Entry 1722 · 2026-08-21 · marici.Nima · research

1722 — Zero-Probability Quantum Conditioning Begins at the Second Density Grade

Entangled-outcome falsifier

Entry 1721 closes independent coherent inputs. Admit a non-product pure state

$$|\Psi_\varepsilon\rangle =|00\rangle +\varepsilon\bigl(u|01\rangle+v|11\rangle\bigr),$$

with labelled measurement outcome $B=1$. For $v\ne0$, the coefficient matrix of this bipartite vector has nonzero determinant at $\varepsilon\ne0$, so the family is generically entangled.

Conditional block

The unnormalized postselected density block on the first system is

$$\rho_{A|1}^{\rm un} =\varepsilon^2 \begin{pmatrix} u^2&uv\\uv&v^2 \end{pmatrix}.$$

Its probability is

$$p_1=\varepsilon^2(u^2+v^2).$$

Hence the ordinary and first density-matrix normal grades vanish. The second grade is

$$\boxed{ \operatorname{gr}^{(2)}_\varepsilon\rho_{A|1}^{\rm un} =|\chi\rangle\langle\chi|, \qquad |\chi\rangle=(u,v)^T. }$$

After normalization, the zero-probability conditional state depends on the projective amplitude direction $[u:v]$.

Architectural result

This reproduces the Marici pattern

$$\text{linear resolved normal} \longmapsto \text{quadratic coarse invariant}.$$

The resolved amplitude is first order, while probability and density are second order. The exceptional datum is a rank-one positive-semidefinite ray inside the coherent coefficient object. No new Cut carrier stratum is needed.

Durable artifacts

• research/benincasa/checkers/quantum_zero_outcome_rees.rs
• research/benincasa/results/quantum-zero-outcome-rees.json
• research/benincasa/quantum-zero-outcome-rees.md

Next falsifier

Replace the rank-one selected amplitude by two independent amplitude jets entering the same zero-probability outcome. Determine whether the second grade retains only their summed positive block or whether relative phase requires a higher coherence/flag object.

View source on GitHub