The detection and characterization of quantum coherence is of fundamental importance both in the foundations of quantum theory as well as for the rapidly developing field of quantum technologies, where coherence has been linked to quantum advantage. Typical approaches for detecting coherence employ {\it coherence witnesses} -- observable quantities whose expectation value can be used to certify the presence of coherence. By design, coherence witnesses are only able to detect coherence for some, but not all, possible states of a quantum system. In this work we construct protocols capable of detecting the presence of coherence in an {\it unknown} pure quantum state
∣ψ⟩. Having access to
m copies of an unknown pure state
∣ψ⟩ we show that the sample complexity of any experimental procedure for detecting coherence with constant probability of success
≥2/3 is
Θ(c(∣ψ⟩)−1), where
c(∣ψ⟩) is the geometric measure of coherence of
∣ψ⟩. However, assuming access to the unitary
Uψ which prepares the unknown state
∣ψ⟩, and its inverse
Uψ†, we devise a coherence detecting protocol that employs amplitude-amplification {\it a la} Grover, and uses a quadratically smaller number
O(c(∣ψ⟩)−1/2) of samples. Furthermore, by augmenting amplitude amplification with phase estimation we obtain an experimental estimation of upper bounds on the geometric measure of coherence within additive error
ε with a sample complexity that scales as
O(1/ε) as compared to the
O(1/ε2) sample complexity of Monte Carlo estimation methods. The average number of samples needed in our amplitude estimation protocol provides a new operational interpretation for the geometric measure of coherence. Finally, we also derive bounds on the amount of noise our protocols are able to tolerate.
This study presents a theoretical analysis of three-photon interference in a six-port Mach-Zehnder Interferometer (6p-MZI) system, which incorporates two identical ideal six-port optical beam splitters (tritters) and phase modulators. We investigate interference patterns produced by hybrid input photon configurations that combine Fock state and coherent states, exploring the effects of both relative phase differences and average number of coherent photon state at the input ports. Our analysis demonstrates that the phase positions of constructive interference peaks systematically shift in response to the average number of coherent photons
(∣α∣2) at the input ports. When
∣α∣2 increases from 0 to 0.5, the constructive interference peaks at output ports 2 and 3 undergo symmetric shifts of up to
π/3 radians relative to the extreme interference pattern of output port 3 at
5π/3. The results highlight the high sensitivity of interference patterns to both relative phase differences and coherent state photon amplitudes. These findings suggest novel approaches for controlling three-photon interference patterns in 6p-MZI systems through precise manipulation of phase and low-amplitude coherent photon states. This work advances our understanding of multi-photon interference in the quantum regime, with potential applications in quantum information processing, metrology, and fundamental quantum optics studies.