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We investigate whether a trade-off relation between the diagonal elements ofthe mean square error matrix exists for the two-parameter unitary models withmutually commuting generators. We show that the error trade-off relation whichexists in our models of a finite dimension system is a generic phenomenon inthe sense that it occurs with a finite volume in the spate space. We analyze aqutrit system to show that there can be an error trade-off relation given bythe SLD and RLD Cramer-Rao bounds that intersect each other. First, we analyzean example of the reference state showing the non-trivial trade-off relationnumerically, and find that its eigenvalues must be in a certain range toexhibit the trade-off relation. For another example, one-parameter family ofreference states, we analytically show that the non-trivial relation alwaysexists and that the range where the trade-off relation exists is up to about ahalf of the possible range.

Error trade-off relations for two-parameter unitary model with commuting generators