Experimental demonstration of the equivalence of entropic uncertainty with wave-particle duality

Abstract

Wave-particle duality is one of the most notable and counterintuitive features of quantum mechanics, illustrating that two incompatible observables cannot be measured simultaneously with arbitrary precision. In this work, we experimentally demonstrate the equivalence of wave-particle duality and entropic uncertainty relations using orbital angular momentum (OAM) states of light. Our experiment uses an innovative and reconfigurable platform composed of few-mode optical fibers and photonic lanterns, showcasing the versatility of this technology for quantum information processing. Our results provide fundamental insights into the complementarity principle from an informational perspective, with implications for the broader field of quantum technologies.

Fig. 1.. Different experimental configurations for complementarity measurements.

(A) MZ interferometer with a TBS to recombine the paths and a modulator applying a relative phase shift ${\mathrm{\varphi }}_x$ between the two arms. (B) TBS adjusted to equal transmission and reflection coefficients, yielding full interferometric visibility. (C) TBS adjusted to complete transmission or reflection, equivalent to it being removed. In this case, full path information is available; thus, no interference can be observed. (D and E) For the distinguishability measurements, each path is individually blocked, and the detection events are recorded for any setting of the TBS. In both cases, no interference pattern is possible.

Fig. 2.. Experimental setup for probing the EUR over a |OAM₊₁〉 qubit.

The experiment consists of three main parts: the $\mid {\text{OAM}}_{+1}\rangle$ mode source, the unitary transformation, and the measurement stage (or TBS). An amplitude modulator (AM) and an attenuator (ATT) adjust the average number of photons per pulse to $\mathrm{\mu }=0.2$. In addition, the $\mid {\text{OAM}}_{+1}\rangle$ state can be encoded by applying the appropriate forked diffraction grating to the spatial light modulator (SLM) and also using a 4f system (see Materials and Methods). The amplitude and phase profiles of the ${\text{OAM}}_{+1}$ state are shown, following propagation through the FMF with an InGaAs infrared camera with the laser source unattenuated. Then, the unitary transformation stage performs a mapping from spatial to path information, yielding $\mid \mathrm{\psi }\rangle =\frac{1}{\sqrt{2}}(\mid 0\rangle +e^{i{\mathrm{\varphi }}_x}\mid 1\rangle )$, where a rotation to the state can be applied through the phase modulator ${\mathrm{\varphi }}_x$. The TBS is described by an SI containing an optical delay and a phase modulator ${\mathrm{\varphi }}_s$, which controls the transmissivity and reflectivity of the BS, which determines the maximum obtained visibility. On the other hand, to measure the distinguishabilities ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$, it is necessary to block the paths before the SI with electro-optical ATTs.

Fig. 3.. Dynamic change between OAM wave and particle behavior.

Recorded detections at D1 and D2 with a continuous triangle waveform applied to ${\mathrm{\varphi }}_x$ and ${\mathrm{\varphi }}_s$ periodically changed between 0 (particle) or $\mathrm{\pi }/2$ (wave). Error bars are the SD considering Poissonian statistics of the single-photon detection process.

Fig. 4.. Single counts for detectors 1 and 2 as a function of ϕx, varying the phase shift ϕs applied in the SI.

In the first column, both arms of the interferometer are open, while the second and third columns show the single count when paths $0$ or $1$ are blocked, respectively. In each subplot, the error bars were calculated through error propagation taking into account the Poissonian statistics of the recorded individual counts. The integration time for each point is 0.8 s. Last, the solid and dashed lines represent the best-fit curves obtained by minimizing the mean square error between the experimental data and the fit.

Fig. 5.. Experimental equivalence between WPDR and EUR.

The solid, dashed, and dot-dashed curves are respectively the theoretical values, as a function of the Sagnac phase ${\mathrm{\varphi }}_{\mathrm{s}}$, of the EUR, minimized max-entropy related to the random variable $W$ (wave), and min-entropy related to the random variable $Z$ (particle). The black, blue, and red circles with their error bars are obtained, for each value of ${\mathrm{\varphi }}_{\mathrm{s}}$, by measuring the input distinguishability $D$ and the interferometric visibility $V$, to evaluate the entropies via Eqs. 3 and 6. The black, blue, and red diamonds with their error bars are obtained by taking the measured probabilities related to each variable and applying the entropic definitions as given by Eqs. 8 and 9. Error bars were calculated using error propagation assuming Poissonian statistics for the recorded number of detection events.