Calibration-Independent Certification of a Quantum Frequency Converter

We report on a method to certify a unitary operation with the help of source and measurement apparatuses whose calibration throughout the certification process needs not be trusted. As in the device-independent paradigm our certification method relies on a Bell test, but it removes the need for high detection efficiencies by including the single additional assumption that non-detected events are independent of the measurement settings. The relevance of the proposed method is demonstrated experimentally with the certification of a quantum frequency converter. The experiment starts with the heralded creation of a maximally entangled two-qubit state between a single $^{40}$Ca$^+$ ion and a 854$\,$nm photon. Entanglement preserving frequency conversion to the telecom band is then realized with a non-linear waveguide embedded in a Sagnac interferometer. The resulting ion-telecom photon entangled state is characterized by means of a Bell-CHSH test from which the quality of the frequency conversion is quantified. We demonstrate the successful frequency conversion with an average certified fidelity of $\geq 84\,\%$ and an efficiency $\geq 3.1\times 10^{-6}$ at a confidence level of $99\,\%$. This ensures the suitability of the converter for integration in quantum networks from a trustful characterization procedure.


Introduction-
The enabling technologies for the realization of networks capable of linking quantum systems together have been identified [1][2][3].This includes quantum frequency converters -nonlinear processes in which a photon of one frequency is converted to another frequency whilst preserving all other quantum properties.A converter acts as a quantum photonic adapter allowing one for example to interface high-energy photonic transitions of quantum matters with lower-energy photons better suited for long-distance travel.Together with quantum storage and processing devices, quantum frequency converters enable a range of new technologies using quantum networks, from distributed quantum computing [4], quantum-safe cryptography [5], enhanced sensing [6,7] and time keeping [8].
A natural question arising in view of this integration potential is how to certify the functioning of a quantum frequency converter independently of contingent details, i.e. without the need to know an exhaustive physical model of its inner functioning or to assume that the certification equipment (source and measurements) is well calibrated and remains perfectly calibrated for the whole duration of the certification procedure.Recent works have demonstrated that the quantum nature of a number of channels can be witnessed with assumptions on the source calibration but without any trust on the measurement apparatus [9][10][11][12].However, it would be desirable to quantify the quality of the device.Indeed, an ideal certification method should ensure the usability of the converter for all future purposes.A radical solution to this task is offered by the method of device-independent (DI) characterization, also known as self-testing [13], where the physical implementation of a device is inferred from the correlations observed in a Bell-type experiment [14].The device-independent approach relies on the separation and independence between the apparatuses at hand, but makes no assumption on their internal modeling.As far as we know, only two self-tests have been fully implemented experimentally to date, both related to state certification [15,16].The main reason for this scarcity is that device-independent certification is very demanding regarding the efficiency of measurement apparatuses [17].This requirement has been circumvented in a number of experimental state certifications based on post-selected Bell inequality violation -by considering only the statistics observed from detected events [18][19][20].The question of what remains device-independent in certifications using post-selections has not been discussed in these experimental realizations.
In this article, we provide an accessible method to certify trustworthily unitary operations.Inspired by the device-independent certification techniques presented in Refs.[21,22], our method certifies the quality of a unitary operation from a single Bell test but without requiring high detection efficiencies to be implemented.Precisely, we assume that the physical process responsible for the occurrence of no-detection events is independent of the choice of the measurement setting, but impose no further restriction on it [23].Therefore, no-detection events may still depend on the state being measured or on devices' calibrations in an arbitrary way.This natural assumption allows us to substantially reduce the complexity of unitary certification by removing the need for high overall detection efficiencies without requiring to trust the calibration of the certification devices.We use this tool to realize the first calibration-independent certification of a unitary -a state-of-the-art polarization-preserving quantum frequency converter (QFC) [24][25][26][27].We employ a trapped-ion platform as source of light-matter entanglement between an atomic Zeeman qubit and the polarization state of a spontaneously emitted photon [28].A frequency conversion based on a highly-efficient difference frequency generation process in a nonlinear waveguide embedded in a polarization Sagnac interferometer connects the system wavelength at 854 nm to the telecom C-band at 1550 nm [25].A Bell-CHSH test [29] is finally performed after the frequency conversion, using the ion-telecom photon entangled state.We demonstrate the successful frequency conversion with an average certified fidelity of ≥ 84% and a probability to get a telecom photon detection conditioned on a successful ion state readout of 3.1 × 10 −6 at a confidence level of 99%.
Source, QFC and measurement apparatus modelling-We start by providing an "a priori" quantum model of several devices involved in the setup.The desired models rely on minimal assumptions on the internal functioning of the devices, which nevertheless have enough physical insight to describe the process of frequency conversion.
A QFC can be represented by a channel between two physical systems identified as its input and output.In our case, these are the photonic modes entering the QFC device from the source and exiting it towards the detectors.These modes are filtered to ensure that their frequencies lie in the desired bandwidth ω i and ω f respectively.We can associate to these photonic modes two Hilbert spaces A and H (f ) A , that encompass all the degrees of freedom necessary to describe the emission of the source in addition to the frequencies.To describe the quality of the QFC to be characterized, we are only interested in how the device maps the photonic states received from the source to the state it sends to the detector: Here B(H A ) stands for the set of bounded operators on the Hilbert space H (i) A (similarly for B(H (f ) A )).All auxiliary outputs can be safely ignored, while auxiliary input systems are to be seen as part of the device1 .The completely positive and trace-preserving (CPTP) map QFC in Eq. ( 1) is unknown and our goal is to provide a recipe to characterize it.
In addition to the QFC itself, our setup involves an entanglement source preparing a state shared between two parties called Alice (A) and Bob (B).Alice's system is carried by the electromagnetic field used to characterize the QFC, which is associated to Hilbert space H (i) A introduced above.The physics of Bob's system is irrelevant for the purpose of the QFC characterization because the QFC resides entirely on Alice's side.Its state spans a Hilbert space H B .We denote the state produced by the source by The state obtained after applying the converter on Alice's side reads Finally, the form of the quantum model for the measurement apparatus is needed in order to describe the occurrences of the measurement results.We introduce two possible measurements M (f ) A and M B which act on the system of Alice after the converter and the system of Bob, respectively.Ideally, the measurements should have binary inputs x, y = 0, 1 and binary outputs a, b = 0, 1.In practice however, a third outcome a, b = ∅ is possible corresponding to a no-click event.Each of the measurements is given by two POVMs with three elements each, such as with the operators a|x and M b|y acting on H A and H B respectively.
Weak fair-sampling assumptions-Following the results presented in Ref. [23], we now introduce an assumption on the measurement structure which allows us to relax the requirement on the detection efficiency inherent to device-independent certification.Consider for example the measurement M B specified by the POVM elements M b|y with settings y and outcomes b including a no-click outcome b = ∅.The measurement M B satisfies the weak fair-sampling assumption if i.e. the occurrence of the no-click outcome is not influenced by the choice of the measurement setting 2 .Under this assumption, M B can be decomposed as a filter R B acting on the quantum input (a quantum instrument composed of a completely positive (CP) map and a failure branch that outputs y = ∅) followed by a measurement M B with unit efficiency (without b = ∅ output) [23], that is Assuming that Bob's measurement fulfills the weak fair-sampling assumption, we can focus on the data postselected on Bob's successful detection only.This data can be associated to an experiment where a probabilistic source prepares a state conditional on the successful outcome of Bob's filter R B .
We can therefore only consider the experimental runs where the state (i) is prepared and Bob's detector clicks.
Assuming that Alice's measurement also fulfills the weak fair-sampling assumption, that is we perform a similar decomposition for the final state corresponding to post-selected events for both sides.The success rate P succ (R ] of the filtering R A is given by the probability to observe a click event on Alice's detector, conditional on the click event seen by Bob (defining (i) ) Goal-To set our goal, we first specify what a frequency converter is expected to do.While changing the carrier frequency from ω i to ω f , an ideal QFC should not affect any other degree of freedom carrying meaningful information.Therefore, in the case of photons encoding a qubit degree of freedom within their polarization, an ideal QFC should act as the identity on the polarization of a single photon.We thus need to show that, while changing the frequency of the photons, the map (1) is capable of preserving a two-dimensional subspace.
Following Ref. [22], this can be formalized by requiring the existence of two maps V : where the approximate sign refers to a bound on the Choi fidelity between the two maps , where Φ + is a maximally entangled two-qubit state and is the fidelity between two states ρ and σ.Note that in the case where E is the identity map, Choi fidelity takes a particularly simple form We are concerned with non-deterministic frequency converters.More precisely, our goal is thus to compare the actual frequency converter to a probabilistic but heralded quantum frequency converter -a device which behaves as an ideal QFC with a certain probability, and otherwise reports a failure.To do so, we can allow the maps Λ and V to be non trace preserving.The quality of an announced QFC is captured by two parametersthe probability that it works and the error it introduces in this case.These are quantified by the following figures of merit.The success probability (13) captures the efficiency of the converter.The conditional Choi fidelity bounds the error introduced in the state conditional to a successful frequency conversion.Certifying the converter thus consists in establishing lower-bounds on both quantities P succ and F.
Certification-Following Ref. [22], we certify the QFC through the self-testing of the maximally entangled two qubit state Φ + derived in Ref. [30].The latter is based on the Clauser Horne Shimony Holt (CHSH) inequality -a well-known Bell test derived for a setting where two parties Alice and Bob can choose one of two binary measurements at each round.The CHSH score S is given by where a, b = 0, 1 are the parties' measurement outcomes, and x, y = 0, 1 label their measurement setting.In the quantum framework, the correlation P (a, b|x, y) is given by P (a, b|x, y) = tr ρ M A a|x ⊗M B b|y where ρ is the measured state and {M A a|x } ({M B b|y }) are Alice (Bob)'s appropriate POVM elements.We know from Ref. [30] that for any quantum model (ρ, M A a|x , M B b|y ) exhibiting a CHSH score S, there exist local extraction maps Λ A and Λ B such that Notably, the form of the maps Λ A(B) does not depend on the measurement performed by the other party.
This result holds for all quantum states and measurement.When applying it to the quantum model of the filtered state (f ) after the QFC and the binary measurements M (f ) A and M B for instance, it implies that there exist local maps Λ (f ) A and Λ B such that where S is the CHSH score of the binary measurements on the filtered state, i.e. the post-selected CHSH score.
To derive a certificate on the QFC itself rather than of its output state, we need to show that the state before the action of the QFC can be prepared from Φ + with the injection map V A acting on Alice, i.e. (id We show in the Methods that this can be done perfectly, i.e.
with a probabilistic map V A associated to the success rate ] is carried by a qubit at Bob's side.It can therefore be purified to a state of Schmidt rank 2 and any such state can be efficiently obtained from Φ + by a local filter applied by Alice.
Combining the definition of the filtered state (f ) in Eq. ( 9) with Eqs.(17) and (18), we conclude that for the probabilistic extraction map Λ A = Λ (f ) A • R A , the conditional Choi fidelity of Eq. ( 14) is bounded by We emphasize that this bound is valid for all possible underlying state ρ and measurements {M A a|x }, {M B b|y } subject to Eq. ( 5).

It remains to bound the success probability of the map
This map is successful if both the injection map V A and the filter R A are, hence P succ (R A ) can be estimated experimentally using Eq.(10).
Experimental source of entanglement-The experimental setup is sketched in Fig. 1.Our source of entanglement is a trapped-ion quantum network node which creates light-matter entanglement between a Zeeman qubit in a single trapped 40 Ca + ion (Bob) and the polarization state of an emitted single photon at 854 nm (Alice) [31].The photons are coupled to a single-mode fiber via a high-aperture laser objective (HALO) and guided to the frequency converter, which is the device we aim to certify.
The entanglement generation sequence is slightly modifed compared to [31].The relevant level scheme for the state preparation and detection of the Ca ion is shown in Fig 1 .After Doppler cooling, excitation of the ion on the S 1/2 to P 3/2 transition by a π-polarized, 2 µs long laser pulse at 393 nm creates a spontaneously emitted photon at 854 nm.This photon is collected along the quantization axis, thereby suppressing π-polarized photons, and is entangled with the ion in the state with The oscillation with frequency ω L arises from the frequency difference between the |↑ and |↓ states and the asymmetry in the state results from the different Clebsch-Gordan coefficients (CGC) of the transitions between the P 3/2 and D 5/2 Zeeman sublevels.We compensate for this by means of a partial readout of the trappedion Zeeman qubit during the state preparation: a π/2pulse at 729 nm transfers 50 % of the population from |↓ = D 5/2 , m = − 3 /2 to the S 1/2 ground state.A subsequent fluorescence detection with the cooling lasers is a projective measurement of this population in the following way.The fluorescence detection discriminates between population in the S 1/2 -state which results in scattering of photons from the cooling laser, while population in D 5/2 leaves the ion dark.If it yields a bright result, the measurement is discarded, while a dark result leaves the D-state intact and heralds a successful state preparation.Thus, the ion-photon state after a dark result is maximally entangled In this way, maximally-entangled ion-photon pairs are generated at a rate of 720 s −1 and a probability per shot of 0.36 %.
Experimental QFC device-The QFC device transduces the photons at 854 nm to the telecom C-band at 1550 nm via the difference frequency generation (DFG) process 1/854 nm -1/1904 nm = 1/1550 nm in a periodically-poled lithium niobate (PPLN) waveguide [25].The input photons are overlapped with the classical pump field at 1904 nm on a dichroic mirror and guided to the core of the QFC device, an intrinsically phase-stable polarization Sagnac interferometer.The latter ensures polarization-preserving operation since the DFG process is inherently polarization-selective.The interferometer is constructed in a similar way as in [32], i.e. a polarizing beam-splitter (PBS) spatially separates the orthogonal components and a HWP rotates the not convertible horizontal component of input, pump and output fields by 90 • .Both components are subsequently coupled to the same waveguide from opposite directions.The converted photons take the same interferometer paths, are recombined in the PBS, separated from the pump and input photons via another dichroic mirror and coupled to a single-mode fiber.Multi-stage spectral filtering down to 250 MHz suppresses pump-induced background photons stemming from anti-Stokes Raman scattering in the waveguide.The external device efficiency is measured to 57.2 %, independent of the polarization and including all losses between input and output fiber.The QFC-induced background is measured at the operating point to be 24(3) photons/s, being to our knowledge the lowest observed background of a QFC device in this high-efficiency region.
Measurements-To measure the Bell parameter S, we perform joint measurements of the atomic and photonic qubit in the four CHSH basis settings which we choose to lie in the equatorial plane of the Bloch sphere with respect to the basis defined in Eq. (22).
For the atomic qubit, the required basis rotation is implemented by means of a pulsed sequence of two consecutive π-pulses at 729 nm and a radio-frequency (RF) π/2pulse applied on the S 1/2 ground-state qubit with phase φ RF using a resonant magnetic field coil (Fig. 1).The ground-state qubit states are readout by means of two fluorescence detection rounds yielding bright and dark events depending on whether the state is populated or not, respectively.The phase of the atomic qubit underlies the Larmor precession in D 5/2 with ω L .The arrival time t of the photon reveals this Larmor phase up to a constant offset which is calibrated with an independent measurement and kept fixed for all following ones (see next section).
For the photonic qubit we employ a set of a motorized quarter-and a half-wave plate for arbitrary basis rotations and a Wollaston prism to split orthogonally polarized photons.Both outputs are connected to fibercoupled superconducting-nanowire singe-photon detectors (SNSPDs).To fulfill the weak fair sampling assumption we have to balance the efficiencies of both detectors since the error of the post-selected probabilities scales linearly with the imbalance.To this end we use attenuated laser light and adjust the bias current through the SNSPDs to achieve γ = 1 − η snspd1 /η snspd2 ≤ 0.2 % with η snspd2 = 13.5 %.This reduces the deviation of the postselected probabilities from those obtained with a lossless detector to about 1 % [23].
To avoid influences of drifts over the measurements, we consecutively acquire runs of data for 5 seconds in each basis and cascade up to 660 runs.The CHSH score is then obtained using the setting choices in the photonic side and for the ion side.
Experimental results-Figure 2(a) shows a typical time-resolved coincidence histogram between photonic detection events of one of the detectors (readout base x = 0) and bright events of the atomic state readout (base y = 0).As mentioned previously, the oscillations stem from the Larmor precession of the atomic qubit resulting in a time-dependent entangled state, Eq. 22.
From the histograms of all readout bases we calculate the CHSH Bell parameter according to Eq. ( 15), which is consequently also detection-time dependent (see figure 2(b)).Thus, by postselecting coincidences in a certain time window, we perform a readout in the correct CHSH basis.These windows are located at the top of each oscillation, they are calibrated from an independent measurement and kept fixed during the analysis.
To certify the QFC from finite experimental data, we view the multi-round experiment as a sequence of rounds, labelled with i = 1, . . ., n.Note that each round is an experimental trial of atom-photon-state generation, not the previously mentioned runs.At each round i, the final atom-photon state corresponds to some intrinsic CHSH score S i .By virtue of Eqs.(19) the conditional Choi fidelity of the converter at round i satisfies F i ≥ f (S i ), with f given in Eq.( 16).We are interested to bound the average fidelity over all measurement rounds.By linearity of f , this quantity is bounded by f (S), where S = 1 n n i=1 S i is the average CHSH score.A lower bound on S thus lowerbounds F through Eq. ( 16).
To give a clear lower bound on the CHSH score S in presence of a finite number of measurement rounds, we construct a one-sided confidence interval on S. It can be shown that is the tightest such lower bound for a confidence level 1 − α whenever α < 1/4 and n, nT ≥ 2 [33].Note that this conclusion does not rely on the I.I.D. assumption (independent and identically distributed), e.g. it holds true even if the state produced by the setup is not identical at each round.Here, T = n i=1 T i is the experimental mean of the random variables corresponding to the CHSH game where X i /A i (Y i /B i ) is Alice's (Bob's) measurement setting/outcome in round i. Setting α = 0.01, we obtain 99%-confidence lower bounds Ŝ on S for the state produced in the experiment.
Fig. 2(c) show the calculated Bell values Ŝ -obtained from the independent calibration measurement -for different numbers of time windows (located at each oscillation peak) and different window lengths.The optimal values are a tradeoff between a higher number of events favouring better statistics and thus higher Bell values, the signal-to-background ratio which decreases with an increasing number of peaks due to the exponential decay of the photon wavepacket, and phase resolution being ideal for the smallest possible time window.We choose an optimal time window of 8.125 ns (corresponding to 9 time bins) and the first and second oscillation peak.
The final results of the certification are displayed in Fig. 2(d): we see the Bell value Ŝ after each measurement run (in each run we measure all four correlators).We find a converging behavior due to the increasing number of events, which reduces the statistical uncertainty.The remaining fluctuations of the Bell values after QFC are most likely caused by drifts of the unitary rotation of the photon polarization state in the fiber connecting the ion trap and QFC setup.After 660 runs (16593 events) we find and an average observed CHSH score of 8T − 4 = 2.65.The latter is in good agreement with our known error sources, namely signal-to-background-ratio (0.04), phase resolution (0.028), atomic coherence and fidelity of the atomic state readout (0.085), and polarization drifts in the long fiber (0.028).From Ŝ, we calculate via Eq.( 16) and Eq. ( 19) a certified conversion fidelity of To conclude, we bound the converter's efficiency.Eq. ( 20) and Eq. ( 10) allow us to bound the success probability of the QFC directly as a function of the number of coincidence detection at Alice and Bob n c and the total number of rounds n as where we used the probability estimator free of the I.I.D. assumption from Ref. [33].With n c = 16 593 and n = 2 640 000 000, we obtain the lower bound at a confidence level 1−α = 99%.The limited overall success probability can be attributed to several factors: the success probability to collect a photon at 854 nm from the ion (0.36 %), the external device efficiency of the QFC, i.e. the probability to get a 1550 nm photon in the output fiber per 854 nm photon at the input of the QFC (57 %), the quantum efficiency of the single-photon detectors (13.5 %), further optical losses in the whole experimental setup (60 %) and the ratio between the postselected time window and the total photon wavepacket (3.9 %).

Conclusion-
We have presented the first recipe leveraging device-independent techniques to certify a unitary operation without assuming that the certification devices are perfectly calibrated.Although not fully device-independent, the proposed recipe is widely tolerant to loss.This is achieved by assuming that the occurrence of no-detection events is independent from the choice of measurement, which is both more general and more realistic than independence from the measured state.We used the calibration-independent method on a state-ofthe-art polarization-preserving quantum frequency converter to demonstrate its performance in term of conversion efficiency and fidelity.The proposed recipe could be used to certify quantum storage and processing devices among others.Given the interesting balance between its practical feasibility and high level of trust, we believe that the method is well suited to become a reference certification technique to ensure the suitability of devices for their integration in quantum networks.
Note added-While writing this manuscript we became aware of a related experimental work by Neves et al. [34].

FIG. 1 .
FIG.1.(a) Experimental setup.Light-matter entanglement is generated between a single trapped 40 Ca + ion and the polarization state of an emitted photon at 854 nm.The photons are collected with a high-aperture laser objective (HALO), coupled to a single-mode fiber and guided to the QFC device.The latter features a PPLN waveguide embedded in a polarization Sagnac interferometer to guarantee polarization-preserving operation.The converted photons pass a series of spectral filters (band-pass filter (BPF), volume Bragg grating (VBG) and etalon) to suppress background stemming from the DFG process.The projection setup at 1550 nm consists of a motorized QWP and HWP, a Wollaston prism to split orthogonally-polarized photons, and two fiber-coupled superconducting-nanowire single-photon detectors (SNSPD).In the lower left part the level scheme of the 40 Ca + ion including the most relevant states and transitions for entanglement generation and quantum state readout is shown.The atomic qubit is encoded in two Zeeman levels (m = −3/2 and m = 1/2) of the metastable D 5/2 -state.

FIG. 2 .
FIG. 2. (a)Time-resolved coincidences between photonic detection events of one of the detectors (readout base A0) and bright events of the atomic state readout from both orthogonal states of the readout base B0.The oscillations with the Larmor frequency of the atomic qubit stem from the timedependency of the entangled state.(b) The average Bell parameter in dependence of the detection time.For the further analysis we select coincidences at the detection times which correspond to a Larmor phase of ωL t = π resulting in the Ψ − Bell state.To obtain a feasible SBR, we only select concidences in time windows around the first two maxima, whose positions were determined from an independent measurement.(c) The optimal Bell value is a tradeoff between number of detected events (favouring large time windows) and phase resolution (favouring small time windows).(d) Bell value after QFC after a certain number of measurement runs based on the coincidences from the optimal time window.