bibkey: rastegin2012partialtrace
authors: Alexey E. Rastegin
year: 2012
title: Relations for Certain Symmetric Norms and Anti-norms Before and After Partial Trace
doi: 10.1007/s10955-012-0569-8
url: https://arxiv.org/abs/1202.3853v3
claim: For finite-dimensional operators, partial trace cannot increase a Ky Fan k-sum beyond the input Ky Fan kn-sum, where n is the traced factor's dimension.
strata_touched: []
license: citation-only
triage: anchor

Ky Fan sums under partial trace

Verified source

Alexey E. Rastegin, Journal of Statistical Physics 148 (2012), 1040-1053; arXiv:1202.3853v3. Proposition 1 proves the partial-trace inequality for a family of unitarily invariant norms. Its p=1 specialization is equation (3.8): the Ky Fan k-sum of the partial trace is at most the Ky Fan kn-sum of the original operator when the traced factor has dimension n. The author notes that this specialization had an earlier proof using Ky Fan’s maximum principle.

Use and boundary

In RT Section 36.1, projection onto a flat target factor gives a normalized diagonal-block sum. Embedding its rectangular coefficient operator in a square operator and applying the p=1 partial-trace bound yields the Ky Fan step of equation (36.5). Compression and the d-fold repeated source Schmidt spectrum are separate steps. The paper does not prove the sector-overlap estimate, the reference-labelled channel test, or the full product-channel optimum.