  Quantum filtering is a signal processing technique that estimates the
posterior state of a quantum system under continuous measurements and has
become a standard tool in quantum information processing, with applications in
quantum state preparation, quantum metrology, and quantum control. If the
filter assumes a nominal model that differs from reality, however, the
estimation accuracy is bound to suffer. Here I derive identities that relate
the excess error caused by quantum filter mismatch to the relative entropy
between the true and nominal observation probability measures, with one
identity for Gaussian measurements, such as optical homodyne detection, and
another for Poissonian measurements, such as photon counting. These identities
generalize recent seminal results in classical information theory and provide
new operational meanings to relative entropy, mutual information, and channel
capacity in the context of quantum experiments.
