'Quantum Cheshire Cat' as Simple Quantum Interference

In a recent work, Aharonov et al. suggested that a photon could be separated from its polarization in an experiment involving pre- and post-selection [New J. Phys 15, 113015 (2013)]. They named the effect 'quantum Cheshire Cat', in a reference to the cat that is separated from its grin in the novel Alice's Adventures in Wonderland. Following these ideas, Denkmayr et al. performed a neutron interferometric experiment and interpreted the results suggesting that neutrons were separated from their spin [Nat. Commun. 5, 4492 (2014)]. Here we show that these results can be interpreted as simple quantum interference, with no separation between the quantum particle and its internal degree of freedom. We thus hope to clarify the phenomenon with this work, by removing these apparent paradoxes.

In a recent work, Aharonov et al. suggested that a photon could be separated from its polarization in an experiment involving pre-and post-selection [New J. Phys 15, 113015 (2013)]. They named the effect 'quantum Cheshire Cat', in a reference to the cat that is separated from its grin in the novel Alice's Adventures in Wonderland. Following these ideas, Denkmayr et al. performed a neutron interferometric experiment and interpreted the results suggesting that neutrons were separated from their spin [Nat. Commun. 5, 4492 (2014)]. Here we show that these results can be interpreted as simple quantum interference, with no separation between the quantum particle and its internal degree of freedom. We thus hope to clarify the phenomenon by ruling out these apparent paradoxes. The concept of a quantum weak value, introduced in 1988 by Aharonov, Albert, and Vaidman [1], has allowed the development of novel and important experimental techniques to study quantum systems [2]. The amplification of small signals beyond technical noise [3,4], the direct determination of quantum states [5] and geometric phases [6], and the characterization of the nonclassical behavior of quantum systems [7] are a few examples of its usefulness. The original paper and some follow-ups also suggested that weak measurements could lead to a new interpretation of quantum phenomena. These somewhat controversial ideas became the target of many discussions as can be seen in Refs. [8][9][10][11].
This controversy has been recently revived by a new set of proposals and experiments suggesting even more radical ways of reinterpreting quantum mechanics. For instance, Aharonov et al. argued that a particular weak measurement setup for photons allowed one to state that "in the curious way of quantum mechanics, photon polarization may exist where there is no photon at all" [12]. The idea was reinforced both in a news & views article of Nature Physics [13] and in an experiment performed with neutrons [14]. The first discusses the results of Ref. [12] concluding that "polarization could be effectively isolated from the photons carrying it", while the second implements an equivalent interferometer to the one proposed in Ref. [12] and argues that "The experimental results suggest that the system behaves as if the neutrons go through one beam path, while their magnetic moment travels along the other" [14]. The phenomenon was nicknamed "quantum Cheshire Cat", in a reference to the cat that is separated from its grin in the novel Alice's Adventures in Wonderland, by Lewis Carroll.
Most of the controversy in trying to extract a new interpretation out of weak measurements lies in the attempt to attach physical reality to the mathematically defined weak values. The calculation of these values involves pre-and post-selected states which, in standard quantum mechanics, can be mathematically correlated but bare no a priori physical correlation in any particular realization of an experiment. The other essential piece to the method is to introduce continuous degrees of freedom as a quantum meter which, in the end, is not considered in the construction of the argument for the new interpretation. In the present work we show that by taking these degrees of freedom into account, both the theoretical predictions and experimental results that motivated the somewhat unusual "Cheshire Cat" interpretation can be explained as simple quantum interference where no detachment between the photon and its polarization or between the neutron and its magnetic moment is actually required. We begin by quickly revisiting the proposal of Ref. [12], then we proceed to explain it with standard quantum mechanics interference and we finally show how a similar approach also explains the experiment described in [14].
In Ref. [12], the authors base their proposal in the interferometer shown in Fig. 1 which is designed in order to prepare the photon in state |Ψ and post-select it in state |Φ , both specified below: where |I and |II correspond to the photon being in each arm of the interferometer shown in Fig. 1, and |H and |V represent its horizontal and vertical polarization components. The left and right circular polarization states, which are the eigenstates of the intrinsic angular momentum operator for the photon, can be written as |± = 1 Interferometer setup from Ref. [12]. The photon reaches a beam splitter (BS1) from side II with polarization H, right after which we have the state |Ψ from Eq. (1), called pre-selection state. After reflection from mirrors on both arms I and II, we want to make the post-selection of the state |Φ from Eq. (2) on detector D1. A half-wave plate (HWP) interchanges H and V polarizations on arm I and a phase shifter (PS) adds a specific phase right after it, so that, after a second beam splitter (BS2) and a polarizing beam splitter (PBS) that transmits only horizontal polarization, we assure that D1 will always click for |Φ and never click for any state orthogonal to it.
are placed and adjusted in a way that a photon in the state |Φ right before HWP will always cause a click on the detector D 1 ; if it is in any state orthogonal to |Φ , it necessarily clicks elsewhere. The adjustments of the devices are described in Ref. [12]. After being prepared in the state |Ψ , the photon interacts with devices positioned in both interferometer paths. These devices act as probes, performing either projective measurements (also called "strong") or weak measurements. Although in Ref. [12] the authors considered them as quantum devices, the description of their quantum state is not included in the paper. We will explicitly represent their quantum states here, since this is the essential piece for describing the phenomenon as quantum interference. A polarization detector is inserted in arm I. Its quantum state is |P 1 in the absence of photons, or changes to |P + 1 (|P − 1 ) if a photon of polarization |+ (|− ) propagates through that arm. Meanwhile, a quantum device to measure the presence of photons is placed in arm II, i.e. a device whose quantum state |P 2 changes to |P + 2 if a photon propagates through that arm, or stays in |P 2 in the absence of photons. In the proposal of Ref. [12] the probe of arm I is a combination of birefringent elements that produces a positive (negative) horizontal displacement for a photon with polarization |+ (|− ). The pointer state |P 1 is thus associated to the center of the beam profile in the horizontal direction. The probe in arm II is a glass plate that displaces the photon beam up. The pointer state |P 2 is thus associated to the center of the beam profile in the vertical direction. If the beam displacements produced by the measurement devices are greater than the beam diameter, then we have a projective measurement where we can associate the vertical displacement with the photon path and the horizontal displacement with the polarization of the photon propagating through arm I. If the beam displacements are much smaller than the beam diameters, then we are in the weak measurements regime. In this case we have, for instance, After the interaction with the measurement devices just described, the composite state of the photon and the pointers is . From now on we will ignore the normalization of states, since it plays no role in pointing out the measurement result. A photon detected in D 1 means that the state |Ψ has been projected on |Φ from Eq. (2), leaving the pointers in the state Note that in the above state there is entanglement between the pointers [15]. If the devices make projective measurements, there are three possibilities for each photon. In the first one, the photon has an up displacement greater than the beam diameter. Then there is no contradiction on assuming that the photon had propagated through arm II. In the second (third) one, the photon has a positive (negative) horizontal displacement greater than the beam diameter. Then there is no contradiction on assuming that the photon had propagated through arm I and had polarization |+ (|− ). But the authors of Refs. [12,14] extend this interpretation to the case where the devices interact weakly with the photon, producing vertical and horizontal displacements much smaller than the beam diameter. They consider that if the average vertical displacement of a set of photons is the same of the wavefunction of one photon eventually propagating through arm II, then this indicates that the photons had propagated through this arm. In the same way, they also assume that if the average horizontal displacement of a set of photons is the same of the wavefunction of one photon eventually propagating through arm I with polarization |+ , then there is a |+ polarization in arm I. This is the origin of the paradox in concluding that "the photon is in the left arm (...) while the angular momentum is in the right arm" [12]. As we show in the following, there is no such paradox. As in Ref. [12], let us describe the pointers by the transverse beam profiles of the photon in the fashion of a Bialynicki-Birula-Sipe photon wave function [16][17][18][19] in the paraxial regime. In this regime, the beam propagation is highly directional, such that its state can be written as the product of a function in the xy plane (with z-dependent parameters) and a function that describes its evolution while propagating in the z direction [20].
The beam polarization can be treated as an independent parameter in this regime. We may consider that the transverse properties of the beam do not vary much with the propagation through the interferometer, such that the wave function can be written as a function of x and y only. In the proposal of Ref. [12], |P 1 corresponds to a wave function f (x) and |P 2 corresponds to g(y), such that |P + 1 , |P − 1 and |P + 2 correspond to the same functions displaced by δ x , −δ x and δ y , respectively. Hence, by defining F (x, y) ≡ f (x)g(y), the state (3) is described by the wave function F 1 (x, y) = 2F (x, y − δ y ) + F (x − δ x , y) − F (x + δ x , y). If the detector D 1 is sensitive to the photon position, after many runs it registers the distribution |F 1 (x, y)| 2 . The pattern showed will depend on how orthogonal these three displaced functions above are, and how their overlap creates interference. In a strong (projective) measurement regime, the interaction displaces each term above the beam diameter, such that their overlap is negligible and there is no interference. Each part of the function is then identifiable on the screen, offering trustworthy information about the measured quantities. The weak measurement will happen when the displaced functions are almost completely overlapping, so that they interfere and may even look like the same function displaced by some other amount. As we discussed before, the reason of the paradoxes in Refs. [12,14] is to consider that the resultant displacement may be read as a measurement in the ordinary sense.
Using the approximations F (x ± δ x , y) ≈ F (x, y) ± δ x ∂F (x,y) ∂x and F (x, y − δ y ) ≈ F (x, y) − δ y ∂F (x,y) ∂y , we end up with F 1 (x, y) ≈ F (x − δ x , y − δ y ). According to the arguments of Refs. [12,14], this result is compatible with the situation where the photon is measured in the left arm of the interferometer (the beam was displaced up by δ y ) and, at the same time, there is positive angular momentum on the right arm of the interferometer (the beam was displaced sideways by δ x ). However, as pointed out in our calculations, the probability of finding the photons at this particular range of positions can be interpreted as simple interference. Note that any quantum continuous variable could have been used as probe. The photon wavefunction is a particularly effective example because it fulfills both the superposition principle and the approximations discussed above. Also note that the same holds true for the Schrödinger wavefunction. Actually, our treatment is suitable to describe even the light intensity distribution on the detector if classical electromagnetic waves are sent through the interferometer.
To make this issue clearer, in Fig. 2(a) we plot 2F (x, y − δ y ) = 2f (x)f (y − δ y ), which is the component of the photon wavefunction that comes from the arm II, with f (x) being a Gaussian function with width W centered on zero. In Fig. 2(b) we plot F (x − δ x , y) − F (x + δ x , y), which is the component of the photon wave- function that comes from arm I. We can see that for δ x = δ y W the major contribution to the pointers' wavefunction F 1 (x, y) comes from 2F (x, y − δ y ). But the term F (x − δ x , y) − F (x + δ x , y) interferes destructively with 2F (x, y − δ y ) for negative x and constructively for positive x, resulting in an overall positive displacement in the horizontal direction for the wavefunction even if this term is small, as can be seen in Fig. 2(c). A similar argument was used in Ref. [21] to present a classical explanation of the experimental results of Ref. [22].
The experimental realization by Denkmayr et al. [14] was similar to the theoretical proposal of Ref. [12] and is depicted in Fig. 3. After going through the first beam splitter and the spin rotators, the neutron is prepared in the state |Ψ n = 1 front of D 1 , but not in front of D 2 , such that the detection of a neutron in D 1 or D 2 is associated, respectively, to the projectorsΠ 1 = (|I + e iχ |II )( I| + e −iχ II|) ⊗ |− −| andΠ 2 = (|I − e iχ |II )( I| − e −iχ II|) ⊗Î s , where χ is a controllable phase andÎ s is the identity operator for the spin degree of freedom.
In the first part of the experiments, Denkmayr et al. place absorbers in each path and see if the detection counts in D 1 reduce. Since D 1 only detects neutrons with spin state |− and |Ψ n is orthogonal to |I |− , the detections should not vary when an absorber is placed in path I and should decrease when the absorber is placed in path II. This behavior is in fact observed [14].
In the second part of the experiments, magnetic fields are applied in the interferometer paths to produce a small rotation of the neutron spin. The phase χ of the projec-torsΠ 1 andΠ 2 is varied and the dependence of detection counts in D 1 and D 2 with χ is tested. When no magnetic field is applied, there can be no interference since in |Ψ n the spin states in each path are orthogonal. Therefore, the counts on both detectors should not depend on χ. When a magnetic field is applied in path I changing the neutron spin state from |+ to a|+ + b|− with |a| 2 + |b| 2 = 1, the |− component of the wavefunction of this path can interfere with the wavefunction of path II, such that the counts in both detectors should depend on χ. When a magnetic field is applied in path II changing the neutron spin state from |− to c|− + d|+ with |c| 2 + |d| 2 = 1, the |+ component of the wavefunction of this path can interfere with the wavefunction of path I, such that the counts in detector D 2 should depend on χ. But since the detector D 1 selects only the |− component of spin, the counts in this detector should not depend on χ. All these predictions are confirmed by the experiments [14].
The behaviors described above led the authors to say that "the experimental results suggest that the system behaves as if the neutrons go through one beam path, while their magnetic moment travels along the other" [14]. But as we have seen here, the results can be explained as simple quantum interference, with no separation between the neutron and its spin.
In our opinion, the paradoxical conclusions that a photon may be separated from its polarization [12,13] or that a neutron can be separated from its spin [14] presented as the "quantum Cheshire Cat" effect are one more apparent paradox that arises whenever we attribute physical reality to the superposition of quantum states. Naive interpretations of delayed choice experiments [23,24] or quantum erasers [25][26][27][28] lead to similar apparent paradoxes. As we have shown here, the predictions of Ref. [12] and the experiments of Ref. [14] can be understood as simple quantum interference, with no separation between the quantum particles and their internal degrees of freedom, and we hope our results provide a better understanding of the phenomenon reinforcing that no interpretation weirder than standard quantum mechanics is required.
We would like to ackowledge Cristhiano Duarte for useful comments on the manuscript. This work was supported by the Brazilian agencies CNPq, CAPES, FAPEMIG, and PRPq/UFMG.