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Quantum monotone metrics induced from trace non-increasing maps and additive noise

Quantum monotone metric was introduced by Petz,and it was proved that quantum monotone metrics on the set of quantum states with trace one were characterized by operator monotone functions. Later, these were extended to monotone metrics on the set of positive operators whose traces are not always one based on completely positive, trace preserving (CPTP) maps. It was shown that these extended monotone metrics were characterized by operator monotone functions continuously parameterized by traces of positive operators,and did not have some ideal properties such as monotonicity and convexity with respect to the positive operators. In this paper, we introduce another extension of quantum monotone metrics which have monotonicity under completely positive, trace non-increasing (CPTNI) maps and additive noise. We prove that our extended monotone metrics can be characterized only by static operator monotone functions from few assumptions without assuming continuities of metrics. We show that our monotone metrics have some natural properties such as additivity of direct sum, convexity and monotonicity with respect to positive operators.