Entanglement is an essential resource for nonclassical information tasks in quantum communication [1,2] and quantum computation [3,4], and has also been applied to nonlinear molecular spectroscopy [5]. However, quantifying entanglement requires nonphysical operations or post-processing of the measurement data [6].

One of the most common quantifications of entanglement is concurrence [7,8], and its determination generally requires quantum state tomography [9,10], which relies on projective measurements to form a set of projective bases on the Hilbert space of the system, providing all the information about the state. The number of projective bases increases exponentially with the system scaleup. When the scale of the system is very large, it is impossible to perform projective measurements. To circumvent this impossibility, several approaches have been proposed to quantify the concurrence, including collective measurements [11] and the interference between two copies of the quantum states [12]. However, such approaches contrapose two-qubit pure states and cannot be directly applied to the case of the complicated mixed state.

Recently, weak measurement technology has been used to directly detect quantum states, including the transverse wavefunction of the photon states [13], the density of a mixed two-qubit state [14], and even entangled states [15]. The weak measurement can extract a small amount of information from a single outcome. Although each outcome of the weak measurement is uncertain, the average value can build a certain value, named weak value, to reveal the state information.

In this study, we experimentally use the wake values to demonstrate the feasibility of entanglement quantification associated with a deep learning two-photon polarization-entangled system. Specifically, we compress the weak value information of two-qubit photonic states into local conditional states, which are illustrated via the spatial distributions of photons. An end-to-end relationship between the concurrence and the distribution of photons is determined by a well-trained convolutional neural network (CNN) [16]. We find that only two projective bases are needed to predict the concurrence, greatly reducing the operation time. Besides, the generalization of this method can quantify the entanglement of high-dimensional multiparticle pure states, and the advantage of this method is exponentially increased.

Here, we consider a two-qubit (A and B) system, which is in a class of quantum states ρAB as [17] ρAB(p,θ)=p|ψθ⟩⟨ψθ|+(1−p)IA/2⊗ρBθ,(1)where |ψθ⟩=cosθ|00⟩+sinθ|11⟩ and ρBθ=TrA[|ψθ⟩⟨ψθ|]. θ∈[0,45°] and p∈[0,1] are two state parameters. These states are of the form X-type, and the parameter p controls the components of pure states and mixed states. Figure 1 (insert) shows the main processes involved in our approach, in which we perform a local projective measurement ΓA on A, where ΓA=||0⟩+|1⟩2⟩⟨|0⟩+|1⟩2|. The information about entanglement is compressed into the local conditional state of B, written as ρBΓA=TrA[ρABΓA⊗IB]Tr[ρABΓA⊗IB].(2)To extract the information about entanglement, one can execute a weak measurement of the Pauli observable σ^x=|0⟩⟨1|+|1⟩⟨0| on B [18,19]. Via post-selecting on the state |k⟩ (k∈{0,1}), the obtained weak value [20,21] can be written as ⟨σ^x⟩⟨k|ΓAw=Tr[σ^xρBΓA|k⟩⟨k|]Tr[ρBΓA|k⟩⟨k|],(3)which is closely related to the density matrix elements of ρBΓA. This value reflects the distance between ρBΓA and the maximally mixed state 12IB, which is visually illustrated in the Bloch sphere in Fig. 1 (see Appendixes A and B for more details). The concurrence of the class states can be expressed as C(ρAB)=(3|⟨σ^x⟩⟨0|ΓAw||⟨σ^x⟩⟨1|ΓAw|−1)⁢|⟨σ^x⟩⟨0|ΓAw||⟨σ^x⟩⟨1|ΓAw||⟨σ^x⟩⟨0|ΓAw|+|⟨σ^x⟩⟨1|ΓAw|.(4)In particular, when ρAB is the pure state, C(ρAB) is reduced to 1−|⟨0|⟨σ^x⟩ΓAw|2, and ΓA is reduced to identity measurement [7].

To conveniently access the weak values, here we introduce orbital angular momentum (OAM) as a pointer, where the real and imaginary parts of ⟨σ^x⟩⟨k|ΓAw are encoded as the coordinate x- and y-positions, respectively, of the central singular point [18,19] of the Laguerre–Gaussian (LG) mode on the local spatial distribution (see Appendix C for more details). This singular center can be extracted by straightforward optimal estimation or the least-square fit method. However, this weak shifting of the singular center is too susceptible to be effectively identified because of the experimental noise. To solve this problem, we attempt to directly extract the concurrence from the local spatial distribution by using a deep learning (DL) method [22], which has been used to solve many-body [23], large-scale quantum tomography [24], quantum state [25], and nonclassical correlation [26] problems. Specifically, we use a CNN to establish end-to-end mapping from the photon distribution to the concurrence, which avoids taking account of the form of the fitting functions or the relations between the weak values and concurrence. To train the CNN, we prepare a large number of two-qubit states ρAB. The corresponding local spatial distributions are used as features, and the concurrences determined from the tomographic data are used as the label. After being trained with a substantial number of distribution-concurrence pairs, the CNN establishes a statistical functional mapping between the local spatial distributions and the concurrence. The concurrence of the training set is labeled using the traditional method [7,8] by reconstructing the density matrix ρAB. The CNN also considers the imperfection of the local spatial distribution, so this method has high accuracy and strong robustness.

As shown in Fig. 1, the CNN consists of two paths: the convolution path and the full connection (FC) path. There are three blocks in the convolution path. The input images are mapped into the feature space by convolution layers and downsampled by max-pooling layers to magnify the weak variance in each block. To fit the concurrence, the output results of the convolution path are flattened into a one-dimensional vector and sent to the FC path, which contains four layers with different neurons. The activation function of each layer is selected to be ReLU [y=max(0,x)] so that the network can fit the nonlinear function. We use the Adam optimizer [27], which can effectively prevent local optimization. The mean squared error (MSE), which is used as the loss function, is defined as 1n∑1n(Cacti−Cprei)2, where n is the total number of samples, Cacti represents the actual concurrence labeled by the traditional method, and Cprei means the predicted concurrence. Some neurons were deleted to prevent overfitting (see Appendix D for more details).

The experimental setup is shown in Fig. 2. A 20 mm long periodic KTiOPO4 (PPKTP) crystal located in the Sagnac interferometer [28] is pumped by a 404 nm continuous-wave diode laser to create a pair of polarization-entangled photons via a type-II spontaneous parametric down-conversion process. These two photons are then sent to Alice and Bob. To generate the entangled state cosθ|HH⟩+sinθ|VV⟩ (where H and V represent horizontal and vertical polarization, respectively), a half-wave plate HWP1 is used to control the parameter θ by rotating the polarization of the pump laser.

On Alice’s side, the photon beam passes through an unbalanced interferometer (UI), first separated into two paths by a beam splitter (BS). Two sufficiently long calcite crystals (CCs) with HWP4 set to 22.5° between them are placed on one of the paths to destroy the coherence between the different polarization components. In contrast, the state in the other path remains unchanged. Two variable filters (VFs) are used to change the relative photon counts in these two paths (N1 and N2), where the parameter p can be written as p=N1/(N1+N2). The two-qubit states ρAB(p,θ) can be prepared when these two paths are combined.

On Bob’s side, the photon passes through a weak coupling system (WCS). A vortex phase plate (VPP) is placed to generate an LG wavefunction ϕl(x,y) with an OAM of l=2. Thereafter, a thin calcite crystal (TBC) with a thickness of 0.7 mm is employed to introduce a weak interaction between the polarization and the momentum of the photon; its optical axis is set in the x–o–z plane, and it is oriented at 42° with respect to the z axis [29]. Combining a pair of HWPs (HWP5 and HWP6) with optical axes set to 22.5°, the interaction Hamiltonian is H^=ζσ^x⊗Px, where Px=−iℏ∂∂x→ represents the momentum operator, ζ is the coupling strength, and σ^x=|H⟩⟨V|+|V⟩⟨H|.

The polarization states on both sides are measured by a group of quarter-wave plates (QWPs), an HWP, and a polarization beam splitter (PBS). The coincidence from tomographic measurement is used as the label of CNN while the photon distributions are used as the input features of CNN. On Alice’s side, the photon is detected by a single-photon detector (SPD), and the signal is sent for coincidence or to trigger the intensified charge-coupled device (ICCD) camera on Bob’s side. In our experiment, we use a coincidence window of 3 ns and an exposure time of 1 s for all measurements. The projective direction on Alice’s side only needed to be selected as ΓA=|D⟩⟨D| (|D⟩=|H⟩+|V⟩2) in this experiment.

On Bob’s side, the photon is detected by an SPD and an ICCD camera in the reflected path and the transmitted path, respectively. The signal from SPD is sent for coincidence to reconstruct the experimental density matrix ρexp (the reflected path of the PBS). The fidelities (Tr[ρAB(p,θ)ρexpρAB(p,θ)]2) between the theoretical physical states ρAB(p,θ) and the experimental states ρexp are more than 92%. The concurrences Cact are calculated from ρexp [7,8]. The photon spatial distributions are detected by the ICCD camera. A small down-rightward deflection (ϕl(x,y)→ϕlH(V)(x−Re[⟨σ^x⟩⟨H(V)|w],y−Im[⟨σ^x⟩⟨H(V)|w]); see Appendix C for more details) is introduced [18,19] after post-selecting the photon state onto the basis |H⟩⟨H| (|V⟩⟨V|). ⟨H(V)|⟨σ^x⟩w can be obtained according to Eq. (3) with ρBΓA=TrA[ρAB|D⟩⟨D|⊗IB]/Tr[ρAB|D⟩⟨D|⊗IB] representing the corresponding local conditional state. The final intensities IH=|ϕlH|2 and IV=|ϕlV|2 of the photon spatial distributions are detected by the ICCD camera (Fig. 3). Obviously, only a single projective base is needed to detect the weak value ⟨H(V)|⟨σ^x⟩w while quantum state tomography (QST) needs 16 projective bases. The total number of photon pairs that contribute to each base is approximately 600,000 (600,000 repetitions) with integral time 1 s. If the number of repetitions is too small, the estimation error will be increased, and more data will be required to train the CNN.

In our experiment, we prepared 415 states, with 349 of the feature-label pairs being randomly chosen to be in the training data and the remaining 66 pairs serving as test data. Some photon distributions with different concurrences in the training set and an example of matrices ρAB(p=0.9,θ=0.81) and ρBΓA are shown in Fig. 3.

We first investigated the convergence of CNN. The performance with training time (epoch) is shown in Fig. 4(a). The weight of the CNN is optimized with the batch size 10 for 200 epochs. MSE (brown line) falls below 0.07 and PCC (green line) reaches 0.92 at the end of the training process, which demonstrates small prediction errors and strong relevance. The CNN learns more and more information concerning the entanglement with increasing epochs and is well trained when the epoch number is 200.

To demonstrate the predicted concurrence (Cpre) accuracy of the trained CNN, we compared it to the concurrence (actual concurrence CQST) calculated by QST. The predicted concurrences (brown dots) for the test data are shown in Fig. 4(b). The point distribution is close to the optimal curve (blue line), which means that the predicted concurrences approximate the actual concurrences. We can also find the errors (uncertainty) between the concurrence Cpre predicted by CNN and the actual concurrence CQST is less than 0.2 (|Cpre−CQST|<0.2). With further optimization of the CNN parameters and an increase in the training data, the error can be further reduced.

The predicted concurrence distributions in the parameter space of p−θ are shown in Fig. 4(c). The contour lines represent the theoretical boundaries of ρAB(p,θ), and the predicted concurrence agrees with the theoretical trend. The (p,θ) values of the dots are calculated by minimizing the value of Tr[ρAB(p,θ)−ρexp]. We further investigated the source of the error shown in Fig. 4(d). With an increase in the training time, the average MSE of the separable states in the test set approaches 10−4 (red bars), while the average MSE of the entangled states in the test set approaches 10−2 (blue bars); these values are reached when epoch is greater than 160. These results imply that the CNN is more accurate when determining separated states.

To estimate how the model is expected to perform in general cases with limited samples, a fivefold cross-validation was implemented with our CNN. The data set was split into five parts, and each part was chosen in turn to be the test set, while the rest were used as the training set sent to the CNN. The mean MSE and PCC values of all five results converged to 0.09 and 0.87, respectively, when the epoch number is 200, as shown in Fig. 4(e).

Figure 4(f) demonstrates that the result is dependent on the amount of data. With an increasing data size, we should be able to predict the entanglement with greater accuracy.

It is worth noting that our entanglement quantification method remains efficient even when scaling to pure states with larger numbers of qubits or higher dimensions. To illustrate this, we present the results of three-qubit pure states and two-qutrit pure states below. The concurrence for multipartite pure states ρP=|ψ⟩⟨ψ| in arbitrary dimensions can be expressed as [30] C(|ψ⟩)=minγi∈γ2(1−Tr[ργi2]),(5)where γ={γi} represents the set of all possible bipartitions {Ai|A¯i} of {1,2,…,n}, and ργi=TrA¯iρP is the reduced density matrix across these bipartitions.

For three-qubit states ρABC, there are three possible bipartitions: γ={{A|B,C}, {B|A,C}, {C|A,B}}. The concurrence of an arbitrary three-qubit pure state |ψABC⟩ can be rewritten as C(|ψABC⟩)=min{2det(ρA),2det(ρB),2det(ρC)}=min{1−|⟨0|⟨σ^x⟩ΓB,Cw|2, 1−|⟨0|⟨σ^x⟩ΓA,Cw|2,1−|⟨σ^x⟩⟨0|ΓA,Bw|2}, where ΓA,B, ΓB,C, and ΓC,A represent the measurement on the corresponding subsystem (A, B), (B, C), and (C, A), respectively. Clearly, using the weak measurement method, only three projective bases are required to quantify the entanglement of any three-qubit pure state. However, 64 projective bases are required for tomography.

In the computational basis {|0⟩,|1⟩,|2⟩}, an arbitrary two-qutrit pure state can be expressed as |ψAB⟩=∑i,j=02aij|ij⟩AB,∑i,j=02|aij|2=1.(6)

By combining Eq. (5) and Eq. (6), we can obtain C(|ψAB⟩)2=4det(ρB01)+4det(ρB02)+4det(ρB12), where ρBmn=ρB(mm)|m⟩⟨m|+ρB(mn)|m⟩⟨n|+ρB(nm)|n⟩⟨m|+ρB(nn)|n⟩⟨n| represents the reduced density matrix of Bob in the qubit subsystem. ρB(mn) is the matrix element of ρB=TrA[|ψAB⟩⟨ψAB|] in the m-th row and n-th column, where m,n∈{0,1,2} and m≠n. Furthermore, it can be easily demonstrated that C(|ψAB⟩)=1−|⟨σ^01⟩⟨0|ΓBw|2+1−|⟨σ^02⟩⟨0|ΓBw|2+1−|⟨σ^12⟩⟨1|ΓBw|2, where σ^mn=|m⟩⟨n|+|n⟩⟨m|, and ΓB represents the identity measurement. Clearly, for any two-qutrit pure state, only three projective bases are needed to quantify the entanglement. However, when employing tomography, 81 projective bases are required.

As for universal N-partite D-dimensional pure state, there are ∑j=1⌈N/2⌉CNj possible bipartitions and ∑k=1⌈D/2⌉CDk possible qubit subsystems; thus, no more than ∑j=1⌈N/2⌉∑k=1⌈D/2⌉CNjCDk projective bases are needed to quantify the entanglement. ⌈N/2(D/2)⌉ represents round N/2(D/2) down to the nearest integer, and CNj(CDk) represents the number of all combinations of taking j(k) distinct elements from N(D) distinct elements. In contrast, tomography requires D2N projective bases. Clearly, the benefits of our method become increasingly apparent as the scale and the number of dimensions of the system increase.

In this study, we established the relations between the concurrence and weak values, where the weak values were encoded into the transverse Laguerre–Gaussian (LG) photon distributions. By detecting the shift of singular center, we can directly get the concurrence. This weak measurement method will greatly reduce the number of projective bases and save the measurement resources. For a class of mixed two-qubit states, two projective bases are sufficient to quantify the entanglement, while 16 bases are required for tomography. Assisted by the DL method, we can achieve an “end-to-end” mapping between the photon distribution and concurrence without considering the form of the mapping functions. Moreover, this method, which takes the whole photon distribution as the input, can avoid estimation errors because it extracts the singular center weak shift.

We also show that the advantages of our method become more pronounced when scaling to larger numbers of qubits or higher dimensions, which demonstrates that our entanglement quantization method exhibits increasingly evident advantages over quantum state tomography as the system scales up. This highlights the potential of our method to efficiently characterize entanglement in large-scale quantum systems.

Acknowledgment. This work was supported by the Innovation Program for Quantum Science and Technology (2021ZD0301400 and 2021ZD0301200), the National Natural Science Foundation of China (61725504, 11821404, and U19A2075), and the Anhui Initiative in Quantum Information Technologies (AHY060300). M. Yang is supported by the Postdoctoral Innovative Talents Support Program (BX20230349) and the Fundamental Research Funds for the Central Universities (WK2030000085). Y. Shikano is supported by JSPS KAKENHI (17K05082, 18KK0079, and 19H05156) and JST PRESTO (JPMJPR20M4). Y. Xiao is supported by the National Natural Science Foundation of China (12004358), the Fundamental Research Funds for the Central Universities (202041012 and 841912027), the Natural Science Foundation of Shandong Province of China (ZR2021ZD19), and the Young Talents Project at Ocean University of China (861901013107).