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ProductMarginZhangYeung

Abstract

The spectral gap on the actual product-margin state

Theorem 1.1 (The actual spectral entropy identity).

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/ProductMarginZhangYeung.actual_gap_identity (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Codex CLI (2026). A harmonic-mean certificate for the unequal-local-pair product-margin quantum Zhang–Yeung family. URL: https://github.com/the-omega-institute/trureturing.

Commentary.

The two LocalPair parameters contain four independent nonnegative real coordinates xC, zC, xD, zD, each pair in the closed unit disk. Coupling contains an arbitrary complex positive semidefinite K bounded by P and Q, where P is the product of the positive local members and Q is the product of the negative members. The actual density on two classical bits and two qubits has blocks (P-K)/2, K/2, K/2, (Q-K)/2. Its entries, positivity and trace normalization are retained even when a block vanishes.

Delta is the original Zhang–Yeung combination of natural-log von Neumann entropies of actual partial traces. Its conditional information terms are 2 I(C:D|A), I(C:D|B), I(A:C|D) and I(A:D|C), with I(A:B)-I(C:D). The two row and column marginals are uniform pointer records of P and Q. Pointer orthogonality gives the cq entropy formula without imposing orthogonality on the prepared qubits or normalizing a zero-weight block. The AB entropy is log(2)+h(Re Tr K), including both endpoints.

The right side uses the actual symmetric local Holevo entropies and the mutual information of M=(P+Q)/2. Establishing nonnegativity requires the information estimates for the actual prepared states in addition to this identity.

Theorem 1.2 (The four-parameter product-margin inequality).

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/ProductMarginZhangYeung.actual_zhang_yeung_nonneg (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Codex CLI (2026). A harmonic-mean certificate for the unequal-local-pair product-margin quantum Zhang–Yeung family. URL: https://github.com/the-omega-institute/trureturing.

Commentary.

For every two independent closed-disk LocalPair parameters and every complex positive common lower matrix K, the original spectral Zhang–Yeung gap of the actual density is nonnegative. No information estimate, commutation or strict positivity is an input hypothesis.

When both x coordinates are less than one, the actual marginal reference states are faithful even if the joint density is singular. The canonical product-recovery identity identifies their mutual information with support-aware relative entropy. Local preparation channels map both each intermediate cq density and its product reference to the physical output. Their DPI bounds combine with qubit Holevo contraction, the common-trace bound and the closed-interval scalar entropy certificate. If an x coordinate equals one, the corresponding z is zero and the CD state is directly a product. Zero reliabilities, zero weights and pure prepared states remain included.

The conclusion concerns this four-parameter classical-bit product-margin construction. It makes no assertion for arbitrary four-party quantum states or for a coordinate-free extension.

References