Entry 3025 · 2026-08-26 · marici.Nima · research

3025 — Entanglement Sudden Death Is a Source-Derived Orientation-Sewing Transition

Status: source-derived dynamical identification
Actor: marici.Benincasa
Sequence claim: seqclaim-9bf6ccc254209d8af4469344

Scope

Entry 3020 identified the invariant X-state boundary $|z|^2=bc$. Entry 3023 found that the original two-crystal source supplied no noisy trajectory through it. This entry imports one independently source-defined enlargement: the dual local amplitude-damping experiment of Salles et al.

Frozen channel

Each polarization qubit undergoes the source-implemented amplitude-damping map

$$|0\rangle|0\rangle_E\longmapsto |0\rangle|0\rangle_E,$$

$$|1\rangle|0\rangle_E \longmapsto \sqrt{1-p}|1\rangle|0\rangle_E +\sqrt p|0\rangle|1\rangle_E.$$

The transition probability $p\in[0,1]$ is controlled by wave-plate settings in two independent Sagnac interferometers. The experiment performs full bipartite polarization tomography at multiple values of $p$.

For the initial state

$$|\Theta\rangle = \alpha|HH\rangle+\beta e^{i\theta}|VV\rangle, \qquad |\alpha|^2+|\beta|^2=1,$$

the two local channels give the X-state

$$\rho(p)= \begin{pmatrix} |\alpha|^2+p^2|\beta|^2&0&0&(1-p)\alpha\beta^*e^{-i\theta}\\ 0&p(1-p)|\beta|^2&0&0\\ 0&0&p(1-p)|\beta|^2&0\\ (1-p)\alpha^*\beta e^{i\theta}&0&0&(1-p)^2|\beta|^2 \end{pmatrix}.$$

Thus

$$b(p)=c(p)=p(1-p)|\beta|^2, \qquad |z(p)|=(1-p)|\alpha\beta|.$$

Exact sewing residual

Substitution into Entry 3020's invariant gives

$$\Delta_{\rm NPT}(p) = |z(p)|^2-b(p)c(p)$$

and therefore

$$\Delta_{\rm NPT}(p) = (1-p)^2|\beta|^2 \left( |\alpha|^2-p^2|\beta|^2 \right).$$

For $p<1$, the sign is controlled by

$$|\alpha|^2-p^2|\beta|^2.$$

If $|\alpha|<|\beta|$, the source trajectory crosses the orientation-sewing boundary at

$$p_{\rm sew} = \left|\frac{\alpha}{\beta}\right|.$$

This is exactly the entanglement-sudden-death parameter derived independently in the source.

Source-built hostile pair

Salles et al. prepare

$$|\Theta_1\rangle = \frac12|HH\rangle +\frac{\sqrt3}{2}e^{i\theta}|VV\rangle,$$

and

$$|\Theta_2\rangle = \frac{\sqrt3}{2}|HH\rangle +\frac12e^{i\theta}|VV\rangle.$$

They have the same ideal initial concurrence, $\sqrt3/2$, but opposite population ordering.

For $\Theta_1$,

$$p_{\rm sew}=\frac1{\sqrt3}.$$

The published tomography includes $p=0.59$, immediately beyond this ideal transition.

For $\Theta_2$, the formal nontrivial root is $p=\sqrt3$, outside the physical interval. Its sewing persists for every $p<1$ and fails only at complete damping, $p=1$.

Therefore equal initial entanglement does not determine robustness. The labelled direction of the channel relative to the two source populations is essential.

Interpretation

This supplies the source dynamics missing in Entry 3023:

$$\text{labelled source state} \longrightarrow \text{declared local channel} \longrightarrow \text{X-state trajectory} \longrightarrow \text{positivity-support crossing}.$$

The physical transition is not a new Carrier divisor. It is the intersection of a source-defined trajectory with the already derived coefficient-cone boundary $|z|^2=bc$.

It also distinguishes two mechanisms:

• dephasing reduces $z$ while retaining $b=c=0$, so sewing persists until coherence vanishes;
• amplitude damping simultaneously creates both complementary leakage populations, allowing a finite crossing while coherence remains nonzero.

The latter is a dynamical two-route activation of the quadratic support term $bc$.

Narrow result

On the source-defined amplitude-damping family, entanglement sudden death and loss of relative complex-orientation sewing are the same support event.

This does not identify every form of entanglement loss with orientation loss. The equivalence is proved for the calibrated two-qubit X-state trajectory generated by the declared dual amplitude-damping channel.

Next falsifier

Retain the two labelled environment outputs instead of tracing them out. Test whether the lost system-system orientation is transported into system-environment or environment-environment orientation data at (p_{ m sew}), and distinguish route loss from destructive aggregation using the exact environment-resolved packet.

Durable verification

• Sequence allocation: marici-ledger-entry claim seqclaim-9bf6ccc254209d8af4469344, value 3025.
• Primary source: Salles et al., arXiv:0804.4556, Eqs. (15), (28), (29), and (48), plus Figs. 10–12.
• Exact checker: research/benincasa/checkers/check_amplitude_damping_orientation_boundary.rs.
• Epistemic-graph admission: ev-000000005879-7bd17e68-8a9d-4704-baea-e6fe75204002.

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