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Following the various statements of [DW16] to their logical conclusion, thisnote explicitly argues the following statement, implicit in [DW16]: forpositive semi-definite operators $C_{1},\ldots,\,C_{L} $, a unitary $V_{C_{i}}$commuting with $C_{i}$, and $p\geq1$, the quantity $$\max_{V_{C_{1}},\ldots,V_{C_{L}}}\left\Vert C_{1}^{1/p}V_{C_{1}}\cdotsC_{L}^{1/p}V_{C_{L}}\right\Vert _{p}^{p} $$ is monotone non-increasing withrespect to $p$. The idea from [DW16] is that by allowing unitary swivelsconnecting a long chain of positive semi-definite operators together, we canestablish such a statement, which might not hold generally without the presenceof the unitary swivels. Other related statements follow directly from [DW16] aswell, being implicit there, and are given explicitly in this note.

Monotonicity of $p$-norms of multiple operators via unitary swivels