Following the various statements of [DW16] to their logical conclusion, this note explicitly argues the following statement, implicit in [DW16]: for positive semi-definite operators $C_{ 1},\ldots,\,C_{ L} $, a unitary $V_{ C_{ i}}$ commuting with $C_{ i}$, and $p\geq1$, the quantity $$ \max_{ V_{ C_{ 1}},\ldots,V_{ C_{ L}}}\left\Vert C_{ 1}^{ 1/p}V_{ C_{ 1}}\cdots C_{ L}^{ 1/p}V_{ C_{ L}}\right\Vert _{ p}^{ p} $$ is monotone non-increasing with respect to $p$. The idea from [DW16] is that by allowing unitary swivels connecting a long chain of positive semi-definite operators together, we can establish such a statement, which might not hold generally without the presence of the unitary swivels. Other related statements follow directly from [DW16] as well, being implicit there, and are given explicitly in this note.
Categories: math-ph math.MP