# Trace moments of induced random density matrices: argument selection

This is the build justification for T574. Independent critique is still required.
The instruction record is `agents/runs/20261028T062448Z-build-skills/manifest.json`
in the run snapshot. Both archived skills have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.

## Quantity and conventions

The single quantity is M_q(N,K) = E Tr(rho^q). A pure state is Haar-uniform
on the unit sphere of C^N tensor C^K; rho is its partial trace over C^K.
Equivalently rho = GG*/Tr(GG*) for independent standard complex Gaussian
entries G_ia, with E|G_ia|^2 = 1. The trace is the ordinary matrix trace,
not divided by N. No real ensemble, Bures measure, averaged logarithm,
quantile, or additional scale parameter is included.

The eigenvalues are nonnegative and sum to one. The two subsystems have the
same nonzero eigenvalues, so we choose 2 <= N <= K and avoid swapped copies.
For N = 1 all positive trace powers equal one; these trivial rows are omitted.
The parameter order is N, K, q. The order q indexes one sequence of trace
moments; it does not select separately named quantities.

## Assessment of every parameter

- **N: discrete_index.** N is the integer dimension of the reduced density
  matrix. All dimensions 2 through 8 occur under the total-dimension bound.
  This includes qubits, qutrits and larger small systems without restricting
  dimensions to powers of two. It is an actual finite-dimensional matrix
  family, not integer samples of a continuously varied coupling.
- **K: discrete_index.** K is the integer dimension of the traced-out
  environment. For each N we include every K from N through floor(64/N).
  This covers balanced, near-balanced and unbalanced bipartitions under one
  budget on the size of the underlying pure-state space. The successive K
  ranges are 2..32, 3..21, 4..16, 5..12, 6..10, 7..9, and 8..8. They contain
  31, 19, 13, 8, 5, 3 and 1 pairs, respectively, totaling 80.
- **q: discrete_index.** The integer power in Tr(rho^q) runs through every
  order from 2 to 8 for each dimension pair. Order 2 is purity; the higher
  orders describe successively higher spectral moments. Orders 0 and 1 give
  N and 1 and remain validation controls instead of repeated trivial rows.
  The cutoff 8 permits exhaustive validation of all Wick contractions:
  sum(q!, q=2..8) = 46,232 permutations, whose cycle counts can be reused
  across dimension pairs. This gives seven consecutive low spectral orders
  before investing in high moment depth for only a few dimensions.

The bounds NK <= 64 and q <= 8 are noncanonical computational choices,
independent of numeral notation. NK is the composite Hilbert-space dimension,
so the product bound has a concrete size meaning. No claim is made that every
pair has a separate famous application. The reference purpose is recognition
and validation of finite-dimensional reduced-state spectral statistics.
There are no continuous exploratory reference groups in this selection.

## Breadth, extent and measurements

There are 560 exact rational entries. The selection covers all dimension pairs
under the product bound at all seven orders, including the Hilbert-Schmidt
slice N = K and the entire qubit family K = 2..32. This broadens subsystem and
environment dimensions together before extending moment depth. The generator
orders N, then K, then q increasingly, starting at M_2(2,2) = 4/5.

The longest stored fraction is 675244/1329890705 at (8,8,8), 17 characters.
The dry-run block estimate is 27.3 KiB (its surrogate parameter keys differ
from the serialized table). Exact values have no rounding error or decimal
precision target; spelling them with 100 decimal digits would weaken them.

The parameter declarations give the full domain N >= 2, K >= N, q >= 2.
The completeness note states precisely the finite product bound and order
range, and explains the coverage and verification tradeoff. The default
`enumerate()` is the sole selection dependency; `checks.py` separately checks
the complete expected set and count.

## Supporting sources actually checked

1. Życzkowski and Sommers, *Induced measures in the space of mixed quantum
   states*, arXiv:quant-ph/0012101v3, section III.1, equations (25), (30)-(32):
   complex partial-trace ensemble, dimension interchange and the N = K
   Hilbert-Schmidt connection. Section III.2, equations (40)-(44): low-order
   trace moments and mean purity.
   https://arxiv.org/html/quant-ph/0012101
2. Nechita, *Asymptotics of random density matrices*, arXiv:quant-ph/0702154v3,
   section 3.2 and section 3.3, Proposition 5, equations (34), (37)-(41):
   normalized Wishart representation and the moment-order recurrence.
   https://arxiv.org/html/quant-ph/0702154
3. Collins and Nechita, *Random matrix techniques in quantum information
   theory*, arXiv:1509.04689v1, section 3.3, Theorem 3.9 and section 4.1.2:
   Wick contractions and finite-dimensional induced ensembles.
   https://arxiv.org/html/1509.04689

These sources justify the mathematical family and conventions. The stated
finite bounds are this table's reference choice, not claimed literature bounds.
The problematic finite sum in Nechita (36) is excluded; see source-notes.md.