We analyze quantum state estimation for finite samples based on symmetry information. The measurement concept used compares an unknown qubit to a reference state. We describe explicitly an adaptive strategy that enhances the estimation fidelity of these measurements.
Quantum state estimation is one of the most extensively studied topics in the field of quantum theory and quantum information. This process is essential in quantum protocols to ensure that the estimated state matches the true state. Typically, the quantum state of identical systems can be estimated from the statistics of measurement outcomes across different measurement bases. In this research, we study adaptive-measurement strategies for quantum state estimation used to identify a pure state of two-level systems and compare the obtained estimated state to the state obtained when the measurement bases are chosen randomly. The adaptive measurements in this study include the maximum-information-gain strategy, and the confirmation strategy. In this work, we simulate the outcome of the measurement on a system and use the Bayesian estimation method to estimate the obtained state. We then adaptively change the basis of the next measurement in accordance with the estimated state and the used adaptive strategy. We find that the maximum-information-gain strategy provides the fastest convergence and the most accurate state estimation compared to the confirmation strategy and the randomly-chosen-measurement-basis strategy, especially in the presence of measurement noise. Keywords: Quantum state estimation, State tomography, Adaptive measurements
We extend the concept of probabilistic unambiguous discrimination of quantum states to quantum state estimation. We consider a scenario where the measurement device can output either an estimate of the unknown input state or an inconclusive result. We present a general method how to evaluate the maximum fidelity achievable by the probabilistic estimation strategy. We illustrate our method on two explicit examples: estimation of a qudit from a pair of conjugate qudits and phase covariant estimation of a qubit from N copies. We show that by allowing for inconclusive results it is possible to reach estimation fidelity higher than that achievable by the best deterministic estimation strategy.
The theory of quantum state estimation is exploited here to investigate the most efficient strategies for this task, especially targeting a complete picture identifying optimal conditions in terms of Fisher information, quantum measurement, and associated estimator. The approach is specified to estimation of the phase of a qubit in a rotation around an arbitrary given axis, equivalent to estimating the phase of an arbitrary single-qubit quantum gate, both in noise-free and then in noisy conditions. In noise-free conditions, we establish the possibility of defining an optimal quantum probe, optimal quantum measurement, and optimal estimator together capable of achieving the ultimate best performance uniformly for any unknown phase. With arbitrary quantum noise, we show that in general the optimal solutions are phase dependent and require adaptive techniques for practical implementation. However, for the important case of the depolarizing noise, we again establish the possibility of a quantum probe, quantum measurement, and estimator uniformly optimal for any unknown phase. In this way, for qubit phase estimation, without and then with quantum noise, we characterize the phase-independent optimal solutions when they generally exist, and also identify the complementary conditions where the optimal solutions are phase dependent and only adaptively implementable.
We herein report the experimental demonstration of adaptive quantum state estimation for totally unknown photonic qubits. Similar to our previous study [R. Okamoto et al., Phys. Rev. Lett. 109, 130404 (2012)], the measurement configuration is updated using the results of each photon detection event so that our method does not require prior knowledge of the total number of samples. The experimental results obtained herein demonstrate both strong consistency and asymptotic efficiency through several rigorous statistical tests. Furthermore, we demonstrate that the experimentally obtained distribution of the states estimated using an adaptive quantum state estimation is significantly different from that obtained by conventional state tomography and agrees well with theoretical predictions.