected triple-total- $τ$ factor has its two paths $∣ 1 ⟩$ and $∣ τ ⟩$ , and the final $τ \otimes τ \to a$ vertex has multiplicity one. Thus the four grouped-tree vectors $∣ x, y; a ⟩$ are admissible tree vectors by [F1] and form an orthonormal family by [F2]. Applying the unitary $G_{a}$ preserves orthonormality, so they define an isometric inclusion $J_{a}$ ; [F3] gives the projector $P_{a} = J_{a} J_{a}^{*}$ and $\dim C_{a} = 4$ .

2.1step 1.1step 1.2algebra∎

The orthogonal decompositions $V_{a} = C_{a} \oplus C_{a}^{⊥}$ then give $\dim C_{1}^{⊥} = 5 - 4 = 1$ and $\dim C_{τ}^{⊥} = 8 - 4 = 4$ by step 1.1. These code copies share the same ordered local triple paths, while their final output charge differs. This supplies the dimensions and path identification used in the two B6 blocks.

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Sources

Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)
Michael H. Freedman, Michael J. Larsen, and Zhenghan Wang, A Modular Functor Which Is Universal for Quantum Computation (standard reference, not scraped)