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Exact finite algebra entropy identities and holography boundaries — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Sector-dependent ancillary entropy

Example

Take two sectors with probabilities $p_{1}, p_{2} > 0$ and $p_{1} + p_{2} = 1$ . Let $H_{1}, H_{2}$ be nonzero finite-dimensional complex inner-product spaces, choose density matrices $ρ_{1}, ρ_{2}$ on them, and choose $q_{1}, q_{2} > 0$ with $q_{1} + q_{2} = 1$ and faithful reference density matrices $σ_{1}, σ_{2}$ on the same spaces. Let the first ancillary space be one-dimensional with $χ_{1} = [1]$ , and let the second be $C^{d}$ for an integer $d \geq 1$ , with $χ_{2} = I_{d} / d$ . The encoded state has entropy

$s (ρ_{A}) = H (p_{1}, p_{2}) + p_{1} s (ρ_{1}) + p_{2} s (ρ_{2}) + p_{2} \log d,$

where $H (p_{1}, p_{2}) : = - p_{1} \log p_{1} - p_{2} \log p_{2}$ is the sector-probability entropy in nats. The sector-probability contribution is $H (p_{1}, p_{2})$ and the ancillary contribution is $p_{2} \log d$ ; when $d = 1$ , the ancillary contribution is zero. This is an exact finite matrix example and asserts no literal black-hole area law.

Example references

Sector entropy and relative entropy identities

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

Daniel Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction, Appendix A.7.2–A.7.3

John Watrous, The Theory of Quantum Information, §§5.2.1–5.2.2

Facts & Assumptions

Given: The two-sector data in the Statement and the finite sector entropy identity.

[F1]

Sector entropy and relative entropy identities: $s (ρ_{A}) = H (p) + \sum_{α} p_{α} (s (ρ_{α}) + s (χ_{α}))$ for finite sectors with fixed faithful ancillary density matrices and a faithful encoded reference.

[F2]

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

Verification

1.1givenF2algebra

The one-dimensional matrix $χ_{1} = [1]$ has the single eigenvalue $1$ , so $s (χ_{1}) = - 1 \log 1 = 0$ by [F2].

1.2givenF2algebra

The eigenvalues of $χ_{2} = I_{d} / d$ are $1 / d$ with multiplicity $d$ , hence $s (χ_{2}) = - d (1 / d) \log (1 / d) = \log d$ by [F2]; for $d = 1$ this is $\log 1 = 0$ .

2.1givenF1step 1.1step 1.2algebra∎

Substitute $s (χ_{1}) = 0$ and $s (χ_{2}) = \log d$ from steps 1.1–1.2 into [F1] with $I = {1, 2}$ . The result is $s (ρ_{A}) = H (p_{1}, p_{2}) + p_{1} s (ρ_{1}) + p_{2} s (ρ_{2}) + p_{2} \log d$ , with the sector-probability and ancillary contributions as stated.