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Fixed-subsystem equivalence for independent finite energy shells

Statement

Fix $q \geq 1$ , real one-site levels $e_{j}$ , finite $β \in R$ , and the ensembles of Additive spectral Hamiltonian and normalized ensembles. Suppose for every $N$ the band width obeys $δ_{N} \geq q \sum_{j = 1}^{q} ∣ e_{j} ∣, δ_{N} / N ⟶ 0.$ Then every band is nonempty. For each fixed integer $k \geq 1$ and $N \geq k$ , define $σ_{N, k}$ by its product-basis entries, for $x, x^{'} \in {1, \dots, q}^{k}$ , $(σ_{N, k})_{x, x^{'}} = \sum_{y \in {1, \dots, q}^{N - k}} (σ_{N})_{(x, y), (x^{'}, y)}$ , with a single empty tail label when $N = k$ . Define $γ_{k}$ to be the diagonal matrix on $V_{k} = (C^{q})^{\otimes k}$ with entries $(γ_{k})_{x, x} = \prod_{a = 1}^{k} p_{x_{a}}$ and zero off the diagonal. Then $∥ σ_{N, k} - γ_{k} ∥_{1} \to 0, \frac{1}{N} s_{N} (σ_{N}) \to - \sum_{j = 1}^{q} p_{j} \log p_{j} .$ Here $∥ D ∥_{1}$ for a Hermitian finite matrix is the sum of absolute values of its eigenvalues, and $s_{N}$ is dimensionless spectral entropy. This is an independent finite-alphabet result; it asserts neither global trace-norm convergence nor an interacting or continuous-model equivalence.

Facts & Assumptions

Given: The preceding finite additive ensemble data and the exact width conditions above.

[F1]

Finite-site microcanonical and canonical block equivalence: under the stated finite alphabet, additive energies, and width hypotheses, every shell is nonempty, each fixed- $k$ shell-label marginal converges in total variation to $p^{\otimes k}$ , and $N^{- 1} \log ∣ Γ_{N} ∣ \to - \sum_{j} p_{j} \log p_{j}$ .

[F2]

Subsystem expectations and reduced finite tensor densities: reduced density expectations are partial-trace pairings, and partial trace preserves density matrices.

[F3]

Factorization of finite additive canonical densities: the canonical density has diagonal weights $\prod_{a = 1}^{N} p_{j_{a}}$ , and its first- $k$ reduction has diagonal entries $\prod_{a = 1}^{k} p_{x_{a}}$ and zero off the diagonal.

[F4]

Dimensionless entropy of a finite density matrix: for eigenvalues $λ_{i}$ of a finite density, $s_{N} = - \sum_{λ_{i} > 0} λ_{i} \log λ_{i}$ .

[F5]

Finite tensor density, observable and measurement language: a bounded observable is a Hermitian matrix with operator norm $∥ A ∥_{o p} = \sup_{∥ x ∥ = 1} ∥ A x ∥$ .

[F6]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: the inner-product pairing obeys $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ .

Proof

1.1givenF1algebra

Apply [F1] to the finite alphabet, levels, $β$ and $δ_{N}$ in the statement. It gives $Ω_{N} = ∣ Γ_{N} ∣ > 0$ and the classical $k$ -label marginal convergence. Since $σ_{N}$ is diagonal with equal weight on each label in $Γ_{N}$ , the coordinate formula for partial trace sums over the last $N - k$ labels and yields the diagonal density whose entries are exactly that marginal.

1.2givenF1F4algebra

By [F4], the eigenvalues of $σ_{N}$ are $1 / Ω_{N}$ with multiplicity $Ω_{N}$ and zero otherwise. The entropy definition therefore gives $s_{N} (σ_{N}) = \log Ω_{N}$ . Dividing by $N$ and applying the entropy limit in [F1] proves the second conclusion.

2.1step 1.1F1algebra

Both $σ_{N, k}$ and $γ_{k}$ are diagonal. For diagonal matrices, the eigenvalues of their difference are their diagonal entry differences; therefore its trace norm is the sum of the absolute marginal differences, which is twice total variation. The convergence in [F1] gives $∥ σ_{N, k} - γ_{k} ∥_{1} \to 0$ for each fixed $k$ .

3.1F2F3F5F6algebra∎

[F3] identifies the canonical $k$ -site state as $γ_{k}$ , so the trace-norm convergence already proved compares the shell reduction with the specified canonical reduction. For every Hermitian local observable $A$ , diagonalizing the Hermitian difference $D = σ_{N, k} - γ_{k}$ gives $∣ Tr (D A) ∣ \leq ∥ A ∥ ∥ D ∥_{1}$ : in that eigenbasis each diagonal expectation of $A$ has absolute value at most $∥ A ∥$ . Thus the same conclusion controls every fixed bounded local observable, including ones not diagonal in the energy basis. By [F5] and Cauchy–Schwarz [F6], the bound uses the declared operator norm. The statement fixes $k$ before the limit and makes no claim about a global trace distance.

Fixed-subsystem equivalence for independent finite energy shells

Statement

Facts & Assumptions

Proof

Depends on

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