Quantum uncertainty, a fundamental concept in quantum mechanics, dictates the impossibility of simultaneously predicting the outcomes of measurements on incompatible observables. Unlike classical uncertainty, which stems from a lack of information about the system's state, the quantum case raises the question of whether it's intrinsically quantum or also influenced by information limitations. Previous research, using entropic formulations of the uncertainty principle and illustrated by guessing games, suggests a significant contribution from this lack of information. This paper experimentally explores this hypothesis through the implementation of two-dimensional and three-dimensional guessing games. The guessing game framework provides an operational interpretation of the entropic uncertainty relation, offering insights into foundational aspects and applications in quantum cryptography. The authors aim to experimentally verify that the quantum uncertainty in these games is largely attributed to the guessing party's lack of access to quantum information determining the key properties of the game. Furthermore, the paper introduces a novel and experimentally efficient method for constructing high-dimensional quantum Fourier transform gates, crucial components in quantum computation, communication, and metrology. These gates are commonly employed in algorithms involving phase estimation, quantum state tomography, and quantum key distribution. The high-dimensional Fourier transform gates are implemented in the path degree of freedom of a single photon using a compact experimental design, which offers advantages in control and stability compared to other degrees of freedom. The paper's structure details the framework of the guessing game, presents experimental results, discusses their implications for quantum cryptography and future research directions, and concludes with a description of the experimental setup and methodology.

The paper starts by reviewing the Heisenberg uncertainty principle, Kennard's rigorous proof, and Robertson's generalized uncertainty relation. It highlights the limitations of Robertson's relation, particularly its state-dependence. Entropic formulations, such as the Maassen-Uffink relation, overcome these shortcomings by providing state-independent bounds and information-theoretic interpretations. The authors then focus on the guessing game framework, proposed by Berta et al., which operationally interprets the entropic uncertainty principle. This framework examines the role of information, both classical and quantum, in contributing to the overall uncertainty. Previous work by Rozpędek et al. suggested that quantum uncertainty, even in some cases entirely, can be attributed to the guessing party's lack of quantum information about the choice of the measured observable. This prior work showed that revealing this information allows for a significant reduction, or even complete elimination, of the uncertainty. The current research builds on this foundation to experimentally test these claims.

The experiment uses a guessing game framework where Bob prepares a quantum state and sends it to Alice. Alice randomly chooses between two incompatible measurements (S and T) and announces her choice to Bob, who then tries to guess Alice's measurement outcome. The core of the experiment involves implementing this game in both two and three dimensions (d=2 and d=3). The choice of measurement is controlled by a quantum register (R), which can be prepared with varying degrees of coherence (γ). This coherence parameter controls the amount of quantum information Bob has about Alice's measurement choice, ranging from complete lack of knowledge (γ=0, classical randomness) to complete knowledge (γ=1). The measurements S and T are chosen to be mutually unbiased bases – the standard and Fourier bases. For d=2, these are the standard and Hadamard bases. The experimental setup encodes the system state (B) and the register state (R) in different degrees of freedom of a single photon: path for B and a path layer (upper/lower) for R. A controlled Fourier gate is applied to correlate the two registers. Alice's measurement is performed on the path degree of freedom, and Bob's measurement, to guess Alice's outcome, is on the path layer. The two-dimensional Fourier gate is constructed using standard optical components. The three-dimensional Fourier gate is implemented using a novel and experimentally efficient method employing a horizontally placed beam displacer (HBD)-half-wave plate (HWP)-HBD structure, which simplifies the original scheme and reduces the complexity of the setup. This scheme reduces the number of interferometers needed for the three-dimensional Fourier transform from six to three, improving stability and robustness. The quality of the three-dimensional Fourier gate is assessed by measuring the probability of detecting the photon in the correct output mode after applying the gate to a superposition state. A detailed error analysis is presented, taking into account the imperfect generation of the register state and dephasing in the interferometers. The single-photon source is generated using spontaneous parametric down-conversion (SPDC). For the d=2 game, the coherence parameter γ is tuned by introducing a quartz plate that introduces dephasing. Quantum state tomography is used to reconstruct the register state and estimate γ. The guessing probability is determined from the detection probabilities at the output ports. For the d=3 game, the experiment focuses on the maximum achievable coherence, and several guessing strategies are explored to optimize the guessing probability.

The experimental results demonstrate a clear relationship between the coherence parameter γ and Bob's guessing probability. For the d=2 game, the observed guessing probability consistently exceeds the maximum achievable probability when γ=0 (classical randomness). This supports the hypothesis that the uncertainty at γ=0 stems from the lack of information about the register state. Moreover, the guessing probability increases with increasing γ, reaching approximately 0.9953 when γ is close to 1. This demonstrates that providing Bob with more information about the measurement process significantly improves his ability to guess Alice's outcome, and for d=2, almost all uncertainty can be attributed to the lack of information. For the d=3 game, the highest achieved guessing probability is approximately 0.9628 at γ ≈ 0.9918. This value exceeds the probability achievable when γ=0, again highlighting the contribution of information to the overall uncertainty. The deviation from the theoretical maximum achievable probability (≈0.9793 for γ=1) is attributed to experimental errors, such as state preparation errors and dephasing within the interferometers, indicating the existence of intrinsic uncertainty in this case. This difference is estimated and shown to be consistent with the experimentally characterized error model, which supports the claim that there exists intrinsic uncertainty in the d=3 game that cannot be removed even with full knowledge of the measurement apparatus state.

The experimental results strongly support the hypothesis that quantum uncertainty, within the guessing game framework, is substantially influenced by the lack of information available to the guessing party. The d=2 game shows that almost all observed uncertainty can be attributed to this lack of information, while the d=3 game reveals the presence of an intrinsic uncertainty component, which is consistent with the theoretical predictions. These findings have significant implications for quantum cryptography, particularly protocols relying on mutually unbiased bases, such as BB84. The security of such protocols relies on the inaccessibility of the purification of the measurement-choice register to eavesdroppers. If an eavesdropper had access to this information, they could effectively eliminate uncertainty, potentially compromising the protocol's security. The efficient method developed for constructing high-dimensional Fourier gates opens new avenues for experimental quantum information processing. The parallel-path structure enhances the setup's stability, offering significant advantages for more complex, high-dimensional implementations. While scalability to higher dimensions faces challenges regarding the physical size of beam displacers and phase stability, potential solutions such as stacking PBSs and implementing active phase stabilization are discussed. Furthermore, the presented setup enables future research in areas like wave-particle duality and generalizations of the guessing game to explore scenarios with multiple measurements.

This work provides crucial experimental evidence for the role of information in quantum uncertainty. It confirms that the lack of quantum information is a major contributor to uncertainty, particularly in low-dimensional systems. Higher-dimensional systems show evidence of intrinsic uncertainty beyond information limitations. The development of a novel, efficient three-dimensional Fourier gate contributes to experimental quantum technologies. Future research could explore generalizations of the guessing game, the connection to wave-particle duality, and overcome scaling challenges for higher-dimensional systems.

The experiment's limitations include the imperfect generation of quantum states and some dephasing within the interferometers. These factors contributed to the deviation between observed and theoretically predicted guessing probabilities, especially in the d=3 game. While the error analysis suggests these errors are well-understood and accounted for, they inevitably limit the precision of the results. The highest achievable coherence (γ ≈ 0.9918) in the experiment was less than perfect, which also introduced a degree of uncertainty in the findings. The study focuses on two and three-dimensional guessing games, which may not fully generalize to all higher dimensions.