Experimental study of quantum uncertainty from lack of information

The study addresses whether quantum uncertainty is entirely intrinsic or can arise from a lack of information, analogous to classical uncertainty. Traditional formulations (Heisenberg, Kennard, and Robertson) quantify uncertainty via standard deviations but suffer from state-dependent lower bounds that can become trivial. Entropic uncertainty relations (Maassen–Uffink) provide state-independent bounds and an information-theoretic interpretation. Using a guessing-game framework, the authors experimentally examine the role of information about the measurement-choice register in uncertainty. They probe how access to quantum information (coherence) about the choice of incompatible measurements affects a party’s ability to predict outcomes, testing d=2 and d=3 scenarios with mutually unbiased bases (standard and Fourier bases). The purpose is to separate uncertainty due to lack of information from intrinsic uncertainty and to validate theoretical predictions that, in d=2, uncertainty can be fully attributed to lack of information, whereas in higher dimensions intrinsic uncertainty remains.

Foundational uncertainty relations were introduced by Heisenberg and formalized by Kennard, and generalized to arbitrary observables by Robertson; however, Robertson’s bound can be state-dependent and trivial for some states. Entropic uncertainty relations (Maassen–Uffink) remedy this by providing state-independent bounds quantified via Shannon entropy and overlap c of observables. The guessing-game perspective, initiated by Berta et al., operationalizes entropic uncertainty by having a player guess outcomes of a randomly chosen measurement from incompatible observables. Prior theoretical work showed that uncertainty in this framework can be attributed to the guessing party’s lack of information about the measurement-choice register and, in certain cases (notably d=2 with full quantum access), can be eliminated. Extensions also connect to quantum cryptography and wave–particle duality, and explore scenarios with more than two measurements where perfect guessing depends on system dimension and choice of measurements.

Experimental platform: heralded single photons at 808 nm generated via type-II SPDC (404 nm pump) in BBO. The system B (the probed state) is encoded in spatial path modes (for d=2: paths 0 and 1; for d=3: paths 0,1,2). The register R (measurement-choice control) is encoded in two path layers: upper (u) and lower (l), realized via polarization-path conversion using vertically and horizontally placed calcite beam displacers (VBD/HBD). A controlled operation applies either identity (upper layer) or Fourier transform (lower layer) to B, implementing a controlled Fourier gate. Register coherence control: The register state ρ_R(γ) is prepared to interpolate between classical randomness (γ=0; maximally mixed over |0⟩,|1⟩) and full quantum coherence (γ=1; |+⟩ state). Coherence γ is tuned by inserting quartz plates (QPs) before the first VBD to implement a dephasing channel that couples polarization to frequency, reducing coherence between |u⟩ and |l⟩. Tomography before the VBD reconstructs ρ_R and γ is estimated by maximizing fidelity with the ideal form ρ_R(γ) (fidelities >0.9995 for all settings). Controlled Fourier gates: For d=2, the control applies identity vs. Hadamard basis measurement. For d=3, the authors implement a compact three-dimensional Fourier operation using an HBD–HWP–HBD module to realize effective single-qubit Ry rotations (variable beam splitter functionality) without Mach–Zehnder interferometers, reducing component count and improving stability. Overall interferometer visibilities exceed 0.98; a parallel-path architecture enhances phase stability against environmental fluctuations. Measurements: After Alice’s controlled operation and measurement on B (in the standard basis), Bob receives/registers R and performs an optimal measurement to guess Alice’s outcome X. Since R and B are encoded in different DoFs of the same photon, simultaneous detection at output ports D_ij corresponds to Bob’s guess i of Alice’s outcome j. Bob’s analyzer (QWP+HWP+PBS) is tuned per strategy to maximize successful guessing probability P_guess. Probe state preparation: For d=2, an optimal probe state for all γ is |ψ⟩=(|0⟩−|1⟩)/√2, prepared with H1 at 11.3° and path sorting via HBDs; polarization is unified (HWP at 45°) and set to (|H⟩+|V⟩)/√2 before VBD to create (|u⟩+|l⟩)/√2 in R. For d=3 at γ≈0.9918, a best-known probe |ψ⟩=α1|0⟩+α2|1⟩+α3|2⟩ with α1=0.0938+0.5786i, α2=0.0109−0.1218i, α3=0.8009 is prepared using H1, H2, and HBDs. Bob’s optimal measurement for d=3 is projective distinguishing two dominant outcomes (POVM M0, M1=0, M2) aligned with outcomes 0 and 2. Data acquisition and analysis: P_guess is computed by summing detection probabilities at correct ports (e.g., D00 and D11 for d=2). Error sources include imperfect register coherence (max γ=0.9918±0.0009), finite interferometer visibility (~0.99) inducing dephasing, and waveplate-induced random phases. Error models (state-preparation and dephasing) are used to account for observed gaps to theoretical predictions. Fourier gate quality is benchmarked by inputting |ψ⟩=(|0⟩+|1⟩+|2⟩)/√3 and measuring correct-output probabilities.

- d=2 guessing game: - With γ>0, 10/11 data points yield P_guess(exp)(γ>0,d=2) > P_guess(max)(γ=0,d=2) = (2+√2)/4, demonstrating that access to quantum information in R reduces uncertainty. - Monotonic behavior observed: increasing γ increases P_guess(exp). For γ up to 0.9810, experimentally increasing γ→γ+δ (0<δ<0.2258) yields P_guess(exp)(γ+δ,d=2) > P_guess(max)(γ,d=2), confirming that more quantum information raises guessing success. - Near-perfect guessing achieved: with γ=0.9918±0.0009, P_guess(exp)=0.9953±0.0003, close to the theoretical optimum P_guess(max)(γ=1,d=2)=1. Residual gap attributed to imperfect γ and ~0.99 interferometer visibility causing dephasing. - d=3 guessing game: - At γ=0.9918, best-known strategy yields P_guess(exp)=0.9611±0.001 (data point “3”). Scanning nearby strategies gives a maximum P_guess(exp)=0.9628±0.0009 (data point “4”) due to noise anisotropy and waveplate-induced random phases. - Results exceed the classical (γ=0) baseline, showing that lack of information significantly contributes to uncertainty in d=3 as well. - Comparison to best-known noiseless bound P_guess(γ=1,d=3)=0.9793 indicates an experimental gap P_gap=0.0182 for data point “3”, explained by γ<1 and characterized experimental errors; nonetheless, a substantial portion of the total uncertainty gap 1−P_exp(γ=0.9918,d=3) is intrinsic, with P_gap / [1−P_exp] = 0.4679, indicating intrinsic uncertainty in d=3. - Controlled Fourier gate performance: - Three-dimensional Fourier gate implemented via HBD–HWP–HBD achieves average correct-output probability 0.9771±0.0006 for equal-superposition input, with interferometer visibilities >0.98, demonstrating a compact and stable approach to high-dimensional Fourier transforms.

The experiments directly address the research question of whether uncertainty in entropic guessing games is intrinsic or due to lack of information about the measurement choice. For d=2, providing quantum access to the measurement-choice register effectively eliminates uncertainty, validating the theoretical claim that d=2 uncertainty is not intrinsic within this framework. For d=3, even with high register coherence, perfect guessing remains impossible; the observed fraction of uncertainty not attributable to lack of information provides empirical support for intrinsic uncertainty in higher dimensions. These findings have implications for quantum cryptography: protocols relying on mutually unbiased bases (e.g., BB84) require that the purification of the measurement-basis coin remain inaccessible to adversaries; otherwise, eavesdroppers with later access to the purification could predict outcomes and compromise security. Technologically, the work advances linear-optical control of high-dimensional systems by demonstrating a compact controlled 3D Fourier transform, an essential primitive for quantum algorithms, communication, and metrology. The parallel-path, beam-displacer-based design enhances phase stability and scalability compared with interferometer-heavy approaches. The analysis highlights practical noise sources (dephasing, state-preparation imperfections) and their impact on optimal strategies, informing future experimental design.

This work experimentally disentangles the roles of intrinsic uncertainty and lack of information in entropic guessing games. In d=2, uncertainty can be fully attributed to lack of information, approaching unit guessing probability when register coherence is near maximal. In d=3, despite high access to quantum information, a significant fraction of uncertainty persists, evidencing intrinsic uncertainty. Additionally, the paper introduces a compact, high-fidelity implementation of a controlled three-dimensional Fourier transform using HBD–HWP–HBD modules, contributing to the photonic toolbox for high-dimensional quantum information processing. Future directions include: extending controlled Fourier transforms to higher dimensions and fully qutrit-controlled operations; implementing active phase stabilization for larger interferometric networks; exploring games with more than two measurements and different incompatible measurement sets; investigating connections to wave–particle duality; and integrating the approach on photonic chips for scalable, stable, high-dimensional quantum processing.

- Imperfect access to full quantum information: maximal achieved register coherence γ=0.9918<1 limits attainable P_guess. - Experimental noise: finite interferometer visibility (~0.99) leads to dephasing; waveplate rotations introduce random phases, affecting some strategies more than others. - Optimal strategy unknown for d=3 with γ>0, preventing definitive claims of optimality across all γ. - Non-demolition measurement of B prior to sending R to Bob is not performed; simultaneous measurements are used instead for practical efficiency. - Scalability constraints: beam displacer size and passive phase stability pose challenges for higher-dimensional implementations without active stabilization. - Study limited to d=2 and d=3 cases and mutually unbiased (standard/Fourier) measurements.