Bipartite energy-time uncertainty relation and quantum error correction for metrology with noise

Noise in quantum metrology reduces the sensitivity to which one can determine an unknown parameter in the evolution of a quantum state, such as time. Here, we consider a noiseless quantum system (a probe, or a clock) prepared in a pure state that encodes some time t, on which we apply an arbitrary noise channel. We show an uncertainty relation stating that the noisy probe’s sensitivity to time trades off exactly with the environment’s sensitivity to the energy of the noiseless probe. We obtain necessary and sufficient conditions for when zero sensitivity is lost after application of the noise channel. These conditions are analogous to the Knill-Laflamme quantum error correction conditions but they are easier to satisfy. I will discuss applications to many-body quantum metrology with strongly interacting particles, where we construct probe states that satisfy our condition and are therefore robust against local noise. For a 1D spin chain with nearest-neighbor interactions subject to amplitude damping noise on each site, we verify numerically that our probe state does not lose any sensitivity to first order in the noise parameter.