Classical model of delayed-choice quantum eraser
CClassical model of delayed-choice quantum eraser
Brian R. La Cour ∗ and Thomas W. Yudichak Applied Research Laboratories, The University of Texas at Austin, P.O. Box 8029, Austin, TX 78713-8029 (Dated: January 12, 2021)Wheeler’s delayed-choice experiment was conceived to illustrate the paradoxical nature of wave-particle duality in quantum mechanics. In the experiment, quantum light can exhibit either wave-likeinterference patterns or particle-like anti-correlations, depending upon the (possibly delayed) choiceof the experimenter. A variant known as the quantum eraser uses entangled light to recover thelost interference in a seemingly nonlocal and retrocausal manner. Although it is believed that thisbehavior is incompatible with classical physics, here we show that the observed quantum phenomenacan be reproduced by adopting a simple deterministic detector model and supposing the existenceof a random zero-point electromagnetic field.
I. INTRODUCTION
Wave-particle duality is one of the oldest and mostperplexing aspects of quantum theory [1]. Although thewave-like nature of light had been well established bythe 19th century, experiments of the early 20th centurybrought about the notion of light as a particle, whatwe now call a photon [2–4]. Maintaining this notion oflight as being composed of discrete particles can, how-ever, be rather paradoxical at times, as numerous realand gedanken experiments have shown [5–10].One particular experiment that has captured recentinterest and attention is the delayed-choice quantumeraser. First conceived by Scully and Dr¨uhl in 1982[11], the quantum eraser is a variant of Wheeler’s de-layed choice experiment in which measurements on oneof a pair of entangled light beams are used to recover aninterference pattern, and hence wave-like behavior, thatwould otherwise be lost with the introduction of which-way path information in a Mach-Zehnder interferometer[12–16]. For Wheeler, the delayed-choice experiment wasan argument for anti-realism , the notion that quantumobjects, such as photons, do not have definite, intrinsicproperties that are independent of the measurement con-text [17]. Some, however, have interpreted the results ofdelayed-choice quantum eraser experiments as evidencefor a form of retrocausality [18].More recently, a series of delayed-choice experimentshas been performed that rule out a certain class of non-retrocausal hidden-variable models described by Chaves,Lemos, and Pienaar [19]. The general model they de-scribe can provide a causal description of the standarddelayed-choice experiment, but it fails to describe a vari-ant of this experiment using variable phase delays in thearms of the interferometer. This variant has been thesubject of recent experimental investigations, which areconsistent with theoretical predictions [20–22]. These ex-periments place certain dimensional restrictions on theclass of non-retrocausal hidden variable models that canbe consistent with theory and observations. ∗ [email protected] In this paper, we revisit Wheeler’s delayed-choice ex-periment, its recent experimental variants, and the moreelaborate quantum eraser experiment within the contextof a simple, physically motivated classical model [23, 24].Although loophole-free experiments have already beenperformed to rule out local realism [25–27], the identifica-tion of precisely which phenomena defy a classical inter-pretation remains an open question and one of practicalrelevance to ensure the security and efficacy of emerge-ment quantum technologies. Our approach is modeledafter stochastic electrodynamics (SED) in assuming areified zero-point field (ZPF) corresponding to the vac-uum state [28, 29]. A significant departure from stan-dard SED in our approach is the introduction of a deter-ministic model of detectors using an amplitude thresholdcrossing scheme. We find that these simple assumptions,combined with standard experimental post-selection anddata analysis techniques, adequately describe the ob-served quantum phenomena.The structure of the paper is as follows: In Sec. II weconsider a simple delayed-choice experiment using weakcoherent light as a notional single-photon source. We re-place the coherent light with a source of entangled lightin Sec. III, within the context of a quantum eraser experi-ment, and demonstrate how post-selection, not causality,is the mechanism whereby path information is effectivelyerased. With these two results established, we revisit thetheoretical arguments of Chaves and Bowles in Sec. Vand argue that their assumptions are overly restrictive.Section VI considers a variant of these experiments usingentangled light sources and shows that these, too, canbe understood within a classical framework. Finally, wesummarize our conclusions in Sec. VII. Figures and nu-merical experiments were created and performed usinga custom simulation tool, the Virtual Quantum OpticsLaboratory (VQOL) [30].
II. SIMPLE DELAYED-CHOICE EXPERIMENT
Consider the Mach-Zehnder interferometer of Fig. 1.A laser (LAS) provides a source of coherent, horizontallypolarized light that is strongly attenuated by a neutraldensity filter (NDF) before entering the first beam split- a r X i v : . [ qu a n t - ph ] J a n FIG. 1. (Color online) VQOL experimental setup for a simpledelayed-choice experiment. The different colors (or shades ofgray) in the beams correspond to different polarizations. ter (BS1). Under our model, the laser light exiting theNDF is represented by a stochastic Jones vector of theform a = (cid:18) a H a V (cid:19) = (cid:18) α (cid:19) + σ (cid:18) z H z V (cid:19) , (1)where α ∈ C describes the mean amplitude and phase ofthe light, σ = 1 / √ (cid:126) ω ), and z H , z V are inde-pendent standard complex Gaussian random variables.(We say that z is a standard complex Gaussian randomvariable if E [ z ] = 0, E [ | z | ] = 1, and E [ z ] = 0.) Notethat z H and z V play the role of hidden variables. Thismodel is mathematically equivalent to the correspondingquantum coherent state | α (cid:105) ⊗ | (cid:105) whose Wigner functionis a bivariate Gaussian probability density function iden-tical to that of a . The effect of the NDF is to ensure that | α | (cid:28) → ) and adown-traveling mode (denoted by ↓ ). In addition, there isa down-traveling vacuum mode that enters the top inputport of BS1, represented by the independent stochasticJones vector b = (cid:18) b H b V (cid:19) = σ (cid:18) z H z V (cid:19) , (2)where z H , z V are independent standard complex Gaus-sian random variables (that are also independent of z H , z V ). The two spatial modes may be representedby a pair of stacked Jones vectors as follows: a H a V −− b H b V BS1 −−→ √ a H + b H a V + b V − − −− a H − b H a V − b V . (3)Note that the second term on the right-hand side is againa standard complex Gaussian random vector, owing tothe unitarity of the beam splitter transformation. The right-traveling beam next undergoes a a transfor-mation via a half-wave plate (denoted by HWP) with afast-axis angle θ ∈ [0 , π/
4] relative to the horizontal axis.It subsequently undergoes a phase delay (denoted by PD)that applies a global phase angle φ ∈ [0 , π ]. Finally, apair of mirrors (denoted by M1 and M2) swap the twospatial modes. The resulting stacked Jones vector afterthese three transformations is now1 √ a H + b H a V + b V − − −− a H − b H a V − b V HWP , PD , M1 , M2 −−−−−−−−−−−→ a (cid:48) H a (cid:48) V −− b (cid:48) H b (cid:48) V , (4)where a (cid:48) H = ( a H − b H ) / √ a (cid:48) V = ( a V − b V ) / √
2, and b (cid:48) H = e iφ √ (cid:104) cos 2 θ ( a H + b H ) + sin 2 θ ( a V + b V ) (cid:105) , (5a) b (cid:48) V = e iφ √ (cid:104) sin 2 θ ( a H + b H ) − cos 2 θ ( a V + b V ) (cid:105) . (5b)In the absence of the second beam splitter (denotedby BS2), the Jones vectors ( a (cid:48) H , a (cid:48) V ) T and ( b (cid:48) H , b (cid:48) V ) T willdetermine if a detection is made on the right-travelingmode (by detector D1) or the down-traveling mode (bydetector D2). We adopt an amplitude threshold crossingscheme as a model to determine whether a given detectormakes a detection (or “clicks”) [23]. Under this scheme,a detector clicks if the amplitude of either the horizontalor vertical polarization component of the impinging beamfalls above a given threshold γ ≥ (cid:18) a (cid:48) H a (cid:48) V (cid:19) P1 −→ (cid:18) a (cid:48) H σ z (cid:48) V (cid:19) , (6)where z (cid:48) V is an independent standard complex Gaus-sian random variable corresponding to the vacuum modeof the notional second input port. Similarly, passagethrough P2 will result in the transformation (cid:18) b (cid:48) H b (cid:48) V (cid:19) P2 −→ (cid:18) b (cid:48) H σ z (cid:48) V (cid:19) , (7)where z (cid:48) V is, again, an independent standard complexGaussian random variableThus, detector D1 clicks, according to this model,if the random variables z H , z V , z H , z V , z (cid:48) V , z (cid:48) V aresuch that they fall with the event set D = (cid:110) | a (cid:48) H | > γ or | σ z (cid:48) V | > γ (cid:111) . (8)Similarly, detector D2 clicks under the event set D = (cid:110) | b (cid:48) H | > γ or | σ z (cid:48) V | > γ (cid:111) . (9)Clearly, the probabilities for these events are independentof the phase angle φ , since only the amplitudes of a (cid:48) H and b (cid:48) H are considered. Thus, no interference effects would beobserved by simply varying φ .One may consider the alternative counterfactual casein which the second beam splitter, BS2, is present. Inthis case, we have a further transformation, a (cid:48) H a (cid:48) V −− b (cid:48) H b (cid:48) V BS2 −−→ √ a (cid:48) H + b (cid:48) H a (cid:48) V + b (cid:48) V − − −− a (cid:48) H − b (cid:48) H a (cid:48) V − b (cid:48) V = a (cid:48)(cid:48) H a (cid:48)(cid:48) V −− b (cid:48)(cid:48) H b (cid:48)(cid:48) V , (10)Detector D1 now clicks under the event D (cid:48) = (cid:110) | a (cid:48)(cid:48) H | > γ or | σ z (cid:48) V | > γ (cid:111) , (11)while detector D2 clicks under the event D (cid:48) = (cid:110) | b (cid:48)(cid:48) H | > γ or | σ z (cid:48) V | > γ (cid:111) . (12)Both a (cid:48)(cid:48) H and b (cid:48)(cid:48) H contain a relative phase term, resultingin a dependence on φ for the probabilities of the twoevents. Thus, the addition or removal of the second beamsplitter may causally create or destroy, respectively, aninterference pattern, even if this action is taken well afterthe light has passed through the first beam splitter.We note this general qualitative behavior is itself un-remarkable and may be observed more directly in theintensities of classical light (for which | α | (cid:29) σ ≈ | a (cid:48) H | ≈ | α | , (13) | a (cid:48)(cid:48) H | ≈ | α | (cid:12)(cid:12) e iφ cos 2 θ (cid:12)(cid:12) . (14)We further note that non-zero values of θ can providewhich-way information, insofar as they may completelydestroy, for θ = π/
2, or merely reduce, for 0 < θ < π/ a (cid:48) H , b (cid:48) H obey a proper complexGaussian distribution with expectation values of E [ a (cid:48) H ] = α √ , (15) E [ b (cid:48) H ] = α √ θ (16)and a common variance of σ . They are furthermoreindependent, owing the unitarity of the transformationsinvolved. Thus, | a (cid:48) H | and | b (cid:48) H | each follow a Rician dis-tribution [32]. Since, furthermore, z V and z V are also independent, the probability of a single click on detectorD1 (and not D2) is given by p ( θ, φ ) = Pr[ D ∩ ¯ D ] = Pr[ D ] (1 − Pr[ D ]) , (17)where Pr[ D ] = 1 − F (cid:18) α √ (cid:19) F (0) , (18)Pr[ D ] = 1 − F (cid:18) α √ θ (cid:19) F (0) , (19)and F ( α ) = Pr[ | α + z H | ≤ γ ] is given by the Marcum Qfunction as [23, 33] F ( α ) = 1 − Q ( √ | α | /σ , √ γ/σ ) . (20)Note that Q (0 , √ γ/σ ) = e − γ /σ .A similar analysis may be used when the second beamsplitter is in place, albeit using a (cid:48)(cid:48) H and b (cid:48)(cid:48) H instead. Theprobability of a single click on detector D1 is now p (cid:48) ( θ, φ ) = Pr[ D (cid:48) ] (1 − Pr[ D (cid:48) ]) , (21)wherePr[ D (cid:48) ] = 1 − F (cid:16) α e iφ cos 2 θ ) (cid:17) F (0) , (22)Pr[ D (cid:48) ] = 1 − F (cid:16) α − e iφ cos 2 θ ) (cid:17) F (0) . (23)In Fig. 2 we have plotted the expected number ofcounts, in excess of the dark counts, for the cases withand without BS2 present and for different values of θ .The number of notional trials was taken to be N = 10 .The laser and NDF were taken to be such that α = 0 . γ = 1 .
95, correspond-ing to a dark count probability of p d = 0 . N p ( θ, φ ) − N p d ;with BS2 in place, they are N p (cid:48) ( θ, φ ) − N p d . The HWPfast-axis angle, θ , was take to be either 0, 30, or 45 de-grees, corresponding to either no, partial, or completewhich-way information, respectively.As expected, there is no interference pattern when BS2is removed. With BS2 in place, there is a strong inter-ference pattern when there is no which-way information(i.e., θ = 0 ◦ ). When partial which-way information isavailable, corresponding to a non-zero vertical compo-nent of the mean polarization, the interference pattern isdiminished but remains discernible. If complete which-way information is available, corresponding to a meanvertical polarization in the upper arm of the interferom-eter, the interference pattern is diminished to the pointof being no longer present. These effects are completelycausal and arise from the interference of classical, albeitstochastic, waves. The particle-like behavior, manifestedby single-detection events, is a consequence of the low in-tensity of the input beam, for which detections on bothdetectors are rare, and the fact that we have removednon-detection events through post-selection.In the parameter regime we have chosen, the interfer-ence pattern matches a scaled and shifted version of thecos ( φ/
2) probability predicted by quantum mechanicsfor a single-photon state. Larger values of α may exhibitdeviations from this prediction, as the corresponding co-herent state | α (cid:105) may no longer be considered a good ap-proximation of a vacuum-plus-single-photon state. (deg) E x pe c t ed C oun t s without BS2with BS2, =0 ° with BS2, =30 ° with BS2, =45 ° FIG. 2. (Color online) Plot of the expected counts, minus darkcounts, for a delayed-choice experiment versus the phase delayangle φ in the upper arm of the interferometer. The dashedline corresponds to when BS2 is removed, and the solid linescorrespond to when BS2 is present. Interference patterns areobserved when there is no which-way information [ θ = 0 ◦ ,red (light gray)] or only partial which-way information [ θ =30 ◦ , blue (dark gray)]. The interference pattern vanishes withcomplete which-way information ( θ = 45 ◦ , black). III. DELAYED-CHOICE QUANTUM ERASER
Wheeler’s delayed-choice experiment can be changedto a quantum eraser experiment by replacing the laserand NDF with an entanglement source (ENT). The ex-periment is illustrated in Fig. 3, where we have also addeda third polarizer and detector, denoted P3 and D3, re-spectively. Unlike the other two polarizers, P3 is rotatedby 45 ◦ so as to admit diagonally polarized light.The entanglement source is modeled classically as atype-I parametric down conversion process for which theinputs states are a pump laser (not shown) and classicallymodeled vacuum modes from the ZPF [24]. The Jonesvectors for the right-traveling ( → ) and left-traveling ( ← )spatial modes of the entanglement source, denoted a and c , respectively, are given by a = (cid:18) a H a V (cid:19) = σ (cid:18) z H cosh r + z ∗ H sinh rz V cosh r + z ∗ V sinh r (cid:19) (24)and c = (cid:18) c H c V (cid:19) = σ (cid:18) z H cosh r + z ∗ H sinh rz V cosh r + z ∗ V sinh r (cid:19) , (25) FIG. 3. (Color online) VQOL experimental setup for adelayed-choice quantum eraser experiment. The entangle-ment source is labeled ENT. where z H , z V , z H , z V are independent standard com-plex Gaussian random variables and r ≥ a and c are statistically dependent for r > a and c isidentical to the Gaussian Wigner function of a four-modeentangled squeezed state [34]. If r (cid:28)
1, this squeezedstate may be approximated as a superposition of a vac-uum state and the entangled Bell state | Ψ (cid:105) = | H (cid:105) ⊗ | H (cid:105) + | V (cid:105) ⊗ | V (cid:105)√ , (26)where the left and right kets in each tensor product in-dicate the spatial modes traveling to the left and right,respectively.As before, b denotes the Jones vector of the down-traveling ( ↓ ) vacuum mode entering BS1 and given by b = (cid:18) b H b V (cid:19) = σ (cid:18) z H z V (cid:19) , (27)where, again, z H , z V are independent standard complexGaussian random variables.The transformations of a and b through the interfer-ometer are identical to those in the simple delayed-choiceexperiment discussed previously, as given by Eqns. (5)and (10). We shall again denote these by a (cid:48)(cid:48) , b (cid:48)(cid:48) and a (cid:48) , b (cid:48) for the cases with and without BS2, respectively.Passage through the polarizers P1 and P2 results in thetransformations (cid:18) a (cid:48) H a (cid:48) V (cid:19) P1 −→ (cid:18) a (cid:48) H σ z (cid:48) V (cid:19) (28)and (cid:18) b (cid:48) H b (cid:48) V (cid:19) P2 −→ (cid:18) b (cid:48) H σ z (cid:48) V (cid:19) , (29)with similar transformations for a (cid:48)(cid:48) , b (cid:48)(cid:48) .Upon passing through polarizer P3, c becomes (cid:18) c H c V (cid:19) P3 −→ c D √ (cid:18) (cid:19) + σ z A √ (cid:18) − (cid:19) , (30)where c D = ( c H + c V ) / √ c and z A is an independent standard complex Gaus-sian random variable corresponding to the missing anti-diagonal component of the ZPF.Individual detection events for D1, D2, and D3 for thecase in which BS2 is not present may now be defined asfollows: D = (cid:110) | a (cid:48) H | > γ or | σ z (cid:48) V | > γ (cid:111) , (31) D = (cid:110) | b (cid:48) H | > γ or | σ z (cid:48) V | > γ (cid:111) , (32) D = (cid:110)(cid:12)(cid:12)(cid:12) c D + σ z A √ (cid:12)(cid:12)(cid:12) > γ or (cid:12)(cid:12)(cid:12) c D − σ z A √ (cid:12)(cid:12)(cid:12) > γ (cid:111) . (33)The events D (cid:48) , D (cid:48) , D (cid:48) , for the case in which BS2 isin place, are defined similarly by replacing a (cid:48) H , b (cid:48) H with a (cid:48)(cid:48) H , b (cid:48)(cid:48) H . (Both c D are z A are, of course, unchanged.)Let us consider detection events with BS2 in place andwith the HWP rotated to θ = 45 ◦ , corresponding to com-plete which-way information. Ignoring the detections ondetector D3, the probability of a single-detection on de-tector D1 is, as before, p (cid:48) ( φ ) = Pr[ D (cid:48) ∩ ¯ D (cid:48) ] . (34)If we post-select on events for which detector D3 alsoclicks, this probability becomes p (cid:48) ( φ ) = Pr[ D (cid:48) ∩ ¯ D (cid:48) | D (cid:48) ] . (35)We do not have a closed-form expression for the jointdistribution of a and c ; however, these probabilities maybe estimated numerically. Taking γ = 1 .
95, as before,and r = 1, we estimated the probabilities p (cid:48) ( φ ) and p (cid:48) ( φ ) for an ensemble of N = 10 realizations. Theresults are shown in Fig. 4. As expected, the interferencepattern is recovered when we post-select on D3 detec-tions.In the language of quantum mechanics, we have“erased” the which-way information by collapsing thestate with a projective measurement at D3. In truth,all we have done is sample from a subensemble of ZPFrealizations for which there is a detection on D3. Clas-sically, there is no causal mechanism by which detectionevents at D3 affect those at D1 or D2. It is merely areflection of the pre-existing correlations between a and c and their modification as a result of post-selection. Ofcourse, it does not matter whether BS2 was inserted be-fore or after the light has passed through BS1, so thereis no need for retrocausal explanations either. (deg) P r obab ili t y without post-selectionwith post-selection on D3 FIG. 4. (Color online) Plot of the probability of a single-detection on D1 with either no conditioning on D3 [ p (cid:48) ( φ ),blue (dark gray) line] or with post-selection of a detection onD3 [ p (cid:48) ( φ ), red (light gray) curve]. IV. QUANTUM-CONTROLLED EXPERIMENTS
In the quantum eraser experiment of the previous sec-tion, the second beam splitter was either present or ab-sent, resulting in the presence or absence of an inter-ference pattern, respectively. A similar experiment wasperformed by Jacques et al. using a quantum random-number generator (QRNG) and a classical switch [5].Ionicioiu and Terno have suggested using a controlledquantum gate in place of a classical switch and, fur-thermore, have argued that this scheme may be used torule out a certain class of hidden-variable models [6, 10].Subsequently, this proposal was implemented experimen-tally using a programmable nuclear magnetic resonance(NMR) device [7], a reconfigurable integrated photonicdevice [8], and a polarization-dependent beam splitter(PDBS) [9]. All experiments showed the expected contin-uum of wave-like and particle-like behavior for differentexperimental settings controlling which-way path infor-mation. In this section, we consider a classical model forthe PDBS experiment.To this end we have implemented the experimentalsetup described in Ref. [9] with a PDBS that behaves asa 50/50 beam splitter for horizontally polarized light andis transparent (except for a swapping of spatial modes)for vertically polarized light. (See Fig. 5.) For Jonesvectors a and b corresponding to light entering from thetop and left, respectively, of the PDBS, the transformedpolarization states are given by a (cid:48) H a (cid:48) V b (cid:48) H b (cid:48) V = √ √
00 0 0 1 √ − √
01 0 0 0 a H a V b H b V . (36) FIG. 5. (Color online) VQOL experimental setup for aquantum-controlled delayed-choice quantum eraser experi-ment using a polarization-dependent beam splitter (PDBS).The PDBS is composed of a beam splitter surrounded byfour polarizing beam splitters that transmit (reflect) horizon-tal (vertical) light, and a beam blocker (black square). Thebeam blocker absorbs ZPF modes entering from the unusedinput ports of the top and left PBS of the PDBS.
As before, the phase delay (PD) is set to an angle of φ ∈ [0 , π ], and we have added a half-wave plate, HWP3,with a fast-axis angle of θ ∈ [0 , π/ π/ | Ψ (cid:48) (cid:105) = 1 √ (cid:104) ˆ A † w cos 2 θ + ˆ A † p sin 2 θ (cid:105) ˆ c † H | (cid:105) + 1 √ (cid:104) ˆ A † w sin 2 θ − ˆ A † p cos 2 θ (cid:105) ˆ c † V | (cid:105) , (37)where | (cid:105) is the vacuum state, ˆ A † w is the “wave” creationoperatorˆ A † w = 12 √ (cid:104) (1+ e iφ )(ˆ a † H +ˆ a † V )+(1 − e iφ )(ˆ b † H +ˆ b † V ) (cid:105) , (38)ˆ A † p is the “particle” creation operatorˆ A † p = 12 (cid:104) e iφ ˆ a † H − e iφ ˆ a † V + ˆ b † H − ˆ b † V (cid:105) , (39)and the operators ˆ a † H , ˆ a † V , ˆ b † H , ˆ b † V , ˆ c † H , ˆ c † V are the creationoperators for the modes corresponding to detectors D1,D2, D3, D4, D5 D6, respectively. In the quantum mechanical description, a single detec-tion on either D5 or D6 results in a collapse of the wavefunction onto one of two subspaces, each of which is asuperposition of wave-like and particle-like states. A sin-gle detection on D1 or D2, say, conditioned on a singledetection on D6 would therefore have a probability of q ( θ, φ ) = cos ( φ/
2) sin (2 θ ) + cos (2 θ ) . (40)Thus, for θ = 0 ( θ = π/
4) we expect fully particle-like(wave-like) behavior. This result is equivalent to thatfound in Ref. [9], which uses θ and α in place of φ and2 θ and conditions on the horizontal, rather than vertical,mode at PBS3 due to opposite PDBS conventions.For our classical model we take our initial states to bethe Jones vectors a , b , and c as defined in Eqns. (24),(27), and (25), respectively. The final Jones vectors forthe light entering PBS1, PBS2, and PBS3 are then foundto be a (cid:48) , b (cid:48) , and c (cid:48) , respectively, where a (cid:48) is given by a (cid:48) H = 1 + e iφ √ a H + e iφ a V − − e iφ √ b H + e iφ b V , (41a) a (cid:48) V = 1 + e iφ √ a H − e iφ a V − − e iφ √ b H − e iφ b V , (41b) b (cid:48) is given by b (cid:48) H = 1 − e iφ √ a H + 12 a V − e iφ √ b H − b V , (42a) b (cid:48) V = 1 − e iφ √ a H − a V − e iφ √ b H + 12 b V , (42b)and c (cid:48) is given by c (cid:48) H = c H cos 2 θ + c V sin 2 θ , (43a) c (cid:48) V = c H sin 2 θ − c V cos 2 θ . (43b)Unlike the quantum description, there is no clear dis-tinction between modal subspaces. For θ = 0, say, asingle detection on D6 will occur when | c V | is large and | c H | is small. Since c and a are statistically correlated,this will tend to occur when | a V | is large and | a H | is smallas well. This, in turn, implies that a (cid:48) H and a (cid:48) V are domi-nated by the particle-like a V term, vice the wave-like a H term, thereby giving rise to more particle-like behavior.However, the effects of the wave-like a H term will notbe completely absent, particularly since, in amplitude, itmay be up to a factor of √ a V term. Thus, some wave-like interference will persist,and a mixture of wave-like and particle-like behavior willalways be exhibited.As before, we may define the individual detectionevents for each detector as follows: D = (cid:110) | a (cid:48) H | > γ or | σ z (cid:48) V | > γ (cid:111) (44a) D = (cid:110) | a (cid:48) V | > γ or | σ z (cid:48) H | > γ (cid:111) (44b)... D = (cid:110) | c (cid:48) V | > γ or | σ z (cid:48) H | > γ (cid:111) , (44f)where z (cid:48) H , . . . , z (cid:48) V are independent standard complexGaussian random variables corresponding to the unusedinput ports of PBS1, PBS2, and PBS3.We may now define the four relevant coincident detec-tion events, C , C , C , and C , as follows: C = D ∩ ¯ D ∩ ¯ D ∩ ¯ D ∩ ¯ D ∩ D (45a)... C = ¯ D ∩ ¯ D ∩ ¯ D ∩ D ∩ ¯ D ∩ D (45d)Finally, the conditional probability of a single detectionon D1 or D2, given a coincident detection with D6, is p ( θ, φ ) = Pr[ C ∪ C ]Pr[ C ∪ C ∪ C ∪ C ] . (46)Our task is now to compare p ( θ, φ ) to q ( θ, φ ).To do this, we generated N = 5 × random realiza-tions for each θ and φ , corresponding 5 s of simulationtime in VQOL. We took γ = 1 .
95 and r = 0 .
25, to bet-ter match the results of Ref. [9]. For these settings wefound an average of 125 coincidences per run, about fiveof which may be considered accidental. In Fig. 6 we haveplotted q ( θ, φ ) as well as estimates of p ( θ, φ ) from ournumerical experiments. We find that the results com-pare favorably with Fig. 4 of Ref. [9] and exhibit a clearmorphing between particle-like and wave-like behavior,although there are deviations from the ideal quantumprediction. We note also the slight vestige of wave-likebehavior at θ = 0 and particle-like behavior at θ = π/ FIG. 6. (Color online) Surface plot of q ( θ, φ ) and point es-timates of p ( θ, φ ) (black circles) based on numerical experi-ments. The small vertical lines represent approximate 95%confidence intervals. No fitting was performed on the data. Given these results, we should like to return to theproof by Ionicioiu and Terno that conformity with thequantum predictions should rule out a certain class ofhidden-variable models, as this would seem to be at vari-ance with our results. This class is specified by three defining characteristics: (1) wave-particle objectivity, (2)determinism, and (3) local independence [6, 10].Our model is clearly deterministic: given a particu-lar realization of all ZPF modes, the detection outcomesare uniquely determined. It is approximately consistentwith wave-particle objectivity, which states that the setof hidden-variable states corresponding to the presenceor absence of a second beam splitter should be disjoint.The notional idea is that there is a hidden variable thatdetermines whether particle-like or wave-like behavior isexhibited. Certainly the sets of hidden-variable states forwhich a detection occurs on either D5 or D6 (i.e., the sets D ∩ ¯ D and ¯ D ∩ D ) are disjoint, but conditioning onthese events provides only approximate conformity withthe ideal two-photon quantum predictions. Even underconditioning, the outcomes are a mixture of particle-likeand wave-like behavior.Finally, local independence is the property that thehidden-variable space may be separated into two sets ofstatistically independent variables controlling, on the onehand, detections at the output of the interferometer and,on the other, the presence or absence of the second beamsplitter. Since the experimental setup is not a properimplementation of the Ionicioiu-Terno scheme, this con-dition cannot be satisfied. Specifically, the right spatialmode of the entanglement source is used both as a controland an input to the interferometer. Although the ZPFmodes, which may be construed as the hidden variables,are all statistically independent, they cannot be uniquelyassociated with either control or input due to the exper-imental setup and, in particular, use of the PDBS. Ofcourse, the presence or absence of the second beam split-ter could be controlled by a separate, independent clas-sical random variable, but this reduces to the quantumeraser experiment of the previous section. V. RELATION TO DIMENSION WITNESS
We now turn to some more recent theoretical and ex-perimental results regarding delayed-choice experiments.Bowles, Quintino, and Brunner have considered the prob-lem of distinguishing quantum systems from classicalcounterfeits in a device-independent manner using a so-called “dimension witness” [35]. In their scheme, theclassical system is assumed to produce, upon a certainstate preparation, a “message” m ∈ { , . . . , d − } , where d is the Hilbert space dimension of the correspondingquantum system. (“Classical,” in their sense, means thatonly finite, digital information is conveyed.) The finalmeasurement outcome is assumed to depend only on themessage m and some given measurement setting.Denoting by p ( x, y ) the probability of a desired out-come for preparation x and measurement y , both takento be integer indexes, Bowles et al. define the matrix W = (cid:18) p (1 , − p (2 , p (3 , − p (4 , p (1 , − p (2 , p (3 , − p (4 , (cid:19) . (47)For a two-dimensional classical system (i.e., one for whichmessages are restricted to one of d = 2 values), theyshow that only det( W ) = 0 is possible. By contrast, atwo-dimensional quantum system (i.e., one for which theHilbert space dimension is d = 2) can achieve values ashigh as | det( W ) | = 1.Chaves et al. have used this construction to design adelayed-choice experiment that can distinguish betweena two-dimensional quantum system and a classical sys-tem, so defined, of the same dimension [19]. One versionof the proposed experiment was recently performed us-ing a single qubit in a polarization-based Mach-Zehnderinterferometer and a pair of phase retarders to providedifferent preparation and measurement settings [20].A simplified version of the experimental setup is shownin Fig. 7. A laser (LAS) generates horizontally polar-ized light, which is attenuated by a neutral density filter(NDF). A half-wave plate (HWP1) with a fast-axis angleof 22 . ◦ is then applied to play the role of a beam splitter.Next, a birefringent phase retarder (PR1) applies a phasefactor e iφ x to the vertical polarization component. Thisconstitutes the preparation stage of the experiment. Themeasurement stage consists of another phase retarder,PR2, that applies a phase factor e iσ y to the vertical com-ponent. A second half-wave plate, HWP2, rotated 22 . ◦ completes the interferometer, and a final polarizing beamsplitter (PBS) and two detectors, D1 and D2, are used todetect horizontal and vertical polarization, respectively.In the actual experiment, σ y was chosen randomly andset only after the light had passed through HWP1. FIG. 7. (Color online) VQOL experimental setup for dimen-sion witness experiment.
In our model, the initial state following the NDF isrepresented, as before, by the stochastic Jones vector (cid:18) a H a V (cid:19) = (cid:18) α (cid:19) + σ (cid:18) z H z V (cid:19) . (48)After passing through the phase retarders and HWP2,the state becomes (cid:18) a H a V (cid:19) MZI −−→ (cid:20) (1 + e i ∆ xy ) a H + (1 − e i ∆ xy ) a V (1 − e i ∆ xy ) a H + (1 + e i ∆ xy ) a V (cid:21) , (49)where ∆ xy = φ x + σ y .For each preparation x and measurement y , we define p ( x, y ) as the probability of obtaining a detection on D1, given that a single detection occurred on either D1 or D2.This probability is given by p ( x, y ) = Pr[ D ∩ ¯ D ]Pr[ D ∩ ¯ D ] + Pr[ ¯ D ∩ D ] , (50)where D = D + ∪ {| σ z V | > γ } (51) D = D − ∪ {| σ z H | > γ } (52)and D ± = (cid:8)(cid:12)(cid:12) (1 ± e i ∆ xy ) a H + (1 ∓ e i ∆ xy ) a V (cid:12)(cid:12) > γ (cid:9) . (53)We similarly define p ( x, y ) as the probability of a singledetection on D2. Note that z H and z V correspond tothe ZPF components entering from the top input portof the PBS and constitute an independent source of ran-domness in the measurement stage.Following the experimenters, we construct W for thefour preparation settings φ = 0, φ = π , φ = − π/ φ = π/ σ = 0, σ = π/
2. Quantum mechanics predicts, for a single pho-ton, that p (1 ,
1) = p (3 ,
2) = 1, p (2 ,
1) = p (4 ,
2) = 0,and p (3 ,
1) = p (4 ,
1) = p (1 ,
2) = p (2 ,
2) = , giving | det( W ) | = 1. Using our model, we take p ( x, y ) = p ( x, y ) and compute the matrix W using σ = 1 / √ γ = 1 .
95, as before, and vary α .In Fig. 8 we have plotted our classical model results for | det( W ) | as a function of the input intensity, as givenby | α | . (For a coherent state | α (cid:105) , this corresponds tothe average photon number.) For 0 < | α | (cid:28)
1, the ZPFdominates and the low coherence results in a small, butpositive, value of | det( W ) | . For | α | (cid:29) | det( W ) | ap-proaches the ideal single-photon prediction of one. Notethat, although the limit | α | → ∞ corresponds to clas-sical light (i.e., light for which the ZPF is negligible),post-selection on single-detection events maintains theparticle-like anti-correlations of the interferometer andallows for agreement with the quantum prediction. Theexperimental value of 0.95 observed in Ref. [20] corre-sponds, in our model, to | α | = 1 . | det( W ) | greater than zero can be obtained, as the “messages”between preparation and measurement in our model arenot limited to discrete values but, rather, can take ona continuum of possible values in C . Agreement withquantum mechanics was further made possible by post-selecting on what would appear to be single-photon de-tections, as was done in the actual experiment. It isclear from this example that an assumption of finite-dimensional classical models is overly restrictive.In our model, the measurement outcomes depend onwhat may be construed as four hidden variables: z H , z V , z H , and z V . The first two, λ = ( z H , z V ), maybe considered as part of the preparation stage, while thelast two, ( z H , z V ) may be considered part of the mea-surement stage. They are independent of and unaffected | | | de t ( W ) | FIG. 8. (Color online) Plot of | det( W ) | versus | α | . Thedashed line indicates the ideal quantum prediction. The red(light gray) dot corresponds to the value of | det( W ) | ob-tained experimentally in Ref. [20]. by the choice of preparation or measurement settings, soit is clear that the model is completely causal. Neverthe-less, it may be useful to examine measurable bounds onretrocausality.To study this question, Chaves et al. have proposedthe use of a retrocausality measure R Y → Λ intended tocapture the apparent retrocausal impact of the measure-ment setting choice on the preparation hidden-variablestate [19]. A hidden-variable model for which R Y → Λ = 0is completely causal, while one for which R Y → Λ = 1 isconsidered strongly retrocausal. Since, in our model, thepreparation hidden variables are independent of the mea-surement settings, R Y → Λ = 0. Chaves et al. show thatone can measure a bound R min such that any hidden-variable model must satisfy R Y → Λ ≥ R min in order tobe consistent with observations. Thus, they conclude, if R min > R min is given in terms of a dimension wit-ness I DW [36]. This, in turn, is defined to be I DW = (cid:12)(cid:12) (cid:104) B (cid:105) + (cid:104) B (cid:105) + (cid:104) B (cid:105) − (cid:104) B (cid:105) − (cid:104) B (cid:105) (cid:12)(cid:12) , (54)where (cid:104) B (cid:105) xy = p ( x, y ) − p ( x, y ). In terms of I DW , R min = max (cid:26) I DW − , (cid:27) . (55)Algebraically, I DW ∈ [0 ,
5] and R min ∈ [0 , ]. It hasbeen shown that I DW ≤ I DW = 1 + 2 √ ≈ . R min = 0 for two-dimensional classical systems,while a two-dimensional quantum system may achieve upto R min = ( √ − / ≈ . I DW for our modelusing the preparation settings φ = 7 π/ φ = 5 π/ φ = π/ σ = π/ σ = 0.As before, we took σ = 1 / √ γ = 1 .
95 while varying α . The results are shown in Fig. 9. We find that the clas-sical bound for two-dimensional models is surpassed for | α | > .
33, with | α | = 0 .
58 giving the value I DW = 3 . | α | > .
58, we find that I DW can exceed even the quantum bound. Such exceedances,while often associated with post-quantum theories, arein fact a well known consequence of post-selection [38].Asymptotically, I DW approaches the algebraic limit offive as | α | → ∞ . The retrocausality bound R min followssimilar behavior. | | I D W FIG. 9. (Color online) Plot of the dimension witness I DW versus | α | . The upper dashed line indicates the ideal, single-photon quantum prediction, while the lower dotted line in-dicates the two-dimensional classical bound. The red (lightgray) dot corresponds to the value of I DW obtained experi-mentally in Ref. [20]. It may seem curious that a manifestly causal modelsuch as we have described would give a non-zero lowerbound for the retrocausality measure. Indeed, this wouldseem to contradict the assertion that R Y → Λ ≥ R min .However, the nonzero values of R min are a direct con-sequence of I DW exceeding the two-dimensional classicalbound. This, in turn, arises from the process of post-selection and gives rise to classical contextuality and,hence, the illusion of retrocausality. Thus, there is nocontraction between our results and the theoretical in-terpretation of R min . VI. PREPARATION VIA HERALDING
The dimension witness has also been considered in arecent experiment by Huang et al. using an entangle-ment source [21]. Instead of an attenuated laser, theexperimenters used heralded detections on a paramet-ric down conversion source of entangled light to produce0the prepared state. As in the Polino et al. experiment ofRef. [20], polarization components are used as a surrogatefor the two paths of an interferometer, and measurementproceeds in much the same way.A simplified version of the experiment is shown in Fig.10. The entanglement source (ENT) is used to preparethe Bell state given by Eqn. (26). The setups for Al-ice and Bob are similar in that each uses a phase re-tarder (PR), a half-wave plate (HWP) set to 22 . ◦ , apolarizing beam splitter (PBS) and a pair of detectorsfor each polarization mode. Alice’s phase retarder (PR1)is set to an angle α x , while Bob’s (PR2) is set to β y . Inthe experiment of Ref. [21], α = π/ , α = π/
2, while β = π/ , β = 0. Also, the distance to Alice was madeshorter than the distance to Bob, ensuring that settingchanges by Alice cannot affect Bob. Coincident detec-tions are determined by the known relative delay. FIG. 10. (Color online) Experimental setup for anentanglement-based dimension witness measurement.
The setup allows Alice to prepare four different statesfor Bob to measure, owing to the two values of α x and thetwo detectors (D3 and D4) used by Alice for heralding.Quantum mechanically, the state entering Alice’s PBS isof the form | Ψ (cid:105) = | H (cid:105) ⊗ | ψ x (cid:105) + | V (cid:105) ⊗ | ψ x (cid:105) , where | ψ x (cid:105) = | H (cid:105) + e iα x | V (cid:105)√ | ψ x (cid:105) = | H (cid:105) − e iα x | V (cid:105)√ . (57)Thus, a single detection on D3 may be construed aspreparing the state | ψ x (cid:105) for Bob, while one on D4 maybe considered to prepare | ψ x (cid:105) . Note that, since | ψ x (cid:105) = | H (cid:105) + e i ( α x + π ) | V (cid:105)√ , (58)we may consider a herald on D4 as equivalent to Alicesetting her phase retarder to the angle α x + π . Following Bob’s phase retarder and half-wave plate,the final state entering his PBS is | ψ (cid:48) jxy (cid:105) = 12 (cid:0) e iφ jxy (cid:1) | H (cid:105) + 12 (cid:0) − e iφ jxy (cid:1) | V (cid:105) , (59)where either φ xy = α x + β y or φ xy = α x + π + β y is used,depending upon whether Alice made a detection on D3or D4, respectively. The theoretical quantum predictionfor the probability of a detection on, say, D1, given asingle detection on either D3 or D4, is q j ( x, y ) = 14 (cid:12)(cid:12) e iφ jxy (cid:12)(cid:12) . (60)A tabulation of the different possible preparation andmeasurement bases is shown in Table I. Basis Herald ( x, y ) φ jxy π/ π/ π π/ π/ π/
43 D4 (1,1) ( π/ π ) + π/ π π/ π ) + π/ π/
45 D3 (1,2) π/
26 D3 (2,2) π/
47 D4 (1,2) 3 π/
28 D4 (1,2) 5 π/ In our classical mode, the entanglement source is mod-eled by the random Jones vectors a and b , where a = (cid:32) a H a V (cid:33) = σ (cid:32) z H cosh r + z ∗ H sinh rz V cosh r + z ∗ V sinh r (cid:33) (61)and b = (cid:32) b H b V (cid:33) = σ (cid:32) z H cosh r + z ∗ H sinh rz V cosh r + z ∗ V sinh r (cid:33) . (62)Local transformations by the phase retarders and half-wave plates yield a −→ a (cid:48) ( x, y ) = 1 √ (cid:32) a H + e iα x a V a H − e iα x a V (cid:33) (63)and b −→ b (cid:48) ( x, y ) = 1 √ (cid:32) b H + e iβ y b V b H − e iβ y b V (cid:33) . (64)Each of the polarizing beam splitters introduces an ad-ditional vacuum mode from the ZPF. For Bob, we shalldenote this is σ ( z H , z V ) T , and for Alice we denote this σ ( z H , z V ) T . The measurement events for a detection1on each detector, regardless of the others, are thereforethe following: D ( x, y ) = (cid:110) | b (cid:48) H ( x, y ) | > γ or σ | z V | > γ (cid:111) , (65) D ( x, y ) = (cid:110) | b (cid:48) V ( x, y ) | > γ or σ | z H | > γ (cid:111) , (66) D ( x, y ) = (cid:110) | a (cid:48) H ( x, y ) | > γ or σ | z V | > γ (cid:111) , (67) D ( x, y ) = (cid:110) | a (cid:48) V ( x, y ) | > γ or σ | z H | > γ (cid:111) . (68)We now consider the four coincident events ( D , D ),( D , D ), ( D , D ), ( D , D ) for each of the possible val-ues of α x and β y . (For now, we drop the explicit depen-dence on x and y .) These events are given as follows: C = D ∩ ¯ D ∩ D ∩ ¯ D (69) C = D ∩ ¯ D ∩ ¯ D ∩ D (70) C = ¯ D ∩ D ∩ D ∩ ¯ D (71) C = ¯ D ∩ D ∩ ¯ D ∩ D (72)The experiment considers coincident events C = C ∪ C (73) C = C ∪ C (74)such that C is the event that Bob gets a click on just D1and Alice gets a click on either D3 or D4 (but not both).From these events, we may define the conditional prob-abilities p ij ( x, y ) = Pr[ C ij | C i ] = Pr[ C ij ( x, y )]Pr[ C i ( x, y )] . (75)Conditioning on coincident events is equivalent to adopt-ing the fair-sampling assumption, which the experi-menters in Ref. [21] have used. In Fig. 11 we plot the nu-merically determined values of p j ( x, y ), the conditionalprobabilities of detections on D1, and compare theseagainst the theoretical quantum predictions q j ( x, y ). Inthe simulation, we used r = 1, γ = 1 . σ = 1 / √ N = 10 random realizations. We find that theagreement between theory and model predictions is com-parable to or better than that found experimentally inRef. [21].Finally, we compute the dimension witness I DW = (cid:12)(cid:12)(cid:12) B + B + B − B − B (cid:12)(cid:12)(cid:12) , (76)where B jxy = p j ( x, y ) − p j ( x, y ) . (77)Figure 12 shows a plot of I DW versus the squeezing pa-rameter r . (All other parameters are the same as thosefor Fig. 11.) We see that for r < . r ≈ . I DW achieves the value3.445 observed in Ref. [21]. For r > . I DW surpassesthe theoretical quantum bound of 1 + 2 √
2. Larger valuesof r result in numerical instability but suggest that I DW continues to increase monotonically. Preparation & Measurement Basis C ond i t i ona l P r obab ili t y Q TheoryHV Model
FIG. 11. (Color online) Plot of the conditional probabilityfor each of the eight preparation and measurement bases, asdefined in Table I. The left blue (dark gray) bars are the theo-retical quantum predictions q j ( x, y ) , and the right red (lightgray) bars are the values of p j ( x, y ) determined numericallyfrom our model. r I D W FIG. 12. (Color online) Plot of the dimension witness I DW versus r . The upper dashed line indicates the ideal quantumprediction, while the lower dotted line indicates the upperbound for a two-dimensional classical model. The red (lightgray) dot corresponds to the value of I DW obtained experi-mentally in Ref. [21]. VII. CONCLUSION
Current optical delayed-choice experiments, even thoseinvolving entangled light, can be understood from astrictly causal, classical perspective. We illustrated thisusing a specific, physically motivated classical model withtwo key elements: (1) a reified zero-point field and (2)a deterministic threshold-based detector model. This2model is not restricted to delayed-choice experiments butis expected to be applicable to a wide range of quantumoptical phenomena, although the precise domain of va-lidity is not yet known.The use of a dimension witness as a tool to distinguishclassical from quantum systems was found to be inad-equate due to its overly restrictive assumption of finiteclassical messaging. The small class of hidden variablemodels that it is capable of ruling out is of no practicalinterest, as a simple classical device with analog mes-saging can easily spoof the witness. Likewise the retro-causality measure, which is functionally related to thedimension witness, was found to provide no evidence forthe presence of retrocausality or other nonclassical be-havior, as nonzero values can easily be reproduced by astrictly causal classical model. We found instead that the post-selection of desiredmeasurement outcomes is critical to reproducing quan-tum behavior and appears to be what gives rise to theapparent causal (or retrocausal) behavior observed indelayed-choice experiments. This is consistent with pastexperimental tests of quantum nonlocality that rely uponthe fair-sampling assumption and thus may be suscepti-ble to the detection loophole. It may be possible thata delayed-choice experiment could be performed thatavoids this detection loophole, but this has not yet beendemonstrated.
ACKNOWLEDGMENTS
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