Experimental quantification of quantum coherence for a set of quantum states

AI summary

Overview

Research area: Quantum information science, specifically quantum coherence theory and experimental quantum optics.

Technical level: Intermediate. The paper assumes familiarity with the Bloch sphere, quantum state tomography, and linear-optics experiments, but its central idea (measuring coherence of a set of states without a preferred basis) is explained clearly enough for graduate students entering the field.

Scope in one sentence: The paper reports a linear-optics experiment that measures basis-independent "set coherence" for two small sets of single-qubit states and argues that the states used in the BB84 quantum key distribution protocol are exactly those with maximal set coherence.

What This Paper Is About

Most measures of quantum coherence depend on a chosen reference basis, and most experiments measure coherence for a single quantum state at a time. This paper takes a recently proposed, basis-independent measure called set coherence — the coherence of an entire collection of states, minimized over all possible measurement bases — and tests it experimentally for the first time in a direct way. The goal is to show that this quantity can be measured reliably in the lab and that it has a concrete use in quantum communication protocols such as BB84.

Key Contributions

1. First direct experimental quantification of basis-independent set coherence. The authors measure the mean robustness of set coherence for two different sets of pure qubit states, rather than for a single state, and compare each result against theory.

2. A Sagnac-interferometer method for extracting set coherence. Rather than performing full quantum state tomography, they show that each single-state robustness term equals the interference visibility of a Sagnac interferometer, so set coherence can be read off from visibility measurements as the basis is scanned.

3. Experimental confirmation of theoretical predictions for n = 2 and n = 3. The measured values, 0.507 ± 0.003 and 0.519 ± 0.003, match the predicted 1/2 and 10√2/27 ≈ 0.524 within experimental error.

4. A new application: set coherence as a figure of merit for BB84. They prove that the four BB84 states have maximal set coherence of 1/2, and that deforming the X basis by an angle θ reduces set coherence according to a simple formula, tracking the known drop in key rate.

Main Findings

• n = 2 case matches theory: For the set {|0⟩, |+⟩}, the theoretical set coherence is 1/2. The experimentally obtained minimum was ℛ₁ᵉˣᵖ(ϱ₁) = 0.507 ± 0.003, found at a half-wave-plate angle α₃ of 68.25°.

• n = 3 case matches theory: For the set {(|0⟩ + √2|1⟩)/√3, |0⟩, (|0⟩ − √2|1⟩)/√3}, the theoretical value is 10√2/27 ≈ 0.524. The measured minimum was ℛ₁ᵉˣᵖ(ϱ₂) = 0.519 ± 0.003, at α₃ = 19.5°.

• Interferometer quality is high: With both internal unitaries set to identity, the Sagnac interferometer reached a visibility of 99.4%, indicating a reliable optical path.

• Set coherence is a periodic function of the measurement basis: Averaged visibility curves for both sets oscillate as the basis-changing half-wave plate H₃ is rotated from 2° to 92° in 2.5° steps, with extra points at 23.25°, 45.75°, 68.25° and 90.75° giving 41 data points for n = 2 and 37 data points for n = 3.

• Known upper bounds are respected: For pairs of pure qubit states the set coherence obeys ℛ₁ ≤ 1/2, and for three states the upper bound is 2/3; the chosen three-state set lies below that bound, consistent with prediction.

• BB84 states are optimal: The set {|0⟩, |1⟩, |+⟩, |−⟩} attains the maximal set coherence of 1/2, and the set coherence of the four-state BB84 ensemble equals that of the two states {|ψ(μᵢ, φᵢ)⟩} alone.

• Set coherence tracks key rate: When the X basis is rotated by θ into a "W basis" {|w⁺⟩, |w⁻⟩}, the set coherence becomes ℛ₁(Φ) = (1/2)|sin(π/2 − θ)|, falling from 0.5 to 0 as θ goes from 0 to π/2 — mirroring the fall of the key rate to zero reported in earlier work.

• Approximation in the three-state preparation: Because of limited wave-plate rotation precision, the angle α₂ for preparing |φ₃⟩ and |φ₅⟩ was approximated (set to ±27.4°), yet the results still agreed with theory.

Methodology in Plain English

The team encodes a qubit in the horizontal and vertical polarization of single photons. Photon pairs are produced by spontaneous parametric down-conversion in a periodically poled KTiOPO₄ crystal (1 × 2 × 20 mm, poling period 10.025 μm, held at 25 °C), which converts 405 nm pump light into 810 nm photon pairs; one photon is detected as a heralding trigger and the other is sent into the interferometer.

The signal photon first passes a wave-plate group that sets the initial state anywhere in the X–Z plane of the Bloch sphere, followed by a half-wave plate H₃ whose angle α₃ defines the reference basis. The photon then enters a Sagnac interferometer built around a 50:50 non-polarizing beam splitter, whose two counter-propagating arms contain different wave-plate combinations implementing the unitaries U_c = −σ_x and U_d = iI. Rotating a phase shifter inside the loop scans the relative phase ω, giving maximum and minimum count rates whose normalized difference is the interference visibility.

The key theoretical step is that, for a single qubit, the robustness of coherence in a given basis equals |⟨φᵢ|U†σ_xU|φᵢ⟩| — exactly the quantity the visibility measures. So instead of reconstructing each density matrix by tomography, the authors simply measure visibilities for each state at many values of α₃, then average over the states in the set and take the minimum over α₃ to obtain the set coherence. For these states the optimal basis is guaranteed to lie in the X–Z plane, so only real rotations need to be scanned.

Why This Matters

Impact on research: This is a direct experimental test of a basis-independent coherence measure, moving set coherence from a theoretical proposal to a measurable laboratory quantity. It also shows that interference visibility can substitute for full state tomography when quantifying coherence, which is attractive for larger systems where tomography becomes expensive.

Real-world applications:

• Quantum key distribution: Set coherence provides a single number that tracks the key rate of BB84 and its variants, potentially serving as a quick diagnostic when evaluating new basis choices.
• Quantum secure direct communication: The paper argues the same basis-selection reasoning applies to choosing transmitting and receiving bases in QSDC.
• Probabilistic quantum cloning: The authors highlight cloning of a set of quantum states as a task where set coherence could be a useful resource parameter.
• Quantum metrology and computing: Coherence is cited as a resource underlying quantum metrology, quantum computing and the study of quantum phase transitions.

Industry relevance: Practical QKD deployments must choose bases and monitor how far a real implementation deviates from the ideal BB84 states. A single, basis-independent metric that degrades in step with the key rate is a potentially useful benchmarking and security-analysis tool for QKD hardware developers.

Future Directions

• Extend to larger and mixed sets: Only two and three pure qubit states in the X–Z plane were tested; whether the visibility method scales to higher-dimensional or mixed-state ensembles is untested.
• Improve preparation precision: The ±27.4° approximation for two of the three-state inputs was forced by wave-plate resolution, so tighter state preparation would remove this source of error.
• Connect set coherence to protocol performance quantitatively: The link between set coherence and key rate is currently argued by analogy with previously published results; a direct side-by-side measurement of both quantities in one experiment would strengthen the claim.
• Apply to other QKD and cloning protocols: The authors suggest extending the analysis to other quantum key distribution schemes and to probabilistic quantum cloning of state sets.

Target Audience

Graduate students and researchers in quantum information and experimental quantum optics, particularly those working on coherence theory, linear-optics implementations of quantum protocols, or QKD security analysis. It is also accessible to protocol designers who want a compact, experiment-friendly metric for comparing sets of encoding states.

Authors’ abstract

We present a direct experimental verification of quantum coherence for a set of quantum states in a basis-independent manner. We find that the theory of quantum coherence for a set of quantum states matches our experimental results perfectly and can be applied to protocols like BB84 and quantum secure direct communication. Using a Sagnac interferometer, we experimentally quantify the quantum coherence for two sets of experimental states. Moreover, we introduce a novel application of set coherence, illustrating that the states employed in the BB84 protocol exhibit maximal set coherence. Our results pave the way for wider applications of quantum coherence for a set of quantum states, including quantum key distribution protocols and probabilistic quantum cloning.