Ordinarily, quantum-measurement based state discrimination methods use Bayesian inference to determine the maximally likely optical input state out of N possible inputs. Each practical single-shot measurement yields a unique vector of N likelihood values (one per each possible outcome) whose significance is not appreciated. Here we show theoretically and experimentally that measured Bayesian likelihood values are equal to observed probabilities of a successful state discrimination. Thus, the vector of Bayesian likelihoods provides significantly more information about the input state than is available in a typical state discrimination measurement.
The Bayesian inverse of a quantum channel
E is a channel
F in the reverse direction of
E that yields time-symmetric correlations for sequential measurements performed on open quantum systems. Such an operational form of time-reversal symmetry for open quantum systems is quite remarkable, as the dynamics of open quantum systems are inherently irreversible due to system-environment interactions. Similar to the Petz map, a Bayesian inverse
F is defined with respect to a fiducial reference state
ρ for the channel
E. However, Bayesian inverses do not always exist, and it is often a non-trivial task to determine the set of states
ρ for which a Bayesian inverse of
E exists. In this work, we solve the general problem of quantum Bayesian inversion for unital channels acting on a single qubit. Our analysis is streamlined by demonstrating that finding a Bayesian inverse for a unital qubit channel may be reduced to finding a Bayesian inverse of a Pauli channel, which is simply a mixture of unitary channels associated with the Pauli matrices. As such, we provide a complete description of when operational time-reversal symmetry is attainable for sequential measurements of a single qubit in the presence of unital noise.
We present an approach to Bayesian mean estimation of quantum states using hyperspherical parametrization and an experiment-specific likelihood which allows utilization of all available data, even when qubits are lost. With this method, we report the first closed-form Bayesian mean estimate (BME) for the ideal single qubit. Due to computational constraints, we utilize numerical sampling to determine the BME for a photonic two-qubit experiment in which our novel analysis reduces burdens associated with experimental asymmetries and inefficiencies. This method can be applied to quantum states of any dimension and experimental complexity.