For any matrix describing a bipartite system, the trace norm is bounded by sqrt(2) times the smaller dimension times the product-observable norm, and sqrt(2) is optimal.
Szarek, and Andreas Winter
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Global vs. Product Observables in Bipartite Quantum Systems: The Sharp Bound
For any matrix describing a bipartite system, the trace norm is bounded by sqrt(2) times the smaller dimension times the product-observable norm, and sqrt(2) is optimal.