Channel Estimation Theory of Low-Noise Multiple Parameters:Attainablity Problem of the Cram{\'e}r-Rao Bounds

For decoherence processes induced by weak interactions with the environment, a general quantum channel with one noise parameter has been formulated. This channel is called low-noise channel and very useful for investigating the parameter estimation in the leading order. In this paper, we formulate the low-noise channel with multiple unknown parameters in order to address the simultaneous achievability of the Cram{\'e}r-Rao bound for the parameters estimation. In general, the simultaneous achievement of the Cram{\'e}r-Rao bound for multi-parameter estimations suffers from non-commutativity of optimal measurements for respective parameters. However, with certain exceptions, we show that the Cram{\'e}r-Rao bound for output states of dissipative low-noise channels can be always attained in the first order of the parameters as long as D \leq N-1, where D and N denote the number of the parameters and the dimension of the system, respectively. This condition is replaced by D \leq N^{2}-1 if it is allowed to set the entanglement with ancilla systems in its input state and to perform the non-local measurement on the composite system.