1. INTRODUCTION

The recent rapid advancement in quantum technologies is sometimes called the second quantum revolution. Among the carriers of quantum states for use in quantum technologies, photons are appealing for their stable coherence at room temperature, ease of precise control, and efficient detection using existing technologies [1,2]. Linear-optic photonic networks with multiple single-photon inputs have been applied for quantum computation [3–6], quantum communication [7,8], and quantum sensing [9,10]. For instance, boson sampling has attracted attention since the calculation of the photon number distribution of single photons with a certain linear photonic network is a classically hard task [11–14]. Furthermore, generated quantum states in multimode–multiphoton systems and the entanglement between two systems will be a useful resource for various tasks, for instance, in all-photonic quantum repeaters [15] or the remote-state preparation of quantum states [16].

Normally, entanglement is evaluated using the estimated state density matrix via quantum state tomography [17]. However, there is a critical problem in this method. The required resources for quantum state tomography increase exponentially as the number of photons or the number of modes of the system increases [18]. The problem has been addressed for a number of specific cases of multiphoton entanglement by the introduction of appropriate criteria for the detection of entanglement, represented by experimentally observable correlation sums known as entanglement witnesses. Entanglement witnesses have been formulated for multi-partite entanglement between individual photons [1,19] and for multiphoton two-mode states [20,21]. However, none of the previous approaches applied to a scalable bipartite system where both the number of photons and the number of modes in each of the two optical subsystems increase jointly. To facilitate the development of scalable multiphoton multimode entanglement, it is therefore necessary to develop the corresponding scalable entanglement criteria based on only a small number of easily realized measurements.

For this purpose, we have recently derived a statistical bound (the aforementioned entanglement witness) that can verify entanglement between two systems with an arbitrarily high number of modes and photons using only the direct detection of photon number distributions between the modes and the photon number distributions detected after using a discrete Fourier transform (DFT) of the modes [22]. For two-photon polarization entangled states, $|{\psi _0}\rangle = (|H\rangle |V\rangle - |V\rangle |H\rangle)/\sqrt 2$, the entanglement can be verified by the observation of perfect anti-correlation in both of two non-orthogonal basis sets: horizontal/vertical polarizations and diagonal/anti-diagonal polarizations. In larger linear-optic networks, the DFT replaces the rotation of polarization as transformation between two mutually unbiased sets of modes. The entanglement can then be verified from just two sets of “classical correlation tables” representing the photon number distributions obtained from the original input modes and from the mutually unbiased interference of these modes in the DFT. The exponential increase in resources required for full tomography is thus replaced with an increase that is polynomial in the numbers of photons and modes.

In this paper, we report the first experimental demonstration of the verification of entanglement between two multimode–multiphoton systems. As a simple but non-trivial example, an entangled state between one photon in three modes and two photons in three modes is investigated. For the experiment, a pair of linear-optic quantum circuits for DFT for three optical modes, where a total of six multiphoton interferences occur at beam splitters (BSs), has to be implemented and stabilized with perfect phase synchronization. We have overcome this difficulty by combining the displaced-Sagnac architecture [4,9,23] and hybrid polarizing BSs [23]. In the experimental result, clear correlation can be observed in both photon number distribution tables with (${\rm{K}}$) and without (${\rm{n}}$) DFT, and the measured sum of the corresponding fidelities is ${F_{\rm{n}}} + {F_{\rm{K}}} = 1.555 \pm 0.018$, which exceeds the lower bound, 4/3, for the verification of entanglement between the two systems with 12.3 standard deviations. Our experimental result paves the way to understanding the coherence and entanglement of multi-partite systems, not only with photons but also with other quantum resources with a large number of degrees of freedom.

2. ENTANGLEMENT VERIFICATION SCHEME FOR MULTIMODE–MULTIPHOTON SYSTEMS

A. General Theory

In quantum optics, it is well known that photon number states are highly non-classical states. One aspect of this non-classical nature is that the delocalization of a single photon between the two output ports of a 50:50 BS generates an entangled state, where the entanglement is represented by the coherent superposition of the vacuum and the single-photon state in each mode. It is therefore possible to generate scalable entanglement by simply splitting the input beams from $M$ single-photon sources into $2M$ beams, which are then separated into two locations A and B. Using the photon number basis of the $2M$ modes, this entangled state can be written as

Importantly, this superposition can be multiplied out into a superposition of product states of multiphoton multimode states at location A and the corresponding multiphoton multimode states at location B. The quantum state in Eq. (1) thus describes bipartite entanglement between the collective multimode–multiphoton system in A and the collective multimode–multiphoton system in B. Note that this kind of entanglement exists even after the total photon numbers in A and in B are determined. It is therefore possible to observe entanglement in any distribution of $N$ photons in A and $M - N$ photons in B as long as $1 \le N \le M - 1$.

The experimental challenge posed by these entangled states is that the practical consequences can be seen only in correlated multiphoton interference effects within local systems A and B. It is possible to make these effects observable by using a mode DFT [22]. The DFT is realized by equally interfering all $M$ input modes with a mode-dependent phase shift defined by the product of the input mode index $m$ and the output mode index $k$. The transformation of multiphoton coherences can then be described by the relation between the creation operator $\hat b_k^\dagger$ of the DFT output modes with the creation operators $\hat a_m^\dagger$ of the input modes used to define the photon number basis of Eq. (1):

It has been shown that the multiphoton coherences of an $N$-photon–$M$-mode system in the input basis are characterized by the sum of the $N$ output mode indices ${k_i}$ for the $i$th photon observed in the output modes of the DFT [22]. From the entangled state in Eq. (1), the sum of the mode indices ${k_i}({\rm{A}})$ in system A and the mode indices ${k_i}({\rm{B}})$ in system B must be a multiple of the mode number $M$. If ${N_{\rm{A}}}$ photons are detected in the $M$ modes of system A and ${N_{\rm{B}}}$ photons are detected in the $M$ modes of system B, the entanglement of $|\psi \rangle$ guarantees that the outcomes satisfy the relation [22]

As explained in the introduction, entanglement can be verified by observing strong correlations between at least two complementary properties that are related to each other by quantum superpositions. In the present case, these correlations between complementary properties are represented by the anti-correlated distribution of photons in the input modes shown in Eq. (1) and the condition for the sums of output mode indices of the DFT shown by Eq. (3). In the absence of entanglement, the uncertainty principle puts a strict limit on correlations between the two complementary properties. This limit can be expressed in terms of the sum of the fidelities with which the intended correlations are observed. The experimental fidelity ${F_{\rm{n}}}$ is defined as the probability of finding exactly one photon in each input mode $m$, either in A or in B. Mathematically, this condition can be expressed by $M$ sums over the photon numbers ${n_m}({\rm{A}})$ and ${n_m}({\rm{B}})$,

For separable states, the highest possible value of the fidelity sum ${F_{\rm{n}}} + {F_{\rm{K}}}$ is achieved when ${F_{\rm{n}}} = 1$ for photon number distributions with ${d_c}$ different possible $k$ values, so that the probability of satisfying Eq. (3) is $1/{d_c}$ [22]. The sum of the fidelities therefore serves as an entanglement witness, where entanglement is successfully verified if the experimental data exceeds the bound

Note that if $M$ is a prime number, ${d_c}$ is always equal to $M$. A detailed discussion of the value of ${d_c}$ is given in Ref. [22].

B. Example: One Photon in Three Modes and Two Photons in Three Modes

The density matrix of the entangled state in Eq. (1) can be decomposed into sub-matrices of the cases that have exactly $N$ photons in A and $M - N$ photons in B ($N = 1$ to $M - 1$). To understand the essence of the proposed method, it is useful to limit the analysis of the entanglement to the specific distribution of input photons between systems A and B. In the following, as a simple but non-trivial example, we perform the experiment for the three-mode case $(M = 3)$ and characterize the entangled state obtained whenever two photons are detected in A $(N = 2)$ and one photon is detected in B, shown in Fig. 1. The ideal output state in the input mode basis is then

This state describes entanglement between the six-dimensional Hilbert space of two photons in the three modes of A with the three-dimensional Hilbert space of a single photon in the three modes of B. For a state in this combination of the two Hilbert spaces, there are 18 possible outcomes, but only three of them contribute to the fidelity ${F_{\rm{n}}}$, which is given by

Here, ${P_{{\exp}.}}({p_{\rm{A}}},{p_{\rm{B}}})$ is the pattern probability represented by the photon number distribution in A and B. If DFTs are applied to both A and B [Fig. 1(b)], the sum of the two mode indices ${k_i}({\rm{A}})$ in A should be related to the single-mode index $k({\rm{B}})$ of the photon in B, so that

There are two possible solutions for each value of

 

Fig. 1. (a) Schematic setup for evaluating ${F_{\rm{n}}}$ of the entanglement between two photons in three modes (A) and one photon in three modes (B) without DFTs. BS, beam splitter. (b) Schematic setup for evaluating ${F_{\rm{K}}}$ of the entanglement between A and B with DFTs.

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For separable states, the maximal fidelity sum is ${F_{\rm{n}}} + {F_{\rm{K}}} = 4/3$, achieved by product states that satisfy one of the two conditions and randomly satisfy the other condition with a probability of 1/3. Entanglement is verified if

3. EXPERIMENTAL DEMONSTRATION

We designed and constructed a three-mode DFT circuit. Figure 2(a) shows the three-mode DFT, consisting of linear-optical devices, namely, balanced BSs ($R = 1/2$), an unbalanced BS ($R = 1/3$), and phase shifters. Figure 2(b) shows a schematic diagram of the optical quantum circuit. Each of the three input photons is reflected and transmitted by the 50:50 BSs. For the evaluation of ${F_{\rm{n}}}$, the photons in the three optical modes in A and B are measured directly by single-photon detectors. To evaluate ${F_{\rm{K}}}$, the three optical modes in A and B are transformed by the DFT circuits, and the output photons are measured by the detectors. To evaluate the entangled state in Eq. (8), we analyzed the events where there were two photons in A and one photon in B. Note that the three phase shifters on the input side shown in Fig. 2(a) are not necessary because the phase shifters affect only the overall phase for the entanglement.

 

Fig. 2. (a) Three-mode DFT circuit using linear optics. (b) Schematic illustration of the evaluation of ${F_{\rm{n}}}$ and ${F_{\rm{K}}}$. (c) Implementation of the setup using the polarization degree of freedom. (d) Implementation of the setup in (c) using displaced Sagnac architecture and a hybrid beam splitter (HBS). BS, beam splitter; UBS, unbalanced beam splitter (reflectance of 1/3); PS, phase shifter; PBS, polarizing beam splitter; HWP, half-wave plate.

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In order to realize the optical circuit in Fig. 2(b), we need to stabilize the path length differences of the complicated interferometer consisting of the three BSs and the two DFT circuits. To solve this difficulty, we implemented a compact optical circuit using the polarization degree of freedom [Fig. 2(c)] and the displaced Sagnac architecture [4,9,23] with a hybrid BS (HBS) [23] consisting of an unbalanced BS ($R = 1/3$) and a mirror [Fig. 2(d)]. In this setup, all six optical modes pass though the same optical components inside the interferometer, and the optical circuit is extremely robust against drafts and vibrations of the optical components.

Figure 3 shows the experimental setup. A femtosecond pulsed laser (Tsunami with second-harmonic generation, Spectra Physics, repetition rate of 82 MHz and a central wavelength of 390 nm) is used to pump a beta barium borate (BBO) crystal (type-I, thickness of 1.6 mm). The injected pump laser pulses generate photon pairs via spontaneous parametric down conversion (SPDC), in the forward direction. After passing through the BBO crystal, the pump laser is reflected by a mirror and also generates photon pairs in the backward direction [24,25]. The generated photons pass through the narrow band pass filters (center wavelength of 780 nm, FWHM of 2 nm), and are coupled into polarization maintaining fibers. One of the photons in the backward pair is detected by a single-photon detector (SPCM-AQR, Excelitas Technologies), and the output electric signal is used as a trigger. The other three photons, namely, both of the photons in the forward pair and one of the photons in the backward pair, are diagonally polarized by half-wave plates (HWPs), becoming superposition states in either horizontal or vertical polarization. To evaluate ${F_{\rm{K}}}$, the three photons are injected into the displaced Sagnac interferometer corresponding to the DFT circuits for the three optical modes in both polarization modes, as shown in Fig. 3. Note that in our experiment, the visibility of the Sagnac interferometer measured using single photons was 99.92%.

 

Fig. 3. Experimental optical setup for the evaluation of ${F_{\rm{n}}}$ and ${F_{\rm{K}}}$. BBO, beta barium borate; BPF, band pass filter; HWP, half-wave plate; BS, beam splitter; HBS, hybrid beam splitter; FPBS, fiber polarization beam splitter; FBS, fiber beam splitter; APD, avalanche photodiode.

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Fig. 4. Photon number distributions in the output modes (a),(c),(e) without DFT and (b),(d),(f) with DFT. In the graphs showing the results obtained with DFT, the sums of the ${k_i}$ are indicated for each photon number distribution, with $k({\rm{A}}) = {k_1}({\rm{A}}) + {k_2}({\rm{A}})$ (mod 3). Graphs on top: (a),(b) ideal correlations of the pure state; graphs in the middle: (c),(d) experimental results; and graphs on the bottom: (e),(f) results of a numerical simulation of the error sources. Note that the vertical axis in the graphs of the experimental results [(c),(d)] show the actual number of counts obtained in the experiments, and the raw data are shown in Supplement 1.

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The photon number distributions in six optical modes [three spatial modes in either horizontal (system A)/vertical (system B) polarization] were obtained by nine single-photon detectors. At each of the three output modes, the polarization modes were first converted to the spatial modes by the fiber polarization BS (FPBS). Then, the horizontal polarization (system A) component was detected by two cascaded single-photon detectors after a fiber BS (FBS) in order to discriminate the photon numbers up to two, and the vertical polarization (system B) was detected by a single-photon detector. Note that the intrinsic 50% loss of the detection efficiency for the events where two photons are in the same spatial mode in system A is compensated for when the output photon number distribution is calculated.

To evaluate ${F_{\rm{n}}}$, the three diagonally polarized photons were reflected by a flip mirror at three points in front of the DFT circuit (points are indicated by “F”), and then the measurement was performed using nine single-photon detectors. In both of the above cases (with and without DFT circuits), the photon number distributions were obtained by four-fold coincidence events (two detection events among six detectors in system A, one detection event among three detectors in system B, and the trigger signal) using a time-to-digital converter (DPC-230, Becker & Hickl GmbH).

4. RESULTS

A. Verification of Entanglement

Figure 4(c) shows the experimentally obtained correlation tables between the photon number distributions for systems A and B without a DFT circuit. The table for the ideal case is shown in Fig. 4(a). In the ideal table, the probabilities for $|011{\rangle _{\rm{A}}}|100{\rangle _{\rm{B}}}$, $|101{\rangle _{\rm{A}}}|010{\rangle _{\rm{B}}}$ and $|110{\rangle _{\rm{A}}}|001{\rangle _{\rm{B}}}$ are all 1/3, and the probabilities for the other cases are zero. The experimental result [Fig. 4(c)] is in good agreement with the theoretically expected distribution. From this experimentally obtained table, we can calculate ${F_{\rm{n}}} = 0.843 \pm 0.011$. Hereafter, we assume Poisson statistics for evaluating the uncertainty. Figure 4(d) shows the experimentally obtained correlation tables between the photon number distributions for systems A and B with DFT circuits. The table for the ideal case is shown in Fig. 4(b), showing that the probabilities for $|200{\rangle _{\rm{A}}}|100{\rangle _{\rm{B}}}$, $|020{\rangle _{\rm{A}}}|010{\rangle _{\rm{B}}}$, $|002{\rangle _{\rm{A}}}|001{\rangle _{\rm{B}}}$ $|011{\rangle _{\rm{A}}}|100{\rangle _{\rm{B}}}$, $|101{\rangle _{\rm{A}}}|010{\rangle _{\rm{B}}}$, and $|110{\rangle _{\rm{A}}}|001{\rangle _{\rm{B}}}$ are 2/9 or 1/9, and the probabilities for the other cases are zero. The experimental result [Fig. 4(d)] is in good agreement with the theoretically expected distribution. From this table, we can calculate ${F_{\rm{K}}} = 0.712 \pm 0.014$. The sum of the fidelities is ${F_{\rm{n}}} + {F_{\rm{K}}} = 1.555 \pm 0.018$, which clearly exceeds the upper bound of the separable states 4/3 with a 12.3 standard deviation. As we have seen, the existence of entanglement in the given three-photon, six-mode state is successfully verified using just two sets of correlation measurement tables and the calculated fidelities ${F_{\rm{n}}}$ and ${F_{\rm{K}}}$.

In the experimental results shown in Figs. 4(c) and 4(d), there are some unexpected detection events [e.g., $|200{\rangle _{\rm{A}}}|100{\rangle _{\rm{B}}}$ in Fig. 4(c) and $|020{\rangle _{\rm{A}}}|100{\rangle _{\rm{B}}}$ in Fig. 4(d) for which the theoretical detection probability is zero], and slight deviations in the frequency from the ideal cases. We think these are due mainly to excess photon pair emission events from the sources and spatiotemporal mode mismatches of photons [26]. For a detailed estimation of the amount of these error sources, see Supplement 1. We numerically simulated the possible outcomes under the errors taking the excess photon pair emission events up to three pairs, the spatiotemporal mode mismatch of photons, non-ideal reflectivity of balanced/unbalanced BSs in DFT circuits, and non-uniform quantum efficiencies of detectors and non-ideal beam splitting ratios of FBSs, following the method in Ref. [27]. We performed numerical simulations, shown in Figs. 4(e) and 4(f), which are in good agreement with the experimentally obtained results. The estimated fidelities of these numerical simulations are ${F_{\rm{K}}} = 0.731$ and ${F_{\rm{n}}} = 0.872$, which are also in good agreement with the measured fidelities, suggesting that our error analysis is reasonable. From these results, the 16% error for ${F_{\rm{n}}}$ was caused mainly by excess photon pair emission from the SPDC source and the non-uniform quantum efficiency of the detectors. For the 29% error for ${F_{\rm{K}}}$, the spatiotemporal mismatch of the photons and non-ideal beam splitting ratio of the BSs in DFT contributed almost equally to the excess photon pair emission and non-uniform detection efficiencies.

B. Separable State

Finally, the experimental results for a separable state of the three-photon, six-mode state are shown in Fig. 5. For this experiment, we used exactly the same experimental setup as before, but intentionally used distinguishable photon inputs by changing the arrival times of the photons at the input to the DFT circuits. Since the temporal separation of the photons leads to a suppression of multiphoton interference effects, the experimentally obtained distribution with DFT circuits [Fig. 5(b)] does not show the characteristic correlations of the entangled state [Fig. 4(d)]. Instead, the result is similar to the theoretical expectation for independent single-photon interferences in the DFT shown in Fig. 5(a). The experimentally observed fidelity of the correlation between ${k_i}$ sums drops to ${F_{\rm{K}}} = 0.344 \pm 0.014$. Since time delays do not change the correlations observed without the DFT, the results in Fig. 4(c) can be used to evaluate the sum of the fidelities, ${F_{\rm{n}}} + {F_{\rm{K}}} = 1.187 \pm 0.018$, which is 8.1 standard deviations below the upper bound of 4/3 for separable states. Here too, the numerical simulation taking the effect of the excess photon pair emission events from the sources and the spatiotemporal mode mismatch of photons shown in Fig. 5(c) and the calculated fidelity ${F_{\rm{K}}} = 0.333$ is in good agreement with the experimental results.

 

Fig. 5. Experimental results for the output with DFT obtained with an input simulating a separable state between multiphoton–multimode systems. The arrival times of the photons are mutually delayed by path lengths of 300 µm, which is sufficiently large to avoid mutual overlap of photonic wave packets with coherent length of 134 µm calculated from the wavelength bandwidth (2 nm). Top graph: (a) ideal correlation table; middle graph: (b) experimental results (raw data shown in Supplement 1); graph (c): numerical simulation of the effects of errors.

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5. CONCLUSION

In conclusion, we have reported the experimental demonstration of direct and efficient verification of entanglement between two multimode–multiphoton systems, using just two sets of classical correlation tables by using DFT circuits. As a simple but non-trivial case, an entangled state between one photon in three modes and two photons in three modes was investigated, and a clear correlation was observed in the photon number distribution tables, both with (${\rm{K}}$) and without (${\rm{n}}$) DFT, and the measured sum of the two fidelities was ${F_{\rm{n}}} + {F_{\rm{K}}} = 1.555 \pm 0.018$, which exceeds 4/3, the upper bound for the separable, with a 12.3 standard deviation, successfully demonstrating verification of the entanglement in the given three-photon, six-mode state. Furthermore, we performed experiments using separable states of photons and confirmed that the sum of the experimentally obtained fidelities, $1.187 \pm 0.018$, did not exceed the upper limit of 4/3.

Our results successfully demonstrate that the entanglement between two high-dimensional systems realized by multiple photons in multiple modes can be performed with just two sets of correlation table measurements using DFTs, which is a dramatic reduction in the required resources compared to the normally used quantum state tomography. In principle, our scheme is scalable since DFT circuits for any number of modes $M$ can be constructed using linear optics [28]. For future implementation in larger scale systems, an on-chip photonic quantum circuit is a promising candidate [29–31]. The demonstrated scheme can be applied not only to photonic systems but also to other quantum resources with high-dimensional degrees of freedom.