PHYSICAL REVIEW A 108, L060201 (2023)
LetterMisinference of interaction-free measurement from a classical system
Valeri Frumkin*

and John W. M. Bush†

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Received 23 June 2023; accepted 15 November 2023; published 12 December 2023)
Interaction-free measurement is thought to allow for quantum particles to detect objects along paths they
never traveled. As such, it represents one of the most beguiling of quantum phenomena. Here, we present aclassical analog of interaction-free measurement using the hydrodynamic pilot-wave system, in which a droplet
self-propels across a vibrating fluid surface, guided by a wave of its own making. We argue that existingrationalizations of interaction-free quantum measurement in terms of particles being guided by waveforms allow
for a classical description manifest in our hydrodynamic system, wherein the measurement is decidedly notinteraction-free.
DOI: 10.1103/PhysRevA.108.L060201

Interaction-free measurement in quantum mechanics seemingly allows one to obtain information about the quantum stateof an object without its being “disturbed” by the measurement
process. Similar ideas, such as negative result measurement,
can be traced back to Renninger [1] and Dicke [2]. However,the currently accepted notion of interaction-free measurement is due to Elitzur and Vaidman [3], who argued thatthe presence of an object in an interferometer could modify
the interference of a photon, even without any direct interaction between object and photon. The authors considereda Mach-Zehnder interferometer with a bomb placed along
one of its arms (Fig. 1). The bomb has a highly sensitive
trigger, so that any photon passing through it will detonate thebomb; however, if the bomb malfunctions, the photon passes
undisturbed. Since a functioning or malfunctioning bomb isequivalent to its being present or absent in the interferometer,
we adopt the latter description. The standard rationale for theeffect is as follows [3]. As the photon enters the first beam
splitter B1 , its wave function is split in two, with each part
traveling along one arm of the interferometer. In the absenceof a bomb, the two wave packets recombine at the second
beam splitter B2 , in which case the photon will be detected
at detector D1 with probability 1. If the bomb is present, 50%of the time the particle follows path 1 and detonates the bomb;otherwise, it follows path 2. As it does so, path 1 is blocked
by the bomb, thus, interference between the two wave packets
at B2 is prevented, and the particle is detected at D1 or D2
with equal probability. Thus, if a particle is detected at D2 ,it provides the experimenter information about the presenceof a live bomb positioned along a path that it never traveled.
After many realizations of the experiment in which the bombis present 50% of the time, the experimenter has a 25% chanceof detecting the particle at D2 and so the bomb along path 1.
Interaction-free measurement has been studied in various
contexts, both theoretically [4–13], and experimentally [9,14–22]. However, there is still no widely accepted consensus on*†valerafr@mit.edu
bush@math.mit.edu

2469-9926/2023/108(6)/L060201(6)the precise meaning of the term “interaction-free” [23]. Indeed, the extent to which an experiment such as that proposed
by Elitzur and Vaidman is interaction-free depends on theparticular interpretation of quantum mechanics one chooses
to adopt. Notably, the Copenhagen interpretation of quantum
mechanics provides no description of particles travelingalong well-defined paths, so it is at odds with the standard
description of the effect, which relies explicitly on the notionof particle trajectories. An interpretation that does allow one todescribe real particle paths is Bohmian mechanics, also known
as the de Broglie–Bohm pilot-wave theory [24–26]. According to Bohmian mechanics, particles move under the influenceof the wave function, with a velocity equal to the quantum

FIG. 1. A schematic of the Elitzur-Vaidman bomb experiment. A
particle emitted from a source S passes through a beam splitter B1 , atwhich point its associated wave (a wave function [3] or a pilot wave[4]) is split in two. The wave is then recombined at a beam splitter
B2 , and the particle continues toward the detectors. In the absence ofa bomb, the particle will be detected at D1 100% of the time, while
in its presence, the particle will be detected at D2 25% of the time.
A detection event at D2 indicates the presence of a live bomb along
a path the particle never took. Red and blue dashed lines indicate
possible paths taken by a particle emitted from S in the absence andpresence of the bomb, respectively.

L060201-1©2023 American Physical Society

VALERI FRUMKIN AND JOHN W. M. BUSH

PHYSICAL REVIEW A 108, L060201 (2023)

FIG. 2. Surreal trajectories in quantum mechanics and pilot-wave hydrodynamics [39]: (a) A variant of the interferometer setup consideredby Englert, Scully, Süssman, and Walther (ESSW) [40]. An incoming wave packet is split by a beam splitter B and reflected by the mirrors
M1 and M2 . The wave packets interfere in the region I and then move towards the detectors D1 and D2 . The blue path represents the particle
trajectory anticipated by ESSW, while the red path is that predicted by Bohmian mechanics, the so-called “surreal trajectory.” The red dot
represents the one-bit, which-way detector employed in the weak-measurement experiments of Mahler et al. [41]. (b) In the associatedhydrodynamic analog [39], droplets reflect off submerged barriers, indicated in white. When the setup is symmetric, the droplet enters theright or left channel with equal probability and is then reflected off the associated barrier. Its subsequent deflection away from the systemcenterline results in a real surreal trajectory. Twenty such trajectories are shown. (c) When one of the barriers is removed, the symmetry of thesystem is broken. The walking droplet is then reflected away from the remaining barrier, resulting in the trajectory that one might expect.

velocity of probability in the standard quantum formalism.
Both the particle and its pilot wave are assumed to be real
physical objects. The Bohmian version of the Elitzur-Vaidman
experiment was first presented by Hardy [4], who positedthe following physical picture. At the first beam splitter, theparticle takes one of the two paths in the interferometer, while
its pilot wave is split in two. One part of the pilot wave carries
the particle along its path, while the other, so-called empty
wave, proceeds along the other path without the particle. Bothcomponents of the pilot wave recombine at B2 , producing thestandard outcome. If a bomb is present along the path thatthe particle did not take, it blocks the empty wave, preventing
it from interfering with the particle’s pilot wave at B2 , thus
enabling the particle to arrive at detector D2 with nonzero
probability. Note that according to both this physical picture
and that proposed by Elitzur and Vaidman [3], the experimentcan be considered “interaction-free” only in the sense thatthe particle does not interact with the bomb: The waveform
propagating along path 1 interacts with and is altered by thebomb without detonating it. We proceed by demonstratingthat such detection of an object by a particle that interacts
with the particle only through its associated waveform may
likewise be achieved in a classical system. We then argue thatthe term “interaction-free” is misleading, both for the classicalsystem under consideration and its quantum counterpart.
A classical pilot-wave system was discovered by Couder
and Fort in 2005 [27], and consists of a millimetric droplet
bouncing and self-propelling across the surface of a vibrating
liquid bath through an interaction with its own wave. This system has provided the basis for a growing list of hydrodynamic
quantum analogs [28,29], including single- and double-slit
diffraction and single-particle interference [30], quantizationof orbital states [31,32], the emergence of wavelike statistics in corrals [33,34], Friedel oscillations [35], superradiance[36], and hydrodynamic spin lattices [37]. As the bouncing
droplets are the sources of their own pilot-wave field, the accompanying physical picture is more closely aligned with therelativistic double-solution pilot-wave theory proposed in the
1920’s by de Broglie [38] than with the de Broglie–Bohm theory [29]. Nevertheless, the hydrodynamic pilot-wave system
has captured certain features predicted by Bohmian mechanics, including surreal particle trajectories [39], unexpectedtrajectories predicted by Bohmian mechanics to arise in aninterferometer owing to the interference of empty waves withthe particle’s pilot wave (see Fig. 2). While surreal trajectories were thought to be unphysical [40], Mahler et al. [41]reported that experimental trajectories inferred from weakmeasurement are consistent with the predictions of Bohmianmechanics. As noted by Vaidman [42], the phenomenon ofsurreal trajectories is related to that of interaction-free measurement, in that both rely on interference effects between
a particle-carrying pilot wave and an empty wave. Here, webuild upon the experimental setup used to realize surreal trajectories in pilot-wave hydrodynamics in order to demonstratethat when a particle is accompanied by a pilot wave, an analog
of interaction-free measurement can be realized in a classicalsystem.
Our experimental system consists of a circular bath filledwith a 7.0 ± 0.3 mm deep layer of silicon oil with surface
tension σ = 0.0209 N/m, viscosity 20 cSt, and density ρ =0.965×10−3 kg/m3 . The height of the submerged topography is 6.0 ± 0.2 mm, so the oil depth in the overlaying
shallow regions is approximately 1 mm. The bath is vibrated vertically using an electromagnetic shaker with forcing
F (t ) = γ cos(2π f t ), frequency f = 80 Hz. and peak vibrational acceleration γ = 3.95g, where g is the gravitational
acceleration. The Faraday threshold, at which the free surface
destabilizes to a pattern of subharmonic Faraday waves, wasγF = 4.05g. The electromagnetic shaker was connected to thebath by a steel rod coupled with a linear air bearing, thus
insuring spatially uniform vibration [43]. Two accelerometers placed on opposite sides of the bath allowed for controlof the vibrational acceleration amplitude to within ±0.002g.
We visualized the droplet trajectories and guiding pilot wave

L060201-2

MISINFERENCE OF INTERACTION-FREE MEASUREMENT …

PHYSICAL REVIEW A 108, L060201 (2023)

FIG. 3. Interaction-free measurement in pilot-wave hydrodynamics: (a) A schematic illustration of the topography used in the experiment.
The upper half of the left barrier (orange) plays the role of a “bomb.” (b) In a symmetric setup, the droplet enters the right or left channel
with equal probability. If the droplet goes to the left channel, it detonates the “bomb,” and if it goes to the right, it is deflected away from thesystem centerline, resulting in a “surreal” trajectory [39]. Thus, the droplet will always be detected on the right side of the setup. Twenty suchtrajectories are shown. (c) If the bomb is removed, the symmetry of the system is broken, the droplet’s pilot wave does not interact with thebomb, and surreal trajectories are suppressed. Thus, the droplet will always be detected on the left side of the setup. If the bomb is present 50%of the time, there is 25% chance of the droplet being detected on the right, and so the bomb on the left. The scale bar represents the Faraday
wavelength λF = 4.84 mm.

using a semireflective mirror angled at 45 ◦ relative to thebath, and a charge-coupled-device (CCD) camera mounted
directly above the setup. We used a diffuse-light lamp, positioned horizontally in front of the mirror, to illuminate thebath, yielding images with bright regions corresponding tohorizontal regions of the bath surface, specifically extrema or
saddle points.
The topographical configuration used in our experiments is
depicted in Fig. 3(a). A walking droplet is launched towards
a submerged rhombus that acts as a beam splitter, forcing thedroplet along one of two adjoining channels with equal probability. We consider the upper part of the left channel to be our“bomb,” an object whose presence we seek to detect without
the droplet interacting with it directly. While the droplet islocalized at all times, its pilot wave is spatially extended
and so interacts with the geometry of its environment. Thisdelocalization of the pilot wave accounts for the different drop
behaviors in the different geometric configurations. Whenthe “bomb” is present in the system [Fig. 3(b)], the setup is
identical to that used to realize surreal trajectories [39]. If
the droplet takes the left channel, it hits the bomb and so
effectively detonates it. At this point, the droplet trajectory
is considered to be terminated. If, however, the droplet takes
the right channel, its pilot wave interacts with its environmentso as to deflect the droplet toward the right, along a surreal
trajectory. Thus, if the bomb is present, the droplet will either
detonate it, or be detected on the right side of the setup(see Supplemental Video 1 [44]). The droplet follows a surreal
trajectory followed only when the boundary geometry of the
setup is right-left symmetric [39]. When the bomb is absent[Fig. 3(c)], this symmetry is broken, and the surreal trajectoryis precluded. If the droplet takes the right path, it is reflected,
then continues in a straight line towards the left. If the droplet
takes the left path, there is nothing there to reflect it, so it
continues along a straight path to the left. Thus in the absenceof the bomb, the droplet will always be detected on the left
side of the setup (see Supplemental Video 2 [44]). Aftermany realizations of the experiment in which the bomb waspresent 50% of the time, the experimenter has a 25% chanceof detecting the droplet on the right side. Such a detection
indicates that the bomb was present in the left channel, eventhough the droplet took the right path and so never interacted
directly with it.
Discussion. Our experiment demonstrates that if particles are accompanied by guiding waveforms, the statisticalbehavior that has led to the inference of interaction-freemeasurement in quantum mechanics may be achieved in aclassical system. Notably, not all interpretations of quantum
mechanics need appeal to interaction-free measurement. In
their original paper, Elitzur and Vaidman write that according
to the many-worlds interpretation [45], “since all worlds takeplace in the physical universe, we cannot say that nothing
has touched the object. We get information about the object
without touching it in one world, but we pay the price ofinteracting with the object in the other world” [3]. In Bohmianmechanics, while the particle does not interact with the bomb,
its pilot wave is blocked by it, so the measurement is notstrictly interaction-free. In the context of the Copenhagen
interpretation of quantum mechanics, before measurement,
one can only talk about the particle’s wave function, which
is a nonlocal object that is determined by the configuration of
the interferometer as a whole. Thus, in the absence of a bomb,
the wave function can only collapse at detector D1 , while inits presence, the wave function has a nonzero probability ofcollapsing at D2 . According to this interpretation, the wave
function is considered to be a mathematical construction forcomputing probabilities rather than a physical field; thus, one
could argue that the effect is indeed interaction-free. However, a similar argument can be applied quite generally in
the Copenhagen picture, according to which the measurement
process is marked by the collapse of the wave function, andthe probability of a particular outcome is prescribed by theconfiguration of the system as a whole. For example, according to the Copenhagen interpretation, any measurement of

L060201-3

VALERI FRUMKIN AND JOHN W. M. BUSH

PHYSICAL REVIEW A 108, L060201 (2023)

one half of an entangled pair will yield information about theother half without interacting with it. Thus, all such measurement outcomes may be considered interaction-free, rendering
the notion superfluous. If, however, one thinks of particles
in the Elitzur-Vaidman experiment as physically localized
wave packets traveling in an interferometer, a more accurate
description of the effect would be “measurement-free interaction,” since the presence of the bomb is presumed not tocollapse the wave function.
It is important to note the differences between interactionfree measurement in quantum mechanics, and the analogous
statistical inference made in the classical system presented
here. First, the quantum wave function is a nonlocal object
that is determined by the entire configuration of the experimental setup. As a result, interaction-free measurement canbe realized (at least in theory) with an interferometer thathas arms of unequal length, provided the lengths of the twoarms (B1 M1 B2 and B1 M2 B2 in Fig. 1) differ by nλ, and that
this length difference is smaller than the coherence length of
the interferometer. In our system, the pilot wave is affectedby the global geometry, just as a standing field of Faraday
waves is affected by boundaries in confined geometries [46].
The form of the pilot wave is affected by the totality of theboundary geometry only if its spatial extent is sufficiently
large. Consequently, changing the configuration of the setup(for example, by either increasing its size or decreasing thesystem memory) will serve to suppress the surreal trajectories and so nullify the effect. Second, in the Bohmianversion of the Elitzur-Vaidman experiment, the bomb blocks
the empty wave, thus eliminating interference. In our setup,the situation is reversed: The presence of the “bomb” promotes interference of the pilot wave, resulting in a surreal
trajectory. This configurational difference does not alter the
key statistical behavior common to the two systems. Thirdand most importantly, whatever the case may be in its quantum counterpart, the statistical inference made in our systemshould in no way be taken as evidence of interaction-freemeasurement.
All current attempts to rationalize interaction-free measurement rely on a physical picture where localized waveparticle objects (e.g., traveling wave packets, or Bohmianparticles with their pilot and empty waves) travel along thearms of the interferometer. Our study demonstrates that such
localized wave-particle descriptions of interaction-free measurement can also be achieved in a classical system. We notethat the validity of such descriptions can be tested directly
along the lines first suggested by Renninger [47], by realizing
the Elitzur-Vaidman bomb experiment while increasing thelength of one of the interferometer arms by an amount nλ,where n is an integer and λ the wavelength of the photon.
If one represents the photon as a wave packet that is split in
two at B1 , then for sufficiently large n, the two wave packets

will not recombine at B2 since they travel paths of different
length at the same speed. Likewise, in the Bohmian picturesuggested by Hardy [4], the interference of the particle’s pilot
wave with an empty wave at B2 would be eliminated. Thus, thephoton would have a nonzero probability of being detectedat D2 even in the absence of a bomb. If the effect does not
persist for arbitrary large n, it would validate these localizedwave-particle descriptions and, when considered in light ofour results, suggest that interaction-free measurement is amisnomer. If the effect does persist for arbitrary large n (as one
might expect), this experiment would invalidate the localizedwave-packet description of the problem. One would then be
obliged to adopt a nonlocal view, for example, that offered
up by the Copenhagen interpretation or by the second-orderformulation of Bohmian mechanics, according to which particles are guided by a quantum potential prescribed by thewave function [24–26]. Since the wave function is a nonlocal
object, so too is the quantum potential; thus, the path taken bya Bohmian particle is determined by the entire configurationof the interferometer.
As was the case in the hydrodynamic analog of surreal
trajectories, the hydrodynamic pilot-wave system captures
a feature of quantum mechanics correctly predicted byBohmian mechanics, without having to invoke the quantum
potential. The role of the quantum potential in guiding theparticle in Bohmian mechanics is played by the droplet’s
guiding wave, which is generated by the droplet but interacts
with the system geometry in such a way as to produce thesurreal trajectories. In neither Bohmian mechanics nor oursystem is the measurement really interaction-free. In Bohmianmechanics, the quantum potential changes form according to
the presence or absence of the bomb; in our system, the local,
particle-generated pilot wave does likewise.
Finally, given the numerous applications proposed forinteraction-free measurement [10,48–50], it is noteworthy thatnone of these requires the nonlocal features of the wave function for their implementation, only its interference properties.
It is thus worth considering whether some of these applications might be implemented in the hydrodynamic pilot-wave
system. A classical macroscopic implementation of theseideas could potentially allow one to bypass the difficulty ofdecoherence inherent in all quantum systems, and so conceivably serve as a physical platform for quantum-inspiredclassical computing.

The authors gratefully acknowledge the financial supportof the National Science Foundation Grant No. CMMI2154151.
The authors declare no competing interests.

[1] M. Renninger, Messungen ohne Störung des Meßobjekts,
Z. Phys. 158, 417 (1960).
[2] R. H. Dicke, Interaction-free quantum measurements: A paradox?, Am. J. Phys. 49, 925 (1981).

[3] A. C. Elitzur and L. Vaidman, Quantum mechanical interactionfree measurements, Found. Phys. 23, 987 (1993).
[4] L. Hardy, On the existence of empty waves in quantum theory,
Phys. Lett. A 167, 11 (1992).

All data are available in the main text or the Supplemental
Material [44].

L060201-4

MISINFERENCE OF INTERACTION-FREE MEASUREMENT …

PHYSICAL REVIEW A 108, L060201 (2023)[5] G. Mitchison and S. Massar, Absorption-free discrimination between semitransparent objects, Phys. Rev. A 63, 032105 (2001).
[6] L. C. Ryff, Null-result detection and Einstein-Podolsky-Rosen
correlations, Found. Phys. 44, 58 (2014).
[7] X. Zhou, Z.-W. Zhou, M. J. Feldman, and G.-C. Guo, Nondistortion quantum interrogation using Einstein-Podolsky-Rosen
entangled photons, Phys. Rev. A 64, 044103 (2001).
[8] S. H. Simon and P. M. Platzman, Fundamental limit on“interaction-free” measurements, Phys. Rev. A 61, 052103(2000).
[9] A. C. Elitzur and S. Dolev, Nonlocal effects of partial measurements and quantum erasure, Phys. Rev. A 63, 062109(2001).
[10] A. Karlsson, G. Björk, and E. Forsberg, “Interaction” (energy
exchange) free and quantum nondemolition measurements,
Phys. Rev. Lett. 80, 1198 (1998).
[11] P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A.
Kasevich, Interaction-free measurement, Phys. Rev. Lett. 74,
4763 (1995).
[12] S. Thomas, C. Kohstall, P. Kruit, and P. Hommelhoff, Semitransparency in interaction-free measurements, Phys. Rev. A 90,
053840 (2014).
[13] E. H. du Marchie van Voorthuysen, Quantum-mechanical indirect measurements, Found. Phys. Lett. 10, 563 (1997).
[14] P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M.
Kasevich, Experimental realization of interaction-free measurements, Ann. N.Y. Acad. Sci. 755, 383 (1995).
[15] E. H. du Marchie van Voorthuysen, Realization of aninteraction-free measurement of the presence of an object in alight beam, Am. J. Phys. 64, 1504 (1996).
[16] M. Hafner and J. Summhammer, Experiment on interactionfree measurement in neutron interferometry, Phys. Lett. A 235,
563 (1997).
[17] T. Tsegaye, E. Goobar, A. Karlsson, G. Björk, M. Y. Loh, andK. H. Lim, Efficient interaction-free measurements in a highfinesse interferometer, Phys. Rev. A 57, 3987 (1998).
[18] A. G. White, J. R. Mitchell, O. Nairz, and P. G. Kwiat,“Interaction-free” imaging, Phys. Rev. A 58, 605 (1998).
[19] P. G. Kwiat, A. G. White, J. R. Mitchell, O. Nairz, G. Weihs, H.
Weinfurter, and A. Zeilinger, High-efficiency quantum interrogation measurements via the quantum Zeno effect, Phys. Rev.
Lett. 83, 4725 (1999).
[20] S. Inoue and G. Björk, Experimental demonstration of
exposure-free imaging and contrast amplification, J. Opt. B:
Quantum Semiclass. Opt. 2, 338 (2000).
[21] A. E. Turner, C. W. Johnson, P. Kruit, and B. J. McMorran,Interaction-free measurement with electrons, Phys. Rev. Lett.
127, 110401 (2021).
[22] J. Peise, B. Lücke, L. Pezzé, F. Deuretzbacher, W. Ertmer, J.
Arlt, A. Smerzi, L. Santos, and C. Klempt, Interaction-free measurements by quantum Zeno stabilization of ultracold atoms,
Nat. Commun. 6, 6811 (2015).
[23] L. Vaidman, The meaning of the interaction-free measurements,
Found. Phys. 33, 491 (2003).
[24] D. Bohm and B. J. Hiley, The Undivided Universe: An Ontological Interpretation of Quantum Theory (Routledge, London,1993).
[25] P. R. Holland, The Quantum Theory of Motion: An Account of thede Broglie–Bohm Causal Interpretation of Quantum Mechanics(Cambridge University Press, Cambridge, UK, 1993).

[26] D. Dürr and S. Teufel, Bohmian Mechanics: The Physics andMathematics of Quantum Theory (Springer, Berlin, 2009).
[27] Y. Couder, S. Protière, E. Fort, and A. Boudaoud, Walking andorbiting droplets, Nature (London) 437, 208 (2005).
[28] J. W. M. Bush, Pilot-wave hydrodynamics, Annu. Rev. Fluid
Mech. 47, 269 (2015).
[29] J. W. M. Bush and A. U. Oza, Hydrodynamic quantum analogs,
Rep. Prog. Phys. 84, 017001 (2021).
[30] Y. Couder and E. Fort, Single-particle diffraction and interference at a macroscopic scale, Phys. Rev. Lett. 97, 154101(2006).
[31] E. Fort, A. Eddi, A. Boudaoud, J. Moukhtar, and Y.
Couder, Path-memory induced quantization of classical orbits,
Proc. Natl. Acad. Sci. USA 107, 17515 (2010).
[32] S. Perrard, M. Labousse, M. Miskin, E. Fort, and Y. Couder,Self-organization into quantized eigenstates of a classical wavedriven particle, Nat. Commun. 5, 3219 (2014).
[33] D. M. Harris, J. Moukhtar, E. Fort, Y. Couder, and J. W. M.
Bush, Wavelike statistics from pilot-wave dynamics in a circularcorral, Phys. Rev. E 88, 011001(R) (2013).
[34] P. J. Sáenz, T. Cristea-Platon, and J. W. M. Bush, Statistical projection effects in a hydrodynamic pilot-wave system,
Nat. Phys. 14, 315 (2018).
[35] P. J. Sáenz, T. Cristea-Platon, and J. W. M. Bush, A hydrodynamic analog of Friedel oscillations, Sci. Adv. 6, eaay9234(2020).
[36] V. Frumkin, J. W. M. Bush, and K. Papatryfonos, Superradiant droplet emission from parametrically excited cavities,
Phys. Rev. Lett. 130, 064002 (2023).
[37] P. J. Sáenz, G. Pucci, S. E. Turton, A. Goujon, R. R. Rosales,
J. Dunkel, and J. W. M. Bush, Emergent order in hydrodynamic
spin lattices, Nature (London) 596, 58 (2021).
[38] L. de Broglie, Interpretation of quantum mechanics by the double solution theory, Ann. Fond. Louis de Broglie 12, 399 (1987).
[39] V. Frumkin, D. Darrow, J. W. M. Bush, and W. Struyve, Realsurreal trajectories in pilot-wave hydrodynamics, Phys. Rev. A
106, L010203 (2022).
[40] B.-G. Englert, M. O. Scully, G. Süssmann, and H. Walther,
Reply to Comment on “Surrealistic Bohm trajectories”,
Z. Naturforsch. A 48, 1263 (1993).
[41] D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J.
Resch, H. M. Wiseman, and A. Steinberg, Experimental nonlocal and surreal Bohmian trajectories, Sci. Adv. 2, e1501466(2016).
[42] L. Vaidman, The reality in Bohmian quantum mechanics or
can you kill with an empty wave bullet?, Found. Phys. 35, 299(2005).
[43] D. M. Harris and J. W. M. Bush, Generating uniaxial vibration with an electrodynamic shaker and external air bearing,
J. Sound Vib. 334, 255 (2015).
[44] See Supplemental Material at http://link.aps.org/supplemental/
10.1103/PhysRevA.108.L060201 for Supplemental Video 1
shows the behavior of the walking droplet in our “hydrodynamic interferometer”, in the presence of the bomb, while
Supplemental Video 2 shows its behavior in the absence of thebomb.
[45] H. Everett, “Relative state” formulation of quantum mechanics,
Rev. Mod. Phys. 29, 454 (1957).
[46] A. Kudrolli, M. C. Abraham, and J. P. Gollub, Scarred patterns
in surface waves, Phys. Rev. E 63, 026208 (2001).

L060201-5

VALERI FRUMKIN AND JOHN W. M. BUSH

PHYSICAL REVIEW A 108, L060201 (2023)[47] M. Renninger, Zum wellen-korpuskel-dualismus, Z. Phys. 136,
251 (1953).
[48] G.-C. Guo and B.-S. Shi, Quantum cryptography based oninteraction-free measurement, Phys. Lett. A 256, 109 (1999).

[49] M. Czachor, Quantum cryptography with polarizing interferometers, Phys. Lett. A 257, 107 (1999).
[50] G. Mitchison and R. Jozsa, Counterfactual computation,
Proc. R. Soc. London Ser. A 457, 1175 (2001).

L060201-6