Experimental Sample-Efficient and Device-Independent GHZ State Certification

The gist: This work experimentally demonstrates, for the first time to our knowledge, the DI certification of a single copy of a four-partite GHZ state, completely free of the IID assumption (Page 1).

Motivation and Requirements

The certification of entangled quantum states is one of the most important quantum information primitives because entangled states are very often the difficult resource for quantum information applications (Page 1). When entangled states are intended for use for a given application, their certification should meet three key requirements: First, it must output a state that can be used for a given application, without additional assumptions (Page 1). Second, certification should ideally not rely on complete trust in the measurement devices as it should be at least to some degree device-independent (Page 1). Lastly, the certification process must account for possible memory effects thereby avoiding the assumption that the source produces independent and identical copies in every round of the experiment (Page 1).

Device-Independent Certification Protocol

The objective of our certification protocol is to quantitatively assess the proximity of a state, σc, generated by an uncharacterized source to the target state GHZ⟩ = (HHHH⟩ + V V V V ⟩)/√2 (Page 1). In a device-independent scenario where measurements are uncharacterized and conform to a Bell scenario, we only have access to input-output correlations (Page 2). The approach involves testing the violation of a Bell inequality which self-tests the target state: a high Bell violation implies that all N − 1 measured copies are close to the target state (Page 2). To address uncertainty inherent in treating all measurement devices as black boxes, we propose using extractability as an appropriate metric, defined as fidelity optimized over all possible local isometries (Page 2). The certified extractability is conditional on the outcomes of the performed measurements, characterizing the conditional state: σ˜c = Tr1,c−1,c+1,…N [(⊗ N−c k=1 Mokik)σN] (Page 2).

Performance Metrics and Bounds

The goal is to claim with a confidence level 1 − δ that the extractability of the conditional state, σ˜c, from the target state, GHZ⟩, is bigger than some value 1 − η (Page 2). This is written as: Ξ(˜σc, GHZ⟩) = maxΦF(Φ[˜σc], GHZ⟩) ≥ 1 − η (Page 2). The protocol relies on robust selftesting statements based on a Bell inequality to establish a lower bound on the extractability of the underlying quantum state, Ξ(˜σc, GHZ⟩) ≥ sβ + µ (Page 9). The Mermin operator displays the tightest bound from a robust self-testing perspective, suggesting that it is likely the most appropriate operator for the certification protocol (Page 6).

Experimental Implementation and Results

The experiment uses a compact and high-performance four-party GHZ state source based on spontaneous parametric down conversion (SPDC) in a layered-Sagnac interferometer configuration (Page 4). The experimental results show, for the first time to our knowledge, the experimental device-independent certification of a four-qubit GHZ state with an extractability of Ξ(˜σc, GHZ⟩) ≥ 0.896 for a total of 4643 verified samples and a confidence level of 1 − δ = 0.99 (Page 4). This demonstration is only possible due to the high-fidelity states produced with our experimental setup, yielding an average success probability of P = 0.973 (Page 4). The analysis shows that by tuning the confidence level, we can reduce the number of samples required to achieve the same certification level (Page 5).

Conclusion and Practical Implications

This work reinforces the validity of the fully DI certification of quantum states as a valuable fundamental resource for a wide range of quantum information applications (Page 5). It emphasizes the practicality of our protocol in providing a rigorous framework in which a finite number of samples can yield meaningful results without further assuming identical and independent distribution for all produced copies (Page 5). The results show that while the theory is able to characterize the few-copies regime, demanding confidence levels and extractability requirements can only be achieved for relatively large samples (Page 5). The ability to experimentally demonstrate the fully DI certification of such a high extractability level paves the way to the reliable and robust use of quantum information systems in practical, real-world settings (Page 5).

Methods Summary

1. The protocol involves generating N copies of a quantum state, σ = Σ i σ i (Page 8).

2. The verifier rolls an N-faced die to determine the set of states Sc to be preserved and certified, while the remaining states constitute the verification set Sv (Page 9).

3. Measurement settings for each copy are determined by randomly generating inputs ik,j for each player k in the verification set (Page 9).

4. The overall pass rate P is calculated as Nwin/Nv after gathering scores from all copies in the verification sample (Page 9).

Key Operators

The paper compares three Bell operators: F. Baccari et al. [19], Q. Zhao et al. [13], and the Mermin operator (Eq. (4)) (Page 6). The Mermin operator is chosen because its quantum bound, βQ, saturates the algebraic bound, leading to a maximum success probability of pQM = 1 (Page 6). The analysis suggests that the Mermin operator outperforms the others in terms of sample efficiency and overall performance regarding how close we can certify a state with respect to the target state (Page 6).

Experimental Setup Details

The polarization-entangled state is generated through entanglement fusion of two Bell pairs, each produced via type-II spontaneous parametric down conversion (SPDC) within a periodically-poled KTP crystal (Page 10). Polarization entanglement is achieved by pumping the crystal from two opposing directions and then interfering the two paths using a Polarizing Beam Splitter (PBS), realized with a Sagnac interferometer (Page 10). A GHZ state of the form GHZ⟩ = (HHHH⟩+e iδ V V V V ⟩)/√2 is generated conditioned on fourfold coincidences (Page 10). The fidelity with respect to the GHZ state is F = ⟨GHZ ρexp GHZ⟩ squared = (94.15 ± 0.21)% for an acquisition time of 150 s per basis (Page 6).

Data Collection and Analysis

The experiment was conducted for three different pump power settings, associated with different state generation rates: 6 Hz, 46 Hz and 101 Hz (Page 7). For each configuration, more than 4 × 10 5 states were collected over a fixed acquisition window of 15 seconds per randomly selected classical input (Page 7). Two methods for analyzing the resulting data are considered: selecting one random outcome from the full set of recorded outcomes for each input, or considering the full dataset resulting from decomposition of each classical input into as many inputs as the total number of recorded outputs within the 15-second acquisition window (Page 7). The certified extractability converges towards the self-testing bound only due to a drastic increase of copies to N ∼ 4 × 10 5 (Page 7).

Self-Testing Bounds Summary

The analysis involves finding s, µ ∈ R such that Ξ(σ, GHZ⟩) ≥ sβ + µ (Page 9). The optimal bound is determined by the minimum value of s and corresponding µ satisfying the imposed condition for all αi (Page 9). The Mermin operator displays the tightest bound from a robust self-testing perspective (Page 6). Table I summarizes numerical self-testing bound parameters for the three different operators considered (Page 6). The lower bound on extractability provided by the DI quantum state certification protocol depends on the maximum probability of winning with a quantum strategy, pQM (Page 9). The corrected expression for c is c = 1/2˜cβalgebraic serving as a fundamental factor in establishing a link between the self-testing bound and the nonlocal game (Page 9).

Experimental Details Summary

The experimental setup uses a Ti:Sapphire laser with an average power of 3.4 W emitting 2 ps pulses at a wavelength of 775 nm (Page 4). The spatial mode is shaped into a Gaussian profile, and the pump pulses are split into two parallel beams using a spatial multiplexer (Page 4). A Sagnac interferometer is used to achieve polarization entanglement in the Sagnac loop (Page 10). The final GHZ state generation involves post-selecting events resulting in each photon occupying a different spatial port of the FPBS, conditioned on fourfold coincidences (Page 10). The fidelity is estimated using Quantum State Tomography based on linear regression and fast maximum likelihood estimation (Page 6).

Acknowledgments

We thank Uta Isabella Meyer and Henrique Silvério for fruitful discussions and technical support (Page 10). We acknowledge financial support from the European Union’s Horizon 2020 framework programme under the Marie Sklodowska Curie innovation training network project AppQInfo, Grant No. 956071 (LdSM), the Horizon Europe research and innovation programme under the project QSNP, Grant No. 101114043 (ED), the European Research Council Starting Grant QUSCO, Grant No.

1. No IID assumption certification allows for the certification of a single copy of a four-qubit GHZ state, completely free of the IID assumption, enabling robust verification in real-world experimental settings where perfect source control is unattainable.

2. The protocol leverages extractability as an appropriate metric, defined as fidelity optimized over all possible local isometries, to provide a metric for certification where the fidelity between states cannot serve as a standard metric in this setting.

3. AI systems can now perform device-independent certification of quantum resources by using a high Bell violation implies that all N − 1 measured copies are close to the target state to infer the quality of an unmeasured copy.

4. The system can quantify the required sample size by solving for N in equation (3), which links confidence level, desired extractability, and the Kullback-Leibler divergence between winning probabilities: N ≥ ln δ / (ln 1 − µ + µeD(pQM−ϵ1pQM−ϵ2)).

5. The system can select the optimal Bell operator for certification by comparing bounds from Table I, choosing the Mermin operator because it outperforms the others and provides a significant advantage in terms of sample efficiency, by almost two orders of magnitude.