Probing quantum coherence at a distance and Aharonov-Bohm non-locality
Sebastian Horvat, Philippe Allard Guerin, Luca Apadula, Flavio Del Santo
PProbing quantum coherence at a distance
Sebastian Horvat, Philippe Allard Guérin,
1, 2
Luca Apadula,
1, 2 and Flavio Del Santo
1, 2, 3 Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria Institute for Quantum Optics and Quantum Information (IQOQI),Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria Basic Research Community for Physics (BRCP) (Dated: April 30, 2020)In a standard interferometry experiment, one measures the phase difference between two paths by recombin-ing the two wave packets on a beam-splitter. However, it has been recently recognized that the phase can alsobe estimated via local measurements, by using an ancillary particle in a known superposition state. In this work,we further analyse these protocols for different types of particles (bosons or fermions, charged or uncharged),with a particular emphasis on the subtleties that arise when the phase is due to the coupling to an abelian gaugefield. In that case, we show that the measurable quantities are spacetime loop integrals of the 4-vector potential,enclosed by two identical particles or by a particle-antiparticle pair. Furthermore, we generalize our considera-tions to scenarios involving an arbitrary number of parties performing local measurements on a general chargedfermionic state. Finally, as a concrete application, we analyse a recent proposal by Marletto and Vedral [1].
I. INTRODUCTION
One ubiquitous feature that distinguishes quantum andclassical theory is the superposition principle. In order todetect the presence of a quantum superposition (coherence),one has to perform measurements of at least two incompat-ible observables and, in fact, it is sufficient to measure therelative phase between the two (or more) states supposedly insuperposition, in addition to the absolute value of the ampli-tudes. This can for example be achieved by an interferomet-ric experiment, whose simplest implementation is the Mach-Zehnder interferometer. There, a single particle is preparedin superposition of two spatially separated paths (by sendingit through a beam-splitter). The phase difference between thepaths can be measured by recombining the paths at a sec-ond beam-splitter and by collecting the statistics of detectorsplaced at the output ports.A celebrated variant of this interferometric experiment al-lows to detect a phase that is due to the interaction betweena charged quantum particle (e.g. an electron) and the elec-tromagnetic potential, known as the Aharonov-Bohm (A-B)effect [2–4]. In particular, this experiment features a Mach-Zehnder interferometer that encircles a solenoid such that,despite the electromagnetic field being zero everywhere onthe paths visited by the electron, there is a measurable phasethat is directly proportional to the magnetic flux through thesurface crossed by the solenoid. It thus seems that a rela-tive quantum phase can only be measured indirectly by re-combining the two paths in a closed interferometer, and ar-guably this limitation looks even more dramatic in the case ofAharonov-Bohm-like phases, since they depend on the valueof the total electromagnetic flux contained in a closed region.Despite these indications, a number of works have proposedprotocols to detect quantum superpositions without needingto reinterfere the beams, i.e. by using only local operationsand classical communication (LOCC) [1, 5–9].In this work, we further analyze these kinds of protocols,characterizing their domain of applicability to different typesof particles (bosons or fermions, charged or uncharged) andemphasize the constraints imposed by superselection rulesand gauge symmetries in determining which observables are measurable. After a brief analysis of uncharged bosons, wefocus on uncharged fermions for which the parity superselec-tion rule forbids certain measurements. Nevertheless, one cancircumvent this limitation by using an ancillary system as aresource; indeed, we show that an arbitrary number of par-ties sharing a generic fermionic pure state, can perform fullstate tomography by means of an ancillary state and LOCC.We then proceed with the case of electromagnetically chargedparticles and explain issues arising from gauge coupling andgauge invariance. We show that in protocols involving an an-cillary system and local measurements, the measured phasecorresponds to the net phase picked up around a closed loopin spacetime. Thus, even if it is true that one can extract infor-mation about the surrounding electromagnetic field by meansof only LOCC conducted at distant locations (i.e., withoutrecombining the paths in an interferometer), its value is stillequal to the electromagnetic flux through the hyper-surfaceenclosed by the paths travelled by the particle(s) in space-time. We then generalize the latter result to an arbitrary num-ber of sources and parties and show that all information aboutthe gauge field that the parties can gather from LOCC canbe reconstructed from bipartite loop integrals for all pairs ofsources and parties. In the final section, we apply our consid-erations to the scenario proposed in [1, 8], which deals withthe time-dependent Aharonov-Bohm effect.
II. LOCAL MEASUREMENTS OF THEINTERFEROMETRIC RELATIVE PHASE
In this section we review some protocols allowing to mea-sure locally the relative phase acquired along two arms of aninterferometer, without having to recombine the two paths.The setup we consider is illustrated in Figure 1 and is similarto the settings studied in Refs. [1, 7, 8] . Two experimenters,Alice and Bob, reside at two distant locations and a quantumparticle is sent in an equal-weighted superposition of the twopaths towards the parties. Most of the following applies toany particle type (boson or fermion, charged or uncharged);when specificities arise we will point them out explicitly. Weassume that the situation can be described semi-classically, a r X i v : . [ qu a n t - ph ] A p r '
Despite the above-mentioned issues, a possible solution tothe apparent impossibility of locally measuring phase differ-ences of fermions has been proposed in Refs. [1, 5]. The ideais to bypass the superselection rule by using an ancillary parti-cle in a known superposition state as a resource which enablesthe phase measurement. The protocol proceeds as follows.Let us assume that Alice and Bob already possess an ancil-lary particle in spatial superposition of their locations and thatthe phase difference between the two components of the wavefunction is null. After some time, the second particle is sentin spatial superpostition of the two paths, of which we wantto measure the relative phase difference. Upon reaching Aliceand Bob, the total state of the two-particle system is | ψ (cid:105) = 12 (cid:16) a † A + a † B (cid:17) (cid:16) a † A + e i ∆ ϕ a † B (cid:17) | (cid:105) , (5)where the mode corresponding to the ancillary particle is in-dicated by subscript 1 and the mode corresponding to the sec-ond one –whose relative phase needs to be estimated– by sub-script 2. Both Alice and Bob now possess two “wave pack-ets”, one arising from each particle. Furthermore, supposethey possess local beam-splitters allowing them to locally in-terfere their wave packets, and detectors allowing to measurethe outcome statistics at the output ports of the beam-splitters.In order for the parties to be able to perform the measure-ments, the particles must have the same parity and charge be-cause of the associated superselection rules. We thus choosethe particles to be identical. Now, suppose that Alice and Bobperform their measurements and discard the results if theydetect either zero or two particles: the postselected quantumstate of interest is then (prior to the interference through thebeam-splitter) | ψ (cid:105) P S = 1 √ (cid:16) a † B a † A + e i ∆ ϕ a † A a † B (cid:17) | (cid:105) , (6)The beam-splitters, followed by measurements at the outputports, implement local projective measurements of the form As we will discuss in detail in Section III, for charged particles this as-sumption is not as innocent as it may look, for it corresponds to fixing agauge. Π ± = |±(cid:105) (cid:104)±| acting on | ψ (cid:105) P S , where |±(cid:105)
A/B = 1 √ (cid:16) a † A /B ± a † A /B (cid:17) | (cid:105) . (7)Note that the outcome probabilities do not depend on whetherthe particles are fermions or bosons, because we are only con-cerned with the subset of events in which a single particle isfound at each of the two locations. The outcome statistics en-ables Alice and Bob to reconstruct the phase shifter’s phase ∆ ϕ using only LOCC despite issues caused by the parity su-perselection rule.The latter considerations were concerned with the case of twoparties sharing one particle. In Appendix A, we providea generalization to a scenario involving N parties sharing ageneral fermionic uncharged state, which can now involve anarbitrary number of excitations. Notice that the parity supers-election rule allows the state to be in a superposition of stateswith different numbers of fermions (albeit with equal parity).We show that, analogously to the single-particle case, the par-ties can fully reconstruct an arbitrary fermionic pure state us-ing LOCC and a common global state as a resource. The pro-tocol essentially involves the usage of an auxiliary referencestate prepared by an external party and local beam-splitteroperations performed by the N parties. Even though the par-ties’ measurements are local, the overall process involves thepreparation of a global (entangled) state which cannot be gen-erated by local means; this is in accord with the results ofRefs. [17, 18]. III. CHARGED PARTICLES AND THEGAUGE-DEPENDENCE OF RELATIVE PHASES
Let us now turn to charged particles, for which we alreadypointed out that measurements of the type in expression (4)are prohibited by the charge superselection rule. Further-more, in the general protocol of Section II A, we had to as-sume, in Eq. (5), that it is possible to prepare the ancillary par-ticle in a known state. As we will emphasize in this section,since the phase acquired along a path is gauge-dependent, thecondition of Eq. (5), namely that the ancillary state has zerorelative phase, is not a gauge-independent property. In thefollowing we show how the protocol can nevertheless be usedto provide local measurements of gauge-invariant phases, i.e.phases that are acquired by integrating along a closed loop inspacetime. Hereinafter, we will focus on charged fermions,and, as a matter of simplicity, we will consider the particularcase of electrons. Consider once again the setup of Fig. 1.The phase difference is given by Eq. (1), with Lagrangian L = 12 m (cid:126) ˙ x − e(cid:126) ˙ x · (cid:126)A + eV ( (cid:126)x ) . (8)We emphasize that setting the gauge potential to zero, even inthe absence of external electric and magnetic fields, amountsto fixing a gauge. Hence, the accumulated phase on the twopaths now necessarily depends also on the electromagneticpotential: | ψ (cid:105) = 1 √ (cid:16) e ie (cid:82) γA A µ dx µ | A (cid:105) + e ie (cid:82) γB A µ dx µ e iβ | B (cid:105) (cid:17) , (9) where β is the gauge-invariant "mechanical" phase differ-ence, i.e. due to the kinetic term m (cid:126) ˙ x in the Lagrangian;whereas A µ is the 4-vector potential, i.e. (in the units where c = 1 ) A µ = ( V, (cid:126)A ) , and we use a metric with signature (+ , − , − , − ) . Therefore, the phase difference between thetwo paths now reads ∆ ϕ = β + e (cid:90) ( γ B − γ A ) A µ dx µ , (10)which is an explicitly gauge-dependent quantity. We thus seewhy it is necessary to introduce the charge superselection rulewhich forbids us to implement the projectors of Eq. (4): ifthis rule did not exist we would be able to measure physicallymeaningless gauge-dependent quantities! This argument hastwo main consequences: (i) the principle of gauge invarianceimplies the charge superselection rule and prohibits this typeof local tomographic protocols for a delocalized electron.Furthermore, (ii) the acquired phase difference between thetwo paths is not an observable since it is a gauge-dependentquantity: it might be equal to the mechanical phase shift β only in a specific gauge.On the other hand, note that if the electron’s wave packetsare reinterfered on a beam-splitter as it happens in a standardAharonov-Bohm experiment, the phase difference betweenthe two paths connecting the two beam-splitters reduces to ∆ ϕ = β + e (cid:73) A µ dx µ , (11)where the loop integral is performed around the whole inter-ferometer. The phase is now a gauge-independent quantityand it depends on the distribution of electromagnetic currentsin spacetime. A specific case is a regular Mach-Zehnder inter-ferometer, where, since there are no currents, the loop integralvanishes and, as expected, the total phase difference is equalto the mechanical phase shift β only. A. Measuring phase differences at a distance
We now show how, as in the protocol of Section II A, wecan exploit an auxiliary identical particle (another electron) inorder to circumvent the limitation imposed by the charge su-perselection rule and gauge invariance. The crucial differencewith respect to the previous case is that, as already empha-sized, there is an inevitable coupling to the gauge potentialregardless of the surrounding electromagnetic sources, whichimplies that even the ancillary particle necessarily acquires agauge-dependent phase difference which may be null only ina specific gauge (i.e. we cannot assume that Alice and Bobcan prepare the ancillary particle in a known state withoutfixing a gauge). After the particles are sent to the parties, the A similar argument has already been invoked for example in [19]; for amore formal treatment of the relationship between the charge superselec-tion rules and gauge invariance, see [20]. two-particle state is | ψ (cid:105) = 12 (cid:18) e ie (cid:82) γA A µ dx µ a † A + e ie (cid:82) γB A µ dx µ a † B (cid:19)(cid:18) e ie (cid:82) γA A µ dx µ a † A + e ie (cid:82) γB A µ dx µ e iβ a † B (cid:19) | (cid:105) , (12)where the mode corresponding to the first particle is indicatedby subscript 1 and the second one by subscript 2, and thepaths γ A/B / are defined in Fig. 2. Alice and Bob again per-form measurements with local beam-splitters and postselecton the one-particle subsector: | ψ (cid:105) P S = 1 √ (cid:16) a † A a † B + e i ∆ ϕ a † A a † B (cid:17) | (cid:105) , (13)where the cumulative phase difference is now given by ∆ ϕ = β + ϕ A + ϕ B − ϕ A − ϕ B , (14)with ϕ A/B / ≡ e (cid:90) γ A/B / A µ dx µ . (15)As in Section II A, the quantity ∆ ϕ can be inferred from thestatistics of the detection clicks at the two beam-splitters, re-spectively situated at Alice’s and Bob’s locations.In Figure 2 we portray the paths of the two particles in aspacetime diagram and show that they enclose a closed path.Therefore, the additional phase accumulated because of theinteraction with the electromagnetic potential is gauge invari-ant. If there are no surrounding currents or fields, the loopintegral vanishes and only the mechanical phase shift β re-mains. On the contrary, if the particles are surrounded by anarbitrary distribution of currents, the loop integral does notgenerally vanish and can depend explicitly on the trajectorytravelled by the ancillary particle (even if it is not acted on byclassical forces).Had the two particles different charges e and e , the totalphase ∆ ϕ would involve the sum of two paths which do notcompose into a spacetime loop integral as in (14) and wouldthus not be a gauge invariant quantity. However, in that case,the required measurement would not be possible since non-identical particles do not interfere with each other. Therefore,we see a consistency between gauge invariance, superselec-tion rules and the operational attainability of the required lo-cal measurements.In Appendix B, we analyse a variation of the above ex-periment introduced by Aharonov and Vaidman in [5], whichinstead of two identical particles (e.g. two electrons) involvesa particle-antiparticle pair (e.g. an electron and a positron).We show that the same quantity ∆ ϕ from Eq. 14 can be es-timated by measuring photons resulting from the annihilationof the electron with the positron.From these considerations we draw the following conclu-sions:• loops (in spacetime) can be closed by states involvingtwo identical particles or by a particle-antiparticle pair,i.e. by two different excitations of the same quantumfield (contrary to the standard Aharonov-Bohm effectwhere the loop is closed by a single electron); t
In Section II A and in Appendix A we saw the possibil-ity of performing state tomography of an arbitrary unchargedfermionic state using LOCC and an ancillary system. Now,we want to analyse a similar scenario in the case of chargedparticles. However, since the concepts of “primary” and “an-cillary” systems are not well defined in the presence of gaugecoupling, here we ask a different question. Suppose that anarbitrary number of parties perform local measurements onan arbitrary state of multiple charged particles: what are thespacetime loop integrals that the outcome probabilities de-pend on? Can all probabilities be reconstructed from loopssimilar to the one depicted in Fig. 2? In Appendix C weshow that this is indeed the case. More precisely, suppose thatwe have d sources emitting single identical charged fermionsat d spacetime points. Each source prepares a single parti-cle in an arbitrary superposition of spatial trajectories (againusing the semi-classical approximation). Moreover, suppose S
The traditional interpretation of the Aharonov-Bohm ef-fect [2–4] is that the electric and magnetic fields are in generalnot sufficient for describing the physics of certain (quantum)scenarios and that the gauge-theoretic electromagnetic poten-tial is in fact indispensable. Going against the received view,Vaidman has argued that the Aharonov-Bohm phase can beexplained without the introduction of potentials if one takesinto account the quantum nature of the solenoid [6]. A weak-ness in his treatment is that it relies on an instantaneous in-teraction between the solenoid and the electron. Marletto andVedral [1] –followed by further developments by Saldanha[8]– have recently addressed this problem, concluding thatthe Aharonov-Bohm phase is acquired locally. In this sectionwe follow up on these recent developments by applying theanalysis from the previous section to the specific case wherethe only source of electromagnetic fields is a solenoid witha time-dependent current. As before, the goal for Alice andBob is to locally measure the phase difference acquired bythe electron through the interaction with the gauge potential.The scenario studied in Ref. [1] is a special case of thegeneral situation in Fig.2. During the journey of the ancillary particle towards the parties, the flux through the coil is setto zero. After the two wave packets of the ancillary particlearrive at the location of the parties, they are trapped usingsome external field. The current (and thus the flux) in thecoil is then increased slowly until it reaches a stationary state,following which the second particle is sent to the parties andthe local measurements of (7) are performed. The detectionprobabilities depend on ∆ ϕ as defined in (14), where (in theCoulomb gauge) the vector potential in the region outside thecoil is given by (cid:126)A ( (cid:126)r, t ) = Φ( t )2 π ( x + y ) (cid:126)r × ˆ z, (16)where Φ( t ) is the magnetic flux through the solenoid at time t , and (cid:126)r is the position vector of the particle (with the origin ofthe coordinates centered in the solenoid). Note that the elec-tric field is not zero outside the solenoid at all times, becauseof the time-dependence of the current in the solenoid. Sincethe charge density is zero everywhere and at all time, we havethat the scalar potential is zero in the Coulomb gauge, andthus the phase that the ancillary particle accumulates due tothe solenoid during the time when it is trapped in Fig. 2 iszero (in this gauge). Furthermore, when the ancillary electronis moving, there is no current in the coil and consequently no4-vector potential is interacting with the charge, so the contri-bution of the solenoid to the phase accumulated along γ A /B is zero. Hence, we are left with the phase acquired by the sec-ond particle ∆ ϕ = e (cid:90) ξ (cid:126)A · d(cid:126)l, (17)where ξ is the path shown in Figure 3. Using the expres-sion (16) for the vector potential and switching to polar coor-dinates yields ∆ ϕ = (cid:90) θ Φ f π dθ (cid:48) = Φ f π θ, (18)where Φ f is the final flux through the solenoid and θ is theangle defined in Figure 3. The probabilities for the measure-ment described in Section III A depend on ∆ ϕ and thus theoutcome statistics allows to estimate ∆ ϕ using LOCC.It is tempting to interpret the calculation that we have justdone, in particular Eq. (17), as showing that the phase is lo-cally accumulated along the path ξ . However, this appar-ent localisation of the phase accumulation is merely a con-sequence of our (arbitrary) choice to work in the Coulombgauge. In other gauges, the phase accumulated by the ancil-lary particle is not necessarily zero; only the full loop integralis gauge-independent. Notice that, even though the particlesseemingly enclose a spatial region with no electromagneticfields, the measured quantity ∆ ϕ corresponds to the space-time loop integral which is non-zero due to the induced elec-tric field (which arises due to the time dependence of thecurrent) piercing through the hypersurface enclosed by thespacetime loop. IV. CONCLUSIONS AND OUTLOOK
We have studied a general protocol allowing to locallymeasure the relative phases of the state of a particle that isprepared, by using a beam-splitter, in a superposition of dif-ferent spatial locations. In order to perform this protocol, thelocalised parties must share a known superposition state ofan ancillary particle. We have shown how, using this knownstate as a resource, one can estimate the relative phase byperforming only local measurements, despite the restrictionsimposed by the parity superselection rule. Moreover, we ex-tended the protocol to general fermionic systems shared byan arbitrary number of parties and we showed how the par-ties can perform full state tomography using LOCC and aglobal state as a resource. We proceeded by addressing thecase of electromagnetically charged particles, where the un-avoidable coupling to the gauge field makes it impossible tocontrol the relative phase of the ancillary state without fix-ing the gauge. We have shown that the protocol neverthe-less measures a gauge-independent quantity, namely a space-time loop integral of the 4-vector potential. The protocol re-quires the two particles to possess equal charges in order toobtain gauge invariant quantities; however, the required mea-surements would be physically impossible on two differentcharges due to the charge superselection rule. Alternatively,one can employ a slightly modified protocol that involvesparticles with opposite charges (e.g. electron and positron)which can annihilate into an uncharged particle (photon): theobtained measurement results yield the same gauge invariantphase as in the former protocol. The latter discussion showsthe tight relation between charge conservation, gauge invari-ance and superselection rules. We then proceeded with thegeneral scenario involving many source and many parties per-forming LOCC and showed that all probabilities arising fromsuch experiments can be reduced to simple combinations ofbipartite loop integrals involving two sources and two parties.Finally, we have applied the protocol to the case of local mea- surements of phases in the time-dependent Aharonov-Bohmeffect, and, in particular, we have demonstrated its applicationto the setups of Refs. [1, 8]. Since all probabilities obtainedin these experiments depend on loop integrals which are ex-plicitly nonlocal quantities, the interpretation that the phaseis acquired locally is not viable, and is apparently manifestedonly in a specific gauge.In this manuscript we focused on scenarios which involvequantum particles coupled to Abelian gauge fields; it wouldbe interesting to see to what extent our analysis can be ex-tended to the non-Abelian case, where the probabilities woulddepend on gauge-invariant functionals of Wilson loop opera-tors. In particular, one should inspect whether the result ofsection III B holds in any gauge theory or only in the Abelianones.
ACKNOWLEDGMENTS
We would like to thank ˇCaslav Brukner and Borivoje Daki´cfor useful discussions and comments that greatly improvedthis paper. F.D.S. acknowledges the financial support througha DOC Fellowship of the Austrian Academy of Sciences(ÖAW). S.H acknowledges support from an ESQ DiscoveryGrant of the Austrian Academy of Sciences (ÖAW). P.A.G.and L.A. acknowledge support from the research platformTesting Quantum and Gravity Interface with Single Pho-tons (TURIS), the Austrian Science Fund (FWF) throughthe projects BeyondC (F7113-N48), and the doctoral pro-gram Complex Quantum Systems (CoQuS) under Project No.W1210-N25 and the financial support from the EU Collabo-rative Project TEQ (Grant Agreement No.766900). They alsoacknowledge a grant from the Foundational Questions Insti-tute (FQXi) Fund and by the ID [1] C. Marletto and V. Vedral, (2019), arXiv:1906.03440.[2] Y. Aharonov and D. Bohm, Physical Review , 485 (1959).[3] Y. Aharonov and D. Bohm, Phys. Rev. , 1511 (1961).[4] M. Peshkin, in
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Let us start by considering a simple generalisation of Fig. 1 to N parties. The particle goes through a beam-splitterwith N output ports, and along each arm it acquires an unknown phase ϕ i , so that the state received by the observers is √ N (cid:80) Ni =1 e iϕ i c † i | (cid:105) , where c † i is an operator that creates a particle at the location of observer i . We would like to perform aprocedure for estimating the phases ϕ i that uses only local operations and classical communication. This can be achieved ifthe parties share a second particle in a known state √ N (cid:80) Ni =1 c † i | (cid:105) . Let each party apply a beam-splitter locally and measureat the click at the output ports, and post-select on cases where the two particles are found at different locations; this happenswith probability − N . Supposing that the particles are found at positions i and j , the probabilities for each of the 4 possibleoutcomes are the same as in the case of Fig. 1, so the above protocol allows to give an estimate for ϕ i − ϕ j . After performingmany rounds it will be possible to reconstruct all the phases ϕ k with a good accuracy. An interesting feature of this protocolis that the post-selection probability goes to one as the number of parties becomes large, which means that in this limit almostall rounds of the experiment yield useful information (in contrast with the two arm case, where half of the rounds have to bediscarded).Turning now to the general case, the most general fermionic state shared by N parties is | ψ (cid:105) = (cid:88) (cid:126)x ,...,(cid:126)x N λ ( (cid:126)x , ..., (cid:126)x N ) e iϕ ( (cid:126)x ,...,(cid:126)x N ) (cid:89) j (cid:16) c (1) † j (cid:17) x j (cid:89) j (cid:16) c (2) † j (cid:17) x j ... (cid:89) j N (cid:16) c ( N ) † j N (cid:17) x NjN | (cid:105) . (A1)In the latter expression the sum ranges over all bit strings, the length of which depends on the maximal number of local modesavailable to each of the parties (the bit string notation automatically implements the fact that there can be no more than oneexcitation per mode). λ ( (cid:126)x , ..., (cid:126)x N ) are real amplitudes, ϕ ( (cid:126)x , ..., (cid:126)x N ) are the mechanical phases that we want to estimate, and c ( k ) † j k denote fermionic creation operators that create one fermion in the j k -th mode of the k -th party.In order to measure the weights λ ( (cid:126)x , ..., (cid:126)x N ) , each party performs a local projective measurement on their local modes,yielding the desired information via λ ( (cid:126)x , ..., (cid:126)x N ) = | (cid:104) ψ | (cid:89) j (cid:16) c (1) † j (cid:17) x j ... (cid:89) j N (cid:16) c ( N ) † j N (cid:17) x NjN | (cid:105) | . (A2)The parties communicate their local results to an external party who estimates the amplitudes λ ( (cid:126)x , ..., (cid:126)x N ) , prepares a uniformsuperposition over states with non-zero amplitude and sends it towards the N parties, who store it in separate modes from theones occupied by the original state. The "copied" state is thus | ˜ ψ (cid:105) = 1 √ M (cid:88) (cid:126)x ,...,(cid:126)x N (cid:89) j (cid:16) ˜ c (1) † j (cid:17) x j ... (cid:89) j N (cid:16) ˜ c ( N ) † j N (cid:17) x NjN | (cid:105) , (A3)where ˜ c ( k ) † j k ≡ c ( k ) † n k + j k , with n k being the maximum number of modes present at the k -th party’s location and M is the totalnumber of non-zero components in the original state (A1).Next, each party interferes each of the original modes with the corresponding copied modes on local beam-splitters, i.e. ∀ k =1 , ..., N and ∀ j k = 1 , ..., n k c ( k ) † j k → √ (cid:16) c ( k ) † j k + ˜ c ( k ) † j k (cid:17) , ˜ c ( k ) † j k → √ (cid:16) c ( k ) † j k − ˜ c ( k ) † j k (cid:17) . (A4)The final joint state, after undergoing the beam-splitter operations, is thus | ψ (cid:48) (cid:105) = √ M (cid:88) (cid:126)x ,...,(cid:126)x N (cid:126)x (cid:48) ,...,(cid:126)x (cid:48) N − / [ (cid:80) nj ( x nj + x (cid:48) nj )] λ x e iϕ x N (cid:89) k =1 n k (cid:89) j k =1 (cid:16) c ( k ) † j k + ˜ c ( k ) † j k (cid:17) x kjk N (cid:89) k (cid:48) =1 n (cid:48) k (cid:89) j (cid:48) k =1 (cid:16) c ( k (cid:48) ) † j (cid:48) k − ˜ c ( k (cid:48) ) † j (cid:48) k (cid:17) x (cid:48) k (cid:48) j (cid:48) k | (cid:105) , (A5)where λ x and ϕ x stand for λ ( (cid:126)x , ..., (cid:126)x N ) and ϕ ( (cid:126)x , ..., (cid:126)x N ) .Finally, the parties perform local projective measurements of their modes in the occupation number basis; the outcome proba-bilities depend explicitly on the required phases: P xy = 1 M (cid:80) n,m ( x nm + y nm ) ] | λ x + ( − s xy e i ( ϕ y − ϕ x ) λ y | , (A6)where x = ( (cid:126)x , ..., (cid:126)x N ) and y = ( (cid:126)y , ..., (cid:126)y N ) are two different bit strings, P xy is the probability of projecting on state (cid:110)(cid:81) Nk =1 (cid:104)(cid:81) n k j k =1 (cid:16) c ( k ) † j k (cid:17) x kjk (cid:105) (cid:81) Nk (cid:48) =1 (cid:104)(cid:81) n (cid:48) k j (cid:48) k =1 (cid:16) ˜ c ( k (cid:48) ) † j (cid:48) k (cid:17) y k (cid:48) j (cid:48) k (cid:105)(cid:111) | (cid:105) , and s xy is an integer that arises due to fermionic anticommu-tation relations. The parties can thus estimate all the unknown phases ϕ ( (cid:126)x , ..., (cid:126)x N ) using LOCC and a shared global ancillarystate. Appendix B: The protocol involving a particle-antiparticle pair
Instead of two identical particles, one can choose the two particles used in the bipartite protocol of Section III to be a particle-antiparticle pair, for instance, an electron and a positron. In this case, Alice and Bob can annihilate their wave packets intophoton pairs and measure the phase difference between the two components by performing local tomography on the resultingphotons in a similar fashion as in section III A. More precisely, the state of the two particles before the annihilation processis the same as in protocol involving two identical particles, up to a minus sign in the phases acquired by the antiparticle (the“ancillary” system is now a positron with charge − e ): | ψ (cid:105) = 12 (cid:18) e − ie (cid:82) γA A µ dx µ b † A + e − ie (cid:82) γB A µ dx µ b † B (cid:19) ⊗ (cid:18) e ie (cid:82) γA A µ dx µ c † A + e ie (cid:82) γB A µ dx µ e iβ c † B (cid:19) | (cid:105) , (B1)where b † and c † are respectively positron and electron creation operators.Next, Alice and Bob let the particles interact and they postselect exclusively on processes which give rise to photons, therebydiscarding the one-particle sector and those processes in which the pairs scatter without annihilating (e.g. the Bhabba scatter-ing). The annihilation processes give rise to photon pairs of different momenta (cid:126)k and (cid:126)k (cid:48) : b † i c † i → a † i,(cid:126)k a † i,(cid:126)k (cid:48) , (B2)where a † i,(cid:126)k denotes a (suitably smeared) creation operator for a photon of momentum (cid:126)k produced at location i . The quantumstate after the interaction and post-selection is thus | ψ (cid:105) = 1 √ (cid:18) a † A, (cid:126)k A a † A, (cid:126)k (cid:48) A + e i ∆ ϕ (cid:48) a † B, (cid:126)k B a † B, (cid:126)k (cid:48) B (cid:19) | (cid:105) , (B3)where the phase difference is now ∆ ϕ (cid:48) = β + ϕ (cid:48) B + ϕ (cid:48) B − ϕ (cid:48) A − ϕ (cid:48) A , ϕ (cid:48) i j ≡ e j (cid:90) γ ij A µ dx µ . (B4)Since the electric charges of the two particles are e = − e and e = e , the phase difference acquired due to the interaction withthe gauge potential is given by a spacetime loop integral of the gauge potential and is equal to Eq. (14). Once Alice and Bobpossess their photons, they can estimate the phase from measurement results of local projections on states |±(cid:105) i = 1 √ (cid:18) ± a † (cid:126)k i a † (cid:126)k (cid:48) i (cid:19) | (cid:105) . (B5)Therefore, we found a different procedure which yields the same gauge invariant phase as the one obtained in the protocolinvolving two identical particles. Appendix C: Local measurements on a general charged fermionic state
Before tackling the fully general case, let us first analyse the following simpler case. Suppose that the scenario involves threesources which emit three electrons in spatial superposition towards three parties labelled as A, B and C. Upon receiving theparticles, the parties perform local unitary transformations on their pertaining wave packets, detect the particles and postselecton the cases in which each party detects one excitation. The postselected state of interest (prior to the local transformations) isthus | ψ (cid:105) P S = 1 √ (cid:88) i,j,k =1 i (cid:54) = j (cid:54) = k (cid:54) = i e iφ ijk c † A i c † B j c † C k | (cid:105) , (C1)where c † are fermionic creation operators (for example c † A i creates the i -th particle at A’s location). The phase φ ijk arises dueto the coupling to the gauge potential A µ (for simplicity we omit the mechanical phases): φ ijk = e (cid:32)(cid:90) γ Ai A µ dx µ + (cid:90) γ Bj A µ dx µ + (cid:90) γ Ck A µ dx µ (cid:33) , (C2)where e.g. γ A i is the trajectory traced by the first particle towards A’s location. The parties apply local unitary transformations c † P i → (cid:88) j =1 U ( P ) ij c † P j , P = A, B, C, (C3)where U ( P ) ij are matrix elements of the transformations.The final state before the measurement is thus | ψ (cid:105) P S = 1 √ (cid:88) lmn (cid:88) i,j,k =1 i (cid:54) = j (cid:54) = k (cid:54) = i e iφ ijk U ( A ) il U ( B ) jm U ( C ) kn c † A l c † B m c † C n | (cid:105) . (C4)Finally, the parties perform local projective measurements; the probability of measuring the state c † A l c † B m c † C n | (cid:105) is P ( A l B m C n ) = 16 | (cid:88) i,j,k =1 i (cid:54) = j (cid:54) = k (cid:54) = i e iφ ijk U ( A ) il U ( B ) jm U ( C ) kn | . (C5)One of the phase differences that appear in Eq. (C5) is for instance φ − φ , which can be expressed as a sum of twobipartite loop integrals, as shown in Fig. 4. The analogous holds for all other outcome probabilities, i.e. all information thatone can gather from local measurements can be reconstructed from bipartite loop integrals. S