Entanglement and Non-Locality in Quantum Protocols with Identical Particles

In the following, we consider a simplified, yet rather illustrative setting consisting of bosons with spatial $S=L,R$ and internal, $\sigma =\uparrow ,\downarrow$, degrees of freedom corresponding to the orthonormal basis: in the single-particle Hilbert space. One can then consider the corresponding mode creation and annihilation operators $a_{S,\sigma }$, $a_{S,\sigma }^{†}$ and proceed to bipartite the whole Bose algebra constructed upon such mode operators into commuting algebras made of functions of the left or of the right mode operators respectively, denoted, in a compact notation, by:

Products $A_L{A}_R$ of observables $A_L\in {\mathcal{A}}_L$ and $A_R\in {\mathcal{A}}_R$ such as, for instance, the boson number operators, are mode-local with respect to the algebraic bipartition $\left({\mathcal{A}}_L,{\mathcal{A}}_R\right)$.

Other choices of commuting algebras, e.g., ${\mathcal{A}}_{\uparrow }={\{a_{S,\uparrow },a_{S,\uparrow }^{†}\}}_{S=L,R}$ and ${\mathcal{A}}_{\downarrow }={\{a_{S,\downarrow },a_{S,\downarrow }^{†}\}}_{S=L,R}$, give rise to different kinds of mode entanglement. In all cases, unlike particles that cannot in general be individually addressed, modes described by commuting algebras can.

Though reduced to a few degrees of freedom, the above setting allows one to discuss the usefulness for the quantum metrology and teleportation of N bosons occupying the states $|L,\uparrow \rangle$ and $|R,\downarrow \rangle$. These N-boson states belong to a subspace spanned by state vectors that, in first quantization, read: where $\mathcal{S}$ is the symmetrization projector and, in second quantization, become Fock number states: satisfying (see (21)):

A generic normalized state vector spanned by the above Fock states reads:

Particular instances of these states can be manipulated and used as resources for quantum protocols; indeed, we propose to focus on the following states as they exemplify several resource states used in quantum information tasks:

With the exception of the Fock number state $|{\Psi }_{\mathrm{Fock}}\rangle$, which, according to (11), is mode separable with respect to the bipartitions $({\mathcal{A}}_L,{\mathcal{A}}_R)$ and $\left({\mathcal{A}}_{\uparrow },{\mathcal{A}}_{\downarrow }\right)$, all other states are mode entangled with respect to them. Moreover, the state $|{\Psi }_{\mathrm{unif}}\rangle$ is maximally entangled as quantified by several entanglement measures, like entanglement entropy and entanglement negativity [61,62].

As a first protocol, let us consider the estimation of the phase-shift $\theta$ implemented by a Mach–Zehnder interferometer acting on an N-boson state $|\Psi \rangle$ as in (25) by shifting the relative phase of the modes $|L,\uparrow \rangle$ and $|R,\downarrow \rangle$. The action of such an interferometer is described by the unitary operator: where $N_{L,\uparrow }=a_{L,\uparrow }^{†}{a}_{L,\uparrow }$ and $N_{R,\downarrow }=a_{R,\downarrow }^{†}{a}_{R,\downarrow }$ count the number of bosons in the modes $|L,\uparrow \rangle$ and $|R,\downarrow \rangle$.

In quantum metrology, the quantum Cramér–Rao inequality lower bounds the best accuracy ${\delta }^2\theta$ reachable in the determination of $\theta$ by the inverse of the quantum Fisher information $F_{|\Psi \rangle }\left(N_{LR}\right)$ relative to the input state $|\Psi \rangle$ and the generator $N_{LR}$ of the transformation [11,12,13,14]. The input state being pure, the quantum Fisher information reduces to (four times) the variance ${\Delta }_{|\Psi \rangle }{N}_{LR}$ of the generator with respect to the state $|\Psi \rangle$. Then:

The best accuracy achievable by classical interferometry is the inverse of the classical Fisher information; it corresponds to the so-called shot-noise, namely to the Fisher information increasing linearly with the number of particles. Whenever the Fisher information exceeds the shot-noise threshold, quantum features, either in the initial state or in the interferometer, need to be at work.

In the case of the metrological task depicted above, the required quantumness is the mode entanglement of the state with respect to the bipartition $\left({\mathcal{A}}_L,{\mathcal{A}}_R\right)$. In fact, according to (9), the unitary operator $U_{\theta }$ in (29) is mode-local with respect to the bipartition $\left({\mathcal{A}}_L,{\mathcal{A}}_R\right)$, for $N_{L,\uparrow }$ and $N_{R,\downarrow }$ commute so that $U_{\theta }$ splits into a product of commuting unitary operators:

It follows that mode entanglement in the initial state is necessary to beat the shot-noise, if the interferometer is mode-local and the particle number is conserved [63,64]. This fact can be best appreciated as follows: if the N-boson state $|\Psi \rangle$ were mode-separable with respect to $\left({\mathcal{A}}_L,{\mathcal{A}}_R\right)$, it would be of the form (11) with ${\mathfrak{f}}_1^{†}={\mathfrak{f}}_L^{†}$ a suitable function of $a_{L,\uparrow }^{†}$ and $a_{L,\downarrow }^{†}$ and ${\mathfrak{f}}_2^{†}={\mathfrak{f}}_R^{†}$ a suitable function of $a_{R,\uparrow }^{†}$ and $a_{R,\downarrow }^{†}$. Then, according to Remark 2 and using (12), one computes: where $|{\mathfrak{f}}_{L,R}\rangle \equiv {\mathfrak{f}}_{L,R}^{†}|\mathrm{vac}\rangle$. It thus follows that:

Indeed, since the total number of bosons is fixed to be N, $|{\mathfrak{f}}_L\rangle$ and $|{\mathfrak{f}}_R\rangle$ are as $|{\Psi }_{\mathrm{Fock}}\rangle$ in (23), hence eigenstates of the number operators $N_{L,\uparrow }$ and $N_{R,\downarrow }$. The vanishing variances are in agreement with the fact that phase-changes and particle number operators are conjugate observables so that perfect knowledge of one of them entails total ignorance about the other one. This same situation occurs with any fixed, finite number of modes, or for distinguishable particles with a fixed and finite single-particle Hilbert space dimension.

For the mode-entangled states in (25), one finds: whence (34) together with (30) yields:

In the case of the specific states in (26)–(28), the Fisher information reads:

It follows that $F_{|{\Psi }_{{\ell }_1,{\ell }_2}\rangle }\left(N_{LR}\right)$ beats the shot-noise whenever $F_{|{\Psi }_{{\ell }_1,{\ell }_2}\rangle }\left(N_{LR}\right)>N$, e.g., for the NOON state when ${\ell }_1=N-{\ell }_2=0$ and $\xi =1/2$. In this case, $F_{|{\Psi }_{0,N}\rangle }\left(N_{LR}\right)=N^2$, reaching the maximum value for the quantum Fisher information, the so-called Heisenberg limit. As for the other states, $F_{|{\Psi }_{\mathrm{unif}}\rangle }\left(N_{LR}\right)$ always beats the shot-noise: indeed, if the phases ${\phi }_{\ell }\propto \ell$, $|{\Psi }_{\mathrm{unif}}\rangle$ is the eigenstate of the phase operator canonically conjugated to $N_{LR}$ [65]. Instead, $F_{|{\Psi }_{\mathrm{coh}}\rangle }\left(N_{LR}\right)$ never beats the shot-noise because $|{\Psi }_{\mathrm{coh}}\rangle$ is a coherent state of the SU(2) group whose elements model linear interferometers and reproduce classical performances. In particular, while Equation (33) shows that entanglement is necessary if the particle number is conserved [63,64], Equation (38) implies that entanglement is not in general sufficient to beat the shot-noise with local interferometers. It is indeed well known also for distinguishable particles that not all entangled states are useful for quantum metrological advantages.

Now, let us consider the unitary operator:

It describes a transformation that is non-local with respect to the bipartition $\left({\mathcal{A}}_L,{\mathcal{A}}_R\right)$ and amounts to a phase-change operated through a rotation of the modes $(L,\uparrow )$ and $(R,\downarrow )$, implementable by means of beam splitters. By acting with $V_{\theta }$ on the state $|{\Psi }_{\mathrm{Fock}}\rangle$ in (23), which we have seen to be mode-separable with respect to the bipartition $\left({\mathcal{A}}_L,{\mathcal{A}}_R\right)$, one can beat the shot-noise limit when $\ell \ne 0,N$ [22,23]. Indeed, one computes: which improves upon the shot-noise at least by a multiplicative constant for $\ell \ne 0,N$, while reaching the Heisenberg scaling $F_{|{\Psi }_{\mathrm{Fock}}\rangle }\left(T_{LR}\right)\sim N^2/2$ if $\ell \sim N/2$. Instead, when $\ell =0,N$, the state $|{\Psi }_{\mathrm{Fock}}\rangle$ is of the coherent form of $|{\Psi }_{\mathrm{coh}}\rangle$ in (28) with $\xi =0,1$, which indeed ensures only classical performances. On the other hand, if $\ell \ne 0,N$, the resource able to provide the entanglement in the output state necessary for quantum-enhanced performances is the non-local action of the interferometer. It is therefore possible to achieve quantum advantages with mode-separable states of N particles independently prepared in two non-overlapping localizations and internal states. Namely, there is no need to entangle the input state if the devices operates non-locally on the modes being used. Indeed, the theory of mode entanglement identifies the entangling power of beam splitters as a quantum resource [64,66,67], as also happens with violations of Bell’s inequalities in similar physical situations [68,69].

As well known, quantum teleportation [70] is a protocol for the transmission of an unknown quantum state to a remote location using an entangled resource state, local operations, and classical communication. In the standard framework of distinguishable particles, states of a specific particle, say particle A, are teleported to a distant one, say particle C. This latter particle is initially entangled with another particle, say particle B, that is close to particle A. The sender performs a projective measurement onto Bell states of the particles A and B; then, according to the obtained result, which is sent to him/her through a classical communication channel, the receiver applies a local operation to particle C, whose state finally turns out to coincide with the initial state of particle A. This protocol has been generalized to three indistinguishable particles by considering a fully symmetrized or antisymmetrized state [71], whereby it was shown that symmetrization or antisymmetrization cannot consistently generate entanglement useful for implementing teleportation.

In many-body systems, indistinguishable particles are not in general confined to different locations, otherwise they could be distinguished, neither can they be individually addressed, e.g., by ascribing them unambiguous properties. A full generalization of the teleportation protocol in a many-body context consists of teleporting superpositions of the occupation number eigenstates of suitable sets of modes in the Fock space, rather than particle states as done in a first quantization approach [72,73,74,75]. This general protocol recovers the standard teleportation protocols involving distinguishable particles when the local particle occupation numbers of these modes are fixed. Indeed, in this case, addressing modes coincides with addressing particles.

In order to be more specific, consider a state of M indistinguishable bosons distributed between the two modes $(X,\uparrow )$ and $(Y,\downarrow )$ with spatial locations X and Y, according to the state: where the mode $(Y,\downarrow )$ plays the role of particle A in the standard setting. The goal of the protocol presented below is to teleport particle states from the mode $(Y,\downarrow )$ to the mode $(R,\downarrow )$, which then corresponds to particle C in the standard setting. Such a mode is accessible to N bosons described by a resource state $|\Psi \rangle$ as in Equation (25) that is thus mode-entangled with respect to the algebraic bipartition $\left({\mathcal{A}}_L,{\mathcal{A}}_R\right)$. In this scheme, the remaining mode $(L,\uparrow )$ plays thus the role of particle B in standard teleportation.

The protocol consists of:

The operations involved in the teleportation protocol are mode-local as they are described in terms of operators relative to the fixed spatial locations Y and L on the one hand and R on the other. The main difference with respect to the metrological setting discussed in the previous section is that the analysis of the teleportation protocol needs the extension of the two-mode subalgebra, ${\mathcal{A}}_L$, to a four-mode one, ${\mathcal{A}}_{YL}={\{a_{Y,\sigma },a_{Y,\sigma }^{†},a_{L,\sigma },a_{L,\sigma }^{†}\}}_{\sigma =\uparrow ,\downarrow }$.

The teleportation performances are measured by the fidelity given by the overlap between the input state $|\Phi \rangle$ and the obtained state of the modes $(X,\downarrow )$ and $(R,\uparrow )$ (as if they were in the same Hilbert space) averaged over all Bell measurement outcomes and over all $|\Phi \rangle$. For two-mode boson states, the teleportation fidelity turns out to be [74,75]:

If the resource state is $|{\Psi }_{\mathrm{unif}}\rangle$, the final state of modes $(X,\uparrow )$ and $(R,\downarrow )$ is the same as the state $|\Phi \rangle$ in (25) for a fraction $(N-M+1)/(N+1)$ of measurement outcomes, while other outcomes result in teleporting only some components of the same state. Thus, the teleportation fidelity approaches one for a large particle number in the resource state:

Teleportation is not exact for all Bell measurement outcomes because of the particle number conservation. When the resource state is $|{\Psi }_{\mathrm{coh}}\rangle$ with $\xi =1/2$, the teleportation fidelity also approaches one when $N\to \infty$:

It is remarkable that coherent states $|{\Psi }_{\mathrm{coh}}\rangle$ provide very good teleportation performances with negligible distortions for large N, although they behave classically for the estimation of interferometric phases. Instead, the resource state $|{\Psi }_{{\ell }_1,{\ell }_2}\rangle$ gives rise to poor teleportation performances, very close to those of mode-separable states $|{\Psi }_{\mathrm{Fock}}\rangle$ in (23), which yields:

This fidelity equals the maximum fidelity for teleporting $(M+1)$-level distinguishable particles; further, note that also the mode $(Y,\downarrow )$ to be teleported by means of the protocol discussed here has $M+1$ orthogonal states. The case of $N=M=1$ is particularly suited to enlighten how mode entanglement is able to capture the role of indistinguishability. Indeed, in such a case, the states $|\Psi \rangle$ and $|\Phi \rangle$ are single-particle states, and the Bell-like measurement outcomes that provide perfect teleportation are associated with projections onto single-particle states. It is therefore clear that correlations with respect to the mode-algebraic bi-partitions, rather than among particles, is the relevant resource (see Appendix B).

The above protocol can also be applied to swap entanglement: the modes $(X,\uparrow )$ and $(R,\downarrow )$ in the initial state are not entangled, while they are after the protocol. If the aim is to distribute as much entanglement as possible, a measure of the entangled shared by the modes $(X,\uparrow )$ and $(R,\downarrow )$ is a better figure of merit than the teleportation fidelity as the latter carries information that is not relevant for the entanglement distribution [74,75].

The no-label approach characterizes the entanglement of N indistinguishable particles with at least two degrees of freedom within a formalism that avoids the use of unphysical particle labels [77,78]; as a consequence, state vectors of particles with only one degree of freedom turn out to be always unentangled. We adopted a reformulation in second quantization that is more convenient especially for many-particle states (see [48,78]). In this approach, the entanglement shared between n particles and the remaining $N-n$ ones in an N-boson state vector $|\Psi \rangle$ is (i) defined with respect to a given choice of a finite-dimensional subspace $\mathcal{K}$ of the single-particle Hilbert space and (ii) evaluated by means of the von Neumann entropy of a matrix ${\rho }^{\left(n\right)}$ that is derived from the N-particle projector $|\Psi \rangle \langle \Psi |$ as follows. An orthonormal basis $\{|{\psi }_j\rangle {\}}_{j=1}^p$ is selected in the subspace $\mathcal{K}$; then, a selective measurement is performed based on the set of operators ${\prod }_{j=1}^p{a}_{{\psi }_j}^{n_j}$ with all possible choices of occupation numbers of the corresponding modes such that ${\sum }_{j=1}^p{n}_j=n$, transforming $|\Psi \rangle \langle \Psi |$ into:

As regards ${\rho }^{\left(n\right)}$, two facts need be noticed:

In order to make a comparison between the mode and the no-label approaches to identical particle entanglement in the case of the metrological task, let $\mathcal{K}$ be the two-dimensional subspace of ${\mathbb{C}}^4$ spanned by the vectors $|L,\downarrow \rangle$ and $|L,\uparrow \rangle$. This latter is the usual choice in the applications of the no-label approach, which aims at addressing quantum correlations between localized particles. The matrix ${\rho }^{\left(n\right)}$ results then:

In the case of states $|\Psi \rangle$ of the form (25), one finds: whence:

In this case, the matrix ${\rho }^{\left(n\right)}$ is thus a pure state, and therefore, its von Neumann entropy is zero for any n. As a consequence, according to the no-label approach, the states $|\Psi \rangle$ are not entangled with respect to any particle partitioning.

Within the no-label approach, as much as the notion of entanglement, also that of locality depends on the single-particle subspace $\mathcal{K}$. In particular, single-particle operators whose measurement projects single-particle states onto $\mathcal{K}$ are deemed local [77,79]. Then, it follows that the unitary operator $U_{\theta }$ in (29) that implements the action of the interferometer discussed in the metrological context is local the product of the two spatially local phase-shifts in (31). The puzzle here is thus as follows: in the mode entanglement approach, the metrological quantum advantage is due to the entanglement of the state relative to the algebraic bipartition with respect to which the interferometer action is instead local. On the contrary, in the no-label approach, neither the state is entangled nor the interferometer is non-local, yet the quantum advantage is there as predicted by the behavior of the Fisher information, which is only due to the structure of the state $|\Psi \rangle$ and of the unitary operator $U_{\theta }$. Experimental evidence that the shot-noise limit is beaten would then force one to accept that, in the no-label approach, quantum metrological advantages might occur without either quantum entanglement or quantum non-locality during any step of the protocol.

Analogously, in the no-label approach, operators of the form: and O, a single particle operator, in first quantization, or: in second quantization, are local. Such structures can be straightforwardly generalized so that n-particle local operators have the form:

Therefore, the measurements in the teleportation protocol involve local observables in the L and Y positions, and thus act locally on the states $|\Psi \rangle$. Furthermore, also the last step of the teleportation protocol is local as it acts unitarily with respect to the mode $(R,\downarrow )$ and possibly on nearby auxiliary modes. Like in the case of phase estimation with local interferometers, the teleportation protocol interpreted within the no-label approach is apparently implementable without either entanglement or quantum non-locality, quite a paradoxical conclusion.

Another particle-based approach to entanglement in systems of indistinguishable particles, sometimes called particle entanglement, defines as separable states those where all particles are in the same single-particle state $|\psi \rangle$, namely ${|\psi \rangle }^{\otimes N}$ in first quantization, or ${\displaystyle \frac{{\left(a^{†}\right)}^N|\mathrm{vac}\rangle }{\sqrt{N!}}}$ for any mode-creation operator $a^{†}$ in second quantization [80,81,82,83].

This approach is a straight extension of the standard theory of entanglement to identical particles. Indeed, unlike in Section 2, symmetrization is taken as a legitimate source of entanglement. For instance, in this way, all Fermionic state vectors turn out to be entangled. Within this approach, we show that quantum advantages as teleportation emerges without particle entanglement, for they are obtainable from particle-separable states. It then follows that this particular type of particle entanglement is not consistent with the operational point of view that asks for entanglement in order to achieve quantum advantages.

As a first example, let us consider the teleportation protocol discussed in Section 3.2, and set $N=M=1$. Then, the resource state (25) reduces to $|\Psi \rangle =\left(a_{L,\uparrow }^{†}+a_{R,\downarrow }^{†}\right)|\mathrm{vac}\rangle$, and thus to the states $|{\Psi }_{{\ell }_1=0,{\ell }_2=1}\rangle$, $|{\Psi }_{\mathrm{unif}}\rangle$ and $|{\Psi }_{\mathrm{coh}}\rangle$ in (26)–(28) with $N=1$. The state $|\Psi \rangle$ is mode-entangled, but as a single-particle state, it is trivially not particle-entangled. On the other hand, the protocol uses projectors onto single-particle states that provide probabilistically perfect teleportation, but, as such, cannot generate particle entanglement. Therefore, the particle entanglement so far considered is not necessary for teleportation.

As a second instance where paradoxical results emerge adopting the above definition of particle entanglement, let us consider a slightly different teleportation protocol that aims at distributing entanglement at distant locations X and R, as done in entanglement swapping. The protocol hinges upon an input state of coherent form, and upon the resource state $|{\Psi }_{\mathrm{coh}}\rangle$ as in (28). Both of these states are particle-separable according to the above definition of particle entanglement. Let us take as the protocol global initial state the combination $|{\Phi }_{\mathrm{coh}}\rangle |{\Psi }_{\mathrm{coh}}\rangle$; notice that this state is not particle-separable since it is not the product of the same single-particle states, despite the fact that $|{\Phi }_{\mathrm{coh}}\rangle$ and $|{\Psi }_{\mathrm{coh}}\rangle$ can be prepared independently at distant locations without any interactions between the corresponding particles. In addition, let us replace the Bell-like measurement with the following generalized measurement: with outcomes $i\in \{0,1\}$ and: where $0\le \eta \le 1$ and $\omega$ an arbitrary phase. Note that only $E_1$ can generate entanglement as $E_0$ projects onto a particle-separable state. Nevertheless, by discarding the “1” outcome and concentrating on the “0” one, as the outcome of the protocol, one ends up with the state: with probability:

For $N=M=n$, one explicitly finds:

This final state turns out to be in general entangled with respect to the untouched modes $(X,\uparrow )$ and $(R,\downarrow )$. Therefore, the above swapping protocol probabilistically generates entanglement starting from a state, $|{\Phi }_{\mathrm{coh}}\rangle |{\Psi }_{\mathrm{coh}}\rangle$, which, as already stressed, can be prepared using only local operations, quite a striking paradox.

Such an inconsistency originates from the impossibility of reconciling the above definition of particle entanglement with any underlying notion of identical particle locality (see Remarks 1 and 3).

A resource theory associated with the notion of particle entanglement here discussed has been recently proposed [83]. In general, resource theories describe quantum resources beyond entanglement [10]; they are based on defining resourceless states, called free states, and resourceless operations, called free operators, as a subset (sometimes a proper subset as in the case of resource theories of entanglement [76,84] and quantum coherence [85]) of operators that cannot transform free states into resourceful ones. It turns out that the resource theory proposed in [83] identifies states of the form $|{\Phi }_{\mathrm{coh}}\rangle |{\Psi }_{\mathrm{coh}}\rangle$ as resourceful, but without specifying any notion of particle-local operations. Instead, in discussing entanglement theory, the notion of locality is crucial in the definition of free operations, which typically consist of local operations and classical communication [76].