  Generation and manipulation of many-body entangled states is of considerable
interest, for applications in quantum simulation or sensing, for example.
Measurement and verification of the resulting many-body state presents a
formidable challenge, however, which can be simplified by multiplexed readout
using shared measurement resources. In this work, we analyze and demonstrate
state retrodiction for a system of optomechanical oscillators coupled to a
single-mode optical cavity. Coupling to the shared cavity field facilitates
simultaneous optical measurement of the oscillators' transient dynamics at
distinct frequencies. Optimal estimators for the oscillators' initial state can
be defined as a set of linear matched filters, derived from a detailed model
for the detected homodyne signal. We find that the optimal state estimate for
optomechanical retrodiction is obtained from high-cooperativity measurements,
reaching estimate sensitivity at the Standard Quantum Limit (SQL). Simultaneous
estimation of the state of multiple oscillators places additional limits on the
estimate precision, due to the diffusive noise each oscillator adds to the
optomechanical signal. However, we show that the sensitivity of simultaneous
multi-mode state retrodiction reaches the SQL for sufficiently well-resolved
oscillators. Finally, an experimental demonstration of two-mode retrodiction is
presented, which requires further accounting for technical fluctuations of the
oscillator frequency.
