The Correlation Blind Spot of Local Marginals

Abstract

Complete local marginals leave every cross-factor correlation direction unread.

In the real Hermitian tensor model with factor dimensions $m,n$, write $L$ for the sum of the canonical traceless local sectors, $C$ for the sector traceless in both factors, and $H_0$ for the full traceless space. Sector dimensions and orthogonality are real.

Independently, $\rho$ is the canonical two-qubit Bell density for 00 and 11, and $\sigma$ their equal diagonal mixture. Here $\ge 0$ means positive semidefinite; partial-trace subscripts name the factor traced out.

Theorem 1.1 (Complete local data omit the full correlation sector).

$$\begin{array}{c}\forall m,n\in \mathbb{N},\\ m\ge 1\wedge n\ge 1\wedge mn>1\Rightarrow \\ L+C=H_0\wedge \\ \dim (L)=(m^2-1)+(n^2-1)\wedge \\ \dim (C)=(m^2-1)(n^2-1)\wedge \\ \frac{\dim (C)}{\dim (H_0)}=\frac{(m^2-1)(n^2-1)}{m^2{n}^2-1}\wedge \\ L\perp C\wedge \\ \rho \ge 0\wedge \mathrm{Tr}(\rho )=1\wedge \mathrm{rank}(\rho )=1\wedge \\ \sigma \ge 0\wedge \mathrm{Tr}(\sigma )=1\wedge {\sigma }^2\ne \sigma \wedge \\ {\mathrm{Tr}}_B(\rho )={\mathrm{Tr}}_B(\sigma )\wedge \\ {\mathrm{Tr}}_A(\rho )={\mathrm{Tr}}_A(\sigma )\wedge \\ \rho \ne \sigma .\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/LocalMarginalCorrelationBlindSpot.local_marginal_correlation_blind_spot (✓ std3). ∎

Source. Repository-derived.

Commentary.

The correlation sector is orthogonal to all local directions. The ratio gives its share of the traceless space.

The fixed witness has identical local marginals but different global matrices: complete local data need not determine cross-factor correlations.

References

• Truth anchor: D5/S3/Quantum/Entanglement/LocalMarginalCorrelationBlindSpot.local_marginal_correlation_blind_spot
• Dependency: D5/S3/Quantum/Entanglement/BellPureStateMixedMarginal
• Dependency: D5/S3/Quantum/Entanglement/BipartiteSectorDecomposition