Weak Measurement and Bohmian Conditional Wave Functions Travis Norsen

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11-2014 Weak Measurement and Bohmian Conditional Wave Functions Travis Norsen

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For more information, please contact [email protected] arXiv:1305.2409v2 [quant-ph] 7 Oct 2013 † ∗ sncl upre n upeetdb h ahrdif- rather Lundeen the of by work supplemented recent ferent – and character supported an epistemic) nicely having merely as is to understood opposed quantum be (as the must which ontological function to wave according or Bar- – state Pusey, [3] of Rudolph theorem and recent rett, the me- example, quantum For in important chanics. questions an foundational become exploring has for [2], tool in reviewed recently and [1] aea(oml tog esrmn fsm te ob- also other some may of one servable measurement pointer, strong) the (normal, to a After make weakly couples times averaging. many system process and the the systems) repeating identically-prepared by for (on this up but made ambiguous; be can quite remains registration pointer’s with stronger, associated were value coupling observable the the register if unambiguously position, would whose pointer a we h nlmaueethsoutcome has reading measurement pointer’s final the the of value (when average the limit, coupling et n esasse nstate in system a lets one ment, [5]). also see but – tomography in- approaches state in quantum reconstructive “direct” in or volved is indirect (This the us- to techniques. measured” contrast measurement “directly weak be ing can function wave quantum lcrncades [email protected] address: Electronic lcrncades [email protected] address: Electronic h oino wa esrmn” rtitoue in introduced first measurement”, “weak of notion The h rcdr osa olw.I ekmeasure- weak a In follows. as goes procedure The B ˆ n otslc nteotoe nteweak- the In outcome. the on post-select and A ˆ r ugse n discussed. sub-system and of suggested dependence are non-local – the arrangemen reveal Experimental thereby “conditiona also matrix. Bohmian density the reduced sub- standard circumstances, a ma measuring appropriate (uniquely) directly mechanics under for Bohmian procedure which oper correspon measurement-based plausibly function sub-system mechanical wave a quantum appropr Thus, system a with of mechanics. resu function applied, Bohmian wave the is of system, technique multi-particle function” same wave entangled) (perhaps this measu a if “directly from that be po shown function can wave is particles) the precisely, identically-prepared (more of particle a of function ihtewa opig oee,the however, coupling, weak the With . ekMaueetad(oma)CniinlWv Functions Wave Conditional (Bohmian) and Measurement Weak twsrcnl one u addmntae experimental demonstrated (and out pointed recently was It .INTRODUCTION I. ilCne,10FeigusnRa,Psaaa,N 08854- NJ Piscataway, Road, Frelinghuysen 110 Center, Hill hsc eatet mt olg,Nrhmtn A01060, MA Northampton, College, Smith Department, Physics eatet fMteaisadPiooh,RtesUniver Rutgers Philosophy, and Mathematics of Departments tal. et | ψ 4 hwn htthe that showing [4] i opewal to weakly couple b stereal the is ) Dtd coe ,2013) 8, October (Dated: adStruyve Ward rvsNorsen Travis is[0.Adi a eetypitdotb Braverman by out pointed recently was trajecto- it Bohmian And 2-slit [10]. of images ries iconic the with beautiful gruent con- The indeed are experiment. 2-slit trajectories experimentally-reconstructed the trajec- in average photons the for reconstruct tories to scheme this plemented atcesvlct seas 7 ].Steinberg 8]). [7, also (see velocity particle’s t aey enn h eoiyo atcea cer- time a at at weak position the particle between its difference a of the of of value particle terms velocity in quantum position the certain a tain defining of experimen- a namely, trajectory to that – the approach out determining pointed operational tally [6] plausible Wiseman naively Bohmian involves questions mechanics. foundational probe to ments identi- of ensemble a(n photon(s). of prepared) function cally wave transverse the Lundeen o h atclrcase particular the For au spootoa otepril’ aefunction: wave particle’s the to proportional is value ato h hr,cmlx wa value” “weak complex) (here, the of part ftepitrscnuaemomentum. conjugate pointer’s measurements via the accessible of also is part imaginary whose A ˆ + nte eeteapeo h s fwa measure- weak of use the of example recent Another nteshm nrdcdb Lundeen by introduced scheme the In ˆ = dt π e”uigwa esrmn.Hr it Here measurement. weak using red” x ilspeieyteBhinepeso o the for expression Bohmian the precisely yields – ti rcsl h ocle “conditional so-called the precisely is lt sse yec ebro nensemble an of member each by ssessed tt ucin ndsatinterventions distant on functions state = st eosrt hsbhvo and – behavior this demonstrate to ts ytmsdniymti hudyield, should matrix density system’s est arx sopsdt the to opposed as matrix” density l h tal. et st h aua ento fasub- a of definition natural the to ds π ˆ | y yLundeen by ly) x x aeps-eeto,t n particle one to post-selection, iate toaitmto o enn the defining for method ationalist ih i e osbe iial,aweak- a Similarly, possible. kes W p x x 4 sdti ehiu odrcl measure directly to technique this used [4] | = and h p h h π ˆ x A 09 USA. 8019, h B ˆ ˆ | x p x i p i x W b ˆ = ih x p W | x ψ USA. sity, tal. et x etu aeta h weak the that have thus we 0 = =0 p = i | x ψ ete aethat have then We . h ∼ i b h t ∗ | = b httewave the that A ψ | n h togvleat value strong the and ˆ † ψ ( | e ψ x i − ) i . ip ψ ˜ x x/ ( p ~ x tal. et ψ ) ( x tal. et ) . n lets one , 9 im- [9] (3) (2) (1) 2 and Simon [11] that such measurements, if performed on y one particle from an entangled pair, should allow an empirical demonstration of the non-local character of the Bohmian trajectories. λa3 Following Braverman and Simon, the goal of the present work is to address the following seemingly natural question: what happens if the Lundeen et al. technique, {X(0+),Y (0+)} for “directly measuring” the wave function of a parti- + cle, is applied to a particle which does not, according χ1(x, 0 ) to ordinary quantum theory, have a wave function of its b λa2 own, because it is entangled with some other particle(s)? The answer turns out to be that, under suitable conditions, the “directly measured” one-particle wave function corresponds exactly to the so-called “conditional wave λa1 function” of Bohmian mechanics [12] . Since this is un- {X(0−),Y (0−)} doubtedly an unfamiliar concept to most physicists, we χ1(x, 0−) review it in Section II before explaining, in Section III, b this central claim and suggesting an experimental setup x in which it should be demonstrable. Section IV then L outlines a parallel result, regarding density matrices, especially appropriate for the (spin) states of particles with FIG. 1: Illustration of the collapse of the Bohmian CWF discrete degrees of freedom. Section V offers conclusions, during an energy measurement (Aˆ = Hˆ ) on a particle in a focusing especially on questions surrounding the claimed box. The initial two-particle wave function Ψ(x,y, 0−) = observability of non-locality. ψ(x)φ0(y) has support in the blueish region of the configuration space. Since this initial state factorizes, the CWF at t = 0− is (up to a multiplicative constant) just χ1(x) = II. BOHMIAN CONDITIONAL WAVE Pn cnψn(x), indicated with the bolded lower curve. The + FUNCTIONS Schrödinger evolution from 0− to 0 produces a two-particle wave function with localized islands of support in the configuration space, indicated by the yellowish regions. Each of Consider for simplicity a system of two spin-0 particles the yellowish blobs is a Gaussian in y (centered at one of the (masses m1 and m2, coordinates x and y) each moving in possible post-interaction pointer positions λan) multiplied by one spatial dimension. According to ordinary quantum one of the energy eigenfunctions (here ψn(x) ∼ sin(nπx/L)). mechanics (OQM) the wave function Ψ(x,y,t), obeying And so if (for example, as shown) the actual configuration an appropriate two-particle Schrödinger equation, pro- point {X(0+),Y (0+)} ends up in the support of the yellowish vides a complete description of the state of the system. blob at y = λa2 – something which will occur with proba- 2 According to Bohmian mechanics (BM), however, the de- bility |c2| with random initial configuration {X(0−),Y (0−)} in accord with the QEH – then the post-interaction CWF of scription provided by the wave function alone is decid- + edly incomplete; a complete description requires speci- particle 1 will be χ1(x, 0 ) ∼ ψ2(x) (as shown). fying in addition the actual particle positions X(t) and Y (t). For BM the wave function Ψ(x,y,t) obeys the usual Schrödinger equation, while X(t) evolves according to This is the obvious and natural way to construct a “single particle wave function” given the resources that BM ~ ∂ ∂ dX(t) Ψ∗ ∂x Ψ Ψ ∂x Ψ∗ provides. (OQM, with fewer resources at hand, provides = − (4) dt 2m1i Ψ∗Ψ no such natural – or even an unnatural – construction.) x=X(t),y=Y (t) What makes this definition natural is that the evolution and similarly for Y (t). It is a joint property of the time- law for the position X(t) of particle 1, Equation (4), can evolution laws for the wave and particles that, if the par- be re-written in terms of particle 1’s CWF as follows: 2 ticle positions X and Y are random and ψ -distributed ∂ ∂ | | dX(t) ~ χ∗ χ1 χ1 χ∗ at some initial time (this is the so-called quantum equilib- = 1 ∂x − ∂x 1 . (6) 2 rium hypothesis, QEH), they will remain ψ distributed dt 2m1i χ1∗χ1 | | x=X(t) for all times. This so-called “equivariance” property is crucial for understanding how BM reproduces the statis- It is thus appropriate to think of χ1(x, t) as the guiding- tical predictions of OQM [12]. or pilot-wave that directly influences the motion of par- The Bohmian “conditional wave function” (CWF) – ticle 1. for, say, the first particle – is simply the (“universal”) It is important to appreciate that χ1(x, t) depends on wave function Ψ(x,y,t) evaluated at y = Y (t): time in two different ways – through the t-dependence of Ψ and also through the t-dependence of Y . Thus, χ1(x, t)= Ψ(x,y,t) . (5) in general, χ (x, t) does not obey a simple one-particle |y=Y (t) 1 3

Schrödinger equation, but obeys instead a more com- wave function Ψ is completely unitary. See Figure 1 and plicated pseudo-Schrödinger equation [12, 13]. In par- its caption for an illustration. ticular, it is easy to see that, under the appropriate We have here explained the idea of (and one important measurement-like circumstances, χ1(x, t) will collapse. and perhaps surprising property of the dynamical evolu- Suppose for example that particle 1 has initial wave function of) Bohmian CWFs as if the particle of interest were tion interacting with the particle or particles constituting a measuring device. That is of course the crucial kind of ψ(x)= c ψ (x) (7) n n situation if one is worrying about the so-called quantum Xn measurement problem. But more important for our pur- where the ψn(x) are eigenstates of some observable Aˆ poses here is the fact that the Bohmian CWF (for a sin- with eigenvalues an. And suppose that particle 2 is the gle particle) is perfectly well-defined at all times for any pointer on an Aˆ-measuring device, initially in the state particle that is part of a larger (multi-particle) quantum

2 2 system. Indeed, BM only really provides a solution of the y /2w φ0(y) e− . (8) measurement problem because it treats “measurements” ∼ as just ordinary physical interactions, obeying the same Now suppose the particles experience a (for simplicity, universal dynamical laws as all interactions. It should impulsive) interaction thus be clear that, according to BM, collapses (like the one we just described happening as a result of an interac- Hînt = λδ(t) Aˆ pˆy. (9) tion with a measuring device) will actually be happening The usual unitary Schrödinger evolution of the initial all the time, as particles interact with each other. It is wave function Ψ(x, y, 0−)= ψ(x)φ0(y) then takes it into the goal of the following analysis to show how this feature of the Bohmian theory can be experimentally manifested Ψ(x, y, 0+)= c ψ (x)φ (y λa ). (10) n n 0 − n using weak measurement. Xn That is, the two-particle wave function after the interaction can be understood as an entangled superposition of III. DIRECT MEASUREMENT OF SINGLE terms, each of which has particle 1 in an eigenstate of Aˆ PARTICLE WAVE FUNCTIONS and particle 2 in a new position that registers the corresponding value an. (Note that we assume here that λ is sufficiently large that the separation between adjacent Let us then turn to the main result of the present pa- per. Suppose we carry out the Lundeen-type “direct mea- values of λan is large compared to the width w of the pointer packet. This is thus a “strong” measurement.) surement of the wave function” procedure on one particle of a two-particle system. As a reality check, suppose to From the point of view of OQM, Equation (10) exhibits the standard problem of Schrödinger’s cat: instead of begin with that the two-particle system has a factorizable quantum state resolving the superposition of distinct a-values, the measuring device itself gets infected with the superposition. Ψ = ψ φ (12) In OQM (where there is nothing but the wave function | i | i| i at hand) one thus needs to introduce additional dynamical (“collapse”) postulates to account for the observed where the first and second factors on the right refer to (apparently non-superposed) behavior of real laboratory particles 1 and 2 respectively. The Lundeen-type proce- equipment. dure involves post-selecting on the final momentum px of In BM, however, there is no such problem. The ob- the particle whose wave function we are trying to mea- servable outcome of the measurement is not to be found sure (here, particle 1). Let us also post-select on the final in the wave function, but instead in the actual position position Y of particle 2 [14]. It is then straightforward Y (0+) of the pointer after the interaction. It is easy to calculate that to see that (with appropriate random initial conditions) ~ 2 Y φ p x x ψ e ipxx/ ψ(x) this will, with probability cn , lie near the value λan px,y=Y x − | | πˆx W = h | ih | ih | i = (13) which indicates that the result of the measurement was h i Y φ px ψ ψ˜(px) + h | ih | i an. Furthermore, it is easy to see that if Y (0 ) is near the value λan, then the CWF of particle 1 will be (up to which is, as expected, identical to Equation (2). a multiplicative constant) the appropriate eigenfunction: If, however, particle 1 is in a general, entangled state with particle 2, as in χ (x, 0+) ψ (x). (11) 1 ∼ n That is, the CWF of particle 1 collapses (from a super- Ψ = dx′dy′Ψ(x′,y′) x′ y′ (14) position of several ψns to the particular ψn which cor- | i Z | i| i responds to the actually-realized outcome of the measurement) as a result of the interaction with the measur- then the operational determination of particle 1’s wave ing device, even though the dynamics for the “universal” function (post-selected on the final strongly-measured 4 position Y of particle 2) yields (a) FT Lens px px,y=Y px x x, Y Ψ λ/2 x or πˆx W = h | ih | i D1 ψ1 λ/4 h i px, Y Ψ | i

h | i D2 ipxx/~ e− Ψ(x, Y ) z = ′ (15) λ ipxx /~ ψ2 sliver PBS dx′Ψ(x′, Y ) e− | i 2 R Restricting, as before, our attention to the cases in which the ﬁnal measured momentum px is zero, we have that

px=0,y=Y πˆ Ψ(x, Y )= χ (x) (16) (b) h xiW ∼ 1 where the right hand side is precisely the Bohmian CWF y φ1 for particle 1. Note that we have tacitly relied on the fact | i non-contextual that, for Bohmian mechanics, position is a φ1 / φ2 (“hidden”) variable. Thus the ﬁnal position measure- | i | i

φ2 ment on particle 2 simply reveals, for Bohmian mechan- | i ics, the actual pre-existing location Y of that particle. In short, the two Y s in the analysis – the one representing BS C B A the outcome of the final position measurement of particle 2, and the one, used in the definition of the Bohmian CWF, representing the actual position of particle 2 – are, FIG. 2: Schematic diagram of the proposed experiment. for Bohmian mechanics, the same. Frame (a), following [4], shows photon 1 propagating to the right and undergoing first a weak measurement (effected by So far we have basically ignored the issue of the exact the λ/2 sliver), then a px = 0 measurement/post-selection, timing of the various measurements on the two-particle and finally the readout of the weak measurement. Frame (b) system. Let us then examine this in the context of a shows photon 2 propagating to the left. The two-photon state 1 somewhat concrete example. Consider two photons pre- 1 1 2 2 1 2 is |Ψi = √2 (|ψ i|φ i + |ψ i|φ i) where |ψ i and |ψ i have pared in some kind of entangled state (to be specified distinct (transverse, i.e., x) spatial profiles – for example, as shortly) and propagating in roughly opposite directions. shown here, perhaps |ψ1i has support only for x> 0 while |ψ2i Let the variable x refer to the transverse spatial degree has support only for x < 0. For photon 2, |φ1i and |φ2i are of freedom of photon 1 (propagating, say, to the right) initially overlapping spatially but are distinct in some prop- and the variable y refer to the transverse spatial degree of erty (such as energy or polarization) which allows the beams freedom of photon 2 (propagating to the left). We imag- to be separated, by some kind of (removable) beam-splitter ine a setup like that reported in [4] in which the weak (BS), as shown. Three different possible detection planes – measurement is effected using an extremely narrow half- A, B, and C – for the final measurement/post-selection of the wave plate (“λ/2 sliver”) and the p = 0 post-selection transverse position Y of photon 2 are shown and discussed in x the main text. is effected by accepting only those photons which pass a narrow slit downstream from and along the axis of a Fourier Transform lens. The remaining photons are then passed through an appropriate quarter (or half) wave overlap spatially, but are distinct in some way (for exam- plate; the imbalance between the two polarization states ple, they could be orthogonally polarized, or could have then yields the real (or, respectively, imaginary) part of different energies) that allows the two parts of the beam the transverse wave function at the location of the λ/2 to be separated by some type of beam splitter (BS). See sliver. See [4] for details. Figure 2(b). We now consider the possibility that each photon that Let us now consider several different spatial locations enters the device is entangled with a second photon: for, and time-orderings involving, the final measurement 1 of the position Y of photon 2. Ψ = ( ψ φ + ψ φ ) . (17) To begin with, let us first imagine that the √ 1 1 2 2 | i 2 | i| i | i| i measurement/post-selection on photon 2’s transverse co- It is important (for the proper functioning of the wave ordinate y occurs (in, say, the lab frame) before the weak function measurement) that ψ1 and ψ2 have the same measurement on photon 1 and at a plane like A in Fig- (say, linear) polarization. But| i let them| i have distinct ure 2 (i.e., before photon 2 has passed any beam split- (transverse) spatial profiles – for example, and most sim- ter). According to OQM, this measurement of Y will ply, suppose that ψ1 has transverse spatial support just collapse the 2-particle wave function and leave photon 1 above the z-axis (i.e.,| i for x> 0) while ψ has transverse with a definite (non-entangled) wave function of its own. | 2i spatial support just below the z-axis (x< 0). See Figure Because φ1 and φ2 overlap at A, however, the po- | i | i 2(a). sition measurement gives no information about φ1 vs. 1 | i As to photon 2, suppose that (at least initially) φ φ2 and so leaves photon 1 in the state ( ψ1 + ψ2 ). 1 | i √2 | i | i and φ have identical transverse profiles and completely| i And this of course coincides with the predicted result of | 2i 5

y at plane C (such that this measurement takes place well after the weak measurement on photon 1 has already gone fully to completion). It is trivial to see that the results of the weak measurement will again be

χ1(x,t1) given by Equation (19) – that is, dramatically diﬀerent (“collapsed”) wave functions for photon 1 will be found

b depending on whether particle 2 is (later!) found in x the support of φ1 or φ2 . From the point of view χ1(x,t2) of OQM, however,| i it is| ratheri difficult to understand why, prior to any actual measurement (meaning here b an interaction involving macroscopic amplification) that would trigger a collapse, one should find collapsed one-particle wave functions. On the other hand, this is perfectly natural from a Bohmian point of view: FIG. 3: Inthe (x,y) configuration space, the entangled two- as sketched in Figure 3 the conditional wave function photon wave function initially has support in the blueish re- (CWF) for particle 1 collapses as soon as the packets gion and the CWF for photon 1 – χ1(x,t1) – looks something like the two-hump curve shown. The passage of photon separate in the two-dimensional configuration space. This happens when the components φ and φ are 2 through the beam-splitter (BS) separates the two-photon | 1i | 2i wave function into the two yellowish islands. The actual con- split at the BS; no actual position measurement is figuration point (shown here as a black dot) ends up (depend- required. (Note that the claim here is not that OQM ing on its random initial position for each photon pair) in one makes the wrong predictions. Undoubtedly it makes of the two yellowish islands, and the CWF at t2 will thus have the right predictions – indeed we have used nothing collapsed to either hx|ψ1i or hx|ψ2i. (The latter case is shown but orthodox quantum ideas to calculate what should here.) be observed in the experiment. The point is rather only that, from the OQM perspective, it is at best obscure what one ought to expect the operationalist the direct measurement protocol: for measurement/post- determination of photon 1’s “one-particle wave function” selection at A to yield at times when photon 1 remains entangled with photon 2. Whereas, from the Bohmian point of view, px=0,y=Y ψ1(x) Y φ1 + ψ2(x) Y φ2 the results are the obvious, natural thing.) πˆ = h | i h | i x W ˜ ˜ h i ψ1(0) Y φ1 + ψ2(0) Y φ2 The proposed experiment, then, should involve a fixed h | i h | i 1 detection plane, like plane C in the Figure, at a greater (ψ1(x)+ ψ2(x)) (18) ∼ √2 optical distance from the two-particle source than that of the measuring apparatus for particle 1. The removable where we have used the fact that, for detection at the A beam splitter BS should, on the other hand, ideally be plane, Y φ = Y φ . h | 1i h | 2i slightly closer to the two-particle source than the parti- On the other hand, if the measurement/post-selection cle 1 apparatus. With the BS in place, the arrangement on photon 2 is performed at plane B (after photon 2 has would be like that shown in the Figure and the “direct passed the beam splitter, but still before the measure- measurement” on photon 1 would yield (after appropriate ment protocol has been carried out on photon 1) then post-selection on Y ) collapsed photon 1 wave functions according to OQM the wave function of photon 1 will (even though the actual position measurement on photon collapse to either ψ1 (if Y is found in the support of 2 would occur well after photon 1 was already measured). y φ ) or ψ (if |Y isi found in the support of y φ ). h | 1i | 2i h | 2i On the other hand, with the BS removed, the situation This again coincides with the expected results of the di- would be equivalent to detecting photon 2 at plane A rect measurement: for detection at the B plane, one or (except that this too would only occur later, after the the other of Y φ and Y φ will be zero. We will thus h | 1i h | 2i measurements on photon 1 had occured) and the mea- find that surement of photon 1 would reveal the uncollapsed wave

x function. One could thus in some sense observe the col- πˆ p =0,y=Y ψ (x) (19) h xiW ∼ m lapse of the Bohmian CWF of photon 1 as a direct result of the insertion (perhaps at space-like separation) of the for Y supp( y φm ). For∈ both ofh | thosei two scenarios, ordinary QM at- BS into the path of photon 2. tributes a one-particle wave function to particle 1 at the This proposed setup should be realizable in practice time in question; this wave function coincides with the along the following lines. Type II spontaneous paramet- Bohmian CWF and the expected results of the weak mea- ric down-conversion yields a pair of photons in an entan- surement technique. In short, there is nothing surprising gled polarization state ( H H + V V ) /√2 with the or interesting here from the point of view of OQM. individual photons being| coupledi| i into| i| single-modei opti- Consider, however, a scenario involving cal fibres. The first photon should then be split by a measurement/post-selection of photon 2’s position polarizing beam splitter (PBS), with, say, the H com- | i 6 ponent being shunted into the +z direction with x > 0 matrix: and the V component passed through a λ/2 plate (ro- | i tating its polarization to H ) before being shunted into πîj W = Tr[ˆπij ρˆ]= j ρˆ i . (24) h i h | | i the +z direction with x| < i0. This prepares photon 1 Of course, for i = j,π ˆ is not a Hermitian operator, as suggested in Figure 2(a) and the subsequent measure- 6 ij ments may then be carried out exactly as in [4]. The sec- so measuring it – even weakly – raises some questions. ond photon can be directly shunted into the z direction But the worrisome matrix elements can be re-expressed so that the two orthogonal polarization components− have in terms of weak values of perfectly reputable operators identical and perfectly overlapping transverse spatial pro- by introducing post-selection, e.g., files, as in Figure 2(b). The overall (transverse spatial ρ = H ρˆ V and polarization) two-photon state after such prepara- VH h | | i = P (D) πˆ D P (A) πˆ A (25) tion can thus be written h HH iW − h HH iW 1 Ψ = ( +H H + H V ) (20) where D = ( H + V )/√2 and A = ( H V )/√2, | i √2 | i|∅ i |− i|∅ i and P (|Di) = |Diρˆ D| iand P (A)| = i A ρˆ|Ai −are | respec-i h | | i h | | i where + indicates that the transverse state has support tively the rates of successful post-selection on the D and A states. See [15] for further details. | i for x> 0, indicates that the support lies in x< 0, and | i indicates that− the support is centered at x = 0. (It is also∅ Consider now a two-particle system in state of course understood that photon 1 is moving in the +z direction and photon 2 in the z direction.) The perfect ψo = dy ψi,j (y) i 1 j 2 y 2 (26) − | i Z | i | i | i correspondence between Equations (20) and (17) should Xi,j be clear. Note that for this type of implementation, the (generic) “BS” in Figure 2(b) can be a standard PBS. where, as before, i, j H, V are one-particle polar- ∈ { } ization eigenstates, and y 2 is a position eigenstate of particle 2. (We suppress,| fori simplicity, the position de- IV. DENSITY MATRICES gree of freedom of particle 1; recall that it may be used as the pointer variable to weakly measure the polarization In the Lundeen et al. procedure for measuring the wave density matrix.) The two-particle state thus has density function, it is the particles’ spatial degree of freedom that operator is probed, with the polarization serving as the pointer. ρˆ = ψ ψ (27) But the polarization state of a particle can also be mea- | 0ih 0| sured using weak measurement techniques, with the po- in terms of which one can define the reduced density ma- sition degree of freedom (or in principle some other ex- trix (RDM) of particle 1 by tracing over the degrees of trinsic degree of freedom) playing the role of the pointer. freedom associated with particle 2: For example, in a scheme recently proposed and demonstrated by Lundeen and Bamber [15, 16] (see also [17]) red ρˆ = dy Tr2 [ y ρˆ y ]= dy y, j ρˆ y, j . (28) the polarization density matrix of a particle can be mea- 1 Z h | | i Z h | | i sured as follows. For a photon in a mixed state described Xj by density operatorρ ˆ, the weak value of an observable Aˆ The RDM is of course the standard way of defining the is given by “state” of a (perhaps-entangled) subsystem. b Aˆρˆ b As discussed above, for systems with complex-valued Aˆ b = h | | i (21) h iW b ρˆ b wave functions, Bohmian mechanics allows one to de- h | | i fine the conditional wave function of a sub-system, in where b is the final post-selected state, as in Equation terms of which the guidance law for the particles com- (1). In| thei case of a pure state,ρ ˆ = ψ ψ , Equation prising the sub-system can be re-expressed. For sys- (21) reduces to (1), whereas for a genuinely| ih mixed| state, tems with discrete (spin, polarization) degrees of free- (21) is the appropriate weighted average. In the event of dom, however, the Bohmian CWF for each one-particle no post-selection, a further averaging gives sub-system would carry the discrete indices for all particles in the system. It thus cannot really be regarded as a ˆ ˆ A W = Tr Aρˆ . (22) “wave function for a single particle”. In such situations, h i h i Bohmian mechanics thus follows ordinary quantum me- The Lundeen/Bamber procedure can then be most sim- chanics in defining the state of the sub-system in terms of ply understood as follows. Defining operators an appropriate density matrix. But as was first pointed out by Bell [18], the correct Bohmian particle trajecto- πîj = i j (23) | ih | ries (needed to reproduce the statistical predictions of (with i, j H, V ) on the two-dimensional polarization ordinary QM) cannot be expressed in terms of the usual Hilbert space∈{ for a} photon, one sees that their weak val- reduced density matrix. Instead, one needs to introduce ues correspond to the entries in the polarization density the “conditional density matrix” (CDM), which involves 7 tracing over the discrete indices associated with particles y = 0. (The spatial degrees of freedom of particle 1, and outside the sub-system in question, but then evaluating the non-transverse spatial degrees of freedom of particle the spatial variables (again, associated with particles out- 2, are suppressed for simplicity.) By means of a polarizing side the sub-system in question) at the actual locations beam splitter that can be inserted (or not) in the path of the Bohmian particles. (See [19] for a detailed discus- of particle 2, the two-particle state may (or may not) be sion.) Thus, for the system introduced just above, we transformed into would have + ψ2 = φ H 1 H 2 + φ− V 1 V 2 /√2 (32) ρˆcond = Tr [ Y ρˆ Y ] . (29) | i | i| i | i | i| i | i 1 2 h | | i where φ± is (say) a Gaussian displaced in the y- Note that a sub-system will in general possess both a direction| byi an amount that is larger than its width.± RDM and a CDM, but that these will not in general be The crucial point is then that, for both ψ1 and ψ2 , equal: the RDM can be understood as the average of all the RDM for particle 1 is | i | i possible CDMs. (If we had normalized the CDM, then 1 the RDM would be given by the average of all possible red /2 0 ρˆ1 1 . (33) CDMs, weighted by the usual quantum probability for −−−−−−−−→H , V basis 0 /2 y = Y .) | i | i Now, what should happen if one performs the Lun- The Bohmian CDM for particle 1 will be (proportional deen/Bamber procedure for directly measuring the po- to) this same matrix if the state is ψ . But if the beam | 1i larization density matrix of one photon from an entan- splitter is inserted such that the state is ψ2 , the CDM gled pair, also – as in the previous section – post-selecting will have “collapsed”, being now proportional| i to either on the final position Y of the second particle? It is easy to see that under such conditions the weak value of an cond 1 0 0 0 ρˆ1 OR (34) operator Aˆ (acting just on the particle 1 Hilbert space) −−−−−−−−→H , V basis 0 0 0 1 is | i | i depending on whether Y supp( y φ+ ) or Y ˆ Y ˆ ∈ h | i ∈ A W = Tr1,2 Y A ρˆ Y supp( y φ ). h i h | | i − h i Accordingh | i to Bohmian mechanics, one of these two = Tr Aˆ Tr [ Y ρˆ Y ] 1 2 h | | i possibilities is realized – and the particle 1 CDM col- h i lapses accordingly – as soon as particle 2 traverses the ˆ cond = Tr1 A ρˆ1 . (30) polarizing beam splitter (should it be inserted). Further- h i more, once the H and V components of the particle The last line is identical to Equation (22) except that 2 beam are split| apart,i | thei actual particle position Y the (one-particle) density matrixρ ˆ is replaced by the will not, according to the theory, change. So the actual Bohmian CDM from Equation (29). One thus expects measurement of Y (for the purpose of post-selection) can that the operational procedure sketched above – in which wait as long as is desired – for example, until after the ˆ A =π îj from Equation (23) – should yield the Bohmian weak measurement procedure on particle 1 has been car- conditional density matrix (and not the usual reduced ried out. Still, when the dust settles and all the data is density matrix) as the directly-measured one-particle properly binned up, OQM predicts that it is a collapsed density matrix. density matrix corresponding precisely to the Bohmian Of course, just as with the setup discussed in the previ- CDM, that should be revealed by the direct measure- ous section, this result is not very interesting or surprising ment of particle 1’s state. if the post-selection-basing measurement of particle 2’s position Y occurs prior to the weak measurement procedure on particle 1. For then, the measurement of Y V. DISCUSSION will have collapsed the two-particle state such that the ordinary RDM and the Bohmian CDM coincide. If the procedure of [4] for making a “direct measure- On the other hand, if one arranges for the measurement ment of the quantum wave function” is applied to one of Y to occur only after the procedure on particle 1 has particle from an entangled pair (and regarded as a plau- gone to completion, it is quite interesting indeed that the sible operationalist definition of the “single particle wave procedure should yield the Bohmian CDM as opposed to function” for such a particle) the result, with suitable the ordinary RDM. Consider for example the following post-selection on the other particle, is precisely the “con- setup, very much in the spirit of the one proposed in the ditional wave function” (CWF) of Bohmian mechanics – previous section. A two-photon system is prepared in the that is, the natural theoretical concept of a “single par- state ticle wave function” that Bohmian mechanics (uniquely) 0 makes possible. Similarly, the results of applying a re- ψ1 = φ ( H 1 H 2 + V 1 V 2) /√2 (31) | i | i | i | i | i | i lated procedure – for directly measuring the density ma- where φ0 , referring to the transverse spatial degree of trix associated with a single particle – to one parti- freedom| ofi particle 2, is (say) a Gaussian centered at cle from an entangled two-particle system, should yield 8 the Bohmian “conditional density matrix” (CDM) as And then, second: It would be easy to get the impres- opposed to the more standard reduced density matrix sion from the way things were put above that by insert- (RDM). These results are particularly interesting when ing (or refraining from inserting) the BS in the path of the weak measurement (that reveals the state of parti- photon 2, one can instantaneously affect the observable cle 1) is carried out prior to the strong position mea- CWF/CDM of the distant particle 1. That is, it would surement on particle 2 on which post-selection will be be easy to get the impression that one could (in principle, based. Thus, in the same way that Braverman and Si- if impractically) send a superluminal signal by running mon [11] have suggested that one can “observe the non- many copies of the experiment in parallel (so that the locality of Bohmian trajectories with entangled photons” many trials required to build up sufficient data all occur one should also be able to observe the nonlocal depen- simultaneously). But this, of course, should be impossi- dence of Bohmian single-particle states (wave functions ble (whether one believes in BM or OQM or any other and density matrices) on distant interventions such as such empirically-equivalent theory). the insertion (or not) of the BS in Figure 2. There are two points to be understood here, one rather Indeed, from the perspective suggested earlier, in obvious and one more subtle. The obvious point is that which each particle’s CWF (or CDM) is regarded as the Alice (on the right) must learn the outcome of Bob’s po- object which directly guides or pilots the Bohmian parti- sition measurement (on the left) before she can know how cle, the present work can be seen as digging yet one level to properly bin her data. And this information will have deeper beyond Braverman/Simon: instead of merely ob- to be sent to her through a “classical” (i.e., here, sub- serving how the particle trajectories change as a result luminal) communication channel. So it is already clear of some distant interventions, one may also observe how that no actual superluminal signalling will be possible. the “field” responsible for those changes itself changes. The more subtle, and more interesting, point is that A successful experimental demonstration of this effect the statistical relationship between Alice’s and Bob’s would thus in some sense reveal the non-local character measurement results is actually independent of the exact of Bohmian mechanics in an unprecedentedly fundamen- temporal sequence of the measurements. This is certainly tal way. not surprising from the point of view of relativity, given But to formulate things in these ways is to invite sev- that Alice’s and Bob’s measurements may well occur at eral possible misunderstandings, so let us clarify a couple spacelike separation. But it is somewhat surprising from of points. the point of view of Bohmian mechanics, which involves First, as Einstein [20] famously remarked: “Whether a hidden (but dynamically relevant) privileged reference you can observe a thing or not depends on the theory frame. which you use. It is the theory which decides what can Essentially for reasons of drama, we have described the be observed.” In particular, the claims above should only setup above, in the allegedly interesting cases, so that the be understood to mean that from the point of view of temporal sequence is as follows: first Bob (or his assis- Bohmian mechanics, the experimental procedures out- tant) decides whether to insert or not insert the BS into lined here can be understood as observations of (for ex- the path of particle 2; then Alice’s measurement proto- ample) the interesting dynamical behavior of single par- col on particle 1 occurs; and then finally Bob measures ticle CWFs. From the point of view of orthodox QM, on the final transverse position Y of particle 2. From the the other hand, the same procedures would evidently not point of view of Bohmian mechanics, then, we may say be regarded as genuine observations of anything. The the following. If this is the true temporal sequence – in situation is thus completely parallel to that surround- the dynamically privileged reference frame posited by the ing the reconstructed particle trajectories in [9]. The theory – then things develop causally in the way we have result of the empirical procedure coincides with certain suggested: the insertion of the BS (if and only if it is things that are posited to actually exist by one candi- inserted) causes the two-particle wave function to divide date theory, Bohmian mechanics. But another candidate in the configuration space as shown in Figure 3 and thus theory, which does not posit those things, also has no causes the CWF (or CDM) for photon 1 to collapse; the trouble accounting for the results. Indeed, insofar as the (here, subsequent) measurement protocol by Alice then two theories are empirically equivalent, the results can- simply reveals the true CWF/CDM of photon 1 at the not really count as evidence for or against either theory. time of that measurement; the final measurement/post- Nevertheless, the expected results remain somehow more selection by Bob then plays the (dynamically) purely pas- natural from the Bohmian point of view! The situation sive role of revealing, to Bob, what was already physically can perhaps be summed up like this: if one accepts, from definite, in order that the already-acquired data can be the outset, that empirically measured weak values corre- properly binned. spond to some physically real features of the system in But since the privileged frame is, for Bohmian me- question, then the predicted results would be trivial to chanics, hidden, it is entirely possible that the “true” reconcile with Bohmian mechanics but difficult to recon- temporal sequence (i.e., the temporal sequence in the cile with ordinary QM. But of course, whether one should privileged frame) is instead as follows: Alice’s measure- accept such a premise is, to put it mildly, highly contro- ment protocol on photon 1 occurs first; then comes Bob’s versial. (assistant’s) decision to insert the BS or not, followed 9 by measurement of the transverse position Y . If this is derlying privileged reference frame. The statistical pat- the “true” temporal sequence, the statistics will be un- terns in the data will – presumably – remain the same as changed, but the causal story will be somewhat differ- the “true” temporal sequence is varied between the two ent. To wit: instead of the passage (or not) of photon possibilities discussed here, even as the Bohmian causal 2 through the BS affecting the CWF/CDM of photon story changes rather dramatically. 1, now it will be the measurement protocol on photon It does, however, remain absolutely valid to say that – 1 which (at least sometimes) affects the CWF/CDM of provided one adopts the Bohmian point of view – the ex- photon 2 and thus influences where, for a given Y , it perimental setup suggested here would allow for a direct will go, should it encounter the BS. Concretely, there will empirical observation of the non-local dependence of a exist possible initial conditions for the 2-particle system single particle’s (Bohmian) CWF/CDM on distant inter- which have the following property: had the measurement ventions. It’s just that – compared to the way we initially protocol on particle 1 not been carried out, particle 2 explained things – it is rather ambiguous whether one is would definitely have gone “up” at the BS (and would observing the effect, on particle 1’s CWF/CDM, of in- hence have been found with Y > 0), but given that the serting or not inserting the BS in front of particle 2 ... or measurement protocol on particle 1 was carried out, par- instead observing the effect, on particle 2’s CWF/CDM, ticle 2 instead went “down” at the BS (and was hence of carrying out the weak measurement protocol on par- found with Y < 0). ticle 1. It thus remains appropriate to conclude that a With this second possible “true” temporal sequence, it realization of the proposed type of experiment would be is no longer really the case that the weak measurement quite interesting and would in particular help bring the protocol on particle 1 is simply revealing the structure of Bohmian CWF/CDM into the light of experimental re- particle 1’s CWF at the time of the measurement. In- ality. stead, the measurement on particle 1 may actively affect the state (in particular, the CWF or CDM) of particle 2, making the subsequent post-selection on particle 2’s position rather less benign, less passive. And this makes Acknowledgments clear in principle why, despite the presence in the theory of a dynamically privileged reference frame in which in- stantaneous action-at-a-distance occurs, one is not only W.S. acknowledges support of a grant from the John prevented from sending signals faster than light, but also Templeton Foundation. The opinions expressed in this prevented from putting any experimental limits on the publication are those of the authors and do not necessar- speed of the laboratory with respect to the presumed un- ily reflect the views of the John Templeton Foundation.

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