Fundamental noisy multiparameter quantum bounds

Quantum multiparameter estimation involves estimating multiple parameters simultaneously and can be more precise than estimating them individually. Our interest here is to determine fundamental quantum limits to the achievable multiparameter estimation precision in the presence of noise. We present a lower bound to the estimation error covariance for a noisy initial probe state evolving via a noiseless quantum channel. We then present a lower bound to the estimation error covariance in the most general form for a noisy initial probe state evolving via a noisy quantum channel. We show conditions and accordingly measurements to attain these estimation precision limits for noisy systems. We see that the Heisenberg precision scaling of $1/N$ can be achieved with a probe comprising $N$ particles even in the presence of noise. In fact, some noise in the initial probe state or the quantum channel can serve as a feature rather than a bug, since the estimation precision scaling achievable in the presence of noise in the initial state or the channel in some situations is impossible in the absence of noise in the initial state or the channel. However, a lot of noise harms the quantum advantage achievable with $N$ parallel resources, and allows for a best precision scaling of $1/\sqrt{N}$. Moreover, the Heisenberg precision limit can be beaten with noise in the channel, and we present a super-Heisenberg precision limit with scaling of $1/N^2$ for optimal amount of noise in the channel, characterized by one-particle evolution operators. Further, using $\gamma$-particle evolution operators for the noisy channel, where $\gamma>1$, the best precision scaling attainable is $1/N^{2\gamma}$, which is otherwise known to be only possible using $2\gamma$-particle evolution operators for a noiseless channel.