648 Hilbert-space dimensionality in a biphoton frequency comb: entanglement of formation and Schmidt mode decomposition

Qudit entanglement is an indispensable resource for quantum information processing since increasing dimensionality provides a pathway to higher capacity and increased noise resilience in quantum communications, and cluster-state quantum computations. In continuous-variable time–frequency entanglement, encoding multiple qubits per photon is only limited by the frequency correlation bandwidth and detection timing jitter. Here, we focus on the discrete-variable time–frequency entanglement in a biphoton frequency comb (BFC), generating by filtering the signal and idler outputs with a fiber Fabry–Pérot cavity with 45.32 GHz free-spectral range (FSR) and 1.56 GHz full-width-at-half-maximum (FWHM) from a continuous-wave (cw)-pumped type-II spontaneous parametric downconverter (SPDC). We generate a BFC whose time-binned/frequency-binned Hilbert space dimensionality is at least 324, based on the assumption of a pure state. Such BFC’s dimensionality doubles up to 648, after combining with its post-selected polarization entanglement, indicating a potential 6.28 bits/photon classical-information capacity. The BFC exhibits recurring Hong–Ou–Mandel (HOM) dips over 61 time bins with a maximum visibility of 98.4% without correction for accidental coincidences. In a post-selected measurement, it violates the Clauser–Horne–Shimony–Holt (CHSH) inequality for polarization entanglement by up to 18.5 standard deviations with an S-parameter of up to 2.771. It has Franson interference recurrences in 16 time bins with a maximum visibility of 96.1% without correction for accidental coincidences. From the zeroth- to the third-order Franson interference, we infer an entanglement of formation (Eof) up to 1.89 ± 0.03 ebits—where 2 ebits is the maximal entanglement for a 4 × 4 dimensional biphoton—as a lower bound on the 61 time-bin BFC’s high-dimensional entanglement. To further characterize time-binned/frequency-binned BFCs we obtain Schmidt mode decompositions of BFCs generated using cavities with 45.32, 15.15, and 5.03 GHz FSRs. These decompositions confirm the time–frequency scaling from Fourier-transform duality. Moreover, we present the theory of conjugate Franson interferometry—because it is characterized by the state’s joint-temporal intensity (JTI)—which can further help to distinguish between pure-state BFC and mixed state entangled frequency pairs, although the experimental implementation is challenging and not yet available. In summary, our BFC serves as a platform for high-dimensional quantum information processing and high-dimensional quantum key distribution (QKD).

Here we report a BFC generated by filtering the signal and idler outputs from a cw-pumped, type-II quasi-phase-matched (245 GHz FWHM phase-matching bandwidth), frequencydegenerate SPDC source using a fiber Fabry-Pérot cavity with 45.32 GHz FSR and 1.56 GHz FWHM linewidth. This BFC's timebinned/frequency-binned Hilbert space dimensionality is at least 324, based on the assumption that our BFC is a pure state. Moreover, when combined with its post-selected polarization entanglement, the BFC's dimensionality doubles to at least 648, implying it has a 6.28 bits/photon classical-information capacity. We characterized this BFC in a variety of ways. First, we observed 61 periodic revivals of Hong-Ou-Mandel (HOM) interference with visibility up to 98.4% without correction for accidental coincidences 59 . Second, we demonstrated this BFC's spectral correlations across five frequency bins. Next, by switching to a BFC produced with the same SPDC source and a 5.03 GHz FSR, 0.46 GHz FWHM linewidth cavity, we got spectral correlations over 19 frequency bins but only seven HOM-interference recurrences. This result is in keeping with the scaling associated with Fourier-transform duality, i.e., more time bins imply fewer frequency bins and vice versa. In additional tests of the 45.32-GHz-cavity source we first did a post-selected measurement that violated the Clauser-Horne-Shimony-Holt (CHSH) inequality for polarization entanglement by up to 18.5 standard deviations with an S-parameter of up to 2.771. Then we witnessed Franson interference recurrences in 16 time bins with a maximum visibility of 96.1% without correction for accidental coincidences. From the zeroth-to third-order Franson interference, we inferred an entanglement of formation (E of ) up to 1.89 ± 0.03 ebits-where two ebits is the maximal entanglement for a 4 × 4 dimensional biphoton-as a lower bound on the 61 time-bin BFC's highdimensional entanglement. To further characterize time-binned/ frequency-binned BFCs we performed Schmidt mode decompositions of BFCs generated using cavities our three (45.32, 15.15, and 5.03 GHz FSR) cavities. These decompositions confirm the time-frequency scaling from Fourier-transform duality that was seen earlier in our spectral correlations and HOM-interference recurrences. For the 45.32 GHz cavity, the resulting Hilbert space dimensionality of the time-binned/frequency-binned BFC was at least 324. Augmented by this state's post-selected polarization entanglement, this state's dimensionality doubles to at least 648, which represents an ≈ 7.5 × improvement over our prior studies 35 and is equivalent to a more than 13 qubit computational space that can encode 6.28 bits/photon classical-information capacity in a biphoton-based communication link.
Although standard perturbation theory predicts that our BFCgeneration procedure produces high-purity states, we further describe the theoretical analysis and modeling from conjugate Franson interferometry. Characterized by the state's joint-temporal intensity (JTI), it can help to directly distinguish between the desired pure-state BFC and a mixed state of entangled frequency pairs (which has the same joint spectral intensity), although the experimental implementation is challenging and not yet available. In summary, our 13 qubit high-dimensional BFC can serve as a platform for hybrid time-frequency quantum key distribution (QKD), time-frequency cluster-state quantum computation, and high-dimensional encoding in quantum networks.

HOM dip recurrences
Our experimental setup is illustrated in Fig. 1a. The SPDC source used a type-II quasi-phase-matched, periodically-poled KTiOPO 4 (ppKTP) waveguide, integrated in a fiber package for high fluence and efficiency 60 . It was pumped by a 658 nm wavelength Fabry-Pérot laser diode, stabilized by self-injection locking. Our SPDC entangled photon source was designed to generate orthogonally-polarized, frequency-degenerate signal-idler photon pairs at 1316 nm with 245 GHz FWHM phase-matching bandwidth 61 . Three high-dimensional BFCs 35 were created by sending the signal-idler photon pairs through one of three Fabry-Pérot fiber cavities, whose FSRs are 45.32, 15.15, and 5.03 GHz, with FWHM linewidths of 1.56, 1.36, and 0.46 GHz, respectively. Each fiber cavity was mounted on a modified thermoelectric assembly with ≈ 1 mK temperature-control stability. A stabilized tunable reference laser at 1316 nm was used to align each cavity's spectrum to the SPDC's degenerate frequency. The resulting BFC biphoton-as predicted by standard perturbation theory with the signal-idler differential group delay suppressed, see for example 62 -can be expressed as: Here: b a y H and b a y V are creation operators for horizontally and vertically polarized photons; ΔΩ is the cavity FSR in rad s -1 ; Ω is the detuning of the SPDC's biphotons from frequency degeneracy; 2N 0 þ 1 is the number of cavity lines passed by an overall bandwidth-limiting filter; f 0 Ω ð Þ ¼ sinc AΩ ð Þ is the SPDC source's phase-matching function, where A ¼ 2:78=πB PM with B PM being the FWHM bandwidth; and f Ω À mΔΩ ð Þis the single frequencybin profile defined by the cavity's Lorentzian transmission lineshape with FWHM linewidth 2Δω, viz., The signal and idler photons were cleanly separated by a polarizing beam splitter (PBS) in our type-II SPDC configuration, so that the BFC was generated without post-selection. Using the temporal wavefunction, the BFC state can be rewritten as: A fiber Bragg grating (FBG) of 346 GHz bandwidth and a longpass filter is used to spectrally select the BFC's 2N 0 þ 1 spectral modes in Eq. (1) and to filter out remaining pump photons. The BFC wavefunction in Eq. (3) implies that HOM-interference recurs at relative delays corresponding to integer multiples of the fiber cavity round-trip time 35,63,64 , which we experimentally verified as follows. The orthogonally-polarized entangled photon pairs were divided by a PBS and directed to two arms of the HOM interferometer. A fiber polarization controller (FPC) in one arm of the interferometer alternates the idler photon polarization to match that of the signal photon at the 50:50 fiber coupler. A tunable free-space optical delay line with insertion loss smaller than 0.02 dB over its 220 mm travel range is used to vary the relative delay between the signal and idler photons for HOMinterference. After the HOM interferometer, coincidences are recorded with two superconducting nanowire single-photon detectors (SNSPDs, ≈ 85% detection efficiency).
The HOM experimental results in Fig. 1b are measured with the 45.32 GHz FSR fiber cavity by scanning the relative optical delay between the biphotons from −340 to +340 ps with respect to the central dip. A pump power of 2 mW is chosen to avoid the multipair emissions that decrease two-photon interference visibility 35,65 . The fringe visibility of the quantum interference, V n for the nth dip is ½C max À C min n ð Þ=C max , where C max is the maximum coincidence count and C min n ð Þis the nth dip's minimum coincidence count. Figure 1b left inset zooms in on the central bin whose visibility is 98.4% before subtracting accidental coincidences, and 99.9% after they are subtracted. Here we note that the visibility of the central HOM dip must exceed 70.7% to be quantified as quantum biphoton interference 66 and, as the temporal delay between signal-idler increases from center dip, the HOM dips' visibilities decrease according to the fiber cavity's Lorentzian profile as described by the theory in Supplementary Discussion I. Moreover, we note that the variation of the central HOM dip's visibility between raw and subtracted data is small (1.5%), indicating that measurement noise is quite modest at the central HOM dip (edge dips are getting close to this noise limitation). The base-to-base width of the central dip-i.e., the relative optical delay difference between the left and right edges of the central HOM dip's triangular shape-is fitted to be 3.86 ± 0.30 ps, which agrees well with the reciprocal of our 245 GHz phase-matching bandwidth, as predicted by theory. We obtain HOM-interference recurrences for a total of 61 time bins within our setup optical delay scanning range, which is a significant advance over our prior studies 35 . The measured repetition time of the recurrences was 11.03 ps, which corresponds to half the repetition period of the BFC 63 , and agrees well with our theoretical modeling in Supplementary Fig. 1. The visibility of the dip recurrences decreases exponentially (see Fig. 1b right inset) due to the Lorentzian lineshape of the BFC frequency bins. In particular, with the 45.32 GHz cavity's 11.03-ps bin spacing, our setup's ≈ 640 ps scan range allows us to observe 61 time bins given the visibility decay associated with that cavity's 1.56 GHz Lorentzian linewidth. A narrower linewidth and broader scan range would yield even more measurable time bins. Note that in addition to the HOM-interference recurrences, Franson interferometer and entanglement of formation is needed to help certify that we have generated the high-dimensionality BFC, as predicted by theory 67,68 .

Frequency-bin correlations
Whereas HOM-interference recurrences arise from the periodic peaks in the BFC's time-domain wavefunction from Eq. (3), the BFC's frequency-bin correlations arise from the correlation structure inherent in its frequency-domain wavefunction from Eq. (1). Inasmuch as those wavefunctions comprise a Fouriertransform pair, we expect there will be time-frequency duality between HOM-interference recurrences and frequency-bin correlations, i.e., the more time bins there are the fewer frequency bins there will be and vice versa. To demonstrate that behavior, we measure spectral correlations between different signal-idler frequency-bin pairs. In the experiments presented in Fig. 2, we use either the 45.32 GHz FSR cavity or the 5.03 GHz FSR cavity. Each frequency-bin pairs are selected by a pair of tunable narrowband filters. For the 45.32 GHz cavity measurements in Fig. 2b, c, the filter has a 300 pm bandwidth; for the 5.03 GHz cavity results in Fig. 2d, the filter has a 100 pm bandwidth. In Fig. 2b, c, these frequency-bins range from −2 to +2, with 0 denoting frequency degeneracy. This figure shows that the BFC exhibits the energy-conservation and frequency correlation, based on Eq. (1). In addition, we investigate the impact of multi-pair emissions on the signal-idler frequency bin cross-talk, as shown in Fig. 2c. At ≈ 4 mW pump power, the strongest frequency-bin cross-talk increased by 5.4 to 6.31 dB compared to the ≈ 2 mW pump power case shown in Fig. 2b. Figure 2d shows the greater number of BFC frequency bins obtained 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°C) limited the number of measurable frequency bins, there are now many more frequency bins compared to the case in Fig. 2b. We also note that higher signal-idler frequency-bin cross-talk is observed in the 5.03 GHz cavity due solely to the 100 pm bandwidth of our filter pairs, which spans several of that cavity's FSRs. We have measured and analyzed the frequency-bin (spectral-correlation) and time-bin (HOM-interference recurrence) subspaces for the BFCs we generated with the 5.03, 15.15 35 , and 45.32 GHz cavities (measured HOM-interference recurrences for the 5.03 GHz cavity are shown in Supplementary Fig. 3). Theory tells us that the BFC's number of frequency bins N Ω equals 2πB PM =ΔΩ, where B PM ¼245 GHz is the FWHM phase-matching bandwidth of the SPDC source, and its number of time bins N T , within an inverse cavity linewidth, equals π=ΔωΔT. Hence their product satisfies N T N Ω ¼ πB PM =Δω for all three cavities. For the ideal case, in which all the frequency bins are measurable, we find that where the subscripts label the cavity FSRs, owing to the nearly identical linewidths of the two cavities. In contrast, the time-bin and 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 high-dimensional encoding for time-frequency QKD.

Post-selected polarization hyperentanglement measurement
The post-selected BFC state can be described by Eq. (1). Hence, we probe the polarization entanglement and frequencypolarization hyperentanglement by using the experimental setup in Fig. 3a. We couple the 45.32 GHz cavity BFC's outputs into low-loss fiber bench setups and performed the polarization entanglement measurement at the central HOM dip. The coupling loss for the fiber benches are ≈ 1.3 dB and ≈ 1.5 dB, respectively. We measured the post-selected polarization entanglement by recording the coincidence-count rates while changing the angle of polarizer P 2 when polarizer P 1 is set at 45°, 90°, 135°, and 180°. The results are shown in Fig. 3b, where we see that the measured fringes are well fit by sinusoidal curves, having accidentals-subtracted mean visibilities of 89.98 ± 0.62% [calculated using the ðC max À C min Þ=ðC max þ C min Þ visibility definition for sinusoidal fringes]. Here we attribute the non-optimal polarization visibility to imperfect mode matching and limited PBS extinction ratio. Subsequently, by setting the optical delay at 0.7 mm, so that the signal and idler photons have a relative delay of ≈ 4.7 ps which is outside the central HOM dip (width = 3.86 ± 0.30 ps), we measured the polarization entanglement to demonstrate our BFC's post-selected frequency-polarization hyperentanglement. As shown in Fig. 3c, the measured fringes have accidentals-subtracted mean visibilities of 97.96 ± 0.41%. The fitted results for Fig. 3c are used to obtain the correlation functions and the corresponding S parameters. The results are shown in Fig. 3d. We measure the coincidences at the CHSH polarizer angles for the polarization subspace, and then calculate the S CHSH parameter, which is given by 69 : where E φ 1 ; φ 2 ð Þ is the two-photon correlation function at measurement angles of φ 1 and φ 2 , respectively. We choose to optimize S CHSH by using 70 . We find S CHSH to be 2.686 ± 0.037 from those Fig. 2 Quantum frequency correlations of high-dimensional BFCs. a Experimental schematic for the joint spectral intensity measurement for the high-dimensional quantum state. Signal and idler photons are sent to two tunable narrowband filters for the frequency-bin correlation measurement with coincidence counting. b Measured frequency correlations of the 45.32 GHz BFC using filters that had matched FWHM bandwidths of 300 pm and were manually tuned for scans from the −2 to +2 frequency bins from frequency degeneracy. The SPDC source was pumped at ≈ 2 mW for these measurements, which produced relatively high coincidence counts only along the diagonal elements of the correlation matrix. The cross-talk between frequency bins was less than 11.71 dB. c Measured frequency correlations of the 45.32 GHz BFC when the SPDC crystal was pumped at ≈ 4 mW, showing increased signal-idler frequency-bin cross-talk to 6.31 dB. d Higher-dimensional frequency-bin joint spectral intensity measurements for the 5.03 GHz BFC. The filters used here had matched FWHM bandwidths of 100 pm and were temperature tuned for scans from the −9 to +9 frequency bins from frequency degeneracy. The off-diagonal components increase compared to those in Fig. 2a because the effective bandwidth of the tunable narrowband filters spanned multiple FSRs in this demonstration.
correlation values, which violates the CHSH inequality by 18.5 standard deviations. In addition, we estimate the maximum achievable S fringe parameter of 2.771 ± 0.016 from the mean visibility of the entanglement-correlation fringes 30 . The combination of postselected polarization entanglement and HOM-interference is consistent with Eq. (1)'s implication that our BFC gives postselected frequency-polarization hyperentanglement 71 .

Franson fringe recurrences and entanglement of formation
To further characterize and verify our BFC to support its being consistent-as predicted by theory-with high-quality time-frequency entanglement 72-76 , we establish and stabilized a Franson interferometry which consists of two unbalanced Mach-Zehnder interferometers (MZIs), as shown in Fig. 4a. To achieve long-term stability, the MZIs are enclosed in a multilayer thermally-insulated enclosure whose temperature is actively stabilized. In Fig. 4a left inset the long-short path mismatch of each MZI is measured to be ΔT = 4.84 ns, which satisfies the requirement of phase-sensitive quantum fourth-order interference 35 . We use a thermal heater in long path of arm1 to finetune the relative phase shift ΔT 1 between the two MZIs using our 45.32 GHz BFC. In addition, the motorized stage position ΔT 2 is fixed at the center of optimum Franson interference, with the maximum constructive interference shown in Fig. 4a left inset. A zoom-in shows the Franson interference visibility of up to 96.1% (99.1% after subtracting accidental coincidences) as shown in Fig. 4a right inset. Figure 4b shows the measured Franson interference fringes and that the recurrence period for them equals the fiber cavity round-trip time. The recurrences have a 22.09 ps period, and with fringe visibilities that decay according to the cavity Lorentzian lineshape as shown in Fig. 4c and in Supplementary Fig. 5. Moreover, when we move the motorized stage to a ΔT 2 in-between cavity round trips (non-integer) such as 33 ps, we indeed observe no fringes, as also shown in the inset of Fig. 4c. We note that we only measured for ΔT 2 ! 0, limited by the free-space optical delay line in the long path of arm2, which can reach up to 360 ps in our measurements.
To give quantitative lower bounds on the time-energy entanglement of our 45.32 GHz and 15.15 GHz BFCs, we calculate the entanglements of formation (E of ) 17,42,77 by using these state's Franson interference recurrences (zeroth-to the third-order for the 45.32 GHz and 15.15 GHz BFC). In Fig. 4d, we compute (E of ) up to 1.89 ± 0.03 ebits for the 45.32 GHz BFC and (E of ) up to 1.40 ± 0.05 ebits for the 15.15 GHz BFC. The theoretical E of for visibilities V of 0.98 and 1 are also illustrated in the dashed line plots with 2 ebits as the maximum entanglement for a 4 × 4 high-dimensional biphoton. The close match between our results and ideal limits bolsters a nearly nonseparable BFC state generation. Detailed analysis numbers are also noted in Supplementary Table III.

Schmidt mode decompositions in the frequency and time domain
The BFC affords discrete-variable (binned) entanglement in both the frequency and time domains that can be quantified from our frequency-binned correlation measurements and our HOMinterference recurrences by means of Schmidt mode decompositions 78,79 in the two domains. In both cases the relevant quantity In all cases, we measured the coincidence-counting rates at the two outputs while changing polarizer P 2 from 0°to 360°. In all four cases the measured fringes are well fit with sinusoidal curves, having accidentals-subtracted mean visibilities of 89.98 ± 0.62%. c By setting the relative delay to 0.7 mm (≈ 4.7 ps) away from the central HOM dip position, we measured the polarization entanglement of our BFC. The measured fringes have accidentals-subtracted mean visibilities of 97.96 ± 0.41%. d Correlation values needed for the CHSH inequality, performed for results in panel c. The abscissa label (φ 1 ; φ 2 ) denotes the measured polarization bases. The S CHSH parameter was calculated to be 2.686 ± 0.037 from these correlations, which violates the CHSH inequality by 18.5 standard deviations. We obtained the maximal achievable S fringe parameter to be 2.771 ± 0.016 from the mean visibility of the entanglement-correlation fringes. Purple and blue dashed lines denote the classical and quantum boundaries. Error bars represent statistical errors.
is the Schmidt number K 36 , defined as: with λ n f g being the Schmidt mode eigenvalues. For the frequency-binned case, the Schmidt eigenvalues are obtained from the frequency-binned joint spectral amplitude (JSA),ψ n s ΔΩ; n I ΔΩ ð Þ which can be obtained by discretizing the BFC's frequency-domain wavefunctionψ ω S ; ω I ð Þ, where ω S and ω I are the signal and idler detunings from frequency degeneracy. For ease of Schmidt-number analysis, we assume that our BFC is close to a pure state, based on the perturbation theory characterization of a cw-pumped SPDC and the excellent stabilization of our fiber Fabry-Pérot cavities. It is challenging, however, to measure the JSA because such measurements would require reconstruction of the full phase information of the entangled state. Instead, the joint spectral intensity (JSI) can be more readily measured by performing spectrally-resolved coincidence measurements, as shown in Fig. 2. Therefore, we will use our JSI data, viz., ψ n s ΔΩ; n I ΔΩ ð Þ j j 2 , and assume that the JSA satisfies ψ n s ΔΩ; n I ΔΩ ð Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψ n s ΔΩ; n I ΔΩ ð Þ j j 2 q ; as predicted from perturbation theory with the signal-idler differential group delay suppressed. Then, by extracting the Schmidt eigenvalues fλ n gfrom the JSI measurements (i.e., the measured frequency-correlation matrix, such as Fig. 2d), the Schmidt number of the frequency-binned state K Ω , can be obtained as shown in Fig. 5a. This parameter indicates how many frequency-binned Schmidt modes are active in the biphoton state, and therefore describes its effective dimensionality 36 . In particular, extracting the Schmidt eigenvalues fλ n g from the five resonance-pairs data of Fig. 2b Fig. 2d to find the 5.03 GHz cavity's frequency-bin Schmidt number. Using the third panel in Fig. 5a, we obtain K Ω,5 GHz ≈ 11.67 for that cavity. This imperfect Schmidt number mostly due to the resolution bounds of our 100 pm bandwidth filters (the ideal Schmidt number for frequency-binned measurements is calculated in Supplementary Discussion IV), but it still demonstrates the scalability of our high-dimensional frequency-binned BFC. In Fig. 6a we compare the extracted frequency-bin Schmidt eigenvalues fλ n g for our three cavities (detailed calculations of the Schmidt eigenvalues fλ n g and the resulting Schmidt numbers are in Supplementary Discussion IV). Note that we obtain different frequency-binned BFC Schmidt numbers from the same SPDC phase-matching bandwidth by using cavities with different FSRs. From our measurements, the diagonal elements of the spectral-correlation matrix (Fig. 2d) show the decreasing-envelope behavior of the BFC. Hence, with the Schmidt decomposition, we have observed BFC states with Hilbert-space dimensionalities, K Ω × K Ω , having lower bounds of least 16 for the 45. Turning now to the Schmidt number for the time-binned BFC, to proceed in a manner analogous to what we used for the frequency-binned BFC would require knowledge of the binned JTI. However, because our BFC is generated with cw pumping, this binned JTI-under the assumption of a pure-state biphoton-is a diagonal matrix with elements ψðnΔTÞ j j 2 , where ψ τ ð Þ is the BFC's time-domain wavefunction from Eq. (3) and nΔT is the relative delay between the signal and idler photons' nth time bin. We can estimate that binned JTI from our HOM-interference data, as we now explain. By sampling ψ τ ð Þ at τ ¼ nΔT, we get ψðnΔTÞ j j 2 ¼ expðÀ2 n j jΔωΔTÞ P N0 n¼ÀN0 expðÀ2 n j jΔωΔTÞ ; for the JTI. From Supplementary Discussion I, we have that the visibility of the nth HOM dip is: thus making it possible to find the BFC's JTI by inverting the oneto-one relation between n j jΔωΔT and V n . Measuring the binned joint-temporal amplitude (JTA), whose singular-value decomposition is the Schmidt decomposition, is prohibitively difficult. Hence, we assume, as predicted by Eq. (3), that it equals the square-root of the binned JTI, i.e., we use for the time-binned wavefunction of the BFC, viz., its JTA, from which it follows that the time-bin Schmidt mode eigenvalues, fλ n g, are given by: where F ¼ ΔΩ=2Δω is the cavity finesse. The time-binned BFC state's Schmidt number is then found from Eq. (6), which leads to the following theoretical results based on our three cavities' finesses: K T,5 GHz ≈ 5.16, K T,15 GHz ≈ 6.71, and K T,45 GHz ≈ 18.30. By performing a parametric ( n j jΔωΔT) fit of our experimental data to the V n expression in Eq. (9), and then applying the result in Eqs. (11) and (6), we obtain the experimental values K T,5 GHz ≈ 5.11, K T,15 GHz ≈ 6.56, and K T,45 GHz ≈ 18.02, which agree well with theory. All the extracted time-binned Schmidt eigenvalues are shown in Fig. 5b.
For the 45.32 GHz cavity's BFC, the HOM-interference recurrences' lower bound on the Hilbert space dimensionality is therefore K T,45 GHz × K T,45 GHz = 324. Augmenting the time bins with the BFC's post-selected polarization entanglement doubles this dimensionality to at least 648. Furthermore, we find that the product of the time-binned and frequency-binned Schmidt numbers (when all the frequency bins are measurable, and the measurable HOM time bins run from −340 to 340 ps relative delay) is similar for the 45.
which mimics the BFC time-frequency product relation from Eq. (4). Our Schmidt mode analysis demonstrates the effective time-frequency scaling (increase/decrease in number of bins) by using our BFCs. For the three fiber cavities that we measured, we have successfully scaled the time-binned Schmidt numbers from K T,5 GHz ≈ 5.11 to K T,45 GHz ≈ 18.02, limited by the finesses of the fiber cavities. In the frequency-binned subspace we scaled from K Ω,45 GHz ≈ 4.31 to K Ω,5 GHz ≈ 11.67, limited by the temperature tunability of the FBG filter. The scaling of time-frequency dimensionality is complementary, the higher the Schmidt number in time or frequency, the smaller the Schmidt number in its conjugate domain. Multiplying the highest estimated time-binned Schmidt number by its frequency-binned Schmidt number counterpart yields a total Schmidt number of 77.67, which could encode over 12 qubits (K T K Ω × K T K Ω > 2 12 ), with potentially 6.28 bits/photon [from log 2 (K T K Ω ) = 6.28)] classical-information capacity that can be used in high-dimensional QKD.
Pure-state versus mixed state for filtered SPDC outputs To this point we have asserted that our filtering of the signal and idler outputs from a cw-pumped SPDC source generates a nearly nonseparable-state BFC. Toward that end we have reported experimental results consistent with that interpretation: HOMinterference recurrences, frequency-bin correlations, Franson interference recurrences and their inferred entanglements of formation, and Schmidt mode decompositions in the time-bin and frequency-bin subspaces. Our BFC frequency-domain wavefunction's being an even function of detuning implies that its JSI determines its HOM-interference behavior 68 , and in general the BFC's JSI determines its frequency-binned correlations and its Franson interference behavior (see Supplementary Discussion I, II, and V for details). That said, standard perturbation theory, see, e.g., 62 , predicts that a cw-pumped SPDC will emit pure-state (or nearly pure-state) biphotons, and the excellent stabilization of our fiber Fabry-Pérot cavities then implies that our filtered SPDC sources should then emit nearly nonseparablestate BFC biphotons. Our prior experimental work supports the nearly pure-state assertion for a SPDC source's output, see 80 , in which both the JSI and JTI were measured for a pulse-pumped The signal and idler outputs from a filtered SPDC source are applied to a pair of MZIs. One arm of each MZI contains an OFS, i.e., a single-sideband modulator, that imposes frequency shifts þω m and Àω m on the signal and idler, respectively. The MZI's outputs undergo positive (for the signal) and negative (for the idler) dispersions of equal magnitude that, together with the frequency shifts, ensure there is no second-order interference present in the signal-idler coincidence counts. b BFC theoretical conjugate Franson interference has high-visibility recurrences and they occur only when the interferometer frequency offset is kΔΩ for integer k. c EFS theoretical conjugate Franson interference has low-visibility recurrences and they only occur when the interferometer frequency offset is 2kΔΩ for integer k. Supplementary Discussion V describes the details of conjugate Franson interferometry, comparing the BFC and EFS recurrence visibilities for the 15.15 GHz and 5.03 GHz cavities.
source, and 81 , in which an SPDC source was entanglement engineered to produce single spatiotemporal-mode heralded single-photon pulses. Moreover, we also note that there have been several experimental demonstrations of high-dimensional frequency-bin entanglement utilizing the sinusoidally driven phase modulator in recent years [82][83][84][85] . Nevertheless, direct experimental evidence of a BFC state purity is highly desirable. Conjugate Franson interferometry 33 , since it is characterized by the signal-idler state's JTI, can provide such evidence. The configuration for conjugate Franson interferometry is shown in Fig. 6a. Compared to our Franson interferometry setup in Fig. 4a, a pair of optical frequency shifters (OFSs) and dispersion modules are required to implement the conjugate Franson interferometry. In particular, this configuration easily allows the desired BFC pure state to be distinguished from mixed states with the same JSI. For example, Fig. 6b, c compare the visibilities for conjugate Franson interference recurrences of our 45.32 GHz BFC to those of the entangled frequency-pair state (EFS)-i.e., an incoherent mixture of the pure states, for ÀN 0 m N 0 -that has the same JSI (see Supplementary Discussion V for more details). The experimental implementation and stabilization of a conjugate Franson interferometer, challenging currently, can provide a pathway for future exploration of the high-dimensional BFC.

DISCUSSION
In this work we have demonstrated high-dimensionality time-frequency subspaces using a BFC generated by filtering the signal and idler outputs from a cw-pumped SPDC source. For a BFC generated with a 45.32 GHz FSR filter cavity we achieved 61 HOM-interference recurrences, with a maximum visibility of 98.4% (99.9%) before (after) accidental coincidences are subtracted. For a BFC generated with a 5.03 GHz FSR filter cavity, we observed high spectral correlations over 19 frequency bins. All told, for the three cavities we employed, we explored spectral and temporal correlations-and hence their Fourier-transform duality-over cavity FSRs spanning nearly an order of magnitude. We then measured up to 16 Franson interference recurrences, observing a maximum visibility of 96.0% (99.1%) before (after) accidental coincidences are subtracted. Using the zeroth to the third recurrence visibilities allowed us to obtain an E of ≥1.89 ± 0.03 ebits-where 2 is the maximum for a 4 × 4 dimensional biphoton-lower bound on our BFC's entanglement. Via Schmidt mode decompositions, we quantified the entanglement scaling of our BFCs' time-binned and frequency-binned subspaces, comparing measured values with their theoretical counterparts. For example, our 45.32 GHz cavity's post-selected frequency-polarization hyperentangled BFC achieves a timebinned Schmidt number of 18 and a Hilbert-space dimensionality of at least 648, based on the assumption of a pure state, representing an advance of almost an order of magnitude compared to our previous work. With the time-frequency duality and the frequency-polarization hyperentanglement of such a BFC, we infer a computational space of more than 12 qubits, with 6.28 bits/photon that can potentially be encoded for classical-information transmission over a quantum channel using only biphotons. This high-dimensionality time-frequency state encodes multiple qubits from different degrees-of-freedom onto the biphoton pair, and thus further increasing the photon information capacity with applications in high-dimensional quantum information processing, time-frequency cluster-state quantum computation, and high-dimensional QKD.

Experimental setups
For our cw-pumped source, we customized a tunable stabilized selfinjection-locked 658 nm laser. We used a precision laser controller from Vescent Photonics (D2-105) to drive the laser diode. For other parts of experimental setups, we used the following components with all optical parts connected using either single-mode optical fibers or polarizationmaintaining fibers: fiber-pigtailed ppKTP waveguide (AdvR), Fabry-Pérot fiber cavities (Luna/Micron Optics), bandpass filters (O/E Land and Agiltron), single-photon detectors (Photon Spot), and timing electronics (PicoQuant and Swabian Instruments).

Franson interferometry
For the phase-sensitive time-energy quantum interference measurements, we need to stabilize Franson interferometry to prevent the mechanical, acoustic, and thermal noises from the environment. Both MZI arms are shielded in double layers of sealed boxes, and temperature-stabilized with four home-made Peltier modules. The Franson interferometer free-space optical delay line (arm2, labeled in Fig. 4a) is based on a miniaturized linear stage with closed-loop piezoelectric motor control (CONEX-AG-LS25-27P, Newport). The optical insertion loss of double-pass free-space delay line is smaller than 0.4 ± 0.05 dB over the entire 360 ps delay-travel range, providing us the capability to measure Franson revival time bins from multiple cavities round-trip times of our BFC. For sub-femtosecond interference measurements, we utilize a temperature controller to thermally adjust the relative phase between the two arms of our Franson interferometer.

DATA AVAILABILITY
The data and analysis codes used in this study are available from the corresponding authors on request.