We study the average quantum coherence over the pure state decompositions of a mixed quantum state. An upper bound of the average quantum coherence is provided, and sufficient conditions for the saturation of the upper bound are shown. These sufficient conditions always hold for two- and three-dimensional systems. This provides a tool to estimate the average coherence experimentally by measuring only the diagonal elements, which remarkably requires less measurements compared with state tomography. We then describe the pure state decompositions of the qubit state in the Bloch sphere geometrically. For any given qubit state, the optimal pure state decomposition achieving the maximal average quantum coherence as well as three other pure state decompositions are shown in the Bloch sphere. The order relations among their average quantum coherence are invariant for any coherence measure. The results presented in this paper are universal and suitable for all coherence measures.
Measurement-induced phase transitions are a central paradigm in non-unitary quantum dynamics, arising from the interplay between unitary evolution and projective measurements. In this study, we analyze the robustness of measurement-induced Ising order under combined coherent rotation errors and stochastic measurement noise, using echo-protected quantum circuits simulated for system sizes .Our results show that magnetization remains above 0.8 for measurement noise up to in the largest system, demonstrating a substantial enhancement compared to unprotected circuits where the same threshold drops below 0.12. Finite-size scaling confirms consistent crossing of magnetization curves at the critical noise , independent of system size within numerical resolution. Fidelity analysis further reveals that global quantum coherence is more sensitive: decreases to 0.5 near , while magnetization still exceeds 0.7, emphasizing the difference between classical order and quantum state integrity.The derivative of fidelity with respect to measurement noise, exhibits a sharp peak around for , providing a clear numerical indicator of the critical point. The robustness metric highlights an operationally useful ordered phase that persists for quantifying the combined effect of classical correlations and state fidelity. Overall, echo protection improves the threshold by more than 50% compared to unprotected dynamics and stabilizes both classical and quantum observables under realistic noise.These findings demonstrate quantitatively that coherent error mitigation, combined with fidelity-based diagnostics, significantly enhances the stability and detectability of measurement-induced phases on near-term quantum devices
The second- and third-order coherence functions
g(n)(0) (
n=2 and 3) of an exciton-polariton condensate are measured and compared to the theory. Contrary to an ideal photon laser, deviation from unity in the second- and third-order coherence functions is observed, thus showing a bunching effect, but not the characteristics of a standard thermal state with
g(n)(0)=n!. The increase in bunching with the order of the coherence function,
g(3)(0)>g(2)(0)>1, indicates that the polariton condensate is different from a coherent state, a number state, or a thermal state. The measurement of third-order coherence has the advantage, compared to the second-order one, that the difference between a thermal state and a coherent state is more pronounced. The experimental results are in agreement with the theoretical model where polariton-polariton and polariton-phonon interactions are responsible for the loss of temporal coherence.