648 Hilbert space dimensionality in a biphoton frequency comb

Qubit entanglement is a valuable resource for quantum information processing, where increasing its dimensionality provides a pathway towards higher capacity and increased error resilience in quantum communications, cluster computation and quantum phase measurements. Time-frequency entanglement, a continuous variable subspace, enables the high-dimensional encoding of multiple qubits per particle, bounded only by the spectral correlation bandwidth and readout timing jitter. Extending from a dimensionality of two in discrete polarization variables, here we demonstrate a hyperentangled, mode-locked, biphoton frequency comb with a time-frequency Hilbert space dimensionality of at least 648. Hong-Ou-Mandel revivals of the biphoton qubits are observed with 61 time-bin recurrences, biphoton joint spectral correlations over 19 frequency-bins, and an overall interference visibility of the high-dimensional qubits up to 98.4%. We describe the Schmidt mode decomposition analysis of the high-dimensional entanglement, in both time- and frequency-bin subspaces, not only verifying the entanglement dimensionality but also examining the time-frequency scaling. We observe a Bell violation of the high-dimensional qubits up to 18.5 standard deviations, with recurrent correlation-fringe Clauser-Horne-Shimony-Holt S-parameter up to 2.771. Our biphoton frequency comb serves as a platform for dense quantum information processing and high-dimensional quantum key distribution.

time-frequency scaling. We observe a Bell violation of the high-dimensional qubits up to 18.5 standard deviations, with recurrent correlation-fringe Clauser-Horne-Shimony-Holt Sparameter up to 2.771. Our biphoton frequency comb serves as a platform for dense quantum information processing and high-dimensional quantum key distribution.
In quantum communications such as device-independent quantum cryptography, highdimensionality provides a pathway towards increased information capacity per photon [12,19,30,[35][36][37][38], improved security against different attacks [39], and better resilience against noise and error [40,41]. Consequently, advances in high-dimensional encoding of qubits have ranged from Bell-type inequalities for energy-time qudits [20,34,42,43], to on-chip quantum frequency comb generation [32,44], certifying high-dimensional entanglement via two global product bases without requiring large full state tomography [45], and increased noise resilience with larger noise fractions [40,41]. Dimensionality of such robust qubits involved, for example, compressive sensing of joint quantum systems with 65,536 dimensions in the position-momentum degree-offreedom [22], on-chip frequency-bin generation with at least 100 dimensions [32], and 19 timebin qubits in biphoton frequency combs [31], scaling remarkably to even 84th time-bin revivals with a path length difference of 100 m [46]. To date, however, the Hilbert space dimensionality of time-frequency entanglement has always been less than 100 and the Schmidt mode decomposition of such dual time-frequency qubits has not yet been fully explored.
Here we demonstrate and quantify a hyperentangled biphoton frequency comb (BFC) with a time-frequency dimensionality of at least 648. First, we observe periodic time-bin revivals of the high-dimensional Hong-Ou-Mandel indistinguishability interference, scaling over 61 time-bins and with visibility up to 98.4%. Second, with time-frequency duality in the quantum frequency comb, we measure the joint spectral correlations over 19 frequency-bins, witnessing the spectraltemporal high-dimensional entanglement generality across three different cavities that span nearly an order-of-magnitude in mode spacings. Third, with the qubit polarization subspace, we obtain a mean visibility of the entanglement correlation fringes up to 2.771  0.016, along with Bell violation of the high-dimensional qubits up to 18.5 standard deviations. Fourth, we perform the Schmidt mode decomposition analysis of the high-dimensional time-frequency qubits, through the dual and discretized joint spectral-temporal intensities. In the frequency subspace, the Schmidtmode eigenvalues are described and extracted over three different cavities to quantify their effective dimensions, as well as to estimate their bounds. In the temporal subspace, we arrived at the time-bin Schmidt-mode eigenvalues, along with theory-experimental dimensionality quantification and their scaling. Including the polarization subspace, we achieved a Hilbert space dimensionality of at least 648to the best of our knowledge, this represents the highest Schmidt number value observed so far in time-frequency entanglement. Our results support the hyperentangled high-dimensional BFC as a platform for hybrid time-frequency quantum key distribution, time-frequency cluster state computations, and dense quantum information processing. Figure 1a shows the experimental setup. Spontaneous parametric downconversion (SPDC) occurs in a type-II periodically-poled KTiOPO4 (ppKTP) waveguide, integrated in a fiber package for high fluence and efficiency [47]. It is pumped by a 658 nm wavelength Fabry-Pérot laser diode, stabilized by self-injection-locking. The type-II quasi-phase-matching is designed for generating orthogonally-polarized frequency-degenerate signal and idler photon pairs at 1316 nm with 245 GHz full-width-to-half-maximum (FWHM) phase-matching bandwidth [48]. The high-dimensional mode-locked two-photon state [31] is created by passing the signal and idler photons through one of three Fabry-Pérot fiber cavities, whose FSRs are 45.32 GHz, 15.15 GHz, and 5.03 GHz, with FWHM linewidths of 1.56 GHz, 1.36

Time-bin revival subspaces:
GHz, and 0.46 GHz respectively. Each fiber cavity is mounted on a modified thermoelectric assembly with ≈ 1 mK temperature-control stability. A frequency-stabilized tunable reference laser at 1316 nm is used to align each cavity's spectrum to the SPDC's degenerate frequency by matching its second-harmonic to the 658 nm pump wavelength. The BFC state generated in this manner can be expressed as: where ∆Ω is the cavity's FSR in rad s -1 , Ω is the detuning of the SPDC biphotons from frequency degeneracy, 2N+1 is the number of cavity lines passed by an overall bandwidth-limiting filter, and the state's spectral amplitude (Ω − ∆Ω) is the single frequency-bin profile defined by the cavity Lorentzian transmission lineshape with FWHM linewidth 2∆ The signal and idler photons are cleanly separated by a polarizing beamsplitter (PBS) by our type-II SPDC configuration, so that the BFC is generated without post-selection. Using the temporal wavefunction, the BFC's state can be rewritten as: where the exponential decay is slowly varying relative to the sin [  The HOM experimental results in Figure 1b are obtained with the 45.32 GHz FSR fiber cavity by scanning the relative optical delay between the two arms of the HOM interferometer from -340 ps to +340 ps with respect to the central dip. A pump power of 2 mW is chosen to avoid the multi-pair emissions that decrease two-photon interference visibility [31,51]. The fringe visibility of the quantum interference, , for the nth dip is [ max − min ( )]/ max , where max is the maximum coincidence count and min ( ) is the minimum coincidence count of the nth dip.
In the left inset of Figure 1b, the central bin visibility is observed to be 98.4% before subtracting accidental coincidences, and 99.9% after they are subtracted, supporting the good fidelity of our entangled state. The base-to-base width of the central dip is fitted to be 3.86 ± 0.30 ps, which agrees well with our 245 GHz phase-matching bandwidth. We obtained HOM-dip revivals for a total of 61 time-bin recurrences, within the optical delay's scanning range, a significant advancement from our prior studies [31]. The measured repetition time of the revivals is 11.03 ps, which corresponds to half the repetition period of the BFC [49], and agrees well with our theoretical modeling. The visibility of the recurrence dips decreases exponentially (see right inset in Figure 1b)

Frequency-bin revival subspaces:
To further characterize our BFC state, we measure correlations between different signal-idler frequency-bin pairs. In the experiments presented in We note that, through temporal-state and conjugate-state projection [52], observation of both HOM revivals and frequency-bin correlation verifies the BFC's high-dimensional entanglement.
In addition, we investigate the effects of multi-pair emissions on the signal-idler frequency bin cross-talk, as shown in Figure 2c. At  4 mW pump power, the frequency-bin largest cross-talk increases by 5.4 dB to -6.31 dB compared to the  2 mW pump power case shown in Figure 2b. Figure 2d shows the larger number of BFC frequency bins obtained from using the 5.03 GHz FSR cavity and 100 pm bandwidth tunable filters. In this measurement, although the temperature limit of these tunable filters ( 100 ºC) bound the number of frequency bins measurable, there are now many more frequency bins compared to the case in Figure 2b. We also note that higher signalidler frequency-bin cross-talk is observed in the 5.03 GHz cavity due solely to the 100 pm bandwidth of our filter pair, which spans several frequency bins.
We measure and analyze the frequency correlation and time-bin HOM revival subspaces for where the subscripts label the cavity FSRs, owing to the nearly identical linewidths of the two cavities. In contrast, the time-bin frequency-bin product for the 5.03 GHz cavity should be roughly a factor of three higher, owing to its smaller cavity linewidth. We note that this time-bin and frequency-bin tradeoff for any of our three cavities supports the dense encoding of the timefrequency quantum key distribution.  Figure 3c. First, we measure the coincidences at the Clauser-Horne-Shimony-Holt (CHSH) polarizer angles, and then calculate the SCHSH parameter, which is given by [53]:
The SCHSH and Sfringe parameters are in good agreement, indicating that the hyperentangled state is generated with high quality. Thus, in addition to the 61 time bins, the BFC has high-fidelity postselected polarization entanglement, further enabling dense quantum information processing.

Frequency-binned biphoton frequency comb Schmidt number
As alluded to earlier, we regard the BFC as providing discrete-variable frequency-binned and time-binned states, depending on how the BFC is measured. Although the frequency-bin correlation measurements and the revivals of our HOM interferometry provide indications of the potential dimensionality of the discrete-variable states, more precise results can come from examining Schmidt-mode decompositions [55,56], in the frequency-and time-bin basis. In both cases the relevant quantity is that of the Schmidt number K in the specific basis [5], defined as: with { } being the Schmidt-mode eigenvalues.
For the frequency-binned state, these eigenvalues are obtained from the frequency-binned joint spectral amplitude, ѱ( ∆Ω, ∆Ω) which can be obtained by discretizing the BFC's frequency-domain wavefunction ѱ( , ), where ωS and ωI are the signal and idler detunings from frequency degeneracy. We have assumedbased on our high-quality HOM interference and frequency-bin correlation measurementsthat our biphoton comb is a pure state. It is challenging experimentally to extract the joint spectral amplitude because such measurements would require reconstruction of the full phase information of the entangled state. Instead, the joint spectral intensity can be more readily measured by performing spectrally-resolved coincidence measurements, as shown in Figure 2. Therefore, we will approximate the joint spectral amplitude by measuring the joint spectral intensity, |ѱ( ∆Ω, ∆Ω)| 2 , and assuming that: ѱ( ∆Ω, ∆Ω) = √|ѱ( ∆Ω, ∆Ω)| 2 .
Then, by extracting the Schmidt eigenvalues { } from the joint spectral measurements (i.e., the measured correlation matrix, such as Figure 2d), the Schmidt number of the frequency-binned state, Ω , can be obtained as shown in Figure 4a. This parameter indicates how many frequency-binned Schmidt modes are active in the biphoton state, and therefore describes its effective dimension [32]. In particular, extracting the Schmidt eigenvalues { } from the five resonance-pairs data of  For completeness, we use the frequency-bin data from Figure 2d to extract the frequency-bin Schmidt number for the 5.03 GHz cavity. Using the third panel in Figure 4a, we obtain Ω = 11.67 for that cavity. This lower-than-ideal Schmidt number mainly results from the resolution bounds of our 100 pm tunable bandpass filter, but it still demonstrates the scalability of our highdimensional BFC frequency-binned state. In Figure 4a we compare the extracted frequency-bin Schmidt eigenvalues { } for our three cavities.

Time-binned biphoton frequency comb Schmidt number
To estimate the Schmidt number for the time-binned BFC state in a manner analogous to what we used for the frequency-binned BFC state would require knowledge of the discretized joint temporal intensity. Because our BFC is generated with continuous-wave pumping, this discretized joint temporal intensityunder the assumption of a pure-state biphotonis a diagonal matrix with elements |Ѱ( ∆ )| 2 , where ∆ is the relative delay between the signal and idler photons. We can estimate the discretized joint temporal intensity from our HOM interferometry data, as we now explain. By sampling the BFC state's temporal-domain wavefunction from Eq. (3) at = ∆ , we get From Section I of the Supplementary Information, we have: thus making it possible to find the BFC's joint temporal intensity from HOM interferometry by inverting the one-to-one relation between | |∆ ∆ and . Because measuring the discretized joint-temporal amplitude whose singular-value decomposition is the Schmidt decomposition is prohibitively difficult, we assume that it equals the square-root of the discretized joint-temporal intensity, i.e., we use which mimics the BFC time-frequency product relation from Eq. (4) and provides further evidence that our hyperentangled BFC is generated with high fidelity and high Hilbert space dimensionality.

Conclusions
In

Additional information:
The authors declare no competing financial or non-financial interests. All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Information. Additional data related to this paper may be requested from the authors.   correlations of the 45.32 GHz cavity's BFC using filters that had matched full-width-at-halfmaximum bandwidths of 300 pm and were manually tuned for scans from the −2 to +2 frequency bins from frequency degeneracy. The SPDC waveguide was pumped at  2 mW for these measurements, which produced relatively high coincidence counts only along the correlation matrix's diagonal elements. The cross-talk between frequency bins was less than -11.71 dB.