Non-positive measurements aren't beneficial in quantum metrology for unitary encoding, but can be for open schemes

We investigate whether non-positive operator-valued measurements can be beneficial for quantum metrology. For unitary encoding, we show that non-positive measurements offer no advantage over positive ones. Going over to open encoding, we find, however, that non-positive measurements can be advantageous for certain cases, while it may mirror the unitary case - no advantage over positive measurements - for others. For arbitrary open-system encoding, we identify a sufficient condition under which positive measurements suffice to achieve the best precision, and more resource-intensive non-positive measurements offer no extra benefit.