Particle Objectivity Compatible with Determinism and Locality?
ARTICLE
Received 11 Nov 2013 | Accepted 14 Aug 2014 | Published 26 Sep 2014 DOI: 10.1038/ncomms5997 Is wave–particle objectivity compatible with determinism and locality?
Radu Ionicioiu1,2, Thomas Jennewein3,4, Robert B. Mann3,4,5 & Daniel R. Terno6
Wave–particle duality, superposition and entanglement are among the most counterintuitive features of quantum theory. Their clash with our classical expectations motivated hidden-variable (HV) theories. With the emergence of quantum technologies, we can test experimentally the predictions of quantum theory versus HV theories and put strong restrictions on their key assumptions. Here, we study an entanglement-assisted version of the quantum delayed-choice experiment and show that the extension of HV to the controlling devices only exacerbates the contradiction. We compare HV theories that satisfy the conditions of objectivity (a property of photons being either particles or waves, but not both), determinism and local independence of hidden variables with quantum mechanics. Any two of the above conditions are compatible with it. The conflict becomes manifest when all three conditions are imposed and persists for any non-zero value of entanglement. We propose an experiment to test our conclusions.
1 Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, 077125 Bucharest-Ma˘gurele, Romania. 2 Research Center for Spatial Information - CEOSpaceTech, University Politehnica of Bucharest, 313 Splaiul Independentei, 061071 Bucharest, Romania. 3 Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. 4 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. 5 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y6. 6 Department of Physics and Astronomy, Macquarie University, Sydney, New South Wales 2109, Australia. Correspondence and requests for materials should be addressed to D.T. (email: [email protected]).
NATURE COMMUNICATIONS | 5:4997 | DOI: 10.1038/ncomms5997 | www.nature.com/naturecommunications 1 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5997
uantum mechanics is proverbially counterintuitive1,2. For presence of both properties, allowing the (objective) view that, at any many years, thought experiments were used to dissect its moment of time, a photon can be either a particle or a wave. The Qpuzzling properties, while hidden-variable (HV) models WDC experiment uncovers the difficulty inherent in this view by strived to explain or even to remove them1–4. The development of randomly choosing whether or not to insert the second beamsplitter 5,6 quantum technologies enabled us not only to perform several (BS2) after the photon enters the interferometer (Fig. 1a). This former gedanken experiments1,2, but also to devise new ones7–11. delayed choice prevents a possible causal link between the One can gain new insights into quantum foundations by experimental setup and the photon’s behaviour: the photon should introducing quantum controlling devices10–12 into well-known not know beforehand if it has to behave like a particle or like a wave. experiments. This has led, for example, to a reinterpretation11–14 The delayed-choice experiment with a quantum control of Bohr’s complementarity principle15. (Fig. 1c) highlights the complexity of space–time ordering of Wave–particle duality is best illustrated by the classic Wheeler events, once parts of the experimental setup become quantum delayed-choice experiment (WDC)16–18, Fig. 1a,b. A photon systems11. The quantum-controlled delayed-choice experiment enters a Mach–Zehnder interferometer (MZI) and its trajectory is has been recently implemented in several different systems19–23. coherently split by the beamsplitter BS1 into an upper and a lower To ensure the quantum behaviour of the controlling device, one path. The upper path contains a variable phase shift j. A random can either test the Bell inequality23 or use an entangled ancilla19. number generator controls the insertion (b ¼ 1) or removal The theoretical analysis of the quantum WDC involved so far a (b ¼ 0) of a second beamsplitter BS2.IfBS2 is present, the single binary HV l describing the classical concepts of wave/ interferometer is closed and we observe an interference pattern particle. Here, we introduce a full HV description for both the depending on the phase shift j.IfBS2 is absent, the MZI is open photon A and the ancilla. We analyse the relationships between and the detectors measure a constant probability distribution the concepts of determinism, wave–particle objectivity and local independent of j. Thus, depending on the experimental setup, independence of HV in the entanglement-controlled delayed- the photon behaves in two completely different ways. In the case choice experiment. We show that, when combined, these of the closed MZI, the interference pattern suggests that the assumptions lead to predictions that are different from those of photon travelled along both paths simultaneously and interfered quantum mechanics, even if any two of them are compatible with with itself at the second beamsplitter BS2, hence showing a wave- it. We propose and discuss an experiment to test our conclusions. like behaviour. However, if the interferometer is open, since always only one of the two detectors fires, one is led to the Results conclusion that the photon travelled only one path, hence Notation. We use the conventions as in refs 3,12; q(a, b,y) are the displaying a particle-like behaviour. quantum-mechanical probability distributions and p(a, b,...,L) The complementarity of the interferometer setups required to the predictions of HV theories with a HV L. We consider either a observe particle or wave behaviour obscures the simultaneous single HV L, which fully determines behaviour of the system, or refine it as L1, L2 pertaining to different parts of the system. For simplicity, we assume L is discrete; the analysis can be easily 0 abgeneralized to the continuous case. 1 1 D 0 H Hb Quantum system. The system we analyse consists of three qubits: BS a photon A and an entangled pair BC (Fig. 1d). We denote the 2 b=0,1 QRNG measurement outcomes for the photon A as a ¼ 0, 1 and for the + two ancilla qubits as b and c; the corresponding detectors are DA, QRNG 0 DB andffiffiffi DC. Theffiffiffiffiffiffiffiffiffiffiffi system is prepared in the initial state BS1 p p 1 j 0iAð Z j 00iþ 1 À Z j11iÞBC; for Z ¼ 2, BC is a maximally entangled EPR pair. cd MZI Photon A enters a MZI in which the second beamsplitter is D D quantum-controlled by qubit B. The third qubit C undergoes a s A H A y 0 HH 0 H iasy rotation RyðaÞ¼e followed by a measurement in the computational basis. The state before the measurements is cos |0> D DB B ffiffiffi pffiffiffiffiffiffiffiffiffiffiffi + sin |1> p jci¼ð Z cos a j pij0iþ 1 À Z sin ajwij1iÞAB j 0iC EPR pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ð1Þ Z a Z a : DC Àð sin jpij0iÀ 1 À cos jwij1iÞAB j1iC
Figure 1 | The evolution of the delayed-choice experiment. (a)In The counting statistics that result from the particle-like j i¼p1ffiffiðj iþ ij j iÞ j i¼ Wheeler’s classic experiment, the second beamsplitter is inserted or state p 2 0 e 1 and the wave-like state w removed after the photon is inside the interferometer; this prevents the eij=2ðcos j j0iÀi sin j j1iÞ are discussed below (equations (3 16 2 2 photon from changing its mind about being a particle or a wave. The and 4) and Methods). detectors observe either an interference pattern depending on the phase f (wave behaviour), or an equal distribution of hits (particle behaviour). A Constraints on HV theories. Our strategy is to show that q(a, b, quantum random number generator (QRNG) determines whether BS is 2 c) cannot result from a probability distribution p(a, b, c, L) inserted or not. Quantum networks: (b) in the classic delayed-choice of a HV theory satisfying the requirements of wave–particle experiment the QRNG is an auxiliary quantum system initially prepared in objectivity, local independence and determinism. Any viable the equal superposition state jþi ¼ p1ffiffiðj0iþj1iÞ and then measured. The 2 HV theory should satisfy the adequacy condition: namely, it Hadamard gate H is the quantum network equivalent of the beamsplitter; should reproduce the quantum statistics by summing over all 11 (c) delayed choice with a quantum control ;(d) entanglement- HVs L: assisted quantum delayed-choice experiment19. The ancilla C is measured X along the direction À a, equivalent to the application of a rotation RyðaÞ¼ qða; b; cÞ¼pða; b; cÞ¼ pða; b; c; LÞ: ð2Þ eiasy before a measurement in the computational basis. L
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Box 1 | Three classical assumptions. determined by the entanglement between B and C,
Wave–particle objectivity. We define particles and waves according to pðlÞ¼ðZ; 1 À ZÞ: ð8Þ the experimental behaviour in an open, respectively closed, MZI11.A particle in an open interferometer (b ¼ 0) is insensitive to the phase shift in one of the arms and therefore has the statistics This incongruity becomes an impossibility when the photon A and the entangled pair BC are prepared independently. In this pða j b ¼ 0; LÞ¼ð1; 1Þ; 8L 2L : ð3Þ 2 2 p case, their HVs are generated independently; that is, a single HV In contrast, a wave in a closed MZI (b ¼ 1) shows interference L not only has the structure L ¼ (L1,L2), where the subscripts 1 2 j 2 j pða j b ¼ 1; LÞ¼ðcos 2; sin 2Þ; 8L 2Lw: ð4Þ and 2 refer to the photon A and the pair BC, respectively, but the prior probability distribution of HV has a product form. To The sets L and L must be disjoint; otherwise, there are values of L p w realize this condition experimentally, we rely on the absence of that introduce wave–particle duality. Writing Lp,Lw ¼L, the wave/ particle property is expressed by a mapping l:L/{p, w} and the sets the superluminal communication and a space-like separation of À 1 À 1 Lp ¼ l (p), Lw ¼ l (w) are the pre-images of p, w under the the two events. function l. Unlike the typical Bell-inequality scenarios, we have a single measurement setup which involves two independent HV Determinism. The HV L determines the individual outcomes of the distributions. Moreover, by performing the rotation Ry(a) and detection3. Specifically, for the setup of (Fig. 1d) the detection DC sufficiently fast, such that the information about A and L cannot reach the detector D , the detection outcome is pða; b; cjLÞ¼wabcðLÞ; ð5Þ 1 C where the indicator function w ¼ 1, if L belongs to some predetermined determined only by L2. Since being a wave (particle) is assumed set, and w ¼ 0 otherwise. to be an objective property of A, l ¼ l(L1) is a binary function of the HV L1 only. Local independence. The HV L are split into L1 and L2, and the prior probability distribution has a product structure Contradiction. We show in Methods that for Za0, 1 (these two
pðLÞ¼f ðL1ÞFðL2Þ; ð6Þ cases correspond to an always closed or opened MZI), the for some probability distributions f and F, where the subscripts 1 and 2, requirements of adequacy, wave–particle objectivity, determinism respectively, refer to the photon A and the pair BC. Such bilocal and local independence are satisfied only if variables have been previously considered in ref. 29. cos 2a ¼ 0: ð9Þ We encapsulate the additional classical expectations into three assumptions (see Box 1 for the formal definitions of the concepts This proves our main theoretical result: determinism, local we consider in this section). independence and wave–particle objectivity are not compatible For a given photon, we require the property of being a particle with quantum mechanics for any aa±p/4, ±3p/4. We will later or a wave to be objective (intrinsic), that is, to be unchanged discuss how exactly a HV theory that satisfies the three classical during its lifetime. This condition selects from the set of adequate assumptions is inadequate. HV theories those models that have meaningful notions of 11 particle and wave . For each photon, the HV L should Proposed experiment. In Fig. 2, we show the proposed experi- determine unambiguously if the photon is a particle or a wave, mental setup for the entanglement-controlled delayed-choice thus allowing the partition of the set of HVs L into two disjoint experiment. Two pump pulses (blue) are incident on two non- subsets, L¼Lp,Lw, where the subscript indicates the property, linear crystals and generate via spontaneous parametric down- particle or wave. conversion two pairs of entangled photons (red). One of the The particle (wave) properties are abstractions of the particle photons is the trigger and the other three are the photons A, B, C, (wave) counting statistics in open (closed) MZI, respectively. The with BC being the entangled pair. behaviour of a particle (wave) in a closed (open) MZI is not Photons A and B are held in the lab (with appropriate delay constrained; this allows for significant freedom in constructing lines) and together they implement the controlled MZI. The HV theories. Experimentally, the wave or particle behaviour central element is the quantum switch, which is the controlled- p depends only on the photon and the settings of the MZI: i sy Hadamard gate C(H) ¼ (W#I)C(Z)(W#I), where W ¼ sze 8 . The photonic controlled-Z gate C(Z) is implemented with a pðajb; c; LÞ¼pðajb; LÞ; ð7Þ partially polarizing beamsplitter and is done probabilistically via post-selection24,25. Optical wave plates perform single-qubit rotations (gates H, j and W) on photon A. Photon C is sent for all values of a, b, c and L. through a channel at a distant location, then measured in a By replacing the single-qubit ancilla with an entangled pair, rotated basis. Two independent lasers generate the two photon one can take advantage of both the quantum control and the pairs (Fig. 2 (refs 26,27)); in this case, we can use equation (6) to space-like separation between events. The rationale behind the describe independent probability distributions for L1 and L2. third qubit C is that it allows us to choose the rotation angle a after both qubits A (the photon) and B (the quantum control) are Discussion detected. This is not possible in the standard quantum WDC11, In this section, we consider how exactly a HV theory, which Fig. 1c, where the quantum control B has to be prepared (by satisfies the three classical assumptions, fails the adequacy test. setting the angle a) before it interacts with A. As discussed in The interference pattern measured by the detector DA0 is S/ S/ Methods, there is a unique assignment of probabilities that IA(j) ¼ Tr(rA|0 0|), with rA ¼ TrBC|c c|, the reduced satisfies all the requirements of adequacy, wave–particle density matrix of photon A. The data can be postselected objectivity and determinism. Adopting this assignment, we according to the outcome c resulting in IA|c. The visibility of the reach the same level of incongruity as in ref. 11, since the interference pattern (Methods) is V ¼ (Imax À Imin)/(Imax þ Imin), probability p(l) of photon A being a particle or a wave is where the min/max values are calculated with respect to j.
NATURE COMMUNICATIONS | 5:4997 | DOI: 10.1038/ncomms5997 | www.nature.com/naturecommunications 3 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5997
1 0.75 Trigger A H H 0.5 0.25 B SPDC1 /4 /2 3 /4
DA1 1
A H W W 0.5 D V Delay A0 A 1 DB0 Delay PPBS optical C(Z) 3 /4 0.5 EPR B /2 D source B1 /4
0 Delay SPDC2 Figure 3 | Visibility. The visibilities VA|c ¼ 0 (yellow) and VA|c ¼ 1 (blue), are C calculated in Methods. In HV theories, the visibility does not distinguish PBS between the c ¼ 0, 1 cases. The inset illustrates this for pðpÞ¼Z ¼ 1, with DC0 2 the straight line representing the HV visibility prediction. C(Z) gate Channel Waveplates DC1 Pulsed pump 1 The ancilla qubits B and C are maximally entangled for Z ¼ 2. The final state laser before measurement is given by equation (1). From it, we calculate the quantum statistics q(a, b, c), where each of a, b, and c take the values {0, 1}. The probability Figure 2 | Proposed experimental setup. Two space-like separated pump distribution for c ¼ 0is pulses (blue) generate, via spontaneous parametric down-conversion, two qða; b; c ¼ 0Þ¼ð1Z cos2 a; ð1 À ZÞsin2 a cos2 j; 2 2 ð13Þ pairs of entangled photons (red). The first photon is the trigger and the 1 2 2 2 j 2Z cos a; ð1 À ZÞsin a sin 2Þ: other three the photons A, B, C. Inset: the quantum-controlled MZI. The where the four entries correspond to the values (a, b) ¼ (00, 01, 10, 11). For c ¼ 1, optical delays in the three photon arms, tA, tB, tC can be adjusted to ensure we obtain the desired time ordering of the detection events. 1 2 2 2 j qða; b; c ¼ 1Þ¼ð2Z sin a; ð1 À ZÞcos a cos 2; 1 2 2 2 j ð14Þ 2Z sin a; ð1 À ZÞcos a sin 2Þ: The postselected visibility for c ¼ 0 is (Fig. 3) This in turn yields ð1 À ZÞsin2 a qða; bÞ¼ð1Z; ð1 À ZÞcos2 j; 1Z; ð1 À ZÞsin2 jÞ; ð15Þ V ¼ ð10Þ 2 2 2 2 A j c¼0 Z cos2 a þð1 À ZÞsin2 a qðb; cÞ¼ðZ cos2 a; Z sin2 a; ð1 À ZÞsin2 a; ð1 À ZÞcos2 aÞ; ð16Þ The full (non-postselected) visibility is VA ¼ 1 À Z and gives information about the initial entanglement of the BC pair. On the qðbÞ¼ðZ; 1 À ZÞ; ð17Þ other hand, if one assumes that the HV are distributed according qðcÞ¼ðZ cos2 a þð1 À ZÞsin2 a; Z sin2 a þð1 À ZÞcos2 aÞ: ð18Þ to equation (6) and satisfy the wave–particle objectivity and 1 1 For Z ¼ 2; the probability distributions for b and c are equal. If Z 6¼ 2,BandC determinism, the visibility is independent of c, are no longer maximally entangled and the symmetry between them is broken: a HV HV HV rotation a on C no longer corresponds to a rotation a on B. The conditional VA VA j c¼0 ¼ VA j c¼1 ¼ 1 À f ; ð11Þ probabilities are in contrast with the quantum-mechanical prediction (Fig. 3). qðcjbÞ¼ðcos2 a; sin2 a; sin2 a; cos2 aÞ; ð19Þ Details of this calculation are in Methods. and from Bayes’ rule q(b|c) ¼ q(c|b)q(b)/q(c). This incompatibility between the basic tenets of HV theories and quantum mechanics has two remarkable features. First, the Solution to the three constraints. We now show that it is possible to construct a contradiction is revealed for any, arbitrarily small, amount of HV model that is adequate, objective and deterministic. The unknown parameters entanglement. This test is in sharp distinction with Bell-type at our disposal are 16 probabilities p(a, b, c, l). These probabilities are derived from experiments insofar as our result is free from inequalities. Wave– the underlying distribution p(L) summed over appropriate domains. At this stage, we do not enquire about the connection with the HV L. The probabilities p(a, b, c, particle objectivity, revealed only statistically, is more intuitive l) satisfy seven adequacy constraints, equations (13) and (14), plus the normal- and technically milder than the assumption of sharp values of ization constraint. The adequacy conditions can be written as quantum incompatible observables. Second, in our setup, any two qða; b; cÞ¼pða; b; cÞ¼pða; b; c; pÞþpða; b; c; wÞ: ð20Þ of the classical ideas together are compatible with the quantum- In addition, equation (7) and the standard rules for the conditional mechanical predictions. This fact, and the way we arrived at the probabilities, such as contradiction, invite questions concerning the internal consis- pða; b; c; lÞ 28 pðajb; lÞpðajb; c; lÞ¼ ; ð21Þ tency of classical concepts . pð0; b; c; lÞþpð1; b; c; lÞ imply the existence of four additional constrains, Methods ; ; ; ; ; ; ; Quantum-mechanical analysis. The initial state of photons A, B and C is pð0 0 c pÞ¼pð1 0 c pÞ ð22Þ ffiffiffi pffiffiffiffiffiffiffiffiffiffi p 2 j 2 j j0iAð Z j 00iþ 1 À Zj11iÞBC: ð12Þ pð0; 1; c; wÞsin 2 ¼ pð1; 1; c; wÞcos 2: ð23Þ
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The resulting linear system has a four-parameter family of solutions. However, convention as for xij, yij, zij and uij. For c ¼ 0, we have X X a straightforward calculation shows that for all these solutions p4(a, b, c, l), the 1 2 1 resulting statistics in an open/closed MZI is independent of l, qð0; 0; 0Þ¼2Z cos a 2 zijpij þ xijuijpij; ð39Þ i2Ip ;j2J0 i2Iw ;j2J0 1 1 p4ðajb ¼ 0; pÞ¼p4ðajb ¼ 0; wÞ¼ð2; 2Þ; ð24Þ 2 2 j qð0; 1; 0Þ¼ð1 À ZÞsin a cos 2 2 j 2 j X X j p4ðajb ¼ 1; wÞ¼p4ðajb ¼ 1; pÞ¼ðcos 2; sin 2Þ; ð25Þ 2 yijð1 À zijÞpij þ cos 2 ð1 À uijÞpij; ð40Þ i2Ip ;j2J0 i2Iw ;j2J0 that is, the statistics of DA is determined solely by the state of the interferometer. We can avoid the reintroduction of wave–particle duality using a special X X 1 2 1 solution qð1; 0; 0Þ¼2Z cos a 2 zijpij þ ð1 À xijÞuijpij; ð41Þ i2Ip ;j2J0 i2Iw ;j2J0 psðbjlÞ¼dlpdb0 þ dlwdb1 psðljbÞ; ð26Þ 2 2 j which imposes the b–l correlation (compare ref. 11). As a result, qð1; 1; 0Þ¼ð1 À ZÞsin a sin X 2 X 2 j pðb ¼ 0; l ¼ wÞ¼pðb ¼ 1; l ¼ pÞ¼0; ð27Þ ð1 À yijÞð1 À zijÞpij þ sin 2 ð1 À uijÞpij; ð42Þ i2Ip ;j2J0 i2Iw ;j2J0 and since the probabilities are positive, X X with analogous expressions for c ¼ 1. Adding and subtracting equations (39) and pða; 1; c; pÞ¼ pða; 0; c; wÞ¼0; ð28Þ (41) we obtain, respectively X X a; c a; c 2 zijpij þ uijpij ¼ Z cos a ð43Þ the eight above probabilities are zero individually. The system appears i2Ip ;j2J0 i2Iw ;j2J0 overconstrained, but it still has a unique solution X ð1 À 2xijÞuijpij ¼ 0 ð44Þ psða; b; c; lÞ¼qða; b; cÞpsðbjlÞ: ð29Þ i2Iw ;j2J0 In particular, Adding equations (40) and (42) yields X X X 2 psðlÞ¼ psða; b; c; lÞ¼ðZ; 1 À ZÞ: ð30Þ ð1 À ZÞsin a ¼ ð1 À zijÞpij þ ð1 À uijÞpij; ð45Þ a; b; c i2Ip ;j2J0 i2Iw ;j2J0 which on substitution back into equation (40) results in X ÀÁ Deriving the contradiction. In addition to the partition of L according to the 2 j cos 2 À yij ð1 À zijÞpij ¼ 0: ð46Þ values of l ¼ p, w, we will use the decomposition of the set of HV according to the i2Ip ;j2J0 outcomes of D . The two branches c ¼ 0, 1 correspond to the partition C Four additional equations (giving a total of seven independent equations) are A 2 2 2 L¼L0[L1; ð31Þ obtained for j J1 with cos a sin a. From equation (26), it follows that uij ¼ 0, i A Iw and zij ¼ 1, i A Ip. Hence, for where for LALc, the outcome of DC is c. The assumption of local independence c ¼ 0, only two equations are not automatically satisfied, implies a Cartesian product structure X X 2 2 Z cos a ¼ pij; ð1 À ZÞ sin a ¼ pij: ð47Þ 2 2 1 1 2 2 L¼fL1gÂðL0[L1Þ¼ðLp[LwÞÂðL0[L1Þ; ð32Þ i2Ip ;j2J0 i2Iw ;j2J0 The corresponding equations for c ¼ 1 are of the set of HV, where the subsets depend on the experimental setup. When the X X superscripts 1 and 2 on L are redundant, we may omit them. 2 2 Z sin a ¼ pij; ð1 À ZÞcos a ¼ pij; ð48Þ Now, we show that under the assumptions of adequacy and the three classical i2Ip ;j2J1 i2Iw ;j2J1 assumptions of the wave–particle objectivity, determinism and local independence, it is impossible to derive the solution p(a, b, c, l) with any arrangement of the which are in agreement with q(c), equation (18). probabilities p(L). The probability of the outcome c satisfies Now we use the product structure of the probability distribution, equation (6), X X X : qðcÞpðcÞ¼ pðLÞ¼ pðLÞþ pðLÞ: ð33Þ pðLÞ¼f ðL1ÞFðL2Þij ¼ fiFj ð49Þ L2Lc L2Lp \Lc L2Lw \Lc Using equations (28) and (30), we find that X To simplify the calculations, we enumerate the variables L1,2 by the indices i, j, f ¼ Z: ð50Þ 2 i respectively. The domain Lc corresponds, according to the hypothesis, to the index i2Ip set J of L , and the domains L1 and L1 to the index sets I and I of L , c 2 p w p w 1 Adding the pairs of equations in (47) and (48) and summing over the index i, respectively. In particular, P 0 1 we express the adequacy condition qðcÞ¼ Fj, j2Ic X X X 2 2 @ A Z cos a þð1 À ZÞsin a ¼ Fj; ð51Þ pðlÞ¼ fi; fi ; ð34Þ j2J0 i2Ip i2Iw X i 2 2 for some fi f ðL1Þ. The prior distribution of HV and the domains of summation Z sin a þð1 À ZÞcos a ¼ Fj; ð52Þ can depend on the parameters Z, j and a. j2J1 The putative behaviour of a wave (l ¼ w) in an open (b ¼ 0) interferometer and Za of a particle (l ¼ p) in a closed (b ¼ 1) one is characterized by two unknown but on the other hand, for 0, 1 summing over i in each of these four equations A A separately and using equation (50) we get distributions xij, i Iw and yij, i Ip, respectively X X 2 2 i j Fj ¼ cos a ¼ sin a ¼ Fj: ð53Þ pðajb ¼ 0; L ¼ðL ; L2ÞÞ ¼ ðxij; 1 À xijÞ; i 2 Ip ð35Þ 1 j2J0 j2J1 These equations can be satisfied for any Z only if i j pðajb ¼ 1; L ¼ðL ; L2ÞÞ ¼ ðyij; 1 À yijÞ; i 2 Iw ð36Þ 1 cos2 a ¼ sin2 a; ð54Þ allowing for a possible dependence on a value of L . The remaining two sets of 2 a a ¼ variables are the probability distributions for b conditioned on the values of HVs L: resulting in the contradiction (for arbitrary ) cos 2 0.
L Li ; Lj ; ; ; pðbj ¼ð 1 2ÞÞ ¼ ðzij 1 À zijÞ i 2 Ip ð37Þ Experimental signature. The interference pattern measured by the detector DA is S/ S/ IA(j) ¼ Tr(rA|0 0|), with rA ¼ TrBC|c c| the reduced density matrix of i j photon A. The data can be postselected according to the outcome c resulting in IA|c. pðbjL ¼ðL1; L2ÞÞ ¼ ðuij; 1 À uijÞ; i 2 Iw: ð38Þ The intensity (signal) measured by detector DA for c ¼ 0 (and no post-selection The requirement of adequacy means that the proposed HV theory reproduces on b) is: the quantum statistics given above. For compactness, we refer to the probability of j j 1 2 2 2 j i i IA j c¼0 ¼ Z cos a þð1 À ZÞsin a cos ; ð55Þ having the HV values ðL1 ¼ L1; L2 ¼ L2Þ, pðL1; L2Þ,aspij, using the same 2 2
NATURE COMMUNICATIONS | 5:4997 | DOI: 10.1038/ncomms5997 | www.nature.com/naturecommunications 5 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5997
giving the visibility 10. Arndt, M. & Hornberger, K. Testing the limits of quantum mechanical ð1 À ZÞsin2 a superpositions. Nat. Phys. 10, 271–275 (2014). V ¼ : ð56Þ 11. Ionicioiu, R. & Terno, D. R. Proposal for a quantum delayed-choice A j c¼0 Z cos2 a þð1 À ZÞsin2 a experiment. Phys. Rev. Lett. 107, 230406 (2011). A similar calculation gives the visibility for c ¼ 1 12. Ce´leri, L. et al. Quantum control in foundational experiments. Found. Phys. 44, ð1 À ZÞcos2 a 576–587 (2014). V ¼ : ð57Þ A j c¼1 Z sin2 a þð1 À ZÞcos2 a 13. Tang, J.-S., Li, Y.-L., Li, C.-F. & Guo, G.-C. Revisiting Bohr’s principle of complementarity using a quantum device. Phys. Rev. A 88, 014103 (2013). The full intensity measured by detector D (without postselecting on c)is A 14. Qureshi, T. Modified Two-Slit Experiments and Complementarity. Preprint at I ¼ 1Z þð1 À ZÞcos2 j and the corresponding visibility A 2 2 http://arxivorg/abs/1205.2207 (2012). VA ¼ 1 À Z: ð58Þ 15. Bohr, N. in Albert Einstein—Philosopher-Scientist (ed. Schlipp, P. A.) 200–241
Thus the visibility of detector DA gives information about the entanglement of (The Library of Living Philosophers, 1949). the BC pair. 16. Wheeler, J. A. in Quantum Theory and Measurement (eds Wheeler, J. A. & We now calculate the visibilities predicted by a non-trivial HV theory that is Zurek, W. H.) 182–213 (Princeton Univ. Press, 1984). assumed to satisfy the three classical assumptions. Using equation (26), we rewrite 17. Leggett, A. J. in Compendium of Quantum Physics (eds Greenberger, D., the counting statistics as Hentschel, K. & Weinert, F.) 161–166 (Springer, 2009). X 18. Jacques, V. et al. Experimental realization of Wheeler’s delayed-choice pð0; 0; 0Þ¼1 f F ; ð59Þ 2 i j Gedanken experiment. Science 315, 966–968 (2007). 2 ; 2 i Ip j J0 19. Kaiser, F. et al. Entanglement-enabled delayed choice experiment. Science 338, X 637–640 (2012). ; ; 1 pð0 0 1Þ¼2 fiFj ð60Þ 20. Roy, S., Shukla, A. & Mahesh, T. S. NMR implementation of a quantum i2Ip ;j2J1 delayed-choice experiment. Phys. Rev. A 85, 022109 (2012). X 21. Auccaise, R. et al. Experimental analysis of the quantum complementarity 2 j pð0; 1; 0Þ¼cos 2 fiFj; ð61Þ principle. Phys. Rev. A 85, 032121 (2012). i2Iw ;j2J0 22. Tang, J.-S. et al. Realization of quantum Wheeler’s delayed-choice experiment. X Nat. Photonics 6, 600–604 (2012). 2 j pð0; 1; 1Þ¼cos 2 fiFj: ð62Þ 23. Peruzzo, A. et al. A quantum delayed choice experiment. Science 338, 634–637 i2Iw ;j2J0 (2012). For the product probability distribution above, we get 24. Langford, N. K. et al. Demonstration of a simple entangling optical gate and its P 1 use in Bell-state analysis. Phys. Rev. Lett. 95, 210504 (2005). 2 fiFk ; ; i2I ;k2J 25. Kiesel, N. et al. Linear optics controlled-phase gate made simple. Phys. Rev. pð0 0 jÞ Pp j 1 pð0; 0 j jÞ¼ ¼ ¼ 2 f ð63Þ Lett. 95, 210505 (2005). pðc ¼ jÞ Fk k2Jj 26. Kaltenbaek, R. et al. Experimental Interference of Independent Photons. Phys. P Rev. Lett. 96, 240502 (2006). 2 j 27. Kaltenbaek, R., Prevedel, R., Aspelmeyer, M. & Zeilinger, A. High-fidelity cos 2 fiFk pð0; 1; jÞ Pi2Iw ;k2Jj 2 j entanglement swapping with fully independent sources. Phys. Rev. A 79, pð0; 1 j jÞ¼ ¼ ¼ cos 2ð1 À f Þð64Þ pðc ¼ jÞ Fk 040302(R) (2009). k2Jj 28. Ionicioiu, R., Mann, R. B. & Terno, D. R. Determinism, independence and P objectivity are inconsistent. Preprint at http://arxivorg/abs/1406.3963 (2014). for j ¼ 0, 1 separately, where f ¼ fi. As a result, i2Ip 29. Branciard, C., Gisin, N. & Pironio, S. Characterizing the nonlocal correlations created via entanglement swapping. Phys. Rev. Lett. 104, 170401 (2010). HV HV 1 2 j IA j c¼0 ¼ IA j c¼1 ¼ 2 f þ cos 2ð1 À f Þð65Þ giving Acknowledgements VHV ¼ VHV ¼ 1 À f ; ð66Þ D.R.T. thanks Perimeter Institute for support and hospitality. We thank Lucas Ce´leri, Jim A j c¼0 A j c¼1 Cresser, Berge Englert, Peter Knight, Stojan Rebic´, Valerio Scarani, Vlatko Vedral and for the visibilities in HV theories. Man-Hong Yung for discussions and critical comments and Alla Terno for help with visualization. This work was supported in part by the Natural Sciences and Engineering References Research Council of Canada. R.I. acknowledges support from the Institute for Quantum Computing, University of Waterloo, Canada, where this work started. 1. Wheeler, J. A. & Zurek, W. H. (eds) Quantum Theory and Measurement (Princeton Univ. Press, 1984). 2. Peres, A. Quantum Theory: Concepts and Methods (Kluwer, 1995). Author contributions 3. Brandenburger, A. & Yanofsky, N. A. Classification of hidden-variable T.J. and R.I. conceived the entanglement-controlled protocol. D.R.T. performed the properties. J. Phys. A 41, 425302 (2008). hidden-variable analysis. R.I. and T.J. produced the experimental design. R.B.M. and 4. Englert, B.-G. On quantum theory. Eur. J. Phys. D 67, 238 (2013). D.R.T. analysed the experimental signatures. All authors contributed to the writing of the 5. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum manuscript. D.R.T. coordinated the project. Information (Cambridge Univ. Press, 2000). 6. Bru, D. & Leuchs, G. Lectures on Quantum Information (Wiley-VCH, 2007). 7. Scully, O. M., Englert, B.-G. & Walther, H. Quantum optical tests of Additional information complementarity. Nature 351, 111–116 (1991). Competing financial interests: The authors declare no competing financial interests. 8. Schmidt, L. Ph. H. et al. Momentum transfer to a free floating double slit: Reprints and permission information is available online at http://npg.nature.com/ realization of a thought experiment from the Einstein-Bohr debates. Phys. Rev. reprintsandpermissions/ Lett. 111, 103201 (2013). 9. Marshall, W., Simon, C., Penrose, R. & Bouwmeester, D. Towards quantum How to cite this article: Ionicioiu, R. et al. Is wave–particle objectivity compatible with superpositions of a mirror. Phys. Rev. Lett. 91, 130401 (2003). determinism and locality? Nat. Commun. 5:4997 doi: 10.1038/ncomms5997 (2014).
6 NATURE COMMUNICATIONS | 5:4997 | DOI: 10.1038/ncomms5997 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. DOI: 10.1038/ncomms16198 Publisher Correction: Is wave–particle objectivity compatible with determinism and locality?
Radu Ionicioiu, Thomas Jennewein, Robert B. Mann & Daniel R. Terno
Nature Communications 5:4997 doi: 10.1038/ncomms5997 (2014); Published online 26 Sep 2014; Updated 26 Mar 2018
The original HTML version of this Article had an incorrect article number of 3997; it should have been 4997. This has now been corrected in the HTML; the PDF version of the Article was correct from the time of publication.
NATURE COMMUNICATIONS | 9:16198 | DOI: 10.1038/ncomms16198 | www.nature.com/naturecommunications 1