Activating remote entanglement in a quantum network by local counting of identical particles

TL;DR

This paper demonstrates that the indistinguishability of fermions enables a new method for activating remote entanglement in quantum networks through local particle counting and classical communication, enhancing quantum correlation transfer.

Contribution

It introduces a novel entanglement transfer mechanism leveraging fermionic indistinguishability, which does not require shared history between distant nodes.

Findings

Fermionic indistinguishability enables remote entanglement activation.

Local counting of identical particles can induce nonlocal quantum correlations.

The technique offers a more stable multistage transfer of quantum entanglement.

Abstract

Quantum information and communication processing within quantum networks usually employs identical particles. Despite this, the physical role of quantum statistical nature of particles in large-scale networks remains elusive. Here, we show that just the indistinguishability of fermions makes it possible a new mechanism of entanglement transfer in many-node quantum networks. This process activates remote entanglement among distant sites, which do not share a common past, by only locally counting identical particles and classical communication. These results constitute the key achievement of the present technique and open the way to a more stable multistage transfer of nonlocal quantum correlations based on fermions.

Figures

Equations

$$|{\Psi }_{\mathrm{f},\mathrm{PS}}^{(4)}⟩=|{\Psi }_{\mathrm{M}},{\Psi }_{\mathrm{AB}}^{-}⟩,$$

$$|{\Psi }_{\mathrm{M}}⟩=|\mathrm{M}\uparrow ,\mathrm{M}\downarrow ⟩,|{\Psi }_{\mathrm{AB}}^{-}⟩=\frac{|\mathrm{A}\downarrow ,\mathrm{B}\uparrow ⟩-|\mathrm{A}\uparrow ,\mathrm{B}\downarrow ⟩}{\sqrt{2}}.$$

$$|{\Psi }_{\mathrm{f},\mathrm{PS}}^{(n)}⟩=|{\Psi }_{{\mathrm{M}}_1},{\Psi }_{{\mathrm{M}}_2},\dots ,{\Psi }_{{\mathrm{M}}_k},{\Psi }_{\mathrm{AB}}^{-}⟩,$$

$$|{\Psi }_{\mathrm{b},\mathrm{PS}}^{(4)}⟩=\frac{|{\Psi }_{\mathrm{M}},{\Psi }_{\mathrm{AB}}^{+}⟩+|{\Phi }_{\mathrm{M}}^{+},{\Phi }_{\mathrm{AB}}^{+}⟩-|{\Phi }_{\mathrm{M}}^{-},{\Phi }_{\mathrm{AB}}^{-}⟩}{\sqrt{3}},$$

$$\begin{array}{rl}& |{\Psi }_{\mathrm{AB}}^{+}⟩=\frac{1}{\sqrt{2}}(|\mathrm{A}\downarrow ,\mathrm{B}\uparrow ⟩+|\mathrm{A}\uparrow ,\mathrm{B}\downarrow ⟩),\\ & |{\Phi }_{\mathrm{AB}}^{\pm }⟩=\frac{1}{\sqrt{2}}(|\mathrm{A}\downarrow ,\mathrm{B}\downarrow ⟩\pm |\mathrm{A}\uparrow ,\mathrm{B}\uparrow ⟩).\end{array}$$

$$|{\Psi }_{\mathrm{PS}}^{(4)}⟩=|{\Psi }_{\mathrm{AC}},{\Psi }_{\mathrm{DB}}⟩,$$

$$|{\Psi }_{\mathrm{AC}}⟩=\frac{1}{\sqrt{2}}(|\mathrm{A}\downarrow ,\mathrm{C}\uparrow ⟩+\eta |\mathrm{A}\uparrow ,\mathrm{C}\downarrow ⟩),$$

$$|{\Psi }_{\mathrm{DB}}⟩=\frac{1}{\sqrt{2}}(|\mathrm{D}\downarrow ,\mathrm{B}\uparrow ⟩+\eta |\mathrm{D}\uparrow ,\mathrm{B}\downarrow ⟩).$$

$$\begin{array}{rl}|{\Psi }_{\mathrm{f}}^{(4)}⟩=& \frac{1}{3}(|\mathrm{A}\downarrow ,\mathrm{A}\uparrow ,\mathrm{M}\downarrow ,\mathrm{M}\uparrow ⟩+|A\downarrow ,\mathrm{A}\uparrow ,\mathrm{M}\downarrow ,\mathrm{B}\uparrow ⟩\\ & +|\mathrm{A}\downarrow ,\mathrm{A}\uparrow ,\mathrm{B}\downarrow ,\mathrm{M}\uparrow ⟩+|A\downarrow ,\mathrm{A}\uparrow ,\mathrm{B}\downarrow ,\mathrm{B}\uparrow ⟩\\ & +|\mathrm{A}\downarrow ,\mathrm{M}\uparrow ,\mathrm{M}\downarrow ,\mathrm{B}\uparrow ⟩+|\mathrm{A}\downarrow ,\mathrm{M}\uparrow ,\mathrm{B}\downarrow ,\mathrm{B}\uparrow ⟩\\ & +|\mathrm{M}\downarrow ,\mathrm{A}\uparrow ,\mathrm{B}\downarrow ,\mathrm{M}\uparrow ⟩+|M\downarrow ,\mathrm{A}\uparrow ,\mathrm{B}\downarrow ,\mathrm{B}\uparrow ⟩\\ & +|\mathrm{M}\downarrow ,\mathrm{M}\uparrow ,\mathrm{B}\downarrow ,\mathrm{B}\uparrow ⟩).\end{array}$$

$$|{\Psi }_{\mathrm{f},\mathrm{PS}}^{(4)}⟩=|\mathrm{M}\uparrow ,\mathrm{M}\downarrow ⟩\wedge |{\Psi }_{\mathrm{AB}}^{-}⟩,$$

$${\mathcal{M}}^{(n)}=\left(\begin{array}{ccc}⟨{\alpha }_1\downarrow |{\alpha }_1\downarrow ⟩& \cdots & ⟨{\alpha }_1\downarrow |{\alpha }_N\uparrow ⟩\\ ⋮& \ddots & ⋮\\ ⟨{\alpha }_N\uparrow |{\alpha }_1\downarrow ⟩& \cdots & ⟨{\alpha }_N\uparrow |{\alpha }_N\uparrow ⟩\end{array}\right),$$

$$|{\Psi }_{\mathrm{f},\mathrm{PS}}^{(n)}⟩=|{\Psi }_{{\mathrm{M}}_1},{\Psi }_{{\mathrm{M}}_2},\dots ,{\Psi }_{{\mathrm{M}}_k},{\Psi }_{\mathrm{AB}}^{-}⟩,$$

$$P_{\mathrm{f}}(n)=\frac{1}{2^{n-1}\det ({\mathcal{M}}^{(n)})}.$$

$$⟨{\phi }_1^{\prime },{\phi }_2^{\prime },\dots ,{\phi }_n^{\prime }|{\phi }_1,{\phi }_2,\dots ,{\phi }_n⟩$$

$$:=\sum _P{\eta }^P⟨{\phi }_1^{\prime }|{\phi }_{P_1}⟩⟨{\phi }_2^{\prime }|{\phi }_{P_2}⟩\dots ⟨{\phi }_n^{\prime }|{\phi }_{P_n}⟩,$$

$$|{\Psi }_{\mathrm{b}}^{(4)}⟩=\frac{1}{5}|(\mathrm{A}+\mathrm{M})\downarrow ,(\mathrm{A}+\mathrm{M})\uparrow ,(\mathrm{M}+\mathrm{B})\downarrow ,(\mathrm{M}+\mathrm{B})\uparrow ⟩.$$

$$\begin{array}{rl}|{\Psi }_{\mathrm{b},\mathrm{PS}}^{(4)}⟩=& \frac{1}{\sqrt{6}}(|\mathrm{A}\downarrow ,\mathrm{M}\uparrow ,\mathrm{M}\uparrow ,\mathrm{B}\downarrow ⟩+|\mathrm{A}\downarrow ,\mathrm{M}\uparrow ,\mathrm{M}\downarrow ,\mathrm{B}\uparrow ⟩\\ & +|\mathrm{A}\uparrow ,\mathrm{M}\uparrow ,\mathrm{M}\downarrow ,\mathrm{B}\downarrow ⟩+|\mathrm{A}\uparrow ,\mathrm{M}\downarrow ,\mathrm{M}\downarrow ,\mathrm{B}\uparrow ⟩),\end{array}$$

$$|{\Psi }_{\mathrm{b},\mathrm{PS}}^{(4)}⟩=\frac{|{\Psi }_{\mathrm{M}},{\Psi }_{\mathrm{AB}}^{+}⟩+|{\Phi }_{\mathrm{M}}^{+},{\Phi }_{\mathrm{AB}}^{+}⟩-|{\Phi }_{\mathrm{M}}^{-},{\Phi }_{\mathrm{AB}}^{-}⟩}{\sqrt{3}},$$

$$|{\Psi }_{\mathrm{b}}^{(n)}⟩=\frac{1}{{\mathcal{N}}_b}|{\alpha }_1\downarrow ,{\alpha }_1\uparrow ,{\alpha }_2\downarrow ,{\alpha }_2\uparrow ,\dots ,{\alpha }_N\downarrow ,{\alpha }_N\uparrow ⟩,$$

$$P_{\mathrm{b}}(n)=\frac{3^{\frac{n}{2}-1}}{2^{n-1}\mathrm{perm}({\mathcal{M}}^{(n)})}.$$

$$|{\Psi }_{\mathrm{b},\mathrm{PS}}^{(6)}⟩$$

$${|{\Psi }^{(4)}⟩}_{\mathrm{A}{\mathrm{M}}_2\mathrm{B}}=\frac{|{\Phi }_{{\mathrm{M}}_2}^{+},{\Psi }_{\mathrm{AB}}^{-}⟩+|{\Phi }_{{\mathrm{M}}_2}^{-},{\Psi }_{\mathrm{AB}}^{+}⟩+|{\Psi }_{{\mathrm{M}}_2},{\Phi }_{\mathrm{AB}}^{-}⟩}{\sqrt{3}}.$$

$$\begin{array}{rl}|{\Psi }^{(4)}⟩=& \frac{1}{4}(|A\downarrow ,\mathrm{A}\uparrow ,\mathrm{D}\downarrow ,\mathrm{D}\uparrow ⟩+|A\downarrow ,\mathrm{A}\uparrow ,\mathrm{D}\downarrow ,\mathrm{B}\uparrow ⟩\\ & +|A\downarrow ,\mathrm{A}\uparrow ,\mathrm{B}\downarrow ,\mathrm{D}\uparrow ⟩+|A\downarrow ,\mathrm{A}\uparrow ,\mathrm{B}\downarrow ,\mathrm{B}\uparrow ⟩\\ & +|A\downarrow ,\mathrm{C}\uparrow ,\mathrm{D}\downarrow ,\mathrm{D}\uparrow ⟩+|A\downarrow ,\mathrm{C}\uparrow ,\mathrm{D}\downarrow ,\mathrm{B}\uparrow ⟩\\ & +|A\downarrow ,\mathrm{C}\uparrow ,\mathrm{B}\downarrow ,\mathrm{D}\uparrow ⟩+|A\downarrow ,\mathrm{C}\uparrow ,\mathrm{B}\downarrow ,\mathrm{B}\uparrow ⟩\\ & +|C\downarrow ,\mathrm{A}\uparrow ,\mathrm{D}\downarrow ,\mathrm{D}\uparrow ⟩+|C\downarrow ,\mathrm{A}\uparrow ,\mathrm{D}\downarrow ,\mathrm{B}\uparrow ⟩\\ & +|C\downarrow ,\mathrm{A}\uparrow ,\mathrm{B}\downarrow ,\mathrm{D}\uparrow ⟩+|C\downarrow ,\mathrm{A}\uparrow ,\mathrm{B}\downarrow ,\mathrm{B}\uparrow ⟩\\ & +|C\downarrow ,\mathrm{C}\uparrow ,\mathrm{D}\downarrow ,\mathrm{D}\uparrow ⟩+|C\downarrow ,\mathrm{C}\uparrow ,\mathrm{D}\downarrow ,\mathrm{B}\uparrow ⟩\\ & +|C\downarrow ,\mathrm{C}\uparrow ,\mathrm{B}\downarrow ,\mathrm{D}\uparrow ⟩+|C\downarrow ,\mathrm{C}\uparrow ,\mathrm{B}\downarrow ,\mathrm{B}\uparrow ⟩).\end{array}$$

$$|{\Psi }_{\mathrm{PS}}^{(4)}⟩={\hat{\Pi }}_{\mathrm{ACDB}}|{\Psi }^{(4)}⟩/\mathcal{N},$$

$$\begin{array}{rl}|{\Psi }_{\mathrm{PS}}^{(4)}⟩=& \frac{1}{\sqrt{2}}(|\mathrm{A}\downarrow ,\mathrm{C}\uparrow ⟩+\eta |\mathrm{A}\uparrow ,\mathrm{C}\downarrow ⟩)\\ & \wedge \frac{1}{\sqrt{2}}(|\mathrm{D}\downarrow ,\mathrm{B}\uparrow ⟩+\eta |D\uparrow ,\mathrm{B}\downarrow ⟩),\end{array}$$

$$\begin{array}{rl}|{\Psi }_{\mathrm{PS}}^{(4)}⟩& =\frac{1}{2}[|{\Psi }_{\mathrm{CD}}^{+},{\Psi }_{\mathrm{AB}}^{+}⟩-|{\Psi }_{\mathrm{CD}}^{-},{\Psi }_{\mathrm{AB}}^{-}⟩\\ & +\eta |{\Phi }_{\mathrm{CD}}^{+},{\Phi }_{\mathrm{AB}}^{+}⟩-\eta |{\Phi }_{\mathrm{CD}}^{-},{\Phi }_{\mathrm{AB}}^{-}⟩],\end{array}$$

$$\begin{array}{rl}& |{\Psi }_{\mathrm{IJ}}^{\pm }⟩=\frac{1}{\sqrt{2}}(|I\downarrow ,\mathrm{J}\uparrow ⟩\pm |I\uparrow ,\mathrm{J}\downarrow ⟩),\\ & |{\Phi }_{\mathrm{IJ}}^{\pm }⟩=\frac{1}{\sqrt{2}}(|I\downarrow ,\mathrm{J}\downarrow ⟩\pm |I\uparrow ,\mathrm{J}\uparrow ⟩),\end{array}$$

Full text

I Introduction

New avenues have recently been opened in quantum information and communication by the transfer of quantum state among different separated sites Gisin and Thew (2007). In fact, it allows distributed quantum computing, a quantum internet and tests of quantum mechanics foundations within composite quantum networks Castelvecchi (2018); Pirandola and Braunstein (2016); Humphreys et al. (2018). Usual state transfer procedures employ identical particles (that are elementary subsystems such as atoms, photons, electrons, qubits), where their entanglement plays an essential role and no effect associated to the statistical nature of the particles typically shows up. This is due to the fact that, in these processes, spatial overlap of the wave functions does not occur in the relevant places so that the identical particles are distinguishable and behave like non-identical ones. One may then inquire whether employing identical particles may lead to new features in the context of quantum communication by exploiting indistinguishability as a direct resource. For this to happen, it is required to investigate those situations where particles spatially overlap, so that particle identity implies their indistinguishability. Identity of particles has been shown to be useful for some quantum information protocols (Omar, 2005; Omar et al., 2002; Paunković et al., 2002; Bose et al., 2003; Bose and Home, 2002; Lo Franco and Compagno, 2016; Sciara et al., 2017; Bellomo et al., 2017; Compagno et al., 2018; Lo Franco and Compagno, 2018) and for quantum metrology Benatti et al. (2014); Braun et al. (2018). In this context, one of the problems which remains to investigate is the role played by the quantum statistical nature of identical particles into the mechanisms of entanglement transfer within large-scale networks.

Among the various mechanisms of many-node state transfer, we focus on the entanglement swapping (ES), which is a must for large-scale distribution of quantum information (Scherer et al., 2011; Ye, 2015; Naseri, 2016; Sun et al., 2017a) and is subject of intense experimental interest (Pan et al., 1998; de Riedmatten et al., 2005; Yang et al., 2006; Kaltenbaek et al., 2009; Megidish et al., 2013; Sun et al., 2017b). ES is an intrinsically quantum phenomenon which permits to entangle two particles not sharing a common past, each particle being outside the light cone of the other. ES constitutes a key process to implement quantum communication (Scherer et al., 2011; Ye, 2015; Naseri, 2016; Sun et al., 2017a) and is crucial to build quantum relays and quantum repeaters (Jacobs et al., 2002; Gisin and Thew, 2007). So far, in all the implementations, essential ingredients are the initial creation of entangled pairs and Bell measurements (Hu and Rarity, 2011). In the standard ES process (Zukowski et al., 1993), two entangled particle pairs are initially prepared and a Bell measurement is successively performed on two particles of the different pairs. As a result, the other two particles become entangled even if they never interacted (Jennewein et al., 2001; Branciard et al., 2010). ES has been experimentally realized using identical but distinguishable particles (photons) by applying the usual operational framework for non-identical particles, based on particle addressability (local operations and classical communication (LOCC)) Horodecki et al. (2009). The initial entangled pairs of photons are typically created by spontaneous parametric down conversion (SPDC) (Pan et al., 1998; de Riedmatten et al., 2005; Megidish et al., 2013; Yang et al., 2006; Kaltenbaek et al., 2009). Recently, ES has been successfully achieved in a quantum network, entangling two photons over a distance of 100 km (Sun et al., 2017b). Multiple ES has been theoretically proposed and experimentally realized by extension of the standard protocol (Bose et al., 1998; Goebel et al., 2008; Khalique et al., 2013; Lu et al., 2009). The overall success of the process is influenced by the low creation rate of photon pairs in SPDC (Vamivakas et al., 2004; Dousse et al., 2010; Scherer et al., 2009) and by the inefficiency in the realization of Bell measurements (Hu and Rarity, 2011; Yang et al., 2006; Lee and Jeong, 2013; Yang et al., 2005; Pakniat et al., 2017a; Grice, 2011; Zhou and Sheng, 2015; Sheng et al., 2010).

In this work we present a new process of entanglement transfer in a many-node quantum network, where neither initial entangled particle pairs nor Bell measurements are needed, exploiting indistinguishability of fermions. Although the idea of using indistinguishability of identical particles to generate entanglement is not a new one, the present process shows that remote entanglement among distant sites can be generated by only locally counting particles. These characteristics constitute the key achievement of the present technique. In the presence of spatial overlap, the identity of particles makes them individually unaddressable. Therefore, we employ an approach based on spatially localized operations and classical communication (sLOCC), where single-particle local measurements are made onto assigned spatial regions (Lo Franco and Compagno, 2018; Compagno et al., 2018) (closer to the spirit of quantum field theory). We finally compare this process to that with bosons.

II Basic process with fermions

Experimental techniques to control fermions in quantum networks have been recently developed (Pirandola et al., 2006; Press et al., 2008; Fève et al., 2007; Bocquillon et al., 2013; Rashidi et al., 2018; Matthews et al., 2013; Sansoni et al., 2012). In the following we focus on unveiling remarkable aspects introduced by both fermionic statistics and spatial overlap in the process of remote entanglement distribution.

To this aim, we first describe the basic process which serves as the elementary step for the extension to a many-node quantum network. As displayed in Fig. 1, we take four identical fermions (two pairs), prepared by four independent (space-like separated) sources $\{\mathrm{S}_{i},\ i=1,...,4\}$. Each particle is sent to the corresponding beam splitter $\mathrm{BS}_{i}$. The two sources $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ independently prepare two fermions with opposite pseudospin. Each beam splitter sends the particle with the same amplitude into two separated sites A and M, so that each particle is in the same delocalized spatial mode $\ket{\alpha}=(\ket{\mathrm{A}}+\ket{\mathrm{M}})/\sqrt{2}$. Similarly, sources $\mathrm{S}_{3}$ and $\mathrm{S}_{4}$ generate the fermions of the second pair with opposite pseudospin in the delocalized spatial mode $\ket{\beta}=(\ket{\mathrm{M}}+\ket{\mathrm{B}})/\sqrt{2}$ (right side of Fig. 1). The modes $\ket{\alpha}$ and $\ket{\beta}$ partially overlap in the shared intermediate node M and the I-th node (I $=$ A, M, B) is chosen such that only the localized bound state $\ket{\mathrm{I}}$ is present. The condition that particles are in the chosen localized bound states can be assured by keeping only the cases when local detectors do not measure particles elsewhere and by classically communicating the results. The initially prepared four-fermion state $|\Psi^{(4)}_{\mathrm{f}}\rangle$ can be then formally obtained from $\ket{\alpha\downarrow,\alpha\uparrow,\beta\downarrow,\beta\uparrow}$ by dropping, because of the Pauli exclusion principle, the terms with the same pseudospins in the central node (same spatial state $|\mathrm{M}\rangle$) Orzel et al. (1999). The ultimate scope is to generate entanglement between particles in the far nodes $\mathrm{A}$ and $\mathrm{B}$. This can be achieved by using sLOCC (Lo Franco and Compagno, 2018), which here consist in a post-selection locally counting only one particle of the first pair in A and one particle of the second pair in B and using classical communication among these sites (this counting implies that in the central node M there are two particles). The classical communication allows sites A and B to know when an entangled pair is obtained. Notice that the local counting operation is a free operation with respect to entanglement Plenio and Virmani (2007). Such a post-selection can be implemented utilizing, for instance, absorbtionless particle-counting detectors in A and B, which do not disturb the pseudospin state (Buks et al., 1998; Armour and Blencowe, 2001; Parkins et al., 1995; Bose and Home, 2002). Similar non-demolition measurements are applied in the other post-selections by sLOCC used along the paper.

One then gets the post-selected global state (see Appendix A)

[TABLE]

where

[TABLE]

We have thus obtained a maximally entangled state of two-particle pseudospins over the distant nodes $\mathrm{A}$ and $\mathrm{B}$, despite the latter are always independent and the particles do not share any common past. We stress that if the four particles are not identical, the same post-selection procedure does not give rise to an entangled state. The state of Eq. (1) is obtained with probability (see Appendix B) $P_{\mathrm{f}}(4)=|\langle\Psi_{\mathrm{f,PS}}^{(4)}|\Psi_{\mathrm{f}}^{(4)}\rangle|^{2}=\langle\Psi_{\mathrm{f}}^{(4)}|\hat{\Pi}_{\mathrm{f}}|\Psi_{\mathrm{f}}^{(4)}\rangle=2/9$, where $\hat{\Pi}_{\mathrm{f}}=\sum_{\sigma,\tau=\uparrow,\downarrow}|\mathrm{A}\ \sigma,\mathrm{M}\uparrow,$ $\mathrm{M}\downarrow,\mathrm{B}\ \tau\rangle\langle\mathrm{A}\ \sigma,\mathrm{M}\uparrow,\mathrm{M}\downarrow,\mathrm{B}\ \tau|$ is the projector onto the (AMB)-operational subspace. We remark that this entanglement distribution is reached without entanglement-inducing Bell measurements on the central particles, but just exploiting the indistinguishability of non-interacting fermions in M. In fact, the spatial overlap of fermions in the shared intermediate site $\mathrm{M}$ plays the key role of an entanglement-transfer gate. Schemes for entanglement swapping without Bell measurements have been proposed in contexts where interaction is essential (e.g., cavity QED) (Yang et al., 2005; Xiu et al., 2007; Pakniat et al., 2017a, b).

III Large-scale process with fermions

The basic scheme of Fig. 1 can be straightforwardly iterated to create a remote entanglement transfer in a many-node quantum network. This is achieved by means of $n$ identical fermions and $k=N-1$ shared intermediate nodes $\mathrm{M}_{i}$ ($i=1,\ldots,k$), where $N=n/2$ is the number of particle pairs. As displayed in Fig. 2, each $j$-th pair ($j=1,\ldots,N$) has opposite pseudospins and spatial mode $\ket{\alpha_{j}}$, with $\ket{\alpha_{1}}=(\ket{\mathrm{A}}+\ket{\mathrm{M}_{1}})/\sqrt{2}$, $\ket{\alpha_{N}}=(\ket{\mathrm{M}_{k}}+\ket{\mathrm{B}})/\sqrt{2}$ and $\ket{\alpha_{j}}=(\ket{\mathrm{M}_{j-1}}+\ket{\mathrm{M}_{j}})/\sqrt{2}$ for $j=2,\ldots,k$. The aim is to activate entanglement of particles in the remote far nodes A and B of the network. The modes $\ket{\alpha_{i}}$ and $\ket{\alpha_{i+1}}$ partially overlap in the shared intermediate node $\mathrm{M}_{i}$ and the I-th node (I $=$ A, Mi, B) is taken, as already mentioned above, such that only the localized bound state $\ket{\mathrm{I}}$ is present. Once again the initial $n$-fermion state $|\Psi^{(n)}_{\mathrm{f}}\rangle$ can be formally determined starting from the state $\ket{\alpha_{1}\downarrow,\alpha_{1}\uparrow,\ldots,\alpha_{N}\downarrow,\alpha_{N}\uparrow}$ simply by dropping, because of the Pauli exclusion principle, the terms having the same pseudospins in each intermediate node. After that, by counting one particle in A and one in B (this entails that each node $\mathrm{M_{i}}$ contains two particles) and allowing for classical communication as before, the post-selected global state is

[TABLE]

where $\ket{\Psi_{\mathrm{M}_{i}}}=\ket{\mathrm{M}_{i}\uparrow,\mathrm{M}_{i}\downarrow}$ and $\ket{\Psi_{\mathrm{AB}}^{-}}$ is the maximally entangled (Bell) state of Eq. (2). The probability to obtain the state above is $P_{\mathrm{f}}(n)=|\langle\Psi_{\mathrm{f,PS}}^{(n)}|\Psi_{\mathrm{f}}^{(n)}\rangle|^{2}$ (see Appendix A for its explicit expression). Thus, we have generated entanglement of particle pseudospins between the independent distant locations A and B of the many-node network, starting with independently-prepared identical fermions, with no Bell measurements and only using local counting operations. We remark that all these features make the remote entanglement activation based on identical fermions deeply different from the standard processes of entanglement transfer such as ES.

IV Process with bosons

The basic setup of Fig. 1 can be also thought to be run by identical bosons. We shall show that, in this case, a Bell measurement onto the intermediate site M is eventually required for achieving the desired entanglement transfer, similarly to a standard protocol of ES. The sLOCC framework is now realized by locally counting two particles in the intermediate node M and only one particle in each of the far nodes A and B, also allowing for classical communication among the different sites. From the initial (unnormalized) state $\ket{\alpha\downarrow,\alpha\uparrow,\beta\downarrow,\beta\uparrow}$, one gets the four-boson post-selected state (see Appendix C)

[TABLE]

where $|\Phi^{\pm}_{\mathrm{M}}\rangle=(|\mathrm{M}\downarrow,\mathrm{M}\downarrow\rangle\pm|\mathrm{M}\uparrow,\mathrm{M}\uparrow\rangle)/2$, $\ket{\Psi_{\mathrm{M}}}$ is given in Eq. (2), while the distant sites A and B share the Bell states

[TABLE]

The presence of these three Bell states is a consequence of the fact that bosonic systems admit two-particle states with the same pseudospins in M. As in the standard ES procedure, a joint (Bell) measurement in the shared intermediate node $\mathrm{M}$ determines the entangled state in which the first and the last boson of the network collapse, each outcome occurring with probability $p=1/3$, as seen from Eq. (4). Since each of the three Bell-state outcomes from the joint measurement in M realizes the desired entanglement transfer over A and B, the success probability of the process coincides with the probability of obtaining the post-selected state of Eq. (4), which is $P_{\mathrm{b}}(4)=|\langle\Psi_{\mathrm{b,PS}}^{(4)}|\Psi_{\mathrm{b}}^{(4)}\rangle|^{2}=6/25$. This bosonic protocol can be then extended, analogously to the standard multiple ES, by a cascaded procedure Goebel et al. (2008). The scheme remains that of Fig. 2 with $n$ independently-prepared identical bosons and $k=N-1$ intermediate nodes ($N=n/2$). The sLOCC framework again consists in counting two particles in each intermediate node and one in the distant nodes A and B, also allowing for classical communication of the counting outcomes. One gets the post-selected state $\ket{\Psi_{\mathrm{b,PS}}^{(n)}}$ and performs Bell measurements step by step on each intermediate node $\mathrm{M}_{i}$ ($i=1,...,k$) to transfer entanglement over A and B. The success probability is $P_{\mathrm{b}}(n)=|\langle\Psi_{\mathrm{b,PS}}^{(n)}|\Psi_{\mathrm{b}}^{(n)}\rangle|^{2}$ (see Appendix C).

V Process with separated intermediate sites

Notice that a procedure much closer to the standard ES can be moreover obtained with indistinguishable particles (bosons or fermions) by employing intermediate separated nodes instead of common intermediate ones.

We take a system made of four identical particles (either bosons or fermions), prepared by four independent (space-like separated) sources $\{\mathrm{S}_{i},\ i=1,...,4\}$. Each particle is sent to the corresponding beam splitter $\mathrm{BS}_{i}$, as depicted in Fig. 3. The two sources $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ independently prepare two particles with opposite pseudospin. Each beam splitter sends the particle with the same amplitude into two separated sites A and C, so that each particle is in the same delocalized spatial mode $|\alpha\rangle=(|\mathrm{A}\rangle+|\mathrm{C}\rangle)/\sqrt{2}$. Similarly, sources $\mathrm{S}_{3}$ and $\mathrm{S}_{4}$ generate the particles of the second pair with opposite pseudospin in the delocalized spatial mode $|\beta\rangle=(|\mathrm{D}\rangle+|\mathrm{B}\rangle)/\sqrt{2}$ (right side of Fig. 3). All the nodes are spatially separated (the modes $\ket{\alpha}$ and $\ket{\beta}$ are orthogonal) and the I-th node (I $=$ A, B, C, D) is chosen such that only the localized bound state $\ket{\mathrm{I}}$ is present. The condition that particles are in the chosen localized bound states can be assured by keeping only the cases when local detectors do not measure particles elsewhere and by classically communicating the results. The global four-particle quantum state (Lo Franco and Compagno, 2016) is therefore $|\Psi^{(4)}\rangle=|\alpha\downarrow,\alpha\uparrow,\beta\downarrow,\beta\uparrow\rangle$. From this state it is possible to obtain entanglement in the pseudospin degrees of freedom linked to the spatial overlap of particles in each pair. This is achieved by sLOCC (Lo Franco and Compagno, 2018), which here consist in a post-selection counting only one particle of the first pair in A and one particle of the second pair in B and classically communicating this outcome to each other. This post-selection can be implemented using, for instance, one absorbtionless particle-counting detector in A and one in B, which do not disturb the pseudospin state (Buks et al., 1998; Armour and Blencowe, 2001; Parkins et al., 1995; Bose and Home, 2002). Similar non-demolition measurements are applied in the other post-selections used along the paper. As a result, each node contains one particle and we obtain the state (see Appendix D)

[TABLE]

where $\ket{\Psi_{\mathrm{AC}}}$ and $\ket{\Psi_{\mathrm{DB}}}$ are the two-particle Bell states

[TABLE]

Even if the particles have been independently prepared, as a consequence of sLOCC, the state $\ket{\Psi_{\mathrm{PS}}^{(4)}}$ shows that the pair of particles in $\mathrm{A}$ and $\mathrm{C}$ is maximally entangled in the pseudospin degrees of freedom, as the $\mathrm{DB}$-pair. This state is obtained with probability $P(4)=|\langle\Psi_{\mathrm{PS}}^{(4)}|\Psi^{(4)}\rangle|^{2}=1/4$. At this stage the particles can be distinguished, since they are in spatially separated sites. We stress that for each pair, if the particles are not identical, the same post-selection procedure does not give rise to an entangled state. The structure of the state of Eq. (6) allows to implement the standard protocol of entanglement swapping (ES) (Zukowski et al., 1993): performing a Bell measurement on near central nodes C and D transfers entanglement to the particles in the far nodes A and B. Notice that this procedure does not require, at the preparation stage, two entangled pairs. The present scheme works for both bosons and fermions, also when particles of different pairs are not identical. Moreover, in analogy with the standard ES, it can be naturally iterated by a cascaded procedure Goebel et al. (2008) to realize multistage entanglement swapping with $n=2N$ independently-prepared particles, being $N$ the number of involved particle pairs. This is achieved by using a network with $n-2$ separated central nodes, where each pair of identical particles (either bosons or fermions) is prepared with opposite pseudospins in an equal delocalized spatial mode peaked in correspondence of two separated nodes (as shown for the two pairs in Fig. 3). After obtaining a single particle in each central node and performing Bell measurements step by step onto two central nodes Goebel et al. (2008), one eventually entangles the particles in the extreme nodes of the network with probability $P(n)=1/2^{N}$.

VI Conclusions

In this work we have presented a new conceptual process of entanglement distribution in a large-scale quantum network which is fundamentally activated by indistinguishability of particles. The standard entanglement swapping, that is the renowned process for entanglement transfer with distinguishable particles, necessitates to start from entangled particle pairs and requires final Bell measurements (Zukowski et al., 1993). Compared to this one, the present process, run by identical fermions, enables remote entanglement among distant nodes through the following different aspects: (i) with no distribution of initial entangled pairs and (ii) without performing Bell measurements, because of the natural entanglement due to spatially overlapping identical particles. Therefore, the process only requires local counting of independently-prepared identical particles. The measurement process, when described on the level of particles, looks different for indistinguishable and distinguishable particles.

Besides its conceptual novelty, the key advantage of this process is that it simplifies the task of distributing entanglement, overcoming the drawbacks encountered in the usual entanglement transfer procedures during the initial preparation stage and the final measurement phase. In fact, it skips the use of sources of entangled particle pairs, which are for instance generated by SPDC at the very low rate of about $10^{-2}$ for single laser pulse (Dousse et al., 2010), and also avoids the experimental inefficiency associated to performing Bell measurements (Hu and Rarity, 2011; Yang et al., 2006; Lee and Jeong, 2013; Yang et al., 2005; Pakniat et al., 2017a; Grice, 2011; Zhou and Sheng, 2015; Sheng et al., 2010).

The proposed fermionic process could be, for instance, realized by using quantum dots as sources of single electrons that can be initialized in particular spin states Press et al. (2008), emitted on demand Fève et al. (2007) and directed to quantum point contacts acting as electronic beam splitters Bocquillon et al. (2013); Bäuerle et al. (2018). Single electrons have been also recently shown to be controlled within atomic circuits Rashidi et al. (2018). Moreover, further setups in quantum optics, simulating fermionic statistics using photons and integrated photonics Matthews et al. (2013); Sansoni et al. (2012) could, in principle, represent convenient platforms.

Our results make it emerge once more (Lo Franco and Compagno, 2018) that spatial overlap of identical particles constitutes an operational resource. In addition, they pave the way to a more stable multistage remote entanglement transfer based on fermions, evidencing the effect of quantum statistical nature of particles on quantum information processing.

Acknowledgments. A.C. acknowledges for useful discussions Dario Cilluffo, Mauro Valeri, Andrea Geraldi and Emanuele Polino.

Appendix A Explicit calculations for shared intermediate sites with fermions

The four-fermion global state $|\Psi^{(4)}_{\mathrm{f}}\rangle$ is

[TABLE]

The sLOCC here consists in counting one particle in A and one in B (this entails having two particles in M) and allowing for classical communication of the outcomes. Projecting therefore the above prepared state onto the subspace spanned by the basis $\mathcal{B}_{\mathrm{f}}=\{|\mathrm{A\ \sigma,M\uparrow,M\downarrow,B\tau\rangle}\}$ $(\sigma,\tau=\ \downarrow,\uparrow)$, we find the post-selected state

[TABLE]

which is obtained with probability $P_{\mathrm{f}}(4)=|\langle\Psi^{(4)}_{\mathrm{f,PS}}|\Psi^{(4)}_{\mathrm{f}}\rangle|^{2}=2/9$. The state $\ket{\Psi_{\mathrm{AB}}^{-}}$ is the maximally Bell state of Eq. (25).

The scheme presented for the minimum core with four particles can be extended to the case of $n=2N$ particles, where $N$ is the number of involved particle pairs. The generalized scheme with $k=N-1$ shared intermediate nodes $\mathrm{M}_{i}$ ($i=1,\ldots,k$) is displayed in Fig. 1 of the main text.

Each $j$-th pair ($j=1,\ldots,N$) has opposite pseudospins and (delocalized) spatial mode $\ket{\alpha_{j}}$, with $\ket{\alpha_{1}}=(\ket{\mathrm{A}}+\ket{\mathrm{M}_{1}})/\sqrt{2}$, $\ket{\alpha_{N}}=(\ket{\mathrm{M}_{k}}+\ket{\mathrm{B}})/\sqrt{2}$ and $\ket{\alpha_{j}}=(\ket{\mathrm{M}_{j-1}}+\ket{\mathrm{M}_{j}})/\sqrt{2}$ for $j=2,\ldots,k$. We take as the initially prepared $n$-fermion state $|\Psi^{(n)}_{\mathrm{f}}\rangle$ the one obtained from $\ket{\alpha_{1}\downarrow,\alpha_{1}\uparrow,\ldots,\alpha_{N}\downarrow,\alpha_{N}\uparrow}$ by eliminating, because of the Pauli exclusion principle, the terms with two particles in the same node with the same pseudospin. The normalization constant $\mathcal{N}_{\mathrm{f}}$ of $|\Psi^{(n)}_{\mathrm{f}}\rangle$ can be conveniently expressed as $\mathcal{N}_{\mathrm{f}}=\sqrt{\mathrm{det}(\mathcal{M}^{(n)})}$ where $\mathrm{det}(\mathcal{M}^{(n)})$ is the determinant of the $n\times n$ matrix

[TABLE]

defined in the $n$-dimensional one-particle basis $\{\ket{\alpha_{1}\downarrow},\ket{\alpha_{1}\uparrow},\ket{\alpha_{2}\downarrow},\ket{\alpha_{2}\uparrow},\ldots,\ket{\alpha_{N}\downarrow},\ket{\alpha_{N}\uparrow}\}$.

The sLOCC framework is once again realized by counting one particle in each of the nodes A and B (this entails that two particles are in each node Mi), also allowing for classical communication of the counting outcomes. The post-selected global state results to be

[TABLE]

where $\ket{\Psi_{\mathrm{M}_{i}}}=\ket{\mathrm{M}_{i}\uparrow,\mathrm{M}_{i}\downarrow}$ and $\ket{\Psi_{\mathrm{AB}}^{-}}$ is a maximally entangled Bell state (see Eq. (25)). The probability to obtain the state above, that is the success probability of the remote entanglement transfer process, is given by $P_{\mathrm{f}}(n)=|\langle\Psi_{\mathrm{f,PS}}^{(n)}|\Psi_{\mathrm{f}}^{(n)}\rangle|^{2}$. It is straightforward to show that its explicit expression as a function of the number of fermions is

[TABLE]

Appendix B Scalar products in the no-label formalism

For calculating the success probabilities of the proposed protocols under different configurations, we need to compute scalar products between states of $n$ identical particles. These are obtained by the $n$-particle probability amplitude defined in the non-standard approach (no-label particle-based formalism) here adopted Lo Franco and Compagno (2016); Compagno et al. (2018), whose general expression is

[TABLE]

where $P=\{P_{1},P_{2},...,P_{n}\}$ in the sum runs over all the one-particle state permutations, $\eta=\pm 1$ for bosons and fermions, respectively, and $\eta^{P}$ is 1 for bosons and 1 (-1) for even (odd) permutations for fermions.

Appendix C Explicit calculations for shared intermediate sites with bosons

The four-boson global state $|\Psi^{(4)}_{\mathrm{b}}\rangle$ is

[TABLE]

The basis for sLOCC, corresponding to counting two particles in the shared intermediate mode M and one particle in each mode A and B, is \mathcal{B}_{\mathrm{b}}=\Bigl{\{}\dfrac{|\mathrm{A\ \sigma,M\ \tau,M\ \sigma^{\prime},B\ \tau^{\prime}\rangle}}{\mathcal{N}_{\tau\sigma^{\prime}}}\Bigr{\}}  ($\sigma,\tau,\sigma^{\prime},\tau^{\prime}=\downarrow,\uparrow$) where $\mathcal{N}_{\tau\sigma^{\prime}}=\sqrt{1+\langle\tau|\sigma^{\prime}\rangle}$. The four-boson post-selected state, after projection onto $\mathcal{B}_{\mathrm{b}}$ (also allowing for classical communication), is then

[TABLE]

which is found with probability $P_{\mathrm{b}}(4)=|\langle\Psi^{(4)}_{\mathrm{b,PS}}|\Psi^{(4)}_{\mathrm{b}}\rangle|^{2}=6/25$.

In the M-subspace, the Bell basis for bosons is given by the three states $|\Phi^{\pm}_{\mathrm{M}}\rangle=(|\mathrm{M}\downarrow,\mathrm{M}\downarrow\rangle\pm|\mathrm{M}\uparrow,\mathrm{M}\uparrow\rangle)/2$ and $\ket{\Psi_{\mathrm{M}}}=\ket{\mathrm{M}\uparrow,\mathrm{M}\downarrow}$. The post-selected state $|\Psi^{(4)}_{\mathrm{b,PS}}\rangle$ can be then expressed in terms of this Bell basis as

[TABLE]

where in each term the particles in the distant sites A and B are in a Bell state (see Eq. (25)). Therefore, each outcome of the joint Bell measurement successfully realizes the entanglement swapping over the distant nodes A and B.

The four-particle bosonic protocol can be extended, analogously to the standard ES, by a cascaded procedure Goebel et al. (2008). The scheme is again that of Fig. 2 with $n$ independently-prepared identical bosons and $k=N-1$ intermediate nodes ($N=n/2$ is the number of particle pairs).

The initially prepared $n$-boson state is

[TABLE]

where the normalization constant $\mathcal{N}_{b}=\sqrt{\mathrm{perm}(\mathcal{M}^{(n)})}$, with $\mathrm{perm}(\mathcal{M}^{(n)})$ being the permanent of the matrix of Eq. (10).

By counting two particles in each intermediate node and one in each of the distant nodes A and B, also allowing for classical communication of the counting outcomes, one gets the post-selected state $\ket{\Psi_{\mathrm{b,PS}}^{(n)}}$. Bell measurements are then performed step by step on each intermediate node $\mathrm{M}_{i}$ ($i=1,...,k$) to transfer entanglement over A and B. The type of the final Bell state transferred over A and B will depend on the consecutive outcomes of the cascaded Bell measurements. The success probability of the protocol is obtained by $P_{\mathrm{b}}(n)=|\langle\Psi_{\mathrm{b,PS}}^{(n)}|\Psi_{\mathrm{b}}^{(n)}\rangle|^{2}$. Its explicit expression as a function of the number of bosons, in the cases of Fig. 4, is

[TABLE]

In order to be more explicit concerning the cascaded procedure leading to the Bell states over A and B, we treat the case with $n=6$ bosons ($N=3$ pairs) and two shared intermediate nodes M1, M2. From the initially prepared state $|\Psi^{(6)}_{\mathrm{b}}\rangle$, easily obtained from Eq. (17), the sLOCC framework counting two particles in the intermediate nodes and one particle in each of the far nodes A and B, including classical communication, leads to the post-selected state

[TABLE]

where the relevant Bell states are analogous to those given after Eq. (15) and in Eq. (25). A first Bell measurement has to be performed on the intermediate node M1 in order to entangle bosons in A and M2. Any outcome is good for continuing the protocol. Let us suppose, without loss of generality, that the result of this first Bell measurement is $\ket{\Phi^{-}_{\mathrm{M_{1}}}}$. From $|\Psi_{\mathrm{b,PS}}^{(6)}\rangle$ above, one sees that the remaining four bosons are left into the state $-\ket{\Phi^{-}_{\mathrm{AM_{2}}},\Psi^{+}_{\mathrm{M_{2}B}}}$. This state, suitably normalized and expressed in terms of the Bell basis in the node M2, assumes the form

[TABLE]

It is now clear that a second Bell measurement on the intermediate node M2 has the final effect to transfer a Bell state over the far nodes A and B. Thus, for six bosons, two cascaded Bell measurements realize the desired entanglement swapping protocol. This procedure can be continued analogously for successive steps with more particles.

Appendix D Explicit calculations for separated intermediate sites

Here, we report the calculations regarding the scheme above, depicted in Fig. 3 . The prepared state $|\Psi^{(4)}\rangle$ can be written as the superposition of 16 terms

[TABLE]

In the linear combination of Eq. (21), there are contributions in which two particles occupy the same site. We perform sLOCC in the form of a post-selection counting a single particle in each site A, B and classically communicating their outcomes to each other (notice that a single particle in A entails one particle in C and a single particle in B implies one particle in D). This post-selection corresponds to project the global four-particle state $|\Psi^{(4)}\rangle$ onto the subspace spanned by the spatially localized basis $\mathcal{B}=\{|\mathrm{A\ \sigma,C\ \tau,D\ \sigma^{\prime},B\ \tau^{\prime}\rangle}\}$, by the projector $\hat{\Pi}_{\mathrm{ACDB}}=\sum_{\sigma,\tau,\sigma^{\prime},\tau^{\prime}=\downarrow,\uparrow}|\mathrm{A\ \sigma,C\ \tau,D\ \sigma^{\prime},B\ \tau^{\prime}}\rangle\langle\mathrm{A\ \sigma,C\ \tau,D\ \sigma^{\prime},B\ \tau^{\prime}|}$. The post-selected (projected) state is thus obtained as

[TABLE]

where $\mathcal{N}=\sqrt{\langle\Psi^{(4)}|\hat{\Pi}_{\mathrm{ACDB}}|\Psi^{(4)}\rangle}=1/2$. Its explicit expression is

[TABLE]

that is $|\Psi^{(4)}_{\mathrm{PS}}\rangle=|\Psi_{\mathrm{AC}},\Psi_{\mathrm{DB}}\rangle$ (see Eqs. (6) and (V) above). In Eq. (23) we have used the wedge product $\wedge$ that, in this case of separated sites under sLOCC, coincides with the standard tensor product Compagno et al. (2018). This state is obtained with probability $P(4)=|\langle\Psi_{\mathrm{PS}}^{(4)}|\Psi^{(4)}\rangle|^{2}=1/4$. Notice that in this sLOCC framework, the prepared state $|\Psi^{(4)}\rangle$ can be written as $|\Psi^{(4)}\rangle=\ket{\alpha\downarrow,\alpha\uparrow}\wedge\ket{\beta\downarrow,\beta\uparrow}$, from which one then obtains particle entanglement between (A, C) and (D, B), as evidenced in Eq. (23). This is linked to the concept of indistinguishability as a resource by sLOCC introduced in Ref. Lo Franco and Compagno (2018).

Since the sites (A, C) and (D, B) are separated, the identical particles can be distinguished by their spatial location. Once got the post-selected state $|\Psi^{(4)}_{\mathrm{PS}}\rangle$, the entanglement swapping proceeds following the lines of the standard protocol for distinguishable particles (Zukowski et al., 1993). A Bell measurement is therefore performed on the intermediate nodes (C, D) to obtain entanglement over the far nodes A and B. In fact, one can write

[TABLE]

where

[TABLE]

with IJ$=$AB, CD. The result of the Bell measurement does not depend on the particle statistics, as expected from the fact that the post-selected state describes identical particles in separated spatial regions under sLOCC.

The previous protocol can be straightforwardly extended to multiple entanglement swapping (with $n=2N$ independently prepared particles, being $N$ the number of involved particle pairs), in analogy to the case of distinguishable particles Goebel et al. (2008), by a cascaded procedure with a success probability $P(n)=1/2^{\frac{n}{2}}$.

Appendix E Probabilities of success

We finally compare the efficiency of the protocol for the various cases treated above, given the prepared state before post-selection. We have already seen that, in the case of separated nodes, the success probability for both bosons and fermions is $P(n)=1/2^{\frac{n}{2}}$ (see green squares in Fig. 4). In the case of shared intermediate nodes, the success probabilities $P_{\mathrm{f}}(n)$ for fermions and $P_{\mathrm{b}}(n)$ for bosons decrease as a function of the particle number similarly to $P(n)$, as displayed in Fig. 4 (orange points and blue triangles, respectively). From the experimental viewpoint, one has to take into account that the requirement of Bell measurements further hinders the protocol efficiency (Yang et al., 2006; Hu and Rarity, 2011; Lee and Jeong, 2013; Yang et al., 2005; Pakniat et al., 2017a; Grice, 2011; Zhou and Sheng, 2015; Sheng et al., 2010). Therefore, the fermionic process results in being not only qualitatively different, but also more advantageous from a practical viewpoint than the other procedures that necessarily require Bell measurements.