  Interferometry with quantum light is known to provide enhanced precision for
estimating a single phase. However, depending on the parameters involved, the
quantum limit for the simultaneous estimation of multiple parameters may not
attainable, leading to trade-offs in the attainable precisions. Here we study
the simultaneous estimation of two parameters related to optical
interferometry: phase and loss, using a fixed number of photons. We derive a
trade-off in the estimation of these two parameters which shows that, in
contrast to single-parameter estimation, it is impossible to design a strategy
saturating the quantum Cramer-Rao bound for loss and phase estimation in a
single setup simultaneously. We design optimal quantum states with a fixed
number of photons achieving the best possible simultaneous precisions. Our
results reveal general features about concurrently estimating Hamiltonian and
dissipative parameters, and has implications for sophisticated sensing
scenarios such as quantum imaging.
