1. Introduction

The existence of incompatible observables in quantum theory represents one of the central departures of quantum from classical mechanics, underpinning the Heisenberg uncertainty principle and precluding the actualization of quantum states with, e.g., arbitrarily well-defined position and momentum. In finite-dimensional Hilbert spaces, incompatibility appears in the form of mutually unbiased bases (MUBs) [1,2]. Consider two orthonormal $d$-dimensional bases $\{| {\alpha _m} \rangle \}$ and $\{| {\beta _m} \rangle \}$; they comprise a pair of MUBs if and only if they satisfy $\left | \langle {\alpha _m|\beta _{m'}} \rangle \right |^{2}=\frac {1}{d}$ for all $m,m' \in \{0,1,\ldots,d-1\}$, which implies that measurements in the $\alpha$-basis provide no information about the results of measurements in $\beta$, and vice versa. MUBs are optimal measurements for tomography of noisy quantum states [1,3], expose tampering from eavesdroppers in quantum key distribution (QKD) [4–6], and provide efficient entanglement witnesses [7,8]. One archetypal pair of MUBs are the logical and discrete Fourier transform (DFT) bases: $\{| {m} \rangle \}$ and $\{| {f_m} \rangle \}$, where $| {f_m} \rangle = \frac {1}{\sqrt {d}}\sum _{n=0}^{d-1} e^{-2\pi \mathrm {i} mn/d} | {n} \rangle $. Measurements with both have been utilized extensively in a variety of photonic quantum information experiments. In time-bin encoding, the DFT has been realized with nested delay interferometers, supporting optical frequency division multiplexing [9,10], high-dimensional QKD [11,12], and quantum state tomography [13,14]. In path encoding, DFT operations have been used for on-chip state characterization [15] and multiphoton quantum interference [16,17], and are core elements in a recent proposal to generate Greenberger—Horne—Zeilinger (GHZ) states [18]. And in orbital angular momentum (OAM), spatial light modulators have enabled measurements in DFT bases for tomography [19] and entanglement certification [20,21].

In frequency-bin encoding, the quantum frequency processor (QFP)—a concatenation of alternating electro-optic phase modulators (EOMs) and pulse shapers [22]—has enabled experimental demonstration of DFT gates up to $d=3$, using a three-element QFP (two EOMs and one pulse shaper) [23]. But although theoretical and numerical results [22,24] indicate the QFP’s potential to reach even higher-dimensional DFTs with additional elements, it is unclear whether more efficient DFT constructions are possible with smaller systems, a question of practical importance toward high-dimensional quantum communications and networking protocols.

In this work, we address this question directly and develop explicit designs for efficient frequency-bin DFTs. We find that a fixed-length three-component QFP is sufficient to reach DFT gate performance with fidelity $\mathcal {F}_W>0.9997$ and success probability $\mathcal {P}_W>0.965$ for all dimensions examined ($d\leq 10$). The only requirement as $d$ increases is the addition of one radio-frequency (RF) harmonic per dimension increment in the EOM drive functions, so that $d-1$ total RF tones enable the $d$-point DFT with high $\mathcal {F}_W$ and $\mathcal {P}_W$. As examples of these designs, we experimentally implement parallel $d=3$ DFTs on multiple frequency-bin entangled states, using the measurement results to perform state tomography and quantify entanglement through Bayesian inference. Our results provide a scalable recipe for the construction of high-$d$ frequency-bin DFTs, useful for basic communication tasks in this degree of freedom and particularly well suited to tight bin spacings envisioned in future integrated devices.

2. DFT gate designs

Figure 1 highlights an example scenario leveraging QFP-based DFT operations. A high-dimensional frequency-bin entangled state—produced, e.g., by pumping a microring resonator (MRR) or a periodically poled lithium niobate (PPLN) waveguide—is split and transmitted to two users, each of whom possess a QFP, wavelength-selective switch (WSS) and superconducting nanowire detectors (SNSPDs). By synthesizing either the identity (EOMs off) or complete DFT on the QFP, measurements of the received photon can be performed in either the logical or Fourier bases, respectively. The upper-left inset shows temporal modulation patterns and pulse shaper phases that enable a high-fidelity ($d=10$)-dimensional DFT; if the original biphoton state is maximally entangled, then joint measurement outcomes will be perfectly correlated in matched bases, as exhibited in the upper-right inset for the case of an input state with uniform phase, i.e., $| {\psi } \rangle =\frac {1}{\sqrt {d}}\sum _{k=0}^{d-1}| {k,d-1-k} \rangle _{IS}$, and DFT operations defined according to an increasing-frequency logical basis convention for both photons (see mode definitions in Fig. 1). Correlations in such complementary bases can then be used for entanglement verification or $d$-dimensional two-basis QKD [6].

 

Fig. 1. Representative application for the frequency-bin DFT. Entangled photons are generated and sent to different users, each of whom uses a QFP to apply either the identity or $d$-dimensional DFT prior to frequency-resolved detection. Insets show simulation results for the ten-dimensional case: optimal DFT modulation patterns (upper left) and spectral correlations for an ideal maximally entangled input (upper right). See text for details.

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Mathematically speaking, in terms of logical and DFT basis states, we can define the DFT such that it maps $| {f_m} \rangle$ inputs to $| {m} \rangle$ outputs, which can be written in terms of the input (output) annihilation operators $\hat {a}_m$ ($\hat {b}_m$) as a matrix $\hat {b}_m = \sum _{n=0}^{d-1} (F_d)_{mn} \hat {a}_n = \frac {1}{\sqrt {d}} \sum _{n=0}^{d-1} {e^{2\pi \mathrm {i} mn/d}} \hat {a}_n$, where the operators apply to discrete frequency bins centered at $\omega _n = \omega _0 + n\Delta \omega$. We note that multiple conventions for the forward $F_d$ and inverse $F_d^{\dagger }$ are possible, depending on the phase in the exponent ($\pm \mathrm {i}$) and the physical mapping to frequency bins (i.e., $\omega _n$ defined increasing or decreasing with $n$). Such conventions have no impact on the general behavior of the DFT, yet must be properly accounted for in making quantitative predictions of quantum state measurements. We focus on synthesizing these gates on a three-component QFP, which collectively implements the modal transformation $W$ with elements

We truncate the formal infinite-dimensional space of Eq. (1) to $M=64$ modes for numerical simulation, which we have found sufficiently large to eliminate spurious edge effects. Of these $M$ modes, only $B$ are phase-shifted by the pulse shaper (the remaining $M-B$ are passed with zero applied phase), leaving a total of $B+4P$ independent parameters to optimize: $B$ pulse shaper phases $\phi _k$, and amplitude and phase for each harmonic of each EOM. Using particle swarm optimization [25], we find the parameter settings which minimize the cost function $C=\mathcal {P}_W \log _{10}(1-\mathcal {F}_W)$ as a convenient means to optimize both $\mathcal {F}_W$ and $\mathcal {P}_W$ while penalizing the former more strongly [26]. We repeat such optimization tasks for $B\in \{4,8,\ldots,52\}$ to investigate the required number of pulse shaper channels as well.

As observed in Ref. [23], a lone single-pass EOM cannot mix $d$ frequency bins equally without at least $\frac {d-1}{2d-1}$ of the input energy scattering into modes outside of the $d$-dimensional subspace. However, by cascading multiple EOMs separated by pulse shapers—the QFP—this scattering can be compensated and high-probability, high-fidelity mixing is possible. In the first QFP realization of the $d=2$ ($d=3$) DFT, a solution with fidelity $\mathcal {F}_W=0.9999$ ($\mathcal {F}_W=0.9999$) and success probability $\mathcal {P}_W=0.9760$ ($\mathcal {P}_W=0.9733$) was demonstrated [23]. Interestingly, while the $d=2$ case utilized single-frequency sinewave modulation, the $d=3$ solution required modulation containing both the first and second harmonic. To see if this “add-RF-harmonic” rule represents a trend for DFT gates, here we perform additional design simulations to synthesize $d$-dimensional DFT gates on a three-element QFP, in which we consider $d-1$ RF tones in the optimization procedure. The resulting fidelities and success probabilities for $d\leq 10$ and channel numbers $B$ appear in Fig. 2. For all $d$, solutions with $\mathcal {F}_W>0.9997$ and $\mathcal {P}_W>0.965$ are possible with these resources. To place these fidelities in context, the average error probability for measurements of input states $| {f_m} \rangle$ is less than 0.02% for all solutions here, well below the asymptotic bounds for two-basis QKD protocols: e.g., $\geq$15.95% for $d\geq 3$ [6].

 

Fig. 2. DFT solutions for a three-element QFP, where $d-1$ RF harmonics are available for each dimension $d$. The vertical arrows mark the bandwidth required to reach the final cost function value, up to three significant digits; the fidelity and success probability $(\mathcal {F}_W,\mathcal {P}_W)$ at these points are provided above each plot.

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While the number of elements required for these results is constant (fixed at three), the effective number of modes utilized does increase with $d$, as expressed by the vertical lines in Fig. 2, which mark where the third significant digit of the cost $\mathcal {P}_W\log _{10}(1-\mathcal {F}_W)$ has converged to its limiting value, specifically at $B=12$ ($d=3$), 16 ($d=4$), 20 ($d=5$), 24 ($d=6$), 28 ($d=7$), 32 ($d=8$), 32 ($d=9$), and 36 ($d=10$). The ordered pair $(\mathcal {F}_W,\mathcal {P}_W)$ above each plot shows the specific fidelity and success probability for the solution at this value of $B$, which we hereafter refer to as “minimum bandwidth” DFT solutions. The solution we found here for $d=3$—albeit using a different cost function—is similar to the one we found in Ref. [23], with both the fidelity $\mathcal {F}_W$ and success probability $\mathcal {P}_W$ remaining the same to four significant digits. This bandwidth scaling with dimension is consistent with previous observations of a tradeoff between QFP depth and optical bandwidth [22]. From a practical side, accessing additional bandwidth is frequently preferred to adding components—in terms of cost and loss—so that this fixed-depth DFT design procedure appears quite useful. Incidentally, it also seems that the “$d-1$ RF tone” rule is unique and well defined: for all simulations we have completed, access to either fewer or more RF harmonics leads, respectively, to noticeable reductions or negligible improvements in DFT gate performance.

Assuming that these observations persist to higher $d$, the total number of adjustable parameters required for the frequency-bin DFT seems to scale like $\sim$8$d$: spectral phases for $B\sim 4d$ modes combined with $4(d-1)$ RF amplitudes and phases (accounting for each tone on each EOM). Such $\mathcal {O}(d)$ scaling is slightly better than the $\mathcal {O}(d\log _2 d)$ scaling for the number of beamsplitters for path-encoded DFT designs [27] but less efficient than recently discovered OAM DFTs based on $\mathcal {O}(\sqrt {d}\log _2 d)$ beamsplitters [28]. Accordingly, the required free parameters in our solutions do appear reasonable in view of previous work in other degrees of freedom.

Figure 3 plots the specific EOM modulation patterns $A(t)$ and $B(t)$, resulting Fourier series coefficients $c_n$ and $d_n$ [Eq. (2)], and pulse shaper phases $\phi _k$ [Eq. (1)] for the minimum bandwidth solutions designated by arrows in Fig. 2. The pulse shaper phase shifts display no obvious trends in their spectrum as $d$ increases. In contrast, the temporal phases on the EOMs possess a clear single-peak structure that becomes sharper and grows in amplitude with $d$, coupling successively more frequency bins as evidenced by the wider spread in Fourier series coefficients in Fig. 3(b). Moreover, the second EOM pattern is the time-reversed version of the first [$B(t)=A(-t)$] for all $d$, which is especially interesting in that these modulation functions do not cancel each other out for the case of zero applied pulse shaper phase ($\phi _k=0$): $B(t)=A(-t)$ leads to the relation $d_n=c_{-n}$ ($|d_n|=|c_n|$ and $\arg d_n = \arg c_{-n}$), rather than $d_n=c_{-n}^{*}$ ($|d_n|=|c_n|$ and $\arg d_n = -\arg c_{-n}$) as required for total cancellation. Our attempts to understand this behavior intuitively have proven unsuccessful, although we suspect a useful explanation should be possible. Nevertheless, the practical value of these recipes for frequency-bin DFTs remains clear, particularly toward on-chip integration where tighter mode spacings could make the synthesis of high-order RF harmonics more manageable.

 

Fig. 3. Full QFP solutions for the $d$-point DFT gates with bandwidths indicated by the arrows in Fig. 2. (a) Phase modulation patterns $A(t)$ and $B(t)$ for the first and second EOM, respectively, plotted over a single temporal period $T=2\pi /\Delta \omega$. (b) Amplitude (stem plots, left axis) and phase (scatter plots, right axis) of the corresponding Fourier series coefficients $c_n$ and $d_n$ from Eq. (2). Since $|c_n|=|d_n|$, only one stem plot per dimension is required. (c) Phase $\phi _k$ applied to each frequency bin by the central pulse shaper, where gray shading encloses the computational space from bin 0 to $d-1$.

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3. Experiment

As an application of the DFT for state characterization, we experimentally implement the $d=3$ solution and apply it to a biphoton frequency comb (BFC). While parallel $d=2$ DFTs have been realized [29], as well as a single-photon $d=3$ DFT [23], this is the first example combining the two: i.e., parallel frequency-bin DFTs on frequency-bin qutrits. Our experimental design for $d=3$ resembles the scheme discussed in Fig. 1, where we enlist the PPLN biphoton source and use a pulse shaper to carve a total of three pairs of frequency-correlated, 20 GHz-spaced, $\sim$10 GHz-wide bins; the bin spacing $\Delta \omega /2\pi =20$ GHz facilitates line-by-line shaping, and RF tones at 20 and 40 GHz are required at each of the QFP’s two EOMs. As experimental simplifications due to available equipment, the signal and idler photons are transmitted in the same optical fiber and then modulated by a single QFP programmed with two parallel DFT gates (separated by a 200 GHz guardband), and coincidences are registered by raster scanning signal and idler WSS filters so that only two SNSPDs are required (rather than the ideal of $2d=6$).

A logical basis measurement (EOMs off) of our $3\times 3$ BFC appears in Fig. 4(a). If we define the biphoton state $| {\phi } \rangle \propto | {02} \rangle_{IS} + e^{i\phi }| {11} \rangle_{IS} + e^{2i\phi }| {20}\rangle_{IS}$, then the state carved by the front-end pulse shaper (with flat spectral phase) is ideally $| {\phi =0} \rangle$. By also enlisting the pulse shaper to apply a phase of $\phi$ to signal bin $n=1$ and $2\phi$ to bin $n=0$, any $| {\phi } \rangle$ can be prepared as input into the QFP. The measured output coincidences for $\phi \in \{0,2\pi /3,4\pi /3\}$ after parallel DFTs follow in Fig. 4(b–d): as expected, the results are strongly correlated, with each setting of $\phi$ determining which three pairs of frequency bins are populated. Despite the small number of measurements considered, the observed correlations are sufficient for meaningful inference of the underlying states. Since the prepared states differ only in phase, we can take the logical basis results in (a) as applying to any of the three $\phi$ cases, giving us two sets of nine-outcome measurements for each $\phi$ value, $\mathbb{1}^{(I)} \otimes \mathbb{1}^{(S)}$ and $F_3^{(I)}\otimes F_3^{(S)}$.

 

Fig. 4. Spectrally resolved coincidences after parallel QFP operations. Results correspond to measurements in the logical (a) and DFT bases (b–d) . The latter vary with the phase of the prepared input superposition state $| {\phi } \rangle$.

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One useful metric for a bipartite state is the distillable entanglement $E_D$ [30]. While extremely difficult to determine directly, bounds can be obtained from computable quantities. For example, a lower bound can be established from conditional entropies [8], namely: $E_D \geq \log _2 3 - \mathcal {H}(\mathbb {1}^{(I)}|\mathbb {1}^{(S)}) - \mathcal {H}(F_3^{(I)}|F_3^{(S)})$. Following previous work [29], we can estimate these entropies directly from the raw counts in Fig. 4, by positing some unknown nine-element probability distribution $\mathbf {p}=(p_{00}, p_{01},\ldots,p_{22})$ for each panel and sampling the Bayesian posterior distribution formed by a flat Dirichlet prior on $\mathbf {p}$ and a multinomial likelihood for the observed counts. Doing so, we obtain the $\min E_D$ values in Table 1.

Table 1. Bayesian inference of Fig. 4 data: distillable entanglement bound $\min E_D$, fidelity $\mathcal {F}_\rho$, and log-negativity $E_\mathcal {N}$.

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In addition, $E_D$ can be upper bounded by the log-negativity $E_\mathcal {N}$ [31], which requires the full density matrix for computation. Utilizing Bayesian quantum state tomography [32,33], which returns uncertainties commensurate with the data gathered, we can indeed estimate the full quantum state with these results, irrespective of their informational completeness. Applying the specific Bayesian workflow described in Ref. [34]—which employs a Bures prior distribution and accounts for raster scanning and $\mathcal {P}_W<1$ through a Poissonian likelihood—we find the fidelities for each state as shown in Table 1, defined as $\mathcal {F}_\rho = \langle\phi |\rho |\phi\rangle $. Since the outcomes are strongly correlated, the inferred states have relatively small uncertainty, even with measurements in two bases only. Computing log-negativity $E_\mathcal {N}$, we obtain a complete interval for $E_D$ of approximately $E_D \in [0.4,1.3]$ ebits for the states considered. This range is quite wide; we suspect that the much higher values for $E_\mathcal {N}$ result from the fact it applies quantum state constraints, in contrast to the entropic bound which treats the measurement results as raw probabilities. In other words, quantum state tomography is based on the assumption of a single ground truth state behind all measurement sets involved, as well as a physical model (Born’s rule) connecting this state to the observed outcomes, whereas our entropic calculation views the detection scenarios as isolated probability distributions with no direct connection to quantum measurement theory. It would be interesting to explore how the $E_D$ range may narrow with higher-fidelity results, which are limited here primarily by the resolution of the state preparation and measurement pulse shapers.

Finally, as an aside, we note that one can alternatively view the $\phi$ and $2\phi$ phase shifts imparted by the front-end pulse shaper as part of the measurement process, rather than state preparation, and pool all results of Fig. 4 into a single likelihood to estimate the state before the pulse shaper (ideally $| {\phi =0} \rangle$). Doing so, we find $\mathcal {F}_\rho =0.81\pm 0.02$ and $E_\mathcal {N}=1.32\pm 0.03$, comparable to the values in Table 1, albeit with a slight increase in entanglement possibly resulting from access to correlations in four—rather than just two—bases in the inference process; nonetheless, since these four bases collectively are not mutually unbiased, the information gained per measurement is nonoptimal [1].

4. Discussion

The frequency-bin DFT designs introduced and analyzed here reveal intriguing opportunities for quantum information processing tasks in frequency encoding, including QKD. Indeed, the high success probabilities of our DFT solutions actually address a challenge shared by time-bin DFT measurements as well: the satellite pulses produced by passive delay interferometers lead to an effective measurement probability of $1/d$ [11,12]. The frequency-bin DFTs here with near-unity success thus render our design similar in spirit to more complex active time-bin interferometers [35–37] that can in principle reach unit DFT measurement efficiency. As another application, aligned with the original motivation for the QFP [22], our DFT is precisely the operation required for a quantum interconnect of frequency-disparate matter qubits. Consider $d$ qubits, each in a tuned lambda energy scheme, such that pump-induced excitation from the ground to excited state is accompanied by a emission of a single photon at frequency $\omega _n=\omega _0 + n\Delta \omega$—a distinct value for each qubit $n$. By coupling these photonic modes into a single bus waveguide terminated in a $d$-point DFT, detection of a single photon in output frequency mode $m$ will herald generation of the $W$-like entangled matter state $| {\psi _m} \rangle = \frac {1}{\sqrt {d}} \sum _{n=0}^{d-1} e^{2\pi \mathrm {i} mn/d} | {0\cdots 0 1_n 0 \cdots 0} \rangle$, a spectral version of the Duan–Lukin–Cirac–Kimble (DLCZ) interconnect [38] generalized to $d>2$ qubits as explored previously in the spatial domain [39,40]. And since the frequency-bin version leverages a single spatial mode in a bus waveguide, environmental fluctuations are shared by all interfering bins, minimizing the need for active wavelength-level path stabilization and automatically circumventing an inherent challenge of path-encoded DLCZ-type protocols.

Finally, the fact that DFT gates of increasing dimension are realizable without the addition of more pulse shapers and EOMs makes our design of particular value with near-term technology. Admittedly, at our experimental value $\Delta \omega /2\pi = 20$ GHz, the $d=10$ solution would require the coherent combination of 9 RF tones up to a maximum frequency of 180 GHz—a questionable prospect both in terms of microwave engineering and raw bandwidth. Yet although the minimum spacing $\Delta \omega$ is limited in our case by the resolution of the diffractive pulse shaper, much tighter frequency spacings should be possible with integrated pulse shapers based on MRR add-drop filters [41–43]. For example, at $\Delta \omega /2\pi = 5$ GHz, the maximum modulation frequency for a 10-point DFT drops to 45 GHz, sufficiently low that even direct digital synthesis of the total waveform should be feasible. We therefore envision on-chip QFPs as the most promising route for the high-dimensional frequency-bin mixers discovered here.