The paper gives if-and-only-if characterizations of the trace via inequalities with the spectral geometric mean and shows a fidelity-related inequality does not characterize the trace.
Jozsa,Fidelity for mixed quantum states, J
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Spectral geometric mean and trace characterizations
The paper gives if-and-only-if characterizations of the trace via inequalities with the spectral geometric mean and shows a fidelity-related inequality does not characterize the trace.