Experimental certification of the nonlocal advantages of quantum imaginarity

AI summary

Overview

Research area: Quantum information theory and experimental quantum optics — specifically quantum imaginarity as a resource, quantum steering-like nonlocal advantages, and comparisons with coherence and Bell nonlocality in polarization-encoded photonic qubits.

Technical level: Intermediate. The core idea (local measurements on one half of an entangled pair steer a basis-dependent "imaginarity" measure of the other half) is intuitive, but the criteria, the MUB parametrization, and the GA-SQP optimizer require comfort with quantum resource theory and numerical optimization.

Scope (one sentence): The paper reports an experimental certification of the nonlocal advantage of quantum imaginarity (NAQI) using two families of two-qubit Bell-diagonal states, enabled by a hybrid genetic-algorithm/sequential-quadratic-programming optimizer, and compares the NAQI, NAQC, and Bell-nonlocality regions for Werner states.

What This Paper Is About

Quantum imaginarity treats the imaginary part of a quantum state — with respect to a chosen reference basis — as a physical resource. Wei and Fei had proposed NAQI, a steering-like scenario in which measurements on Alice's subsystem can push the average imaginarity of Bob's conditional states above a classical bound. That proposal had not been experimentally tested. This paper builds the photonic hardware and the optimization machinery to observe genuine violations of the NAQI inequality, and then asks how the NAQI region compares to the NAQC region and the Bell-nonlocality region on the same platform.

Key Contributions

1. First experimental certification of NAQI. Using the l1-norm of imaginarity inequality, the authors observe clear violations (Δ_QI > 0) for binary Bell-state mixtures and Werner states, implemented with polarization-entangled photons from a Sagnac-interferometer PPKTP source.

2. A hybrid GA-SQP optimization algorithm for measurement settings. Because the NAQI parameter must be maximized over Alice's three two-outcome projective measurements and Bob's mutually unbiased bases, the authors combine a genetic algorithm (global exploration, no gradients) with sequential quadratic programming (fast, gradient-based refinement) to avoid local optima and speed convergence. The eight-parameter search is over x = (θ₁,…,θ₄, φ₁,…,φ₄)ᵀ in [0,π]⁴ × [0,2π]⁴.

3. A state-preparation scheme that switches between two state families in one setup. An unbalanced interferometer with a long and short path, variable filters setting p = N₁/(N₁+N₂), a 45° HWP for the Bell-mixture branch, and a decohering module (two calcite crystals with a 22.5° HWP) for the Werner-state branch.

4. A same-platform comparison of NAQI, NAQC, and Bell nonlocality. For the two-qubit Werner states studied, the violating regions are nested: 𝒟_NAQC ⊂ 𝒟_NAQI ⊂ 𝒟_BN.

Main Findings

• NAQI is observed for Bell-state mixtures at nearly all tested p. Seven states ρ_B(p) with p ∈ {0.0, 0.2, 0.4, 0.5, 0.6, 0.8, 1.0} were prepared with average fidelity 0.9760 ± 0.0013. Consistent with theory, NAQI appears for all tested parameters except p = 0.5, where the violation parameter Δ_QI does not exceed the √5 bound.

• NAQI is observed for Werner states only at high p. Six states ρ_W(p) with p ∈ {0.0, 0.2, 0.4, 0.6, 0.8, 1.0} were prepared with average fidelity 0.9890 ± 0.0006. NAQI appears for p = 0.8 and p = 1.0 only.

• Real-valued states can still show NAQI. Although both families are real in the {|H⟩, |V⟩} basis, imaginarity is basis dependent: the l1-norm of imaginarity is evaluated in optimized MUBs {|Υ_k⁺⟩, |Υ_k⁻⟩}, where the states acquire nonzero imaginary off-diagonal elements. Experimentally this quantity is read out as the absolute expectation value of σ_y in those bases.

• Three correlations switch on in a fixed order as p grows. For nine Werner states with p ∈ {0.00, 0.20, 0.40, 0.60, 0.73, 0.77, 0.80, 0.91, 1.00}, all three violation parameters (Δ_QI, Δ_QC, Δ_BN) increase monotonically with p; Bell nonlocality is detected first, then NAQI, then NAQC. The resulting region nesting is 𝒟_NAQC ⊂ 𝒟_NAQI ⊂ 𝒟_BN.

• Experimental precision. Error bars, estimated from Poisson statistics of two-photon coincidences, are on the order of 10⁻³; small theory–experiment discrepancies are attributed to nonideal state preparation and counting statistics.

• Thresholds used. NAQI: N(ρ_AB) > √5. NAQC: the half-sum over j≠k of p(a|j)·C_k^{l1} ≤ √6. Bell nonlocality: the CHSH expression ≤ 2. Violation parameters are defined as the measured quantity minus the respective bound.

Methodology in Plain English

The team starts with a 405 nm continuous-wave laser, sets its polarization to (|H⟩+|V⟩)/√2 with a half-wave plate at 22.5°, and pumps a type-II PPKTP crystal inside a polarization Sagnac interferometer to make polarization-entangled pairs in |ψ⁺⟩. A 45° half-wave plate converts this to |φ⁺⟩, and interference filters remove the pump light.

One photon goes to Alice. The other goes to Bob through an unbalanced interferometer whose two paths differ by far more than the photon coherence time, so they recombine incoherently at a beam splitter and produce a tunable mixture rather than a superposition. Which mixture depends on what sits in the long path: a 45° half-wave plate turns |φ⁺⟩ into |ψ⁺⟩ and yields the binary Bell-state mixture ρ_B(p); two sufficiently long calcite crystals plus a 22.5° half-wave plate fully decohere the polarization and yield the maximally mixed state, so recombination gives the Werner state ρ_W(p). Variable filters in the two paths set the weight p = N₁/(N₁+N₂) from the measured counts.

For detection, motorized quarter- and half-wave plates plus polarizing beam splitters on each side implement the projective measurements, with single-photon detectors and coincidence counting.

The tricky part is choosing which measurements to perform, because the NAQI value is a maximum over Alice's three projective measurements and Bob's three mutually unbiased bases, parametrized by four θ's and four φ's. The authors first run a genetic algorithm — a population of N candidate parameter vectors, keep the best M, crossover the survivors to make N−M offspring, then mutate — iterating until the best solution stops moving or a maximum iteration count is reached. That output seeds sequential quadratic programming, which repeatedly solves a quadratic-programming subproblem (feasible here because the box constraints are linear, so ∇g is the constant matrix (I₈; −I₈) and the constraint Hessian vanishes) and takes a step with a line-search-verified step size. SQP stops when the KKT residuals fall below tolerances ε₁ and ε₂ or when a maximum iteration count is reached. The optimal settings used in the lab are listed in Appendix A, and the code is available at Ref. [49].

Why This Matters

Impact on research. The work converts a theoretical resource-theoretic proposal into a measured phenomenon, showing that imaginarity — like coherence — can be remotely steered and used as a nonclassicality witness in a discrete-variable photonic setting. Placing NAQI, NAQC, and Bell nonlocality side by side on one platform gives an empirical ordering of what each criterion detects for Werner states, and it shows that a state can be "real" in the computational basis yet still display nonlocal imaginarity in suitably chosen bases.

Potential applications (as motivations cited in the paper for quantum imaginarity generally, not demonstrated here):

• Information hiding and related cryptographic tasks.
• Multiparameter metrology.
• Machine learning and pseudorandomness generation protocols.
• Channel discrimination and broadcasting tasks.
• Characterizing outcome statistics in linear-optical experiments and Kirkwood–Dirac quasiprobability distributions.

In addition, the tunable mixture parameter offers a practical dial for engineering correlations to order: for the Werner family, one can select a regime with Bell nonlocality and NAQI but no NAQC, or a regime with all three.

Industry relevance. For photonic quantum technology, the results give an experimentally cheap nonclassicality witness — an absolute σ_y expectation value in a chosen basis — that can be evaluated with wave plates and polarizing beam splitters rather than full tomography. The GA-SQP approach to finding optimal measurement settings is directly reusable wherever a figure of merit must be maximized over measurement bases under experimental constraints.

Future Directions

• Extending NAQI certification beyond qubit–qubit Bell-diagonal states to other state families, higher-dimensional systems, or Gaussian states, whose imaginarity has been treated theoretically but is not tested here.
• Broadening the three-way comparison beyond two-qubit Werner states and beyond the specific l1-norm / relative-entropy criteria used, since the observed nesting 𝒟_NAQC ⊂ 𝒟_NAQI ⊂ 𝒟_BN is established for this family and these criteria specifically.
• Comparing the l1-norm criterion with the relative-entropy NAQI criterion experimentally (only the l1-norm is adopted here, on the grounds that it is more effective at detecting NAQI).
• Systematizing the GA-SQP optimizer — the paper's Appendix B compares it against other algorithms, and a broader benchmarking across state families and figures of merit would clarify when the hybrid is worth its cost.
• Using the tunable mixture knob to prepare states in the intermediate regions (for example, exhibiting Bell nonlocality and NAQI without NAQC) as inputs to specific quantum information protocols.

Target Audience

Quantum information researchers working on quantum resource theories, steering-like nonlocal advantages, and basis-dependent measures of nonclassicality; experimental photonics groups building polarization-encoded entangled-photon sources and measurement stations; and applied-mathematics or quantum-engineering readers interested in hybrid global/local optimization for measurement-setting search. It is also suitable for graduate students who want a worked example of turning a resource-theoretic inequality into a tabletop experiment.

Authors’ abstract

Quantum imaginarity is a distinct resource in quantum information theory, yet its nonlocal properties have not been fully explored. Here, we report an experimental study of the nonlocal advantage of quantum imaginarity (NAQI), in which local measurements on one subsystem can steer the average imaginarity of the conditional states of the other subsystem beyond the corresponding classical bound. The $l_1$-norm of imaginarity inequality is adopted as an experimentally accessible witness. A hybrid optimization algorithm that combines a genetic algorithm with sequential quadratic programming is developed to avoid local optima and accelerate the search for optimal measurement settings. Using polarization-encoded photonic qubits, we prepare two classes of two-qubit Bell-diagonal states and experimentally characterize the imaginarity of the conditional states following local measurements. Clear violations of the $l_1$-norm of imaginarity inequality are observed, providing an experimental certification of NAQI. We further investigate the relationship among NAQI, the nonlocal advantage of quantum coherence (NAQC), and Bell nonlocality based on their respective inequality criteria. For the two-qubit Werner states considered in this work, the regions of states that violate the respective criteria satisfy $\mathcal{D}_{\rm NAQC}\subset\mathcal{D}_{\rm NAQI}\subset\mathcal{D}_{\rm BN}$. Our work provides an experimental realization of NAQI and a comparison of different forms of quantum nonclassicality in a photonic platform.