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In quantum metrology, the Holevo Cram\'er-Rao bound has attracted renewedinterest in recent years due to its superiority over the Helstrom Cram\'er-Raobound and its asymptotic attainability for multi-parameter estimation. Itsevaluation, however, is often much more difficult than that of the Helstromversion, calling into question the actual improvement offered by the Holevo CRBand whether it is worth the trouble. Here I prove that the Holevo bound is atmost thrice the Helstrom version, so the improvement must be limited. Theresult also shows that the Helstrom version remains a pretty good bound evenfor multiple parameters and can be approached asymptotically to within a factorof 3.

The Holevo Cram\'er-Rao bound is at most thrice the Helstrom version