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

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Jones-Wenzl image gives a finite maximally mixed sector

Statement

Under the finite Hilbert representation data, let $r : = rank P > 0$ and $ρ : = P / r$ . An orthogonal projection on $H$ means a complex-linear endomorphism $Q$ satisfying $Q^{2} = Q$ and $⟨ Q x, y ⟩ = ⟨ x, Q y ⟩$ for all $x, y \in H$ . Then $P$ is an orthogonal projection, $ρ$ is a density matrix with eigenvalue $1 / r$ of multiplicity $r$ and eigenvalue $0$ on $\ker P$ , and its entropy is $s (ρ) = \log r$ in nats. Under the published closure convention, the unnormalized diagrammatic closure trace of the algebraic Jones-Wenzl element is ${Tr}_{n} (p_{n}) = [n + 1]_{δ}$ . The proposition records this separately from the ordinary Hilbert-space trace ${tr}_{H} (P) = r$ and does not assert the two values agree. A sufficient additional hypothesis for equality on this element is that the representation preserve these traces, ${tr}_{H} (π (Φ_{n}^{- 1} (X))) = {Tr}_{n} (X)$ for every $X \in {T L}_{n} (C, δ)$ .

Statement references

Finite Hilbert boundary representation of Temperley-Lieb

Jones–Wenzl projectors: annihilation, uniqueness, partial trace, and closure

Direct-sum sector code entropy model

Sector entropy and relative entropy identities

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

Kernel and image of a linear map

Unnormalized diagram closure and partial trace

Jones–Wenzl recursion over unit quantum integers

Joshua Chen, The Temperley-Lieb Categories and Skein Modules

Zhenghan Wang, Topological Quantum Computation

K. Iohara, G. I. Lehrer, and R. B. Zhang, The Jones Quotients of the Temperley-Lieb Algebras

John Watrous, The Theory of Quantum Information

Facts & Assumptions

Given: The data of the finite Hilbert representation definition, including $n \geq 1$ , $δ \in R$ , nonzero denominators $[2]_{δ}, \dots, [n]_{δ}$ , a unital representation with self-adjoint generator images, and $r = rank P > 0$ .

[L1]

Finite Hilbert boundary representation of Temperley-Lieb: ${\hat{p}}_{n}$ is the presentation image of the recursively defined diagram projector, $P = π ({\hat{p}}_{n})$ , every $π (u_{i})$ is self-adjoint, and $r = \dim_{C} im P > 0$ .

[F1]

Jones–Wenzl recursion over unit quantum integers: $p_{1} = 1_{1}$ and $p_{k + 1} = (p_{k} \otimes 1_{1}) - ([k]_{δ} / [k + 1]_{δ}) (p_{k} \otimes 1_{1}) d_{k} (p_{k} \otimes 1_{1})$ .

[F2]

Jones–Wenzl projectors: annihilation, uniqueness, partial trace, and closure: if $[2]_{δ}, \dots, [n]_{δ}$ are units, then $p_{n}^{2} = p_{n}$ and its unnormalized diagram closure trace is $[n + 1]_{δ}$ .

[F3]

Unnormalized diagram closure and partial trace: ${Tr}_{n}$ is the linear functional obtained by closing each top point to the corresponding bottom point and assigning a factor $δ$ to each resulting loop.

[F4]

Direct-sum sector code entropy model: a finite density matrix is self-adjoint, has nonnegative eigenvalues, and has trace one.

[F5]

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

[F6]

Sector entropy and relative entropy identities: for a 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.

[F7]

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

[F8]

Kernel and image of a linear map: $\ker T = {x : T (x) = 0}$ and $im T = {T (x) : x \in H}$ for a linear endomorphism $T$ of $H$ .

Proof

technique · Induct along the Jones-Wenzl recursion index to prove self-adjointness, then use the range-kernel decomposition of an idempotent

1.1L1F2algebra

By [F2], $p_{n}^{2} = p_{n}$ . The map $Φ_{n}$ and the representation $π$ are algebra homomorphisms, so $P^{2} = π ({\hat{p}}_{n}^{2}) = π ({\hat{p}}_{n}) = P$ .

1.2F2F3

The imported projector result [F2] gives the unnormalized closure value ${Tr}_{n} (p_{n}) = [n + 1]_{δ}$ , where [F3] defines ${Tr}_{n}$ by closing the corresponding diagram endpoints and weighting each resulting loop by $δ$ . This is the diagram-algebra functional recorded separately from the Hilbert-space trace.

1.3L1F1base

At recursion index $1$ , $Q_{1} : = π (ι_{1, n} ({\hat{p}}_{1})) = I_{H}$ is self-adjoint because ${\hat{p}}_{1} = 1$ and $⟨ I_{H} x, y ⟩ = ⟨ x, I_{H} y ⟩$ for all $x, y \in H$ .

1.4L1F1ihalgebra

Suppose $1 \leq k < n$ and $Q_{k} : = π (ι_{k, n} ({\hat{p}}_{k}))$ is self-adjoint. By the commuting tower identity in [L1], the diagram recursion [F1] transfers under $Φ_{n}^{- 1}$ and $π$ to $Q_{k + 1} = Q_{k} - c_{k} Q_{k} π (u_{k}) Q_{k}$ , where $c_{k} = [k]_{δ} / [k + 1]_{δ} \in R$ because $δ$ is real and the recurrence defines real quantum integers. For all $x, y \in H$ , self-adjointness of $Q_{k}$ and $π (u_{k})$ , together with $c_{k} \in R$ , gives $⟨ Q_{k + 1} x, y ⟩ = ⟨ Q_{k} x, y ⟩ - c_{k} ⟨ Q_{k} π (u_{k}) Q_{k} x, y ⟩ = ⟨ x, Q_{k} y ⟩ - c_{k} ⟨ x, Q_{k} π (u_{k}) Q_{k} y ⟩ = ⟨ x, Q_{k + 1} y ⟩$ . Thus $Q_{k + 1}$ is self-adjoint.

2.1step 1.3step 1.4inductiondischarge-induction

Applying the finite step 1.4 from base step 1.3 through $k = n - 1$ gives $Q_{n} = P$ , with $P$ self-adjoint in the stated inner-product sense; when $n = 1$ , step 1.3 already gives the conclusion.

3.1F8step 1.1step 2.1algebra

For every $x \in H$ , $x = P x + (I_{H} - P) x$ , with $P x \in im P$ and $P (I_{H} - P) x = 0$ , so $(I_{H} - P) x \in \ker P$ by [F8]. If $y = P u \in im P$ and $z \in \ker P$ , then $⟨ y, z ⟩ = ⟨ P u, z ⟩ = ⟨ u, P z ⟩ = 0$ by step 2.1 and $P z = 0$ . Thus the two subspaces are orthogonal and sum to $H$ , so their intersection is zero and $H = im P \oplus \ker P$ . The idempotent $P$ acts as the identity on $im P$ and as zero on $\ker P$ , so it is an orthogonal projection with eigenvalues $1$ on its $r$ -dimensional image and $0$ on the kernel.

4.1F4F5F6F7step 1.2step 3.1step 2.1algebradischarge-induction∎

By step 3.1 and [F5], the ordinary Hilbert-space trace is ${tr}_{H} (P) = r$ , while step 1.2 gives the separate diagrammatic closure value. The operator $ρ = P / r$ is self-adjoint and has eigenvalues $1 / r$ on $im P$ and $0$ on $\ker P$ . In a basis formed from bases of these subspaces its trace is $r (1 / r) = 1$ , so it is a density matrix by [F4,F5]. Its entropy is $- r (1 / r) \log (1 / r) = \log r$ by [F6,F7]. This concludes the proof using the self-adjointness established by induction in step 2.1.

Jones-Wenzl image gives a finite maximally mixed sector

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources