Experimental quantum key distribution certified by Bell's theorem

Loopholes

• DIQKD requires each node to keep information local; space-like separation alone is insufficient, and the detection loophole still requires treatment as in conventional Bell tests.The locality condition follows from isolation, so no additional space-like-separation condition is needed if local information remains local.Source: METHODS · pp. 5–6

Link disconnection and isolation

• After heralding, ions are shuttled 3 µm from the imaging focus, reducing coupling to the outside by > 10^4 before basis choices and readout.After readout, deshelving and cooling lasers scramble the remaining qubit state for 100 µs (> 10 time constants) before the link is reconnected.Source: METHODS · p. 6
• A fibre switch could disconnect the link; independent laser sources are also possible if qubit frequencies match and heralding photons align relative to their 7 ns duration.The shared laser sources were used for both nodes; the passage describes these alternatives as possible options.Source: METHODS · p. 6

Data handling

• Each node uses independent ARTIQ control; random choices are stored before the experiment, heralds are never post-selected, and private outcome strings stay on local PCs.The classical network carries protocol messages, such as error-correction syndrome data, and coordinates the experiment schedule.Source: METHODS · p. 6

Contributions

• DPN, PD, BCN, GA, DM, and RS built and operated the experimental apparatus.Source: Contributions · p. 9
• DPN and PD led experimental data collection and performed the data analysis.Source: Contributions · p. 9
• JDB and DPN extracted the key from the raw data.Source: Contributions · p. 9
• KI, RLU, and JDB designed the error correction code.Source: Contributions · p. 9
• JDB, EYZT, NS, PS, and RR established the protocol steps and derived the corresponding security proof.Source: Contributions · p. 9
• NS, JDB, DPN, and CJB wrote the manuscript; CJB and DML supervised experimental work, while JDB and NS supervised theoretical work.Source: Contributions · p. 9
• NS and JDB initiated the project, and all authors contributed to discussion, interpretation of results, and the manuscript.Source: Contributions · p. 9

• CJB is a director of Oxford Ionics, while the remaining authors declare no competing interests.Source: Competing interests · p. 9

• Supplementary Information is available for this paper.Source: ADDITIONAL INFORMATION · p. 9

• Table S1 lists commonly used acronyms, symbols, and expressions.Source: Table S1 · p. 11

Experimental details

• Technical improvements since the setup first described in ref. have produced higher-fidelity, more stable remote entanglement and improved autonomous calibration and coordination between nodes.The improvements include beam-power and magnetic-field stabilisation.Source: S2. EXPERIMENTAL DETAILS · p. 11
• The apparatus connects Alice’s and Bob’s trapped-ion nodes through a central heralding station, with the connection points indicated in the setup sketch.The nodes are nominally identical and operate at room temperature.Source: S2. EXPERIMENTAL DETAILS · p. 11, Figure S1 · p. 12
• Separate acousto-optical modulators switch shared laser sources for each node, while FPGA-programmed radio-frequency sources implement measurement-basis choices and DACs can disconnect the photonic link by changing ion position.The sketch illustrates the radio-frequency control for the 674 nm qubit-manipulation laser; DACs also set trap-electrode voltages.Source: Figure S1 · p. 12

A. The two-node ion trap network

• Each node confines one 88Sr+ ion; photons entangled with the ion state are sent through fibres to a central station, where opposite-polarisation coincidences herald remote entanglement.A beam splitter and polarising beam splitters implement a partial Bell-basis measurement; attempts are repeated at 1 MHz.Source: A. The two-node ion trap network · p. 12
• Alice’s and Bob’s independent control systems use Ethernet for coordination and interruptions, as well as exchanging messages for DIQKD post-processing.The PCs and FPGA-based real-time control systems form the trusted classical processing systems available to the parties.Source: A. The two-node ion trap network · p. 12
• After cooling and synchronisation, the nodes repeat entanglement attempts until a herald or timeout; after success, they move the ions from the lens focus before local analysis.Bob selects a key-generation or test round, Alice follows the key-round signal, and both store their measurement outcomes locally.Source: A. The two-node ion trap network · p. 13
• The 180,000-herald dataset yielded 96.0(1)% average entangled-state fidelity, a 1.6 × 10−4 success probability, and a 100 s−1 heralding rate.The four detector-click patterns had fidelities of 96.1(3)%, 96.4(1)%, 95.8(2)%, and 95.8(2)%.Source: Figure S3 · p. 14, A. The two-node ion trap network · p. 14
• The nodes require no optical phase coherence; the entanglement photons instead must be matched in frequency and arrival time relative to their 7 ns duration.The relative phase between local projective measurements is determined by the 14 MHz RF beat note between the 674 nm pulses.Source: A. The two-node ion trap network · p. 14
• Protocol pauses for hardware issues do not affect final security because the number of rounds is fixed in advance and their timing is free.A laser-lock loss or other detected hardware issue can trigger a pause until operation resumes.Source: A. The two-node ion trap network · p. 14

B. Entanglement characterisation

• Remote entanglement in the ground-state manifold was mapped to an optical qubit with a 674 nm π pulse, followed by basis-selecting coherent manipulation or an additional pulse for computational-basis measurements.The additional pulse reduced shelving errors.Source: B. Entanglement characterisation · p. 14
• Tomography measured the joint ion–ion state after each of four heralding detector patterns using an over-complete set of Pauli eigenstate bases to mitigate xy-plane bias from imperfect analysis pulses.A dataset collected one day before link characterisation and key generation supplied the complete input for a diluted fixed-point iteration procedure to obtain the maximum-likelihood estimate.Source: B. Entanglement characterisation · pp. 14, 16
• 96.0(2)% was the average fidelity to the closest maximally entangled state, with a heralding probability of 1.6 × 10−4 and an effective rate of 100 s−1 including periodic recooling.The main-text rate was lower because it included frequent reloading of Alice’s ion trap due to a vacuum leak.Source: B. Entanglement characterisation · p. 16
• The fidelity estimates were roughly consistent with ion–photon tomography; suspected polarization mixing remained uncharacterised, while local manipulation and readout errors were not significant at the achieved fidelity.Randomized benchmarking indicated gate fidelities of ∼5×10−4 per Clifford group element, and readout state discrimination error was < 10−3.Source: B. Entanglement characterisation · p. 16
• Magnetic-field feedback stabilized noise to the nT level, and ion coherence times were not expected to limit performance before 422 nm fibre loss became significant, even with longer delays between nodes.The cited coherence times were measured without dynamical decoupling.Source: B. Entanglement characterisation · p. 16

C. DIQKD primitives

• Alice and Bob use calibrated CHSH settings, computational-basis measurements for key generation, and Alice inverts her outcomes to match the maximally correlated Bell-state convention.Alice measures in the computational basis for X = 0 and applies Rxy(π/2, 0) for X = 1; Bob applies Rxy(±π/4, ϕ) for Y = 0 and 1, with ϕ calibrated per heralding click pattern.Source: C. DIQKD primitives · p. 16
• 1,920,000 characterization rounds yielded empirical probabilities P(AiBi|XiYi), which are tabulated in Table S4.Figure S4 displays the individual round outcomes; the data are analyzed after recombining inputs X, Y and outputs A, B.Source: C. DIQKD primitives · p. 16, Table S4 · p. 16, Figure S4 · p. 17
• Numerical optimization predicted statistically significant QBER and CHSH improvements from slightly tilted measurement axes, but these gains were not observed or conclusively traced to tomography bias.The prediction used maximum-likelihood estimates for the joint ion-ion state.Source: C. DIQKD primitives · p. 18
• Setting strings and the initial secret key came from a commercial QRNG; randomness tests found no obvious defects, and γ = 13/28 was implemented using eight source bits per round-type choice.The inspected output covered about 3 × 10^10 bits and was assessed with dieharder and TestU01 batteries; the power-of-two fraction and bit consumption were chosen to avoid unintended bias between experiment and security analysis.Source: C. DIQKD primitives · p. 18

D. Shuttling-based link isolation

• The experiment isolated ions by shuttling them 3 µm from the fibre-coupling focus using trap-potential modulation, with no resolved-sideband-detected heating within uncertainty ∆n̄ ≈ 0.1.The nearest-neighbour electrodes were held fixed to maintain the trap’s motional-mode structure.Source: D. Shuttling-based link isolation · p. 18
• After heralding, the ions were mapped to optical qubits and the link disconnected before round-type communication; state scrambling preceded reconnection after 100 µs of deshelving and cooling.The sequence includes a 30 µs disconnection delay and an additional 25 µs processing delay before Alice’s local operations.Source: D. Shuttling-based link isolation · p. 18, Figure S5 · p. 19
• The total leakage probability was 2.5 × 10−3, bounding leaked information by h(2.5 × 10−3) = 0.025, below the DIQKD key rate of 0.064; estimated photon leakage was 6 × 10−4 from Alice and 4.4 × 10−3 from Bob.The bound accounts for the marginal probability of a bright measurement on either side; Table S5 reports the detector-count scope used for the leakage estimate.Source: D. Shuttling-based link isolation · p. 18, Table S5 · p. 19

A. General setting of QKD

• QKD security assumes quantum theory, isolation, random inputs, and trusted classical and quantum operations; this experiment explicitly authenticated classical communication rather than assuming it.The passage notes that authentication otherwise consumes an initial secret.Source: A. General setting of QKD · p. 20

B. DIQKD assumptions

• DIQKD relaxes the trusted-quantum-operations assumption while retaining the isolation assumption, which must be independently established and can still be broken by information leakage from a lab.Its security holds on protocol success even if the quantum operations were not performed as expected.Source: B. DIQKD assumptions · p. 20

C. From assumptions to security

• Random inputs grant the Markov condition, while quantum theory, isolation, and input independence yield factorized systems and Born-rule probabilities used to bound entropy from CHSH winning probability.The security proof’s two required conditions follow from the DIQKD assumptions.Source: C. From assumptions to security · p. 21
• The proof remains valid for any state and measurements satisfying the stated quantum probability description, even if an adversary tampers with the state or the devices do not perform their intended operations.This part of the devices is treated as quantum but otherwise untrusted “black boxes” receiving inputs and producing outcomes.Source: C. From assumptions to security · p. 21

D. The DIQKD assumptions in the experiment

• The labs are spatially separated and physically identifiable, with a tensor-product Hilbert space; local computers store protocol information and each lab contains a trapped ion and its control equipment.The labs connect to a central station through two 1.75 m optical fibres.Source: D. The DIQKD assumptions in the… · p. 21
• The experiment uses ion displacement to close each lab’s optical hatch while preserving entanglement, then scrambles the readout state before reconnecting the link as an isolation method.The paper also treats shared laser modes as power supplies compatible with device independence under the setting-independence assumption, and takes assumptions (i), (iii), and (iv) for granted.Source: D. The DIQKD assumptions in the… · pp. 21–22

E. Possible improvements

• Suggested isolation improvements include independent lasers, optical isolators or RF control, and greater fibre disconnection; the current fibre-hatch extinction ratio is ∼10−4.The operation of additional fibre isolation devices could be verified independently of the ion-trap hardware.Source: E. Possible improvements · p. 22
• Space-like separation can rule out some attacks but is difficult and does not address fibre leakage; with proper isolation it is unnecessary, while quantum memories storing ∼10^6 qubits could remove the need for quantum hatches.Even that future demonstration would still rely on the same four assumptions.Source: E. Possible improvements · p. 22

S4. DIQKD PROTOCOL

• Alice and Bob begin with a shared secret K0 and perform n rounds; Bob samples test settings with probability γ, and Alice sets Xi=0 when a round is not a test round.The protocol aims to create a longer shared secret K1 through measurements and post-processing.Source: S4. DIQKD PROTOCOL · pp. 22–23
• After heralded state preparation, Alice and Bob measure according to their settings; Alice sends a syndrome so Bob can reconstruct her outcomes, and Bob computes a parameter-estimation score.The quantum steps repeat for the n rounds.Source: S4. DIQKD PROTOCOL · pp. 23–24
• The protocol validates error correction and Bell violation, applies privacy amplification and authentication, activates the key, and concatenates it with the unconsumed initial key to extend the secret if |K1|>|K0|.A failed hash check, Bell condition, or required authentication check can cause an abort.Source: S4. DIQKD PROTOCOL · p. 24

S5. MODEL OF THE STATISTICS

• The honest-behavior model condenses observed statistics into CHSH violation S and QBER Q, conservatively sets S=2.64 and Q=0.018, and is not required for security.The model does not assume that the underlying state is a Werner state.Source: S5. MODEL OF THE STATISTICS · pp. 24–25

S6. ERROR CORRECTION

• DIQKD error correction must let Bob reconstruct Alice’s outcomes while keeping the public message short, since every communicated bit reduces key secrecy.The code should also tolerate noise specificities, operate efficiently, and approach finite-size Shannon limits.Source: S6. ERROR CORRECTION · p. 25

A. Minimum syndrome length for a binary symmetric channel

• For a binary symmetric channel, the minimum syndrome length depends on sample size and the permitted error-correction failure probability.The section derives a finite-size channel capacity and gives ε = 10^-3 as an example, for which Q^-1(ε) ≃ 3.Source: A. Minimum syndrome length for a… · pp. 25–26

B. Concrete setting

• The experimental setting combines four BSC sources with error δ′ and one with error δ′′ = Q, so the authors encode all channels together in one block.Because the true statistics can deviate from the model, they rely on code universality; the larger block reduces the overhead required for error-free performance.Source: B. Concrete setting · p. 26

C. Practical coding approaches

• The authors consider LDPC, SC-LDPC, and polar codes; polar codes require channel-specific optimization and have worse finite-length scaling, while SC-LDPC codes offer universal capacity achievement asymptotically.SC-LDPC BP thresholds converge to the underlying LDPC maximum-likelihood thresholds as code length grows.Source: C. Practical coding approaches · pp. 26–27

1. LDPC codes

• Protograph-based LDPC codes are built by lifting a small bipartite graph into a longer code, enabling code properties such as BP thresholds to be derived from the protograph.The lifting replaces each protograph edge with a random permutation matrix and each non-edge with a zero matrix.Source: §1 · p. 27

2. SC-LDPC codes

• SC-LDPC codes chain L copies of a protograph and add check nodes according to coupling width w, while their asymptotic BP thresholds converge to the underlying LDPC maximum-likelihood thresholds.They achieve capacity universally as code length grows.Source: C. Practical coding approaches · p. 27, §2 · p. 28

3. Belief propagation decoding

• Belief propagation uses the sparse parity-check matrix for decoding with running time O(Nit · n), and the authors use standard BP because enhanced decoders lack non-erasure-channel analysis.The decoder incorporates each bit’s error probability into its reliability information and iteratively exchanges messages between Tanner-graph nodes.Source: §3 · p. 29

D. Simulations

• At n = 5e6, string recovery is reliable above overhead η* = 0.196, 6% above the asymptotic value ˜η∞ ≈ 0.1847 and 3.7% above ˜η(5e6) ≈ 0.189.Fig. S13 shows that the threshold for error-free correction approaches the asymptotic bound as block length increases; exact bit reliabilities enable error-free transmission at overhead h(δ̂) ≈ 0.204 for n ≥ 1e6.Source: Figure S13 · p. 30, D. Simulations · p. 31
• Construction couples and lifts a protograph with L = 80, then removes low-degree variable nodes and merges pairs of high-degree checks to reach the target length and overhead.Before decoding, the string is shuffled to distribute noisier bits more evenly, and the decoder knows the shuffle pattern to assign reliabilities.Source: D. Simulations · p. 30
• For p* = 0.9, threshold curves across three regimes show similar convergence, supporting the construction’s universality; the Eq. (19) and Eq. (23) curves provide lower and upper bounds.The regimes use γ = 0.08 with Q ∈ {0.015, 0.02}, or γ = 13/256 with S = 2.6507 and Q = 0.0239.Source: D. Simulations · p. 31
• Using bit reliabilities lowers the threshold at every finite block size compared with global symmetric error, despite the latter converging faster to its asymptotic value.For n ≥ 1e6, exact bit reliabilities allow error-free transmission at overhead h(δ̂) ≈ 0.204.Source: D. Simulations · p. 31
• The construction requires smaller overhead than the AFRV bound and comparable overhead to the TSBSRSL bound.These schemes were considered for DIQKD but lack a known efficient decoding algorithm, making them unsuitable for practical use.Source: D. Simulations · p. 31
• The simulated critical threshold is systematically shifted from the Eq. (19) bound because scaling depends more on lifting factor M than on total block length n.Source: D. Simulations · p. 31
• Joint decoding needs only one code and yields a smaller gap than independently decoding the more- and less-noisy bits, assuming similar gap-function behavior for a single BSC.The assumed behavior is confirmed by experiments.Source: D. Simulations · p. 31

E. Practical implementation of the coding scheme

• p*≥99.9% success was observed in both simulated error-correction cases, with 5 failures in 15,000 samples and 3 failures in 15,000 samples, respectively.The syndrome length is chosen with a decoding margin; the reported-parameter case used m = 296,517 for n = 1.5 × 10^6. The demonstration ran a BP decoder on Bob’s node, converging in 250 s on an Intel Core i7-9700 CPU.Source: E. Practical implementation of the… · pp. 31–33

S7. PROTOCOL FEASIBILITY AND COMPLETENESS

• εcom ≤0.01 is achieved by choosing parameters and enough rounds so expected behavior passes error-correction and Bell-test checks with high probability.The syndrome-length choice keeps the error-correction contribution below 0.1%; authentication tags match for honest, reliably transmitted messages. A three-standard-deviation allowance is used for CHSH test-round losses, contributing an error probability well below 1%.Source: S7. PROTOCOL FEASIBILITY AND… · p. 33

A. Trace distance

• Trace distance compares the protocol’s final state with an ideal uniformly random key state in which Eve has no information.The definitions cover sub-normalized states as well as normalized states.Source: A. Trace distance · pp. 33–34

B. Smooth min and max entropies

• Smooth conditional min- and max-entropies are the maximum min-entropy and minimum max-entropy, respectively, within a state’s neighborhood.The neighborhood is defined using purified distance.Source: B. Smooth min and max entropies · pp. 34–35

C. Security definitions

• The protocol is εsnd-sound when it is both εcorr-correct and εsec-secret for all device implementations.Correctness bounds the probability that accepted keys differ; correctness uses almost-universal hashing, while secrecy uses Trevisan privacy amplification.Source: C. Security definitions · p. 35

A. Strongly universal hashing

• VHASH provides almost-Δ-universal hashing, while its one-time-pad-protected tags form an almost-strongly-universal family used for error-correction checks and classical-message authentication.The implementation uses 64-bit VHASH outputs with collision probability εh = 2−61; UVMAC implements the resulting G family.Source: A. Strongly universal hashing · p. 36

B. Privacy amplification

• Trevisan’s quantum-proof strong extractor maps a weak source to an approximately uniform output that is independent of its short uniform seed.It concatenates one-bit extractors using a weak design to limit seed overlap; this implementation combines a block weak design with polynomial hashing based on Reed-Solomon and Hadamard codes.Source: B. Privacy amplification · p. 37
• Secrecy can be guaranteed by quantifying the conditional smooth min-entropy H^εs_min(C|Q) remaining at the end of the protocol.Bounding this entropy is the main focus of the security analysis.Source: B. Privacy amplification · p. 38
• The Trevisan extractor with block weak design and polynomial hashing is a quantum-proof strong randomness extractor.It acts on the classical register C, leaving Q and the uniformly random seed register S unchanged.Source: B. Privacy amplification · p. 38
• The extractor extracts the string’s whole entropy up to a logarithmic correction, and its seed requirement grows only polylogarithmically with n.For key-rate evaluation, the function Υ_b(x) can be replaced by x − 4 log(x).Source: B. Privacy amplification · p. 38

S10. SECURITY STATEMENT

• The protocol’s key length is sound with error at most max(εEA, εPA + 2εs) + 4εh, and the remaining theorem parameters are optimized to maximize it.The analysis fixes εh = 2^-61 because of the use of VHASH; m and ωthr are set using Equations (23)–(25), with n and γ fixed.Source: S10. SECURITY STATEMENT · p. 40
• A positive key rate in the depolarizing model begins at approximately 2.6 × 10^5 rounds.Figure S15 fixes soundness at εsnd = 10^-10 and estimates completeness error εcom ≤ 0.01; its depolarizing-model rate is shown as a function of visibility.Source: Figure S15 · p. 41, S10. SECURITY STATEMENT · p. 41

S11. SECURITY ANALYSIS – PROOF OF PROPOSITION 2

• The security proof analyzes a modified protocol whose added variables and rearranged steps preserve the original protocol’s statistics, following earlier proofs including.The authors state that, to their knowledge, this is the first DIQKD proof directly applicable to a Trevisan extractor.Source: S11. SECURITY ANALYSIS – PROOF OF… · p. 41

A. Modified protocol

• In the modified protocol, quantum states are distributed at the start, while settings and basis revelation occur round by round; the state after parameter estimation matches the original protocol’s relevant state.The modified protocol uses B′ rather than B for parameter estimation, without changing its statistics, and validation and privacy amplification depend only on registers available at that stage.Source: A. Modified protocol · p. 42

B. Events

• The proof defines non-abort as the conjunction of authentication, test, and parameter-estimation events, and tracks communication receipt separately.The virtual-parameter test passes when the relevant count is at most nγ(1 − ωthr).Source: B. Events · p. 43

C. Correctness

• Correctness holds with εcorr = εh by bounding disagreement between Alice’s and Bob’s reconstructed outcomes when the protocol does not abort.Matching error-correction hashes are necessary for protocol success.Source: C. Correctness · p. 43

D. Secrecy

• Secrecy is bounded by handling three probability cases and, in the remaining case, bounding smooth min-entropy with entropy accumulation after accounting for m + 258 bits of public communication.The stated secrecy criterion compares the extracted key and Eve’s side information with a uniform key independent of Eve; Eve’s information includes X, T, the public communication, and her quantum side information.Source: D. Secrecy · pp. 45–46
• The protocol’s CHSH-test steps are modelled as infrequent-sampling bipartite EAT channels, yielding a conditional smooth min-entropy bound from a min-tradeoff function built using gt.The analysis identifies these steps with a standard CHSH test protocol [72, 73].Source: D. Secrecy · pp. 46–47
• Because the EAT conditioning event is not determined by B′XYE alone, the analysis instead applies intermediate results from in a different order.A constant max-tradeoff function permits an unconditioned bound before conditioning on ΩPE′ using Lemma 2, followed by Lemma 3.Source: D. Secrecy · pp. 47–48

3. Bringing the pieces together

• P(ΩPE′) > εEA, together with the derived bounds and Equations (88), (91)–(94), establishes Proposition 2 for case 3.This concludes the proof for that case.Source: §3 · p. 48