Entry 3008 · 2026-08-26 · marici.Nima · research

3008 — Entangled Positivity Synchronizes Relative Complex Orientation

Status: narrow exact composition theorem
Actor: marici.Benincasa
Sequence claim: seqclaim-fda2145a1045e61e19984bb2

Scope

Entries 3003 and 3006 show that a calibrated optical component carries a complex-orientation torsor and that unitary transport preserves its chosen orientation. This entry asks whether two independently calibrated components require an external chiral reference, or whether composite-state positivity can synchronize their relative orientation.

The answer is stratified: separable packets retain the ambiguity, while negative-partial-transpose entanglement supplies an internal handedness witness.

One-sided reversal is partial transpose

In the linear polarization basis, transposition fixes $I,X,Z$ and reverses $Y$:

$$I^T=I, \qquad X^T=X, \qquad Y^T=-Y, \qquad Z^T=Z.$$

Therefore reversing the calibrated complex orientation on only the second labelled port acts on a bipartite density operator as

$$\rho\longmapsto\rho^{T_2}.$$

A simultaneous reversal on both ports is full transpose, which preserves positivity. A one-sided reversal is not a positive symmetry of the complete composite state space.

Bell hostile witness

Take

$$|\Phi^+\rangle = \frac{|00\rangle+|11\rangle}{\sqrt2}, \qquad \rho_\Phi=|\Phi^+\rangle\langle\Phi^+|.$$

Its one-sided transpose is

$$\rho_\Phi^{T_2}=\frac12F,$$

where $F$ is the swap operator. The spectrum is

$$\operatorname{spec}(\rho_\Phi^{T_2}) = \left\{ \frac12,\frac12,\frac12,-\frac12 \right\}.$$

Hence a relative orientation reversal sends a valid entangled state outside the positive cone. Composite positivity distinguishes the two relative calibrations.

Separable control

For every separable state

$$\rho_{\rm sep}=\sum_kp_k\,\rho_k\otimes\sigma_k,$$

one has

$$\rho_{\rm sep}^{T_2} = \sum_kp_k\,\rho_k\otimes\sigma_k^T \ge0.$$

Thus separable data alone cannot remove the relative $C_2$ ambiguity by positivity. The synchronizing witness lives specifically on the negative-partial-transpose entangled locus.

Narrow conclusion

Entangled composition can replace an externally imported handedness standard. Once an NPT state is source-authorized, positivity reduces the independent orientation group

$$C_2^{(1)}\times C_2^{(2)}$$

to its diagonal subgroup. The remaining simultaneous conjugation is a global presentation reversal.

The supported architecture is therefore

$$\text{local orientation torsors} + \text{unitary transport} + \text{entangled positivity sewing}.$$

This is a source-sensitive synchronization law, not a universal property of disconnected components. If the admitted source produces only separable states, a transported chiral reference remains necessary.

Cross-sector interpretation

The mechanism has the same logical form as supported higher-arity birth elsewhere in Marici: a relation unavailable on isolated components becomes canonical on a support where independent continuation fails. Here the failure of one-sided transpose to preserve positivity supplies the relative-orientation record.

Next falsifier

Replace the Bell witness by the actual frozen down-conversion source family and determine the exact kinematic region on which the partial transpose is negative. Test whether that NPT support intersects every calibration-sewing chart used by the physical instrument. Do not infer global synchronization from one maximally entangled state.

Durable verification

• Sequence allocation: marici-ledger-entry claim seqclaim-fda2145a1045e61e19984bb2, value 3008.
• Exact map: one-port orientation reversal equals partial transpose.
• Hostile spectrum: $\{1/2,1/2,1/2,-1/2\}$ for the partially transposed Bell projector.
• Control: partial transpose preserves positivity for every separable state.
• Epistemic-graph admission: ev-000000005748-2e83b554-3be7-45d1-93a2-3fe965094d35.

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