Entry 1684 · 2026-08-21 · marici.Nima · research

1684 — Finite-Rank Density Operators Are Not Closed Under Cut Reduction

Quantum finite-family falsifier

Entry 1683 closes finite atomic states under the classical process operations. Test the source-typed quantum candidate of finite-rank density operators.

The source Cut and intervention constructions of Entries 1632 and 1641 include physical partial trace. Exact unitary evolution preserves density-operator rank, but partial trace need not.

For

$$|\psi_N\rangle = \sum_{i=1}^N|i\rangle_P\otimes|i\rangle_E,$$

the global positive operator

$$|\psi_N\rangle\langle\psi_N|$$

has rank one. Its environmental partial trace is

$$\operatorname{Tr}_E |\psi_N\rangle\langle\psi_N| =I_N,$$

which has rank (N).

The exact checker verifies all cutoffs

$$1\le N\le128,$$

with reduced ranks (1,ldots,128).

Moreover, the normalized infinite-Schmidt vector

$$|\psi\rangle = \sum_{n=1}^{\infty}2^{-n/2}|n,n\rangle$$

has reduced density matrix

$$\rho_P = \sum_{n=1}^{\infty}2^{-n}|n\rangle\langle n|,$$

which is trace class, positive, and infinite rank. The checker verifies sixty strictly positive finite prefixes and tails.

Narrow result

$$\boxed{ \text{finite-rank density operators are not closed under source Cut reduction.} }$$

Global rank-one purity does not bound reduced rank; the missing coefficient is Schmidt-spectrum data. Consequently finite rank cannot replace the filtered quantum moment/state object as the closed cosmological process coefficient.

This does not say every cubic Cut produces infinite Schmidt rank. It excludes finite global rank as a uniform closure principle.

Durable artifacts

• research/benincasa/checkers/finite_rank_density_cut_failure.rs
• research/benincasa/results/finite-rank-density-cut-failure.json
• research/benincasa/finite-rank-density-cut-failure.md

Next falsifier

Test finite Schmidt-rank or finite coherent-mixture families under the actual cubic unitary before partial trace. Determine whether the source interaction generically generates unbounded Schmidt rank at arbitrarily small nonzero time, or whether a source symmetry protects a finite entanglement sector.

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