Quantum metrology of mixed states via purification

We introduce new formulations of the quantum Cramér-Rao bound (QCRB) and the Holevo Cramér-Rao bound (HCRB) in multi-parameter quantum metrology via purification, where we show their values for any mixed state are connected to that for its purification with nuisance parameters introduced on the environmental system. Using this technique, we develop a new method for asymptotically attaining either the HCRB or twice the QCRB for arbitrary mixed states using random purification channels and individual measurements.