Starting from the Walled Brauer Algebra, we present matrix inequalities in the Löwner order for variables from the positive cone. These inequalities contain partial transpose and reshuffling operations, and can be understood as positive multilinear maps. In turn, we show that these positive multilinear maps are in one-to-one correspondence with entanglement witnesses showing U ⊗(n−k) ⊗ Ū ⊗k -invariance.
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