Can Two Quantum Cheshire Cats Exchange Grins?

TL;DR

This paper explores the transient and exchangeable nature of quantum properties in a thought experiment involving two photons, challenging the notion of fixed properties in quantum systems.

Contribution

It introduces a novel thought experiment demonstrating that quantum properties can be decoupled and exchanged between particles, questioning the reality of observables.

Findings

Quantum properties can be temporarily decoupled from particles.

Properties of two photons can be interchanged during recombination.

The belongingness of properties in quantum systems is highly volatile.

Abstract

A common-sense perception of a physical system is that it is inseparable from its physical properties. The notion of Quantum Cheshire Cat challenges this, as far as quantum systems are concerned. It shows that a quantum system can be decoupled from its physical property under suitable pre and postselections. However, in the Quantum Cheshire Cat setup, the decoupling is not permanent. The photon, for example, and its circular polarization is separated and then recombined. In this paper, we present a thought experiment where we decouple two photons from their respective polarizations and then interchange them during recombination. Thus, our proposal shows that the belongingness of a property for a physical system is very volatile in the quantum world. This raises the question of reality of an observable at a much deeper level.

Equations

$$A_w=\frac{⟨{\Psi }_f|A|{\Psi }_i⟩}{⟨{\Psi }_f|{\Psi }_i⟩}.$$

$$|{\Psi }_i⟩=\frac{1}{\sqrt{2}}(i|L⟩+|R⟩)|H⟩),$$

$$|{\Psi }_f⟩=\frac{1}{\sqrt{2}}(|L⟩|H⟩+|R⟩|V⟩),$$

$${\sigma }_z^L={\Pi }_L\otimes {\sigma }_z,$$

$${\sigma }_z^R={\Pi }_R\otimes {\sigma }_z,$$

$${\sigma }_z=|+⟩⟨+|-|-⟩⟨-|,$$

$$|+⟩$$

$$|-⟩$$

$$({\Pi }_L{)}^w=1\text{mbox}and({\Pi }_R{)}^w=0$$

$$({\sigma }_z^L{)}^w=0\text{mbox}and({\sigma }_z^R{)}^w=1.$$

$$|{\Psi }_i⟩$$

$$|{\Psi }_f⟩$$

$$|+⟩$$

$$|-⟩$$

$$|{+}^{\prime }⟩$$

$$|{-}^{\prime }⟩$$

$${\sigma }_z$$

$${\sigma }_{z^{\prime }}$$

$$U_{{\Pi }_L}=$$

$$R({\theta }_g){|0⟩}_m$$

$$R({\theta }_g){|1⟩}_m$$

$${|\Phi ⟩}_m=⟨{\Psi }_f|U_{{\Pi }_L}|{\Psi }_i⟩{|0⟩}_m.$$

$${|\Phi ⟩}_m\propto {|0⟩}_m+g({\Pi }_L{)}^w{|1⟩}_m,$$

$$({\Pi }_L{)}^w=\frac{⟨{\Psi }_f|{\Pi }_L|{\Psi }_i⟩}{⟨{\Psi }_f|{\Psi }_i⟩}=1,$$

$$({\Pi }_R{)}^w=\frac{⟨{\Psi }_f|{\Pi }_R|{\Psi }_i⟩}{⟨{\Psi }_f|{\Psi }_i⟩}=0,$$

$$({\Pi }_{L^{\prime }}{)}^w=\frac{⟨{\Psi }_f|{\Pi }_{L^{\prime }}|{\Psi }_i⟩}{⟨{\Psi }_f|{\Psi }_i⟩}=0,$$

$$({\Pi }_{R^{\prime }}{)}^w=\frac{⟨{\Psi }_f|{\Pi }_{R^{\prime }}|{\Psi }_i⟩}{⟨{\Psi }_f|{\Psi }_i⟩}=1.$$

$$U_{{\sigma }_z^L}=$$

$$({\sigma }_z^L{)}^w=\frac{⟨{\Psi }_f|{\Pi }_L\otimes {\sigma }_z|{\Psi }_i⟩}{⟨{\Psi }_f|{\Psi }_i⟩}=0,$$

$$({\sigma }_z^R{)}^w=\frac{⟨{\Psi }_f|{\Pi }_R\otimes {\sigma }_z|{\Psi }_i⟩}{⟨{\Psi }_f|{\Psi }_i⟩}=1,$$

$$({\sigma }_{z^{\prime }}^{L^{\prime }}{)}^w=\frac{⟨{\Psi }_f|{\Pi }_{L^{\prime }}\otimes {\sigma }_{z^{\prime }}|{\Psi }_i⟩}{⟨{\Psi }_f|{\Psi }_i⟩}=1,$$

$$({\sigma }_{z^{\prime }}^{R^{\prime }}{)}^w=\frac{⟨{\Psi }_f|{\Pi }_{R^{\prime }}\otimes {\sigma }_{z^{\prime }}|{\Psi }_i⟩}{⟨{\Psi }_f|{\Psi }_i⟩}=0,$$

Full text

Introduction.- It is commonly believed that the properties of a physical system cannot be separated from the system itself. This picture of the nature of physical systems, however, does not hold true in the realm of quantum mechanics. A thought experiment, known as the Quantum Cheshire Cat Cheshire , shows that a property such as the polarization of a photon can exist in isolation to the photon itself. Based on a modified version of a Mach-Zehnder interferometer, the Quantum Cheshire Cat demonstrates that a photon and its circular polarization can be decoupled from each other and made to travel separately through the two arms. This echoes the description of certain events in the novel Alice in Wonderland where Alice remarks,“Well! I’ve often seen a cat without a grin,…, but a grin without a cat! It’s the most curious thing I ever saw in my life!” Carroll1865 .

Quantum Cheshire Cat has opened up a new window for understanding of quantum systems Bancal2013 ; Duprey2018 ; Correa2015 ; Atherton2015 . It pertains not only to photons and their polarizations but can, in principle, be observed with any quantum system and its property, such as neutron and its magnetic moment, electron and its charge and so on. Experimental verifications of the phenomenon with neutron as the cat and its magnetic moment as the grin have been conducted  Denkmayr2014 ; Sponar2016 . The phenomenon has also been observed experimentally in the context of photon and polarization Ashby2016 . Further developments on the idea of the Quantum Cheshire Cat include the proposal of a complete Quantum Cheshire Cat CC and twin Quantum Cheshire Cats Twin . The effect has been used to realize the three box paradox Pan2013 and has been studied in the presence of decoherence Richter2018 . Recently, a protocol has been developed using which the decoupled grin of a Quantum Cheshire Cat has been teleported between two spatially separated parties without the cat  Das2019 .

There also exist criticisms of Quantum Cheshire Cat in literature. A comprehensive list of the same can be found in Ref. Duprey2018 . Typically, the critics tend to refute the claim of the disembodiment of the polarization from the photon, using different types of arguments. In Ref. Stuckey2015 , the absence of the polarization in a particular arm, where the corresponding weak value is zero, is dismissed due to a particular form of the intensity obtained at the detector. However, in the analysis, the authors do not consider weak measurements at all, which we discussed, is vital for the observation of Quantum Cheshire Cat. In another work Atherton2015 , the results of the neutron interferometry experiment Denkmayr2014 , demonstrating Quantum Cheshire Cat, is mimicked by using classical beams and observing electric field intensities at the output detectors. The authors strive to show that there is nothing particularly ”quantum” in the Quantum Cheshire Cat. Yet, the reasoning suffers from the same shortcoming of the previous example. The logic of the phenomenon is heavily dependent on defining weak values of quantum observables like spin or polarization. In Ref. Atherton2015 , a precise definition of such quantum observables is found wanting, as one would expect would be the case for classical beams. Another argument is that Quantum Cheshire Cat is just a form of quantum interference of a particular kind of meter states. The problem of this approach is that it talks about the meter state dynamics and does not involve an interaction Hamiltonian, resulting in no weak values Correa2015 . Also the numerical simulations in Ref. Michielsen2015 , the target of criticism is the neutron interferometry experiment, meant to be experimentally demonstrating the effect, and therefore in no way the negative results actually disapprove the disembodiment in the original theoretical proposal.

The premise of the realization of the Quantum Cheshire Cat involves weak measurements and weak values AAV1988 . The development of the concept weak measurements and weak values stemmed from the limitation posed by the measurement problem in quantum mechanics, in acquiring knowledge about the value of an observable of a quantum system. Given a quantum system in a general pure state $\left|{\Psi}\right\rangle$, if a measurement of an observable $A$ is performed, then the outcome can be any one of the eigenvalues $a_{n}$ of $A$. Measurement of $A$ on an ensemble of states, shows that the measurement outcomes are indeterministic and probabilistic, the probability of an outcome $a_{n}$ in any given run being $|\langle a_{n}\left|{\Psi}\right\rangle|^{2}$, where $\left|{a_{n}}\right\rangle$ is the eigenstate, corresponding to the eigenvalue $a_{n}$. After the strong measurement, the state of the system collapses to $\left|{a_{n}}\right\rangle$. As a result, there is no general consensus among physicists as to whether the measurement of an observable actually reveals a property of the system or is an artifact of the measurement process itself. The proponents of weak value tried to circumvent this problem of wavefunction collapse by minimizing the disturbance caused to the initial state. This is achieved by weakly measuring the observable $A$, causing minimal disturbance to the state. We briefly recapitulate the process of weak measurement below.

Consider a quantum system preselected in the state $\left|{\Psi_{i}}\right\rangle$. Suppose an observable $A$ is weakly measured by introducing a small coupling between the quantum system and a suitable measurement device or a meter. A second observable $B$ is thereafter measured strongly and one of its eigenstates $\left|{\Psi_{f}}\right\rangle$ is postselected. For all successful postselections of the state $\left|{\Psi_{f}}\right\rangle$, the meter readings corresponding to the weak measurements of $A$ are taken into consideration while the others are discarded. The shift in the meter readings, on an average, for all such postselected systems is given by the weak value $A_{w}$ AAV1988 . Mathematically the weak value of $A$ is defined as

[TABLE]

The weak value is therefore to be interpreted as the value the observable A, takes between the preselected state $\left|{\Psi_{i}}\right\rangle$ and the postselected state $\left|{\Psi_{f}}\right\rangle$. The measurement of $A$ must be weak to preserve the initial probability distribution for the final postselected state. Weak values via weak measurements have been observed experimentally Pryde2005 . The weak value of an observable can be complex Jozsa2007 or can take up large values that lie outside the eigenvalue spectrum AAV1988 ; Duck1989 ; Tollaksen2010 . The latter has led to the use of weak measurements as a tool for signal amplification Dixon2009 ; Nishizawa2012 . A geometrical interpretation of weak values can be found in Ref. Cormann2016 . Among the myriad applications of weak measurements are observation of the spin Hall effect Hosten2008 , resolving quantum paradoxes Hofmann2010 ; Dolev2005 , quantum state visualization Kobayashi2014 , quantum state tomography Hofmann2010 ; Wu2013 , direct measurement of wavefunction Lundeen2011 ; Lundeen2012 , probing contextuality Tollaksen2007 , measuring the expectation value of non-Hermitian operators Pati2015 ; Nirala2019 and quantum precision thermometry  Pati2019 .

Any probing of position or a component of polarization of the photon for the observation of the Quantum Cheshire Cat effect must necessarily be weak. This is because projective measurements tend to destroy the original state of a system while extracting information. Weak measurements, on the other hand, minimally disturb the system while gaining small information about it. The state cannot be disturbed in a Quantum Cheshire Cat setup as any alteration of the original state will lead to altered probabilities of the postselected state.

In this paper we explore yet another counterintuitive aspect of the Quantum Cheshire Cat. We design a setup where we not only decouple the grin (circular polarization) from the Quantum Cheshire Cat (photon), but also replace it with a grin (circular polarization) originally belonging to another Cheshire Cat (a different photon). Our setup is comprised of two modified and overlapping Mach-Zehnder interferometers. We try to show that a property of a physical system does not uniquely belong to that system in the quantum domain and can be replaced by the same property from another physical system. Following Neils Bohr, the reality of a physical attribute comes into existence only after a measurement. Our result shows that physical attributes are not real and may not belong to a system under certain scenario. It is to be noted that this exchange of polarizations of two photons is not merely an exchange of two states of polarizations as in spin swapping. Here we literally strip the photon of its polarization and then exchange it with another polarization. Thus, our result raises the question of reality of observables at a much deeper level than what Bohr’s famous quote suggests.

The Quantum Cheshire Cat.- In 2013, Aharonov et. al. Cheshire proposed a thought experiment where it is possible to decouple a property of a system from the system itself. The thought experiment can be realized by a scheme that is based on a Mach-Zehnder interferometer and pertains to the separation of a photon from its circular polarization. To begin with, a source sends a linearly polarized single photon towards a 50:50 beam-splitter $BS_{1}$. Let us denote the left and the right paths of the photon, after passing the beam-splitter, as $\left|{L}\right\rangle$ and $\left|{R}\right\rangle$, respectively which are the two orthogonal states of the photon. If the photon, from the source, is initially in a horizontal polarization state $\left|{H}\right\rangle$, after passing through the beam-splitter $BS_{1}$ it goes to the state

[TABLE]

where the relative phase factor $i$ appears as the photon traveling through the left arm is reflected by the beam splitter. In the analysis of the Quantum Cheshire Cat, this state is used as the preselected state. A combination of a half-waveplate (HWP), a phase-shifter (PS), beam-splitter $BS_{2}$, a polarization beam-splitter (PBS) and three detectors $D_{1}$, $D_{2}$ and $D_{3}$ conducts the postselection. Let us call this the postselection block. The required postselected state to observe the Quantum Cheshire Cat phenomenon is given by

[TABLE]

where $\left|{V}\right\rangle$ refers to the vertical polarization state orthogonal to the initial polarization state $\left|{H}\right\rangle$. To appreciate how the postselection works, note that the HWP flips the polarization of the photon from $\left|{H}\right\rangle$ to $\left|{V}\right\rangle$ and vice-versa. The phase-shifter (PS) adds a phase factor of $i$ to the state of the photon passing through it. The beam-splitter $BS_{2}$ allows only a photon in the superposition state $\frac{1}{\sqrt{2}}(\left|{L}\right\rangle+i\left|{R}\right\rangle)$ towards the PBS and never towards $D_{2}$. Next, the PBS always transmits the horizontal polarization $\left|{H}\right\rangle$ and always reflects the vertical polarization $\left|{V}\right\rangle$. Thus only if a state $\left|{\Psi_{f}}\right\rangle$ enters the postselection block, there is a click of detector $D_{1}$. Any clicking of the detectors $D_{2}$ or $D_{3}$ implies a different state entering the postselection block. Therefore, by selecting the clicks of the detector $D_{1}$ alone and discarding all the others, one can postselect the state $\left|{\Psi_{f}}\right\rangle$.

To find out which arm a photon passed through, we must perform measurements of the observables $\Pi_{L}=\left|{L}\right\rangle\left\langle{L}\right|$ and $\Pi_{R}=\left|{R}\right\rangle\left\langle{R}\right|$ in the left and the right arms, respectively. However, we must also ensure that the measurement does not disturb the initial state much, so that the probability statistics for the desired postselected state is minimally altered. To this end the observables $\Pi_{L}=\left|{L}\right\rangle\left\langle{L}\right|$ and $\Pi_{R}=\left|{R}\right\rangle\left\langle{R}\right|$ must be weakly measured by placing weak detectors in the two arms. Similarly, the polarizations can be detected in the left and the right arms by respectively performing weak measurements of the following operators

[TABLE]

where

[TABLE]

the circular polarization basis $\{\left|{+}\right\rangle,\left|{-}\right\rangle\}$ itself defined as

[TABLE]

The weak values of the photon positions are measured to be

[TABLE]

which implies that the photon has traveled through the left arm. The weak values of the polarization positions, on the other hand, are measured to be

[TABLE]

Equations (7) and (8) jointly tell us that the photon traveled through the left arm but its circular polarization traveled through the right arm. This means the two degrees of freedom of a single entity have been decoupled. Thus, the polarization, a property of the photon, can exist independent of the existence of the photon, in that region.

Exchanging Grins of Cats.– In the previous section we have seen that the grin of a cat can be separated from the cat itself under suitable choices of pre and postselection. Here we consider two such Quantum Cheshire Cats and exchange their grins. In terms of physical realization, we take two linearly polarized photons, decouple their circular polarizations and then recouple them with the other photon.

To see this effect let us consider the setup shown in Fig. 3. There are two sources of linearly polarized photons, near the unprimed and the primed halves of the arrangement, on the left and the right, respectively. The input polarization in the unprimed half is $\left|{H}\right\rangle$ while that in the primed half is $\left|{H^{\prime}}\right\rangle$. The setup allows one to prepare a preselected state given by

[TABLE]

where $\left|{L}\right\rangle$ and $\left|{R}\right\rangle$ indicate the states of a photon in the left and right arms of the unprimed half of the apparatus and $\left|{L^{\prime}}\right\rangle$ and $\left|{R^{\prime}}\right\rangle$ indicate the states of a photon in the left and right arms of the primed half of the apparatus. The right arm of the unprimed half is connected to the output of the primed half and the left arm of the primed half is connected to the output of the unprimed half.

Now, the choice of $\left|{\Psi_{f}}\right\rangle$ which is constrained by the need to define the weak values, makes it necessary to chose a rather a complicated post-selection strategy. In the present work, we use a four qubit entangled state for our purpose. This may look slightly complicated procedure and other choices of post-selection with simple realization may be possible, but we leave that for future. The photons and their polarizations are postselected in the state as given by

[TABLE]

This a four qubit maximally entangled state and is one of the cluster states Briegel2001 . As we shall see the choice of this state leads to interesting consequences which is the subject of this paper. The postselection is realized using the following setup. Beam-splitters $BS_{i}(i=2,3,4,5,6,7)$ and $BS^{\prime}_{i}(i=2,3,4,5,7)$, half-waveplates $HWP$ and $HWP^{\prime}$, phase-shifters $PS$ and $PS^{\prime}$, four polarization beam-splitters $PBS$, $PBS^{\prime}$, $PBS_{1}$ and $PBS^{\prime}_{1}$ and detectors $D_{i}(i=1,..,6)$ and $D^{\prime}_{i}(i=1,..,6)$ together constitute the postselection block. It is clear that this postselected state $\left|{\Psi_{f}}\right\rangle$ demands that there is entanglement between the path degrees of freedom of the two halves of the interferometric arrangement. The $HWP$ and the $HWP^{\prime}$ cause the transformations $\left|{H^{\prime}}\right\rangle\leftrightarrow\left|{V^{\prime}}\right\rangle$ and $\left|{H}\right\rangle\leftrightarrow\left|{V}\right\rangle$, respectively, and $PS$ and $PS^{\prime}$ add a phase-factor $i$. Now let the beam-splitters $BS_{2}$ and $BS_{2}^{\prime}$ are chosen such that if a state $\left|{L}\right\rangle\left|{L^{\prime}}\right\rangle$ is incident on $BS_{2}$ or a state $\left|{R}\right\rangle\left|{R^{\prime}}\right\rangle$ is incident on $BS_{2}^{\prime}$, then the photons emerge towards $PBS$ and $PBS^{\prime}$, respectively. The $PBS$ and the $PBS^{\prime}$, once again, allow only polarizations $\left|{H^{\prime}}\right\rangle$ and $\left|{H}\right\rangle$ to be transmitted and other polarizations to be reflected. Now if any other state is incident on $BS_{2}$ and $BS^{\prime}_{2}$, it proceeds towards $PBS_{1}$ or $PBS^{\prime}_{1}$. These polarization beam splitters, once again, allow polarizations $\left|{H^{\prime}}\right\rangle$ and $\left|{H}\right\rangle$ to be transmitted, towards $BS_{3}$ and $BS^{\prime}_{3}$, respectively, and reflect polarizations $\left|{V^{\prime}}\right\rangle$ and $\left|{V}\right\rangle$ towards $D_{6}$ and $D^{\prime}_{6}$. Next, the beam-splitter $BS_{3}$ is so chosen that the state $\left|{L}\right\rangle$ is transmitted towards the mirror $M_{2}$ then $M_{4}$ and finally towards the beam-splitter $BS^{\prime}_{5}$ while any other state is reflected towards $BS_{4}$. In conjunction with this, the beam-splitter $BS^{\prime}_{3}$ is chosen to transmit the state $\left|{R^{\prime}}\right\rangle$ towards the mirror $M^{\prime}_{3}$ and then towards the beam-splitter $BS^{\prime}_{5}$ and reflect any other state towards $BS^{\prime}_{4}$. The beam-splitter $BS^{\prime}_{5}$ receives inputs $\left|{L}\right\rangle$ and $\left|{R^{\prime}}\right\rangle$. It must be chosen such that a state $\left|{L}\right\rangle\left|{R^{\prime}}\right\rangle$ is channeled towards the beam-splitter $BS_{6}$ while any other state is sent to the detector $D^{\prime}_{4}$. On the other hand, the beam-splitter $BS_{4}$ is chosen such that it reflects the state $\left|{L^{\prime}}\right\rangle$ towards the mirror $M_{3}$, leading to the mirror $M_{5}$ and finally onto the beam-splitter $BS_{5}$. From the beam-splitter $BS_{4}$, any state, other than $\left|{L^{\prime}}\right\rangle$, is transmitted to the detector $D_{5}$. Moving onto the beam-splitter $BS^{\prime}_{4}$, it is so chosen that only a state $\left|{R}\right\rangle$ is reflected towards the mirror $M^{\prime}_{4}$, leading to the beam-splitter $BS_{5}$. A state, other than $\left|{R}\right\rangle$ is transmitted to the detector $D^{\prime}_{5}$. The inputs of the beam-splitter $BS_{5}$ are thus $\left|{R}\right\rangle$ and $\left|{L^{\prime}}\right\rangle$. To achieve the desired postselection, we must choose $BS_{5}$ so that it allows only the state $\left|{R}\right\rangle\left|{L^{\prime}}\right\rangle$ towards the beam-splitter $BS_{6}$. Any other state is sent to the detector $D_{4}$. It is to be remembered that the beam-splitter $BS_{6}$ also receives the state $\left|{L}\right\rangle\left|{R^{\prime}}\right\rangle$. We choose the beam-splitter $BS_{6}$ in a way that only a state $\left|{L}\right\rangle\left|{R^{\prime}}\right\rangle-\left|{R}\right\rangle\left|{L^{\prime}}\right\rangle$ is allowed towards the mirror $M_{6}$ and finally towards the beam-splitter $BS^{\prime}_{7}$. A state, other than $\left|{L}\right\rangle\left|{R^{\prime}}\right\rangle-\left|{R}\right\rangle\left|{L^{\prime}}\right\rangle$ is sent towards the detector $D^{\prime}_{3}$ from $BS_{6}$. Now let us go back to the states $\left|{L}\right\rangle\left|{L^{\prime}}\right\rangle$ and $\left|{R}\right\rangle\left|{R^{\prime}}\right\rangle$ proceeding towards the beam splitter $BS_{7}$. This beam-splitter is such that the state $\left|{L}\right\rangle\left|{L^{\prime}}\right\rangle+\left|{R}\right\rangle\left|{R^{\prime}}\right\rangle$ is transmitted towards the beam-splitter $BS^{\prime}_{7}$ and any other state is transmitted towards the detector $D_{3}$. Finally the beam-splitter $BS^{\prime}_{7}$ sends the state $\left|{L}\right\rangle\left|{L^{\prime}}\right\rangle+\left|{R}\right\rangle\left|{R^{\prime}}\right\rangle+\left|{L}\right\rangle\left|{R^{\prime}}\right\rangle-\left|{R}\right\rangle\left|{L^{\prime}}\right\rangle$ towards the two-photon detector $D_{1}$ while any other state is sent towards the detector $D^{\prime}_{1}$.

Thus by selecting the clicks of the detector $D_{1}$ alone, we can successfully postselect the desired state. To appreciate the working of this arrangement, let us define two circular polarization bases

[TABLE]

and

[TABLE]

and two operators

[TABLE]

To find out which arms the two photons passed through, we must perform measurements of the observables $\Pi_{L}=\left|{L}\right\rangle\left\langle{L}\right|$, $\Pi_{R}=\left|{R}\right\rangle\left\langle{R}\right|$, $\Pi_{L^{\prime}}=\left|{L^{\prime}}\right\rangle\left\langle{L^{\prime}}\right|$ and $\Pi_{R^{\prime}}=\left|{R^{\prime}}\right\rangle\left\langle{R^{\prime}}\right|$ in the left and the right arms of the unprimed and the primed halves of the arrangement, respectively. However, we must also ensure that the measurements do not disturb the initial state much, so that the probability statistics for the desired postselected state is minimally altered. To this end, the above observables must be weakly measured by placing weak detectors in the four arms.

We must also ensure that the observed phenomenon is unambiguously quantum and cannot be explained by classical theory which, as we have briefly discussed earlier has raised suspiscion in previous works Atherton2015 . In our setup, the weak detection of an operator, say $(\Pi_{L})^{w}$, begins by weakly coupling the system, in our case, the photon, with a second system or a meter. In terms of physics, this is usually done through an interaction Hamiltonian AAV1988 , resulting in a unitary acting jointly on the system and the meter. The meter consists of a two level system, defined in $\{\left|{0}\right\rangle_{m},\left|{1}\right\rangle_{m}\}$ basis, and initialized in the $\left|{0}\right\rangle_{m}$ state. Following the technique laid out in Ref. Kim2018 , we execute a joint unitary $U_{\Pi_{L}}$ acting on the unprimed photon and the meter as

[TABLE]

Each term of the above unitary is a tensor product of operators, acting respectively on the path degree of freedom of the unprimed photon, its polarization, path degree of freedom of the primed photon, its polarization and the meter. The operator $Z=\left|{0}\right\rangle_{m}\left\langle{0}\right|_{m}-\left|{1}\right\rangle_{m}\left\langle{1}\right|_{m}$. $R(\theta_{g})$ is defined such that

[TABLE]

with $g=4\theta_{g}$. The coupling between the photon and the meter is made weak by tuning for low values of $g$ so that only the terms till the first order in $g$ make significant contribution. Under the postselection, given by Equation (10), the meter is thrown into a state

[TABLE]

Using Equations (9), (10) and (14), Equation (16) assumes the form

[TABLE]

with $(\Pi_{L})^{w}=\frac{\left\langle{\Psi_{f}}\right|\Pi_{L}\left|{\Psi_{i}}\right\rangle}{\left\langle{\Psi_{f}}\right|\Psi_{i}\rangle}=1$ being the weak value of $\Pi_{L}$. Equation (17) is reached by using the knowledge from theory of weak values that for successful postselections, the meter state shows an average shift, proportional to the weak value of the observable being measured AAV1988 ; Kim2018 .

Similarly a weak measurement of $\Pi_{R}$ can be carried out by choosing a global unitary, obtained by substituting $\Pi_{L}$ by $\Pi_{R}$ in Equation (14). A similar reasoning can be followed for the primed projection operators. In addition to Kim2018 , a general framework to construct a joint unitary evolution and perform measurement using weak detectors can be found in Pryde2005 . The weak values obtained are

[TABLE]

Thus the photon at the unprimed input port must have traveled through the left arm of the unprimed half of the setup while the photon at the primed input port must have traveled through the right arm of the primed half of the setup when the postselection of $\left|{\Psi_{f}}\right\rangle$ is done.

Similarly, the polarizations can be detected in the left and the right arms by respectively performing weak measurements of the operators $\sigma_{z}^{L}=\Pi_{L}\otimes\sigma_{z}$, $\sigma_{z}^{R}=\Pi_{R}\otimes\sigma_{z}$, $\sigma_{z^{\prime}}^{L^{\prime}}=\Pi_{L^{\prime}}\otimes\sigma_{z^{\prime}}$ and $\sigma_{z^{\prime}}^{R^{\prime}}=\Pi_{R^{\prime}}\otimes\sigma_{z^{\prime}}$ to detect the path of the polarization of the photons within the designated arms of the interferometer. To understand these operators, let us take the example of say $\sigma_{z}^{R}=\Pi_{R}\otimes\sigma_{z}$. On the unprimed side of the setup, the beam-splitter $BS_{1}$ sends the photon and the polarization it carries into either $\left|{L}\right\rangle$ or $\left|{R}\right\rangle$. The polarization on the unprimed side is acted upon by unprimed operators. Thus to find the polarization on the right arm of the unprimed part, it is necessary to measure the operator $\sigma_{z}^{R}=\Pi_{R}\otimes\sigma_{z}$. A similar reasoning can be reached for the other choices of the operators. To obtain each weak value we once again include a meter and interact it with the relevant photon by means of a global unitary. For example, the unitary for weakly measuring $\sigma_{z}^{L}$ should have the form

[TABLE]

Similarly other unitaries for the weak measurements of the positions of circular polarization for the unprimed and primed photons can be defined.

These weak values turn out to be

[TABLE]

which means that the circular polarization of the photon at the unprimed input port must have traveled via the right arm of the unprimed half of the arrangement and the circular polarization of the photon at the primed input port must have journeyed via the left arm of the primed half of the setup for all final outcomes $\left|{\Psi_{f}}\right\rangle$. Equations (18) and (20) jointly demonstrate that, under the above postselection, the unprimed photon at some stage was at the beam-splitter $BS_{2}$ but its circular polarization was at the beam-splitter $BS^{\prime}_{2}$. Similarly, the primed photon reached the beam-splitter $BS^{\prime}_{2}$ while its circular polarization went to the beamsplitter $BS_{2}$, for the postselected state $\left|{\Psi_{f}}\right\rangle$. In other words, recombination of the photons and their polarizations do occur, but in the case of the above mentioned postselected states, the unprimed photon recombined with the primed polarization while the primed photon recombined with the unprimed polarization. Thus we have exchanged the grins of two Quantum Cheshire Cats.

We would also emphasize on the fact that the above conclusions are based on defining quantum observables and modelling each of their measurement involving a system-meter interaction. Although classical treatment of the phenomenon may yeild strikingly similar results, it would not clarify with precise definitions what quantum variables would be measured and how. This would suggest that the phenomenon we describe is essentially quantum in nature.

It is to be noted that to execute the process of postselection, culminating in the detection of two photons at $D_{1}$, the two photons come together once again. As a result we cannot comment on the whether the circular polarizations are once again exchanged and whether they return to their original photons. But it is equally true that for the successful postselection of the state $\left|{\Psi_{f}}\right\rangle$, each circular polarization at certain stage of the photons’ progress through the interferometer, say at $BS_{2}$ and $BS^{\prime}_{2}$, associate itself with the other photon.

Before we conclude, we would like to comment that the ambiguity of the association of photons with their properties is closely related to the indistinguishability of quantum particles such as photons. It is therefore especially interesting that the present idea does not rely on actual photon bunching effects. However, some important new insights about photon indistinguishability could be gained from the present results by comparing them a bit more closely to related results observed when photon bunching effects are present. In Ref. Hofmann2002 there has been a proposal for the realization of an entanglement filter and this has been realized experimentally in Ref. Okamoto2009 . In that scenario, two photons are also kept on separate outer paths while the inner paths do cross. It should be noted that the entanglement filter is different from our scenario, because the intended effect in those papers is the elimination of single photon occupation in the inner paths by additional photon bunching. However, in our double Cheshire Cat setup we achieve the elimination of two photons arriving at the same side of the interferometer by suitable post-selection. It seems to be equally important for both cases that the indistinguishability of photons makes it difficult to identify photons by keeping track of their spatial modes, as observed in the instances of two photons leaving the optical circuit on the same side. To describe photons as distinct physical objects, we therefore need to make an effort to keep photons in separate modes. The exchange of polarization in the present scenario seems to demonstrate that non-trivial multi-photon quantum interferences can occur without any actual photon bunching. This is probably mediated by the post-selected state. In future, it may be worth exploring how the present exchange of photon polarization relates to the well-established realizations of photon-photon interactions mediated by quantum interferences between the indistinguishable cases of photon exchange and no photon exchange.

Conclusions.– To summarize, we have proposed a thought experiment in which the circular polarizations of two photons can be exchanged using interferometric arrangements. The setup for executing this process is based on the original Quantum Cheshire Cat setup where the circular polarization can be separated from the photon for suitable postselected states. Our method strives to decouple the polarization and the photon, and replace the original polarization with another that was previously associated with a different photon. This polarization in turn associates itself with the second photon. Our Cheshire Cat experiment not only separates a particle from its properties but also causes one particle to acquire a property previously associated with another particle.

The implications for the exchanging of photon polarization are significant. Our thought-experiment allows us to think about the deep insight on the relation between quantum measurement itself and revealing reality of quantum state. Firstly, it challenges the notion that a property must faithfully ‘belong’ to a particular physical system. In the realm of the quantum systems, this ‘belongingness’ is certainly very capricious with properties belonging to independent physical systems getting interchanged. This raises the question of reality of observables as a much deeper level. In our case the photon is decoupled from its circular polarization and then replaced with the circular polarization from a second photon. The second point to note is that entanglement plays a crucial role in the realization of this exchange process. As discussed before, the exchange is successful only when a certain outcome is attained at the end. This so happens that this outcome is an entangled state. It is possible that with different pre and postselected states one can also observe the exchange of grins. In future, it may be worth exploring if one can also exchange grins of two Cheshire cats with a postselected state that is not entangled. This may indeed mean that one could swap the polarizations of two photon without ever interacting. These may provide new insights in foundations of quantum mechanics, quantum information as well as in technological applications.

I Acknowledgements

The authors thank Sreeparna Das for her help with Fig. (2). This work has been funded by J C Bose Fellowship from Department of Science and Technology (DST), India, under Grant No. JCB/2018/000038 (2019-2024).