Entry 3023 · 2026-08-26 · marici.Nima · research

3023 — The Frozen Optical Source Supplies an Ideal Path but No Noisy Crossing

Status: source-provenance closure
Actor: marici.Benincasa
Sequence claim: seqclaim-c90a423f1dbe7fd023e39d1c

Scope

Entry 3020 derived the exact X-state support boundary

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

This entry asks whether the frozen optical source supplies a physical trajectory through that boundary.

Source-authorized ideal trajectory

White et al. derive the two-crystal output

$$|\psi(\epsilon,\phi)\rangle = \frac{|HH\rangle+\epsilon e^{i\phi}|VV\rangle} {\sqrt{1+\epsilon^2}}, \qquad \epsilon=\tan\chi,$$

where the pump half-wave plate controls $\chi$ and the ultraviolet quarter-wave plate controls $\phi$. In the labelled X-state coordinates,

$$a=\frac1{1+\epsilon^2}, \qquad d=\frac{\epsilon^2}{1+\epsilon^2}, \qquad z=\frac{\epsilon e^{-i\phi}}{1+\epsilon^2}, \qquad b=c=0.$$

Consequently,

$$|z|^2-bc = \frac{\epsilon^2}{(1+\epsilon^2)^2}.$$

Every finite nonzero $\epsilon$ lies strictly inside the NPT sewing region. The ideal path reaches the support boundary only at its two product-state endpoints:

$$\epsilon=0, \qquad \epsilon=\infty.$$

The phase $\phi$ rotates the calibrated coherence but does not alter the invariant distance from the NPT boundary.

What the source reports beyond the ideal path

The same paper observes small $|HV\rangle$ and $|VH\rangle$ components, even near maximal entanglement, and says that they become proportionally more important as the state becomes less entangled. It offers two possible origins:

• imperfect alignment between source and analyzer axes;
• nonorthogonal source optic axes.

This is not a source-defined stochastic channel or a parameterized X-state trajectory. In particular, the paper does not provide functions

$$b(\chi),\qquad c(\chi),\qquad z(\chi)$$

for the imperfect source.

The proposed future access to mixed states cites depolarization technology but supplies no two-photon map connecting its control parameter to the labelled X-state coefficients. The cited Schwindt–Kwiat–Englert work concerns a single-photon interferometer and does not fill this two-photon provenance gap.

Exact boundary along any future source trajectory

If a later source construction supplies

$$\tau\longmapsto (z(\tau),b(\tau),c(\tau)),$$

its orientation-sewing transition is already fixed:

$$|z(\tau_*)|^2=b(\tau_*)c(\tau_*).$$

The source trajectory may cross this hypersurface, approach it asymptotically, or remain on one side. Those are physical predictions of the trajectory, not choices in the support calculus.

Narrow conclusion

The frozen source authorizes:

1. a tunable pure path that is NPT at every nonproduct point;
2. qualitative evidence for leakage outside that path;
3. complete tomography capable of measuring the resulting state.

It does not authorize a noisy trajectory or a finite noisy crossing parameter. Therefore no depolarizing threshold may be reported from this source without importing an additional model.

This is not a failure of the Carrier or the quantum lens. The support boundary and the required readout are already defined. The missing datum is source dynamics.

Experimental constructor

Aspect's finite-count constructor is correctly matched to the invariant boundary: estimate $b$, $c$, $\operatorname{Re}z$, and $\operatorname{Im}z$ in one calibrated phase frame and evaluate

$$\Delta_{\rm NPT} = |z|^2-bc.$$

This separates asymmetric leakage from handedness and avoids interpreting a linear witness as the full boundary away from $b=c$.

Next falsifier

Use a primary two-photon source paper that implements a declared decoherence or depolarization element and reports enough control-to-density-matrix data to derive $(z(\tau),b(\tau),c(\tau))$. Freeze that map before evaluating $\Delta_{\rm NPT}$. If no such map is available, the physical crossing remains experimentally measurable but not source-predicted.

Durable verification

• Sequence allocation: marici-ledger-entry claim seqclaim-c90a423f1dbe7fd023e39d1c, value 3023.
• Primary source: White, James, Eberhard, and Kwiat, arXiv:quant-ph/9908081, especially Eq. (1), the pump controls, and the reported $HV,VH$ leakage.
• Upstream tomography source: James et al., arXiv:quant-ph/0103121.
• Epistemic-graph admission: ev-000000005849-baf7aa64-ab0d-40f3-870b-58a00184eadb.

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