1. Introduction

Entanglement is one of the most significant quantum features, playing a principal role in the study of both quantum foundation and quantum information. A given entangled system cannot be characterized as a simple collection of individual subsystems, even when they are space-like separated [1,2]. It is also a crucial resource of various quantum tasks including quantum computation [3], quantum communication [4,5], and quantum sensing [6].

However, while the entanglement between non-identical particles is mathematically well-described as the superposition of different multipartite states [7], its formalism cannot be directly applied to the case of identical particles. A set of identical particles has the exchange symmetry [8], which makes the Hilbert space of the particles non-factorizable, and thus, every state of identical particles becomes a superposed multipartite state in the first quantization language. Hence, one can say that any particles that are identical to each other are entangled even when they are prepared in worlds apart [9], which seems implausible. This is called the non-factorization problem of identical particles [10–12].

To solve this puzzle, we need to discard unphysical entanglement from the physical one. Such unphysical entanglement is merely mathematical and, therefore, it cannot be extracted by any physical setup [13–22]. Indeed, when we say a multipartite system has nontrivial entanglement, the entanglement can be detected by registers that are distinguishable to each other [23,24]. There are several approaches to formally define the entanglement of identical particles to overcome the non-factorization problem of identical particles. The recent crucial formalisms include algebraic approaches [10,25], the second quantization approach [26–28], no-labeling approach [29–31], and the first quantization approach (1QL) [32]. All these formalisms provide criteria to distinguish the mathematical entanglement from physically extractable entanglement by using exchange symmetry. The mathematical connection among the aforementioned approaches is explained in Ref. [33] from the viewpoint of the partial trace.

One of the imperative queries closely related to the above studies on the entanglement of identical particles is: under what conditions can a set of identical particles carry physical entanglement? Answering this question not only provides an insight to understand the fundamental property of entanglement, but also practically helps to construct quantum systems that employ identical particles as information processing resource [34]. It is well-known that such set is able to possess entanglement only when the particles spatially overlap [31,35,36], or more rigorously speaking when non-zero spatial coherence between particles and detectors exists [32]. Hence, one can surmise that the particle indistinguishability and spatial overlap are two key factors for identical particles to be entangled. On the other hand, while theoretical studies on the quantitative relation of entanglement to spatial overlap alone have been executed before [31,32,36], which are experimentally verified in some recent works [37,38] there has been no attempt to examine the effect of both indistinguishability and spatial overlap to the entanglement of identical particles at the same time either theoretically or experimentally.

In this paper, we theoretically predict the quantitative relation of entanglement between two bosons with particle indistinguishability and spatial overlap using recently proposed partial trace technique [29–31], which is then experimentally verified using photons. We show that the amount of entanglement, which is quantified as concurrence, is a monotonically increasing function of both indistinguishability and spatial overlap. The obtained data confirm that two kinds of quantum properties, i.e., particle indistinguishability and spatial overlap, cooperate to generate entanglement.

2. Theory

This section presents a theoretical analysis on entanglement generation among identical particle with respect to the particle indistinguishability and spatial overlap. Since the conceptual distinction between particle identity and distinguishability can be subtle, we briefly explain it in Supplement 1.

We first introduce a formalism for calculating the entanglement between two bosons in the generic system. Then, we derive a formula for the concurrence to investigate the behavior of entanglement with the variation of the distinguishability and spatial overlap.

Suppose that we have a system of two bosons that are in the states ($\Psi _A$, $\Psi _B$). Then, from the exchange symmetry of bosons, the total state $|\Psi \rangle$ can be written as

Since our goal is to clarify the role of indistinguishability and spatial overlap in the generation of entanglement, we need three degrees of freedom to present the particle states, i.e., internal pseudospin (which will be detected at the output modes, defining in which degree of freedom the entanglement will be found), distinguishability function, and spatial distribution function. Hence, $\Psi _i$ ($i=A,B$) are now specified as $\Psi _i = (\psi _i, s_i, \phi _i)$ where $\psi _i$ is the spatial wave function, $s_i$ is the spin value, and $\phi _i$ is the particle distinguishability. In our experimental setup in Section 3., $(s_i,\phi _i)$ are realized as polarizations and temporal modes. However, it is not inherent at all to impose the role of physical quantities in such a way. The role of polartizations and temporal modes can be reversed, or any internal properties for a given physical setup can play the role of ($s_i, \phi _i$).

Considering that the result becomes trivial for $s_A=s_B$, we restrict our case into $(s_A,s_B) = (\uparrow ,\downarrow )$. The total state of two bosons is now presented as

In order for a multipartite system to have physical entanglement, the subsystems should be individually accessible by observers [39]. In addition, we consider the entanglement of general bosons, which conserve particle number superselection rule (SSR) [40], by which massive bosonic states with different particle number distributions cannot be coherent. One of such entanglement measures is the entanglement of particles introduced in Ref. [23], which is defined as

Figure 1 is a conceptual diagram to extract physical entanglement from two bosons. Since we are interested in the case only when two particles are detected at two observers $L$ and $R$, the spatial relations of the particles to the detectors are expressed in the most general form as

 

Fig. 1. Generic setup for detecting entanglement of two identical particles: The spatially overlapped particles have internal spins ($\uparrow$ and $\downarrow$) and distinguishability (represented as two different colors, red and blue, in the figure). Two detectors located at $L$ and $R$ indicate the spins of the particles.

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Since we are supposed to measure only the pseudospin states, the actually measured particle state is that which the distinguishability part of $|\Psi \rangle _{\vec {N} =(1,1)}$ is traced out. By defining the complete orthonormal basis of particle distinguishability as $\{|X_a\rangle\}$ so that

The concurrence of the state $\rho ^{m}_{(1,1)}$, which quantifies the amount of $E_P$, is computed using the definition in Ref. [42,43] as

Equation 12 shows that the concurrence is determined by three variables, $|\alpha _l|$, $|\beta _l|$, and $|\langle\phi _A|\phi _B\rangle|$. The particles are unentangled when they are completely distinguishable, i.e., $\langle\phi _1|\phi _2\rangle=0$, or when one of the particles is detected at only one side, i.e., one of $\{\alpha _l,\alpha _r,\beta _l, \beta _r\}$ is zero. On the other hand, the particles are maximally entangled when the indistinguishability and spatial overlap are both maximal, i.e., $|\langle\phi _1|\phi _2\rangle|=1$ and $|\alpha _l| = |\beta _l|=\frac {1}{\sqrt {2}}$.

3. Experiment

In an optical system, photons are the identical particles and the photon polarization $\{|H\rangle ,|V\rangle \}$ compose the spin $\{|\downarrow \rangle ,|\uparrow \rangle \}$, where $|H\rangle$ and $|V\rangle$ denote the horizontal and vertical polarization states, respectively. Particle distinguishability can be controlled by other degrees of freedom such as momenta, spectra, and temporal modes. Here, it is controlled by the temporal modes of single photons.

The identical photon pairs, centered at 780 nm, were generated at a type-II BBO crystal via spontaneous parametric down conversion pumped by a femtosecond pulse at 390 nm [44,45]. To guarantee the spectral and spatial indistinguishability, two photons were filtered by 3 nm bandpass interference filters (IF) at 780 nm, then collected by single-mode fibers (SMF). Next, they were sent to the experimental setup shown in Fig. 2 in order to investigate entanglement behavior via particle indistinguishability and spatial overlap.

 

Fig. 2. The experimental setup to entangling photons via their spatial overlap and indistinguishability. Pol. : polarizer, H$^{*}$ : half waveplate at $\theta$, BD : beam displacer, H$_{45}$ : half waveplate at 45$^{\circ }$, WP : waveplates, PBS : polarizing beamsplitter, D1-D4 : single photon detectors, CCU : coincidence counting unit. The optical delay $l$ of a photon has been scanned in order to introduce the particle indistinguishability.

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The initial polarization states of the incoming photons $A$ and $B$ were configured by polarizers (Pol.) and half waveplates (HWP, H$^{*}$). Then, its probability amplitudes were divided into two spatial modes by a beam displacer (BD1) which transmits and reflects horizontal and vertical polarization states, respectively. The HWP at $45^{\circ }$ ($\textrm {H}_{45}$) at spatial modes $\alpha _L$ and $\beta _L$ converted the polarization states from horizontal to vertical and vice versa. Therefore, the polarization states of photon $A$ and $B$ became vertical and horizontal, respectively, regardless of the spatial modes. Subsequently, the spatial modes $\alpha _R$ and $\beta _R$ ($\alpha _L$ and $\beta _L$) were combined by BD2 (BD3) which corresponds to the spatial wavefunction overlap. The two-qubit polarization state $\rho _{\textrm {ex}}$ at the outputs of BD2 and BD3 ($L$ and $R$) was reconstructed via two-qubit quantum state tomography (QST) with maximum likelihood estimation. The waveplates (WP) and polarizing beamsplitters (PBS) were utilized to perform various projective measurements for QST [46,47]. The coincidence counts were registered by a home-made coincidence counting unit (CCU) [48].

 

Fig. 3. Coincidence counts with respect to scanning the optical delay. A HOM dip can be observed for the coincidences D12 (experimental data and fitting curve in red) and D34 (experimental data and fitting curve in blue), the visibilities were $V=0.99\pm 0.04$ and $V=0.91\pm 0.05$, respectively. The error bars represent one standard deviation due to Poissonian counting statistics.

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The experimental setup faithfully demonstrates the conceptual diagram of Fig. 1. The degree of spatial overlap at $L$ and $R$ could be adjusted by changing the initial polarization states. By using $|H\rangle$ transmitting polarizers and HWPs at $\theta$, the initial polarization state becomes $\cos 2\theta |H\rangle +\sin 2\theta |V\rangle$. Note that we set the same initial polarization states for the photons $A$ and $B$ in order to investigate the symmetrical spatial overlap case. After BD1, the coefficients of polarization state are transferred to the probability amplitudes of spatial modes, so $\alpha _L=\beta _R=\sin 2\theta$ and $\alpha _R=\beta _L=\cos 2\theta$. Therefore, one can find the degree of spatial overlap becomes

The degree of distinguishability can be tuned by changing one of the photons arrival time at the BDs. This can be done by scanning one of the input photons, see the optical delay $l$ of Fig. 2. Lastly, the successful case in which only one particle is found at $L$ and the other at $R$ can be post-selected by coincidences between $L$ and $R$. Note that one can trace out the indistinguishability wavefunction $\phi _i$ if the coincidence window is larger than the timing difference between $A$ and $B$, so the timing information is not registered. Assuming that the degree of distinguishability is presented as a Gaussian function with the center $l=0$, Eq. (12) can be rewritten with the variables of our optical system as

 

Fig. 4. Concurrence with respect to (a) particle distinguishability, and (b) spatial overlap, respectively. The particle distinguishability is adjusted by the optical delay $l$, and the spatial overlap is controlled by the angle $\theta$ of the H$_{\theta }$. The error bars present the one standard deviation from 100 reconstructed two-qubit states $\rho _{\textrm {ex}}$ by taking into account the Poisson statistics in measured coincidence counts.

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The distinguishability introduced by the optical delay $l$ can be measured by the Hong-Ou-Mandel (HOM) interference at BD2 and BD3. Figure 3 presents two-fold coincidences of D12 and D34 with respect to the optical delay $l$ where Dij stands for the coincidences between Di and Dj. The input polarization state was prepared as $|++\rangle$ where $|+\rangle =\frac {1}{\sqrt {2}}(|H\rangle +|V\rangle )$ in order to measure the HOM interference at BD2 and BD3, simultaneously. The HOM dip at $l=0$ implies that the two photons $A$ and $B$ are indistinguishable. The HOM interferences of D12 and D34, which can be fitted with the Gaussian function with the full width at half maximum (FWHM), are $132\pm 4\;\mu \textrm {m}$ and $137\pm 4\;\mu \textrm {m}$, respectively. These widths coincide with the coherence length of the photon pair which is determined by the bandwidth of IF. The HOM visibility, defined by the relative depth of the dip to the non-interfering counts, was measured as $V=0.99\pm 0.04$ and $V=0.91\pm 0.05$ for D12 and D34, respectively. To investigate the role of particle indistinguishability for generation of entanglement, we have analyzed two-qubit polarization states at various optical delays of $L_0=0\;\mu \textrm {m}$, $L_1=30\;\mu \textrm {m}$, $L_2=60\;\mu \textrm {m}$, and $L_3=300\;\mu \textrm {m}$.

Figure 4(a) presents concurrence as a function of optical delay $l$ at different HWP $\textrm {H}^{*}$ angles $\theta$. The concurrence is calculated from the experimentally reconstructed two-qubit state $\rho _{\textrm {ex}}$ [42,43,49]. It corresponds to the amount of entanglement with respective to particle distinguishability with fixed different degrees of spatial overlap. It clearly shows that concurrence decreases as the optical delay increases, or equivalently the distinguishability of particles increase. The experimental data are fitted with a Gaussian function with the FWHM of $140\;\mu \textrm {m}$, comparable to that of HOM interference.

Figure 4(b) represents concurrence as a function of the HWP $\textrm {H}^{*}$ angle $\theta$ at different optical delays $l$. Note that $\theta =0^{\circ }$ and $22.5^{\circ }$ correspond to non-spatial overlap and complete spatial overlap cases, respectively. It clearly presents that the amount of entanglement increase as the spatial overlap increases. As expected from Eq. (13), the experimental data are fitted with $C=C_0\sin ^{2}{4\theta }$, where $C_0$ corresponds to the maximum concurrence at a given optical delay $l$, and $\theta$ in radian.

4. Discussion

In this work, we have investigated the quantitative relation of the entanglement between identical particles with particle indistinguishability and spatial overlap from the perspectives of theory and experiment. The entanglement concurrence of two oppositely polarized photons decreases with decreasing particle indistinguishability and spatial overlap, which fits in well with the theoretical predictions for the entanglement of identical particles in the generic boson system. While the theory has been developed in the most general format, our linear optical experiment setup can be analyzed with other techniques e.g., Feynman path formalism or quantum walk [44].

In addition, our analysis can be applied to various quantum information processing utilizing identical particles as media of information. For example, the computational difficulty of the boson sampling problem [50] is known to have a close relation with the distinguishability [51,52] and the spatial distribution of photons [53]. We expect that our work provides an insight into understanding the role of the entanglement of identical particles in quantum information processing such as linear optical quantum computing. In addition, our method can also be applied to fermions. They follow the parity SSR [40,54], which makes the entanglement of fermions more complicated as well as intriguing [55,56].