PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Fusion path recursion and entropy

Statement

For a finite fusion transfer count model, let $e_{0} \in R^{I}$ be the coordinate vector equal to $1$ at label $0$ and $0$ elsewhere, and define $v_{n} : = N^{n} e_{0}$ . Then $v_{0} = e_{0}$ , $v_{n + 1} = N v_{n}$ , and the number of length- $n$ paths from $0$ to a label $a$ is $(v_{n})_{a} = (N^{n})_{a 0}$ . Whenever $D_{n} > 0$ , the operator $ρ_{n} = D_{n}^{- 1} I_{C^{D_{n}}}$ is a density matrix and has von Neumann entropy $s (ρ_{n}) = \log D_{n}$ in nats. For the supplied two-label matrix $N = (\begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array})$ , with labels ordered $0, 1$ , one has $D_{0} = 1$ , $D_{1} = 0$ , $D_{2} = 1$ , and $D_{n + 2} = D_{n + 1} + D_{n}$ for every $n \geq 0$ .

Statement references

Finite fusion transfer count model

Sector entropy and relative entropy identities

Direct-sum sector code entropy model

The basis-independent trace of an endomorphism of a finite-dimensional vector space

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm

Daniel Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction

John Watrous, The Theory of Quantum Information

Diagram convention

The diagram displays the two-label matrix only. An arrow $b \to a$ with label $m$ represents $N_{a b} = m$ transitions; the count proof below uses matrix multiplication and does not rely on the drawing.

Facts & Assumptions

Given: The finite set $I$ , label $0$ , matrix $N$ , and counts $D_{n}$ of the Definition, together with the displayed two-label specialization.

[F1]

Finite fusion transfer count model: $N_{a b}$ counts transitions from $b$ to $a$ , $(A B)_{a c} = \sum_{b \in I} A_{a b} B_{b c}$ , $N^{0} = I_{I}$ , $N^{n + 1} = N N^{n}$ , $D_{n} = (N^{n})_{00}$ , and $ρ_{n} = I_{H_{n}} / D_{n}$ is assigned only for $D_{n} > 0$ ; its coordinate inner product is linear in the first argument and is $⟨ x, y ⟩ = \sum_{j} x_{j} \overline{y_{j}}$ . The Definition verifies the inner-product axioms and defines real matrix addition and vector action by $(A + B)_{a b} = A_{a b} + B_{a b}$ and $(A x)_{a} = \sum_{b} A_{a b} x_{b}$ , using $N \subset R$ for the entries of $N$ ; it verifies compatibility of products and actions and distributivity.

[F2]

Direct-sum sector code entropy model: a density matrix on a finite-dimensional complex inner-product space is self-adjoint, has nonnegative eigenvalues, and has trace one.

[F3]

Sector entropy and relative entropy identities: for a finite density matrix $τ$ , $s (τ) = - \sum_{λ_{j} (τ) > 0} λ_{j} (τ) \log λ_{j} (τ)$ , with $0 \log 0 = 0$ and natural logarithms, so entropy is in nats.

[F4]

The basis-independent trace of an endomorphism of a finite-dimensional vector space: the trace is the basis-independent sum of diagonal entries of a finite-dimensional endomorphism.

[F5]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: for $x > 0$ , $\log (1 / x) = - \log x$ and $\log 1 = 0$ .

[A1]

The two-label matrix is $N = (\begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array})$ in the ordered basis $(0, 1)$ .

Proof

1.1givenF1base

For $n = 0$ , the unique path of length zero is the empty path from $0$ to $0$ , and there is no length-zero path from $0$ to any $a \neq 0$ . Thus the number of length-zero paths from $0$ to $a$ is $(I_{I})_{a 0} = (N^{0})_{a 0} = (e_{0})_{a}$ .

1.2F1F2F3F4F5algebra

If $D_{n} > 0$ , then $D_{n}$ is a positive integer and $ρ_{n} = D_{n}^{- 1} I_{H_{n}}$ has eigenvalue $1 / D_{n}$ with multiplicity $D_{n}$ . Its trace is $D_{n} (1 / D_{n}) = 1$ , and it is self-adjoint since $⟨ ρ_{n} x, y ⟩ = D_{n}^{- 1} ⟨ x, y ⟩ = ⟨ x, ρ_{n} y ⟩$ with the real scalar $D_{n}^{- 1}$ , and its eigenvalues are nonnegative; hence it is a density matrix by [F2]. Substitution into [F3] gives $s (ρ_{n}) = - D_{n} (1 / D_{n}) \log (1 / D_{n}) = \log D_{n}$ by [F5].

2.1step 1.1F1algebrainductiondischarge-induction

Suppose for a fixed $n \geq 0$ that the number of length- $n$ paths from $0$ to each $b \in I$ is $(N^{n})_{b 0}$ . Every length- $(n + 1)$ path from $0$ to $a$ has a unique penultimate label $b$ and a final transition copy from $b$ to $a$ , so the number of such paths is $\sum_{b \in I} N_{a b} (N^{n})_{b 0} = (N N^{n})_{a 0} = (N^{n + 1})_{a 0}$ by the stated matrix-product and power conventions. Induction proves $(v_{n})_{a} = (N^{n})_{a 0}$ for every $n \geq 0$ and $a \in I$ ; in particular $v_{0} = e_{0}$ and $v_{n + 1} = N v_{n}$ .

3.1A1F1algebra∎

Under [A1], direct multiplication gives $N^{2} = (\begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array}) = N + I_{I}$ . Therefore $D_{0} = 1$ , $D_{1} = 0$ , and $D_{2} = 1$ . From the power recursion and associativity, $N^{n + 2} = N^{2} N^{n} = (N + I_{I}) N^{n} = N^{n + 1} + N^{n}$ for every $n \geq 0$ ; taking the $(0, 0)$ entry proves $D_{n + 2} = D_{n + 1} + D_{n}$ .

Fusion path recursion and entropy

Statement

Statement references

Diagram convention

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources