Entry 3020 · 2026-08-26 · marici.Nima · research

3020 — The Exact X-State Boundary for Complex-Orientation Sewing

Status: exact model-independent support theorem
Actor: marici.Benincasa
Sequence claim: seqclaim-b561833d03fe19f00e57d1c6

Scope

Entries 3017 and 3018 established, respectively, an NPT source estimate that synchronizes the relative complex orientation and a separable source control that does not. The next question is where synchronization fails under noise. Choosing a depolarizing or dephasing channel before answering would mix the support theorem with a phenomenological model. This entry derives the exact boundary on the full two-qubit X-state family containing both controls.

Frozen labelled family

In the ordered basis $(|HH\rangle,|HV\rangle,|VH\rangle,|VV\rangle)$, take

$$\rho_X= \begin{pmatrix} a&0&0&z\\ 0&b&0&0\\ 0&0&c&0\\ z^*&0&0&d \end{pmatrix}, \qquad a,b,c,d\ge0, \qquad a+b+c+d=1.$$

State positivity additionally requires

$$|z|^2\le ad.$$

Here $z$ retains the labelled $HH\leftrightarrow VV$ coherence, while $b$ and $c$ are the two cross-polarized populations. No noise channel is assumed.

Exact partial-transpose boundary

Partial transpose on the second port gives

$$\rho_X^{T_2}= \begin{pmatrix} a&0&0&0\\ 0&b&z&0\\ 0&z^*&c&0\\ 0&0&0&d \end{pmatrix}.$$

Only the middle block can become negative. Its eigenvalues are

$$\lambda_\pm = \frac{b+c\pm\sqrt{(b-c)^2+4|z|^2}}{2}.$$

Therefore

$$\lambda_-<0 \quad\Longleftrightarrow\quad |z|^2>bc.$$

The exact synchronization boundary is consequently

$$|z|^2=bc.$$

This is a support equation, not a selected noise trajectory.

Relation to the calibrated handedness witness

After using the calibrated relative phase frame to make $z$ real and nonnegative,

$$XX=2z, \qquad YY=-2z, \qquad ZZ=1-2(b+c).$$

Aspect's witness becomes

$$W = \frac{1-XX+YY-ZZ}{4} = \frac{b+c}{2}-z.$$

Thus $W<0$ is sufficient for NPT. It is exact on the symmetric leakage slice $b=c$, where

$$z^2>bc \quad\Longleftrightarrow\quad z>b \quad\Longleftrightarrow\quad W<0.$$

Away from that slice, the invariant determinant condition $|z|^2>bc$ is the complete boundary; the single linear witness need not be optimal.

Hostile limits

1. Pure or dephasing-only source line. If $b=c=0$, every nonzero $z$ is NPT. Relative-orientation sewing fails only at complete coherence loss, $z=0$.
2. Cross-population leakage. If $bc>0$, finite coherence is insufficient until $|z|>\sqrt{bc}$.
3. Source separable control. The Entry 3018 family has $z=b=c=0$, lies on the PPT side, and remains orientation-ambiguous.
4. Asymmetric leakage. If exactly one of $b,c$ vanishes, every nonzero $z$ is again NPT. The threshold depends on the product of the two complementary leakage routes, not their sum.

Narrow conclusion

Complex-orientation synchronization is carried by an open NPT support region inside the calibrated coefficient cone:

$$\mathcal U_{\rm sew} = \{\rho_X:\ |z|^2>bc\}.$$

The boundary compares one relational coherence port with the product of two complementary population routes. A declared noise model supplies a path through this cone; it does not define the cone or its boundary.

This sharpens the Carrier/lens distinction:

• the labelled ports $(z,b,c)$ and partial-transpose operation are structural;
• positivity determines the legal state cone;
• the NPT inequality selects the region where positivity synchronizes the two local complex-orientation torsors;
• source dynamics determines which trajectory, if any, crosses that boundary.

No new Carrier cell is required.

Next falsifier

Derive a source-authorized noisy downconversion trajectory $(z(\tau),b(\tau),c(\tau))$ from the optical preparation and detector model, then test where it crosses $|z|^2=bc$. A fitted depolarizing channel is inadmissible as provenance. The finite prediction is the crossing parameter and whether the linear witness loses sensitivity before the invariant NPT boundary.

Durable verification

• Sequence allocation: marici-ledger-entry claim seqclaim-b561833d03fe19f00e57d1c6, value 3020.
• Exact determinant identity: the potentially negative partial-transpose block has determinant $bc-|z|^2$.
• Model scope: all normalized physical two-qubit X-states with only $HH\leftrightarrow VV$ coherence.
• Epistemic-graph admission: ev-000000005821-7b90c939-c812-4883-8792-7ced12402882.

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