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Given arbitrary $X,Y \in \mathfrak{su}(4)$, I want to maximize either of the following functions:

$\max_{U,V \in SU(2)} \Re(\text{Tr}(X^\dagger (U^{\dagger} \otimes V^{\dagger})Y (U \otimes V)))$

and/or

$\max_{U,V \in SU(2)} \left|\text{Tr}(X^\dagger (U^{\dagger} \otimes V^{\dagger})Y (U \otimes V))\right|^2$

A bound on the max over $SU(4)$ rather than $SU(2)\otimes SU(2)$ can be found in Von Neumman's trace inequality. Is there any similar approach here?

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  • $\begingroup$ What does the dagger mean? It could mean X transpose conjugate (in which case, $X^*$ or ${\overline X}^T$ would convey it more succinctly, although an explanation might still be required), or the Moore-Penrose inverse (the dagger is standardly used for this), or .... What does the Fraktur R mean? If it means the real part of, use (roman) Re, so that everyone will know. And while we're at it, what is the Fraktur su(4): is this supposed to be SU(4)? Or the corresponding Lie algebra? $\endgroup$
    – David Handelman
    Commented Jun 15, 2017 at 0:51
  • $\begingroup$ Hermitian conjugate, yes. R is real part, and yes it's the lie algebra. $\endgroup$
    – Benjamin
    Commented Jun 15, 2017 at 3:30

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