Fast Quantifier of High‐Dimensional Frequency Entanglement through Hong–Ou–Mandel Interference

High‐dimensional frequency entanglement is an enabling resource in quantum technology due to its high information capacity and error resilience. A concise yet efficient method for precisely quantifying its dimensionality remains an open challenge, owing to the difficulties for performing required superposition measurements in energy‐time domains, and the complexity associated with full quantum state tomography that scales unfavorably with dimensions. With the assistance of Hong–Ou–Mandel experiment that performs a Fourier transform between the entangled photons in terms of joint spectral intensities and the quantum interference in terms of biphoton temporal coincidences, the concept of Shannon dimensionality as a fast quantifier of bipartite continuous frequency entanglement is unlocked. This quantitative technique reveals the complete distribution of frequency entanglement but without suffering from any limitation of modal capacity of the detection geometry. These results may significantly facilitate the use of quantum interference for characterizing the high‐dimensional entanglement nature by avoiding some stringent conditions.


Introduction
Harnessing high-dimensional entanglement may provide an essential tool for elevating the performance of advanced quantum applications. [1,2] In particular, in the context of quantum information processing, photon pairs entangled in high dimensions holds promises for enhancing the quantum channel capacities, [3,4] showing more resilience to noise, [5,6] increasing security against eavesdropping, [7,8] and even speeding up certain tasks in photonic quantum computation. [9,10] With increasing dimensionality, high-dimensional entanglement is predicted to violate various quantum inequalities more strongly that deepens our insight of quantum mechanics. [11,12] While several properties DOI: 10.1002/qute.202300012 of photons can be employed to study the high-dimensional entanglement including position-momentum and frequency-time, the latter has attracted great interest in various quantum-enhanced applications such as wavelength-multiplexing quantum networks, [13,14] entanglement-assisted absorption spectroscopy, [15,16] and highprecision quantum metrology. [17,18] It is crucial to have a quantifier of the dimensionality of entanglement as measured in an experiment. [19][20][21][22] While the characterization of entanglement can typically be implemented by using quantum state tomography, [23,24] a concise yet efficient method for quantifying the dimensionality of frequency entanglement is still a formidable challenge, owing to both the difficulty of performing required superposition measurements in the frequency domain, [25] as well as the general challenges associated with performing full quantum state tomography in a large state space. In classical information theory, the number of independent communication channels of a signal is known as the Shannon number in the spirit of Shannon. [26] The signal being the state of a physical system, the Shannon number is also referred to as the number of dimensionality, or the number of modes, of that system. [27,28] Thus, the concept of Shannon dimensionality is introduced to quantify the detectable modes of orbital angular momentum (OAM) entanglement, which is readily implemented by inserting appropriate angular state analyzers in the measurement setup. Unfortunately, this method still faces the ongoing challenge of performing required superposition measurements in the energy-time domains, which cannot be directly applied to measure the dimensionality of frequency entanglement. Additionally, while the Shannon dimensionality of discrete OAM property has been extensively studied, its investigation in the continuous energy-time domains still remains unexplored. Moreover, the OAM measurement by using angular state analyzers is limited by the modal capacity of the detection geometry and did not measure the true spectrum of the twophoton states. The reason for this discrepancy is that only light within small angular sections around diametrically opposed regions were collected, and thus most of the wave function is discarded. [29][30][31] To tackle this issue, an interferometric technique with bucket detectors is used such that the detection geometry does not shape the spectrum nor limits its dimensionality. [32] However, its counterpart in the energy-time domains is still lacking.
www.advancedsciencenews.com www.advquantumtech.com Here, we present a concise yet efficient method to measure the Shannon dimensionality by using Hong-Ou-Mandel (HOM) interference, [33] which reveals the high-dimensional quantum nature of the incident continuous entanglement. To show the versatility of our approach, we use it to experimentally measure the entanglement in the topical photonic spectral basis and temporal basis. We identify four main advantages associated with our method using HOM interference. i) Instead of performing the projective measurements, HOM interferometry provides an alternative method to measure the dimensionality with great advantages in reducing the experimental complexity. ii) The analogous versions of our method can be widely applied in several degrees of freedom, in particular for those have significant challenges in performing the conventional superposition measurements such as energy and time. [25,34] iii) It is obvious in our proof-of-principle experiment that the measurement complexity of HOM interference pattern is independent of the incident high-dimensional frequency entanglement. Thus, our quantitative technique scales favorably with dimensions. iv) As a direct result of the inherent stability of HOM interferometry, [35] our approach promises great robustness against the deleterious noise, which has the potential to be used in the practical applications.

Shannon Dimensionality for High-Dimensional Quantum Nature
To illustrate our theory, we consider the generic task of evaluating the high-dimensional quantum nature of frequency entanglement by Shannon dimensionality. Typically, the highdimensional discrete frequency entanglement can be prepared by using spontaneous parametric down-conversion (SPDC) process as [36] where k represents the number of dimensionality, sk and ik are central frequencies of the paired signal and idler photons, and f ( sk , ik ) is the corresponding spectral amplitude function with ∫ ∫ |f ( sk , ik )| 2 = 1. As it is customary in optics to describe ensembles by means of their mutual coherence function, [37,38] we introduce the spectral coherence function between paired photons as ( s , i ). This indicates that it furnishes an explicit and simple recipe to represent a given two-photon analyzer by a partially coherent field described by a spectral coherence function ( s , i ). Next, we apply the methods of image-analysis theory to determine the participating degrees of freedom of such a field. Then a modal decomposition is possible and ( s , i ) may be expressed as [39] where the pump is assumed to be infinitely narrow-band. The functions u ( ) are the eigenfunctions and the coefficients ( s , i ) are the corresponding eigenvalues. In fact, the field modes are just In the SPDC process that represents the present-day gold standard with respect to fiber coupling efficiency, entangled photon pair rates, and entanglement fidelity, the paired photons are generated by using a nonlinear crystal pumped with a ultranarrow pump laser. Thus, the frequencies of down-converted photons should satisfy the energy conservation as s + i = p , where p represents the central frequency of pump photons. Conversely, if s + i ≠ p , the joint spectral amplitude f ( s , i ) = 0 and makes no contribution to the calculation of Shannon dimensionality. As a direct result, we typically simplify the calculation of spectral coherence function ( s , i ) as a function of the difference frequency Δ = | s − i | as (Δ). Additionally, the paired photons are created simultaneously as an inherent property of SPDC process such that s = i = . In this case, Equation (2) becomes which indicates that can be obtained by performing an inverse Fourier transform on (Δ) as As a Fourier transformation of directly yields the full bi-photon's difference frequency modes distribution, it thus provides for a complete characterization of spectral coherence function. To quantify the number of detectable modes, let us where Δ m is the frequency detuning of well-separated frequency-entangled bins [36,40] This dimensionality can be interpreted as the number of channels available for communication purposes, [28,42] and also refers to as the number of dimensionality, or the detectable number of modes, of that incident frequency entanglement. Here, the Shannon dimensionality and the Schmidt number K have isomorphic forms in the definition equation, K describes the dimension that generates entanglement, and the effective Shannon dimensionality D is the dimension that measures entanglement, which is the joint property of the generating system and the analyzer, theoretically D ≤ K. [31]

Shannon Dimensionality through HOM Interference
The experimental measurement of the Shannon dimensionality can be implemented by using two-photon HOM interference.
More specifically, the discussed high-dimensional frequency entanglement |Ψ⟩ can be considered as the input quantum state of a HOM interferometer. [17,33,36] As a direct result of the imbalance of HOM interferometer, the relative time delay would introduce a wavelength-dependent phase shift as exp(iΔ ), where Δ = | s − i | is the difference frequency of two well-separated center frequency bins. Then these paired photons are incident on a balanced beam splitter from different input ports, whose operation can be expressed as a † This operation transforms the incident quantum state into a superposition state as a consequence of bunching and anti-bunching effects. The detection post-selection at two distinct detectors merely records the anti-bunching event such that the coincidence probability P( ) at the output of the HOM interferometer can be expressed as [17,36,43] Analogously, we assume that f ( s , i ) makes no contributions to the calculation of Shannon dimensionality if 1 + 2 ≠ p . Here we also use the difference frequency to simplify the equation as We find that the HOM interference probability P( ) is relevant to the eigenvalues as = |2P( ) − 1|. By following its definition, we can define the average number of detectable discrete frequency modes or effective dimensionality as This indicates that the Shannon dimensionality of frequency entanglement can be calculated as the inverse of the area underneath the square of the peak-normalized coincidence fringes |2P( ) − 1|. The Shannon dimensionality is a concept from classical information theory in the spirit of Shannon, where the number of independent communication channels of a signal is known as Shannon number. As we emphasize discrete frequency entanglement, the Shannon dimensionality of a single frequency-bin can be considered as D = 1 because only one spectral path exists. Backed by this assumption, the normalized effective dimensionality is obtained as where denotes the spectral bandwidth of single frequency-bin, and thus the numerator represents the effective dimensionality when there is merely one detectable spectral path. The Shannon number is referred to as the number of degrees of freedom, or the number of modes of the quantum state. For an instructive means of understanding, D eff = 1 represents that the paired photons are at a denegerate wavelength, namely a single spectral path, and for the Bell state in frequency domain with D eff = 2, (| 1 ⟩| 2 ⟩ + | 2 ⟩| 1 ⟩)∕ √ 2 describes the 2D frequency entanglement.

Experimental Implementation
While a biphoton frequency comb can be exploited to generate high-dimensional frequency entanglement, this method is relatively complex and with poor robustness to external disturbances, in particular it is not flexible to tune the free spectral range. Here we use two-photon HOM interference as an alternative tool to harness high-dimensional discrete frequency entanglement. As shown in Figure 1, the Sagnac interferometer is used to prepare polarization-entangled photons 2 at a degenerate wavelength of 810 nm, which are then routed into the first HOM interferometer.
Let us first consider that pairs of entangled photons with non-degenerate wavelength are created by using SPDC processing in a nonlinear crystal, which can be written as is the Gaussian spectral amplitude function with ∫ ∫ d s d i |g( s , i )| 2 = 1 that originates directly from the SPDC process. Here we note that the created sinc-like function resulted from the phase matching condition in a ppKTP crystal is approximated by using this Gaussian function for simplifying the mathematical calculation. Then these paired photons arrive at a balanced beam splitter from different input ports, which constitutes a HOM interferometer. The imbalance between two arms of this HOM interferometer T would introduce a wavelength-dependent phase shift as exp(i T). Build on the analogous analysis as discussed above, the coincidence probability at the output of HOM interferometer is obtained as [44] This indicates that photon anti-bunching occurs only when the two-photon frequency detuning satisfies ( s − i )T = (2n + 1) (n is an integer). Since this HOM interference acts as a biphoton spectral filter, the discrete frequency entanglement can be created. Of particular interest is that this method can flexibly harness the number of dimensionality and the single-photon spectral bandwidth of discrete frequency entanglement by concisely tuning the imbalance of HOM interferometer T (see refs. [36,45] for more details). As a direct consequence, the spectral coherence function of frequency-entangled photons can be expressed as It shows that the intensity of the bi-photon component manifests as a cosine oscillation within a Gaussian envelope, the period of which is related to T. [36,45] With prolonging the relative time delay, the number of dimensionality of discrete frequency entanglement would increase as shown in Figure 2a-d.
Then, we exploit polarization anti-correlations to deterministically eliminate noise contributions that arise from detrimental photon bunching in the preparation stage. [36] After this deterministic filtering process, the second HOM interferometer can be used to characterize the effective dimensions of frequency entanglement by measuring their non-classical spatial beating. The resulting interference patterns of the incident high-dimensional frequency entanglement is revealed by scanning the arrival time  The effective dimensionality D eff as a function of the imbalance of the first HOM interferometer T, which determines the dimensionality of the created discrete frequency entanglement. The solid line shows the theoretical prediction, the insets demonstrates the incident high-dimensional discrete frequency entanglement, and the stars represent the experimental results that are calculated by following Equation (10).
theoretical predictions and experimental measurements of Shannon dimensionality D eff (T) are shown in Figure 3. The effective dimension D eff (T = 0) = 1 represents that the spectrally indistinguishable photons are used as the reference for normalization, namely the detectable mode of degenerate photons are referred as a unit. As Figure 2a-d shows that the dimensions of discrete frequency entanglement increases with respect to the prolongment of the relative time delay T, the experimentally measured Shannon dimensionality, correspondingly, increases monotonously as shown in Figure 3. These experimental results agree well with the theoretical predictions, wherein slight deviations can be attributed to the experimental imperfection and the deleterious noise. With the assistance of HOM interference that performs a Fourier transform between the entangled photons in terms of joint spectral intensities and the quantum interference in terms of biphoton temporal coincidences, we experimentally succeeded in quickly quantifying the effective dimensionality of high-dimensional entanglement. Using the built-in image rotator, there is also a simple Fourier relationship between the two-photon interference visibility and the rotation angle, based on which, the high-dimensional entanglement contained in the OAM of entangled photon pairs can be easily analyzed. [32]

Conclusion
We present a fast, robust, concise yet efficient technique to quantify the effective dimensionality of high-dimensional entanglement with great advantages in reducing the experimental complexity, increasing the robustness to the deleterious noise, and showing its versatility in various photonic degrees of freedom. While our experimental realization is based on bulk optics, the overall scheme could also be extended to integrated quantum sources, where the quantifier for evaluating the incident entanglement still poses a significant challenge. [40,46] These results demonstrate that the quantum interference provides a powerful tool for measuring the dimensionality of entanglement, making it ideal for practical implementations of quantum protocols with general high-dimensional photonic quantum entangled states, even under undesired noise conditions.