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Role of Particle Entanglement in the Violation of Bell Inequalities Tomasz Wasak1, Augusto Smerzi2 & Jan Chwedeńczuk1

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Role of Particle Entanglement in the Violation of Bell Inequalities Tomasz Wasak1, Augusto Smerzi2 & Jan Chwedeńczuk1 www.nature.com/scientificreports OPEN Role of Particle Entanglement in the Violation of Bell Inequalities Tomasz Wasak1, Augusto Smerzi2 & Jan Chwedeńczuk1 Entanglement between two separate systems is a necessary resource to violate a Bell inequality in Received: 15 June 2017 a test of local realism. We demonstrate that to overcome the Bell bound, this correlation must be Accepted: 8 January 2018 accompanied by the entanglement between the constituent particles. This happens whenever a super- Published: xx xx xxxx selection rule prohibits coherences between states with diferent total number of particles and thus imposes a constraint on feasible local operations in each sub-system. We show that the necessary entanglement between the particles might solely result from their indistinguishability. We also give an example of both mode and particle-entangled pure state, which does not violate any Bell inequality. Our result reveals a fundamental relation between the non-locality and the particle entanglement. Te “spooky action at the distance” stands out among the most striking consequences of quantum mechanics1. Tis term was coined by Albert Einstein to underline how counterintuitive it is that a seemingly local manipu- lation on one part of a system immediately afects its other distant part without any signal transferred from one subsystem to the other. Such an efect contradicts the postulates of the “local realism”: a pair of spatially separated, entangled particles can show correlations that cannot be reproduced by any local, hidden variable theory. Te non-locality of quantum mechanics can be revealed by a series of inequalities—frst considered by Bell2—for the correlations between the outcomes of local measurements3–19. Te violation of the Bell inequalities was frst observed long ago5–8 and since then large eforts, devoted to improve the experimental techniques20, culminated in the recent “loophole-free” deviation from local realism claimed in19,21,22. Not all entangled states violate a (known) Bell inequality23, but it has been shown that all entangled pure states do violate a Bell inequality24. For illustration, consider a pure state |ψ〉 shared by two parties A and B that are spatially separated (so that all physical manipulations and measurements performed by one party cannot afect the other one by classical communication or local operations): |〉=|φχ〉⊗|〉 ψ ∑ciiA i B , i (1) where the state has been written in Schmidt decomposition on orthonormal bases of the parties A and B, respec- tively. If the state is A-B entangled—which happens when at least two coefcients of this expansion, say ci and ci′, 24 are non-zero—a Bell inequality will be violated by locally coupling |φiA〉 with |φiA′〉 | and |χi〉B with |χi′〉B. However, sometimes local operations and/or measurements are prohibited by some superselection rule (SSR). Te SSR is a restriction imposed on quantum mechanics forbidding coherences between eigenstates of certain observables25,26. For the purpose of this manuscript, the SSR can be formulated as follows: local operations/meas- urements cannot create/detect coherences between states with diferent number of particles. Here, a particle is understood as a discrete object carrying a set of fundamental quantum numbers, such as the charge or the baryon and lepton numbers27. To illustrate the impact of the SSR on the feasible local operations, consider two states localized in A: one 23 which contains a single sodium 23 atom, denoted by | Na〉A and the other with a rubidium 87 atom, denoted by 87 | Rb〉A. Although these states have the same number of atoms, any operation or measurement coupling these states would not preserve the number of baryons. Terefore, such coupling is forbidden by the SSR25,26,28 imposing the conservation of the total number of baryons. Another known example is a single particle in the superimposed state29: 1 |〉ψ =|(1〉⊗ABAB|〉00+| 〉⊗|〉1). 2 (2) 1Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL–02–093, Warszawa, Poland. 2QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, 50125, Firenze, Italy. Correspondence and requests for materials should be addressed to J.C. (email: [email protected]) SCIENTIFIC REPORTS | (2018) 8:1777 | DOI:10.1038/s41598-018-20034-8 1 www.nature.com/scientificreports/ Te SSR formulated above prohibits the local creation or detection of a superposition of the vacuum |0〉A with the state containing one particle |1〉A. From the point of view of physically realizable local operations one can efectively replace the pure state (2) with an incoherent mixture 1 |〉ψψ〈|→=ρˆ |〉11〈| ⊗|00〉〈 |+|〉00〈| ⊗|11〉〈 |. eff 2 ()ABAB (3) ρ Although the state (2) is A-B entangled, due to the SSR the resulting ˆeff is A-B separable (i.e., non-entangled) and as such does not violate any Bell inequality27,30–32. Note that for photons, to which the SSR does not apply, the 19,21,29 local coupling of |0〉A with |1〉A is allowed and indeed the state (2) violates a Bell inequality . Inspired by this example we formulate and prove a general theorem: the restriction imposed on the local oper- ations by the SSR renders not only the A-B entanglement but also the entanglement of particles shared by A and B necessary for the violation of any Bell inequality. We demonstrate that this latter resource might origin solely from the particle indistinguishability33. Results To set the stage and proceed with the proof, we note that the A-B separable states have a general form ρρ=⊗()i ρ ()i ˆˆ∑piA ˆB , i (4) ρ()i ρ ()i where pi’s are the statistical weights. Here, ˆA and ˆB represent the quantum state of the subsystems in A and B, respectively. We will demonstrate that in presence of SSR and in the context of Bell inequalities, the quantum state should also be inspected through the relation between the N particles shared by A and B. Particle-separable states can be written as ρρ=⊗ (1)(…⊗ρ N) ˆˆ∑pii ˆi , i (5) ρ^()n where i is a quantum state of the n-th particle. Te particle-entangled states are those that cannot be written in this way. We show that all particle separable states do not violate any Bell inequality in presence of SSR. In other words, the quantum state shared by A and B must necessarily be particle-entangled to violate any Bell inequality. Note that the SSR imposes restrictions both globally (on the whole state) and locally, in A and in B separately: it prohibits coherences between states with different particle numbers and renders some local operations unphysical. Distinguishable particles. First, we consider a collection of distinguishable particles. Te basic building block of the N-body density matrix (5) is the one-body pure state, which for the i-th particle reads ()A ††()B |〉ψψ=+αβ()ψψˆˆ()ψ |〉0 . ii()i ii (6) ()k † Here ψˆ i creates a quantum of a feld associated with this particle in the region k, |0〉 is the vacuum and 2 2 |α(ψi)| + |β(ψi)| = 1. According to Eq. (5), the density matrix of N particles forming a separable state is an incoherent mixture of the one-body matrices: N ρˆ =…DDψψ11 NNP(,ψψ,)⊗|ψψii〉〈 |. ∫∫ i=1 (7) Here, the joint probability (,ψψ1 …,)N determines the partition of all the particles among A and B. Te symbol ψi is the integration measure over the set of felds ψi. Following the example from Eq. (2), the SSR enforces every |ψi〉〈ψi| to be replaced with ()A ††()A ()B ()B |〉ψψ〈|→=ραψψ||2 ψψˆˆ|〉〈| +|β ψψ||2 ˆˆ〉〈 |.ψ ii ˆeff ()ii() i 00 i ()ii00 i (8) Tis expression, plugged back into (7) gives N ρρ=…⊗. ˆˆeff DDψψ11 NNP(,ψψ,) eff ()ψi ∫∫ i=1 (9) Since the inter-region coherence is washed out already on the particle level of Eq. (8) and the integral over the ρ felds does not introduce any quantum coherence, the efective N-body density matrix ˆeff is both particle- and A-B-separable (for the rigorous proof, see Methods). To conclude, the SSR “transforms” the state (7) into (9), which has the form of Eq. (4), and as such it will not violate any Bell inequality. If the distinguishable particles are entangled, violation of Bell inequality in presence of SSR might be possible. For illustration, consider an electron (e) and a proton (p) forming a particle- and A-B-entangled state 1 |〉ψ =↑eA〉⊗|↑pB〉+ ↓〉eA⊗|↓〉pB , 2 ()(10) SCIENTIFIC REPORTS | (2018) 8:1777 | DOI:10.1038/s41598-018-20034-8 2 www.nature.com/scientificreports/ where the arrows denote the projection of the spin of each particle on some quantization axis. Now, local oper- ations can be executed by coupling |↑e〉A with |↓e〉A and |↑p〉B with |↓p〉B. Terefore, according to the discus- sion below Eq. (1), this state will violate a Bell inequality. On the other hand, take an alternative particle- and A-B-entangled state 1 ψ =↑ep,0↑⊗BA+⊗0,↓↓ep. 2 ()A B (11) It will not violate any Bell inequality, because SSR forbid the coupling of |↑e, ↑p〉A with |0〉A and |↓e, ↓p〉B with |0〉B. Tis second example highlights the fact that when the SSR apply, both the particle and the A-B entanglement are only necessary, but not sufcient to drive the violation of a Bell inequality. Indistinguishable particles. We now turn to bosons (for a comprehensive review of the entanglement cri- teria for bosons, see34–36), for which a separable state can be written, in analogy with Eq. (7), as37,38 ρ =|DPψψψψ〉〈 |. ˆ ∫ () (12) Here ψ is the spin coherent state, which reads 1 ()AB††() N |=ψψ⟩⟩((αβ)(ψψ^^+|))ψ 0 .
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