# Separating the wave and particle attributes of two entangled photons

Yusuf Turek<EMAIL_ADDRESS>Yi-Fang Ren  School of Physics, Liaoning
University, Shenyang, Liaoning 110036, China

(August 28, 2024)

###### Abstract

We put forward a feasible scheme to spatially separate the wave and particle
properties of two entangled photons by properly choosing the pre- and post-
selection of path states. Our scheme, which implements the quantum Cheshire
cat phenomenon for two-photon states, also guarantees that the observation of
wave and particle properties of the two entangled photons always obey the
Bohr’s complementarity principle.

###### pacs:

42.50.Wk,47.50.Ef, 42.50.−p, 03.67.−a

##  I Introduction

The wave-particle duality is one of the most characteristic phenomena featured
by quantum mechanical systems. According to the Bohr’s complementarity
principle (Bohr, 1928; Wootters and Zurek, 1979; Scully  _et al._ , 1991), one
cannot simultaneously observe the wave and the particle behavior of a
particle. In turn, quantum objects display its wave- or particle-like behavior
depending on the experimental technique used to measure them (Wheeler, 1984;
Grangier  _et al._ , 1986; Kurtsiefer  _et al._ , 1997; DÃŒrr  _et al._ ,
1998; Marshall  _et al._ , 2003; Ionicioiu and Terno, 2011; Tang  _et al._ ,
2012; Wang  _et al._ , 2019). The fundamental issues related to wave-particle
duality and complementarity can be exemplified by Young-like double-slit
experiments and their variants, e.g. Wheeler’s delayed choice experiment
(Wheeler, 1978; Englert, 1996; Jacques  _et al._ , 2007, 2008; Ma  _et al._ ,
2012; Manning  _et al._ , 2015). In those studies, the most important
consequence of the wave-particle duality is the observation of quantum
interference on a screen, independently on whether we send photons or
particles onto the double-slit. Even more remarkable is the fact that quantum
interference occurs also when one performs double-slit experiments with a
single quantum object. A description of the experimental results using common
language involves counterintuitive statements like "the particle passes
through both slits at once", which, however, are not providing much insight. A
more striking characterization of the phenomenon may be gained by means of the
Wheeler’s delayed choice experiment.

In single-photon delayed-choice experiments (Jacques  _et al._ , 2007;
Ionicioiu and Terno, 2011), a photon is sent into a Mach-Zehnder
interferometer (MZI) where a balanced beam splitter (BS) splits it into two
traveling paths. The term delayed choice is referring to the fact that one
randomly determines whether to recombine the two paths or not, by a second BS,
only after the photon has entered the MZI. If the second beam splitter is
present, we observe interference fringes, indicating that the photon behaves
as a wave, passing through both arms of the MZI. If BS2 is not there, each
detector at the output fires with probability 1/2, i.e. only one of the two
detectors is clicking at a time. This indicates that the photon traveled along
a single arm i.e. it reveals a particle feature of the photon. With the double
slit experiment, one realizes that photon has both a wave and a particle
nature, and that these are inseparable. Experiments have shown that the above
statements are true for any quantum systems including single electron, atom,
neutron, and so on.

In general, since the wave and particle properties of a quantum object coexist
simultaneously, one may assume that the two attributes cannot be separated
from each other. However, recent works on the topic (Chowdhury  _et al._ ,
2021; Li  _et al._ , 2023) has shown, upon exploiting the idea of quantum
Cheshire cat (Aharonov  _et al._ , 2013), that one may spatially separates of
wave and particle attributes of a quantum object. The Cheshire cat is a
fictitious character in the famous novel “Alice in Wonderland”. The Cheshire
cat has the ability of disappearing in space while its grin is still visible.
In our usual life, we can imagine a cat without a grin, but it beyond our
common sense to accept the existence of a grin without a cat as Alice said :
“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!”.

In 2013, Y. Aharonov and his collaborators introduced (Aharonov  _et al._ ,
2013) a protocol to detach a quantum object from its inherent feature, and
refer to the result as a quantum Cheshire cat. Quantum Cheshire cat may exist
in a quantum system described by two-state vector formalism (Aharonov  _et
al._ , 1964; Aharonov and Vaidman, 1990, 2008). Furthermore, its realization
depends on performing a post-selected weak measurement (Aharonov  _et al._ ,
1988) to extract the weak values (Aharonov and Botero, 2005) of the
observables associated to the quantum object under investigation. In general,
the weak value of an observable is a complex number and it can take arbitrary
large values (Aharonov and Rohrlich, 2005). This feature of the weak value is
usually referred to as an amplification effect of post-selection, and can be
used to amplify tiny but useful information about physical systems (Hosten and
Kwiat, 2008; Dixon  _et al._ , 2009; Starling  _et al._ , 2009, 2010; Pfeifer
and Fischer, 2011; Egan and Stone, 2012; Zhou  _et al._ , 2013; Viza  _et al._
, 2013; Magaña Loaiza  _et al._ , 2014). Besides, weak measurement have been
also employed in quantum state tomography (Lundeen  _et al._ , 2011; Lundeen
and Bamber, 2012; de Gosson and de Gosson, 2012; Fischbach and Freyberger,
2012; Matzkin and Pan, 2013; Malik  _et al._ , 2013; Salvail  _et al._ , 2013;
Di Lorenzo, 2013; Malik  _et al._ , 2014; Hariri  _et al._ , 2019; Pan  _et
al._ , 2019), Hardy’s paradox (Aharonov  _et al._ , 2002; Jiang  _et al._ ,
2018), three box paradox (Albert  _et al._ , 1985; Matzkin and Pan, 2013;
Imran  _et al._ , 2021; Blasiak and Borsuk, 2022) and so on. For more details
of weak measurement technique, we refer the reader to recent reviews of the
field (Kofman  _et al._ , 2012; Svensson, 2013; Sokolovski, 2013; Dressel  _et
al._ , 2014).

After the original QCC proposal (Aharonov  _et al._ , 2013), the phenomenon
has been observed experimentally (Denkmayr  _et al._ , 2014) and received
large attention theoretically (Guryanova  _et al._ , 2012; Matzkin and Pan,
2013; Lorenzo, 2013; Bancal, 2014; Ibnouhsein and Grinbaum, 2014; CorrÃªa  _et
al._ , 2015; Duprey  _et al._ , 2018; Das and Pati, 2019, 2020; Das and Sen,
2021; Aharonov  _et al._ , 2021; Chowdhury  _et al._ , 2021; Hance  _et al._ ,
2023a, b; Zhou  _et al._ , 2023; Ghoshal  _et al._ , 2023) and experimentally
(Atherton  _et al._ , 2015; Ashby  _et al._ , 2016; Liu  _et al._ , 2020; Kim
_et al._ , 2021; Wagner  _et al._ , 2023; Danner  _et al._ , 2023; Li  _et
al._ , 2023).

Among the different studies, in (Das and Pati, 2020; Liu  _et al._ , 2020) it
has been proved that a quantum system may have a physical property erased and
may obtain a new one that is did not initially have. More recently, a protocol
to separate the wave and particle features of a single photon has been
proposed (Chowdhury  _et al._ , 2021) and implemented (Li  _et al._ , 2023).
Furthermore, in (Zhou  _et al._ , 2023), it has been investigated the
possibility to spatially separate two entangled photons from their
polarizations. Since entanglement is a crucial resource for quantum
information processing and communication, a question arises on whether the
separation of wave and particle attributes for entangled systems is possible.
As we will see in the following, our results provide an affirmative answer to
this question.

In this paper, motivated by previous studies (Chowdhury  _et al._ , 2021; Li
_et al._ , 2023; Zhou  _et al._ , 2023), we introduce a feasible theoretical
model to spatially separate wave and particle properties of two entangled
photons. After appropriate pre- and post-selection of the paths of photons, we
can find the wave and particle features of the photons in different places.
Besides their fundamental interest in the understanding of counterfactual
communication, our results may promote the development of quantum metrology.

The remaining part of this paper is organized as follows. In Section II, we
illustrate our motivations and briefly introduce the idea of quantum Cheshire
cat. In SectionIII, we describe in details our proposal and show that by
suitable pre- and post-selection one may realize the separation of wave and
particle attributes of two entangled photons. In the same Section, we also
describe a possible implementation of our scheme on an optical platform.
Finally, in Section IV, we close the paper with some concluding remarks and
outlooks.

##  II  The quantum cheshire cat

In general, an object and its inherent properties always coexist and cannot be
separated at anytime. Aharonov and collaborators found that the polarization
(say, the grin) property can be separated from a photon (say, the cat) itself
in a pre- and post-selected system. Here, we briefly summarize the ingredients
required to generate such a quantum Cheshire cat. Initially, a horizontally
polarized photon enters a Mach-Zehnder interferometer (MZI), and the pre-
selected state of total system prepared is

$|\psi_{i}\rangle=\frac{1}{\sqrt{2}}\left(i|L\rangle+|R\rangle\right)|H\rangle,$
(1)

where $|L\rangle$ and $|R\rangle$ denote the left and right arms of MZI, i.e.
the two possible paths of the photon, and $|H\rangle$ denotes the horizontal
polarization of the photon. By adding and adjusting some optical elements, we
may post-select the paths and polarization of the photon to obtain the
following state vector

$|\psi_{f}\rangle=\frac{1}{\sqrt{2}}\left(|L\rangle|H\rangle+|R\rangle|V\rangle\right),$
(2)

where $|V\rangle$ represents the vertical polarization of the photon. Now, we
measure the position (path) of the photon together with its circular
polarization inside the MZI. In the two-state-vector formalism, the weak value
of an observable $C$ is given by

$\langle
C\rangle_{w}=\frac{\langle\psi_{f}|C|\psi_{i}\rangle}{\;\langle\psi_{f}|\psi_{i}\rangle}.$
(3)

The location of the photon, i.e, the findings of spatial degree of freedom of
the photon, is described by the projectors

$\Pi_{\mu}=|\mu\rangle\langle\mu|,$ (4)

where $\mu\in\left(L,R\right)$, and $\Pi_{L}\left(\Pi_{R}\right)$ denote left
(right) arm of MZI. On the other hand, the circular polarization of the photon
in the left or right arms of MZI can be characterized by the operator,

$\sigma_{z}^{\mu}=\sigma_{z}\otimes\Pi_{\mu}.$ (5)

where
$\sigma_{z}=|\uparrow_{y}\rangle\langle\uparrow_{y}|-|\downarrow_{y}\rangle\langle\downarrow_{y}|$
with
$|\uparrow_{y}\rangle=\frac{1}{\sqrt{2}}\left(|H\rangle+i|V\rangle\right)$ and
$|\downarrow_{y}\rangle=\frac{1}{\sqrt{2}}\left(|H\rangle-i|V\rangle\right)$,
i.e. the eigenstate of the Pauli matrix
$\sigma_{y}=\left(\begin{array}[]{cc}0&\ -i\\\ i&0\end{array}\right)$ with
corresponding eigenvalues $1$ and $-1$).

By taking into account the above pre- and post-selected states
$|\psi_{i}\rangle$ and $|\psi_{f}\rangle$, the weak value of $\Pi_{\mu}$ and
$\sigma_{z}^{\mu}$ are given by

$\langle\Pi_{\mu}\rangle_{w}=\delta_{\mu L},$ (6)

and

$\langle\sigma_{z}^{\mu}\rangle_{w}=\delta_{\mu R},$ (7)

respectively. Here, $\delta_{ij}$ denote the Kronecker $\delta$ function which
defined as

$\delta_{ij}=\begin{cases}1,&i=j\\\ 0.&i\neq j\end{cases}$ (8)

The weak values in Eq.(6) and Eq. (7) indicates that the spatial degree
freedom of the photon is located in the left arm, while its circular
polarization is detected with unit probability in the right arm of the MZI.
This mean that in pre- and post-selected quantum systems one can realize the
separation of the object from its inherent properties, as the Cheshire cat,
whose smile may be discovered at a different place of the cat itself.

##  III Separation of wave-particle properties of two entangled photons

In this section, we describe our proposal for the separation of wave-particle
properties of two-entangled photons. The schematic diagram of our proposed
setup is shown in Fig. 1. It is based on the so-called WP toolbox introduced
in (Rab  _et al._ , 2017) to convert the coherence superposition of the
polarization states to the coherence superposition of the wave and particle
entities.

Figure 1:  Schematic diagram of the scheme to spatially separate the wave and
particle properties of a two-photon entangled state.

In (Rab  _et al._ , 2017) single-photon states and two-photon entangled states
have been analyzed. The generation processes of the wave-particle entangled
state of two separated photons can be treated as parallel doubling of the
single-photon scheme. In particular, upon using a half-wave plate (HWP, not
shown in the figure) on the path of both the photons produced by spontaneous
parametric down-conversion (SPDC), the two-photon state becomes

$|\psi\rangle_{in}=\cos\alpha|V\rangle|V\rangle+\sin\alpha|H\rangle|H\rangle,$
(9)

where $\alpha$ is an adjustable parameter, tunable by the HWP. After this
stage, each photon enters one of two identical WP toolboxes, and the final
output state may be written as (Rab  _et al._ , 2017)

$|\Phi\rangle_{out}=\cos\alpha|W\rangle|W^{\prime}\rangle+\sin\alpha|P\rangle|P^{\prime}\rangle.$
(10)

Here,

$\displaystyle|W\rangle$
$\displaystyle\equiv|Wave\rangle=e^{i\frac{\phi_{1}}{2}}\left(\cos\frac{\phi_{1}}{2}|1\rangle-i\sin\frac{\phi_{1}}{2}|3\rangle\right),$
(11) $\displaystyle|W^{\prime}\rangle$
$\displaystyle\equiv\\!\\!|Wave^{\prime}\rangle=e^{i\frac{\phi_{1}}{2}}\left(\cos\frac{\phi_{1}}{2}|1^{\prime}\rangle\\!-i\sin\frac{\phi_{1}}{2}|3^{\prime}\rangle\right),$
(12) $\displaystyle|P\rangle$
$\displaystyle\equiv|Particle\rangle=\frac{1}{\sqrt{2}}\left(|2\rangle+e^{i\phi_{1}}|4\rangle\right),$
(13) $\displaystyle|P^{\prime}\rangle$
$\displaystyle\equiv|Particle\rangle=\frac{1}{\sqrt{2}}\left(|2^{\prime}\rangle+e^{i\phi_{1}}|4^{\prime}\rangle\right).$
(14)

In the above expressions, $|n\rangle$ and $|n^{\prime}\rangle$ with
$n,n^{\prime}\in\\{1,2,3,4\\}$ being the $n$-th output modes of from the two
WP toolboxes (Rab  _et al._ , 2017), and $\phi_{1}$, $\phi_{1}^{\prime}$,
$\phi_{2}$ and $\phi_{2}^{\prime}$ being controllable phase shifts in the two
WP toolboxes. For more details about this generation of the states in Eqs.
(11-14) we refer to the original paper (Rab  _et al._ , 2017). Notice that
these states represent the capabilities ($|W\rangle$and $|W^{\prime}\rangle$)
and incapabilities ($|P\rangle$ and $|P^{\prime}\rangle$) of the two photons
to produce interference.

Finally, as shown in Fig. 1, one detects the two photons simultaneously by
measuring the "wave operator" characterized by the projectors
$X_{1}=|W\rangle\langle W|$ and $X_{2}=|W^{\prime}\rangle\langle W^{\prime}|$.
The action of those operators is such that only the wave states $|W\rangle$
and $|W^{\prime}\rangle$ are transmitted whereas the particle states
$|P\rangle$ and $|P^{\prime}\rangle$ are reflected. In this way, the state
$|\Phi\rangle_{out}$ is transformed into

$\displaystyle|\psi_{i}\rangle$
$\displaystyle=\cos\alpha|R_{1}L_{2}\rangle|WW^{\prime}\rangle+\sin\alpha|L_{1}R_{2}\rangle|PP^{\prime}\rangle$
$\displaystyle=\cos\alpha|R_{1}L_{2}WW^{\prime}\rangle+\sin\alpha|L_{1}R_{2}PP^{\prime}\rangle.$
(15)

This is our pre-selected state. Here, as quantum Cheshire cat’s notation, we
use the composite system of path and attributes of two photons as $|.\
.\rangle_{path}|.\ .\rangle_{attributes}$. $|L_{i}\rangle$ and $|R_{i}\rangle$
denoting the left and right paths , $i\in\\{1,2\\}$ labels the photons.
$|R_{1}L_{2}\rangle|WW^{\prime}\rangle$ represents $1$st (2nd) photon’s wave
property $|W\rangle$ ($|W^{\prime}\rangle$) at $|R_{1}\rangle$
($|L_{2}\rangle$), and $|L_{1}R_{2}\rangle|PP{}^{\prime}\rangle$ represents
$1$st (2nd) photon’s particle property $|P\rangle$ ($|P^{\prime}\rangle$) at
$|L_{1}\rangle$ ($|R_{2}\rangle$). $|W\rangle$, $|W^{\prime}\rangle$ and
$|P\rangle$, $|P^{\prime}\rangle$ denote, respectively, the wave and particle
states as (Ionicioiu and Terno, 2011; Rab  _et al._ , 2017)

In order to realize the separation of wave and particle attributes of two
entangled photons, the post-selected state should be of the form

$|\psi_{f}\rangle=\frac{1}{\sqrt{2}}\left(|R_{1}L_{2}WW^{\prime}\rangle+|L_{1}R_{2}PP^{\prime}\rangle\right).$
(16)

This means that in our scheme the post-selected state is a special case of the
pre-selected class of states with $\alpha=\frac{\pi}{4}$.

The determination of the post-selected state $|\psi_{f}\rangle$ is crucial for
the implementation of our scheme. As shown in Fig. 1, its verification can
accomplished by an optical setup made of five beam splitters $BS_{i}$
($i=1,2,3,4,5$), two $\sigma^{1224}$operators, the $X_{3}$ and $X_{4}$
operators, three mirrors $M_{i}$ ($i=1,2,3$), and six detectors $D_{i}$
($i=1,2,3,4,5,6$). The post-selection occurs if and only if the detector
$D_{5}$ clicks. On the contrary, if our post-seleceted state is
$|\psi_{f}\rangle$, then, in the optical arrangement of Fig. 1, we can
guarantee only $D_{5}$ give the answer ,“yes”. Let us prove this statement.

At first, the photons in $L_{1}$ and $R_{2}$ arms go through the $BS$ followed
by $\sigma^{1234}$ operator. Here, effect of $\sigma^{1234}$ operator is to
switch the photon paths in the four-dimensional basis
$\\{|1\rangle,|2\rangle,|3\rangle,|4\rangle\\}$. Its matrix form can be
written as (Rab  _et al._ , 2017)

$\sigma^{1234}=\left(\begin{array}[]{cccc}0&\ 1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{array}\right),$ (17)

and its base vectors can be written as

$\displaystyle|1\rangle$
$\displaystyle=(\begin{array}[]{cccc}1&0&0&0\end{array})^{T},|2\rangle=(\begin{array}[]{cccc}0&1&0&0\end{array})^{T},$
(18c) $\displaystyle|3\rangle$
$\displaystyle=(\begin{array}[]{cccc}0&0&1&0\end{array})^{T},|4\rangle=(\begin{array}[]{cccc}0&0&0&1\end{array})^{T}.$
(18f)

The action of $\sigma^{1234}$ is to switch paths as follows (Li  _et al._ ,
2023)

$|1\rangle\stackrel{{\scriptstyle[}}{{\sigma}}^{1234}]{\sigma^{1234}}{\rightleftharpoons}|2\rangle,\
|3\rangle\stackrel{{\scriptstyle[}}{{\sigma}}^{1234}]{\sigma^{1234}}{\rightleftharpoons}|4\rangle.$
(19)

The beam splitter $BS_{1}$ is chosen as to perform the transformations
$|2\rangle\rightarrow\frac{1}{\sqrt{2}}(|2\rangle+|4\rangle)$, and
$|4\rangle\rightarrow\frac{1}{\sqrt{2}}(|2\rangle-|4\rangle)$. In this way,
the particle state can converted to the wave state after the action of
$BS_{1}$ and $\sigma^{1234}$, i.e,

$\displaystyle|P\rangle\overset{BS_{1}}{\longrightarrow}$
$\displaystyle\frac{1}{\sqrt{2}}\left[\frac{1}{\sqrt{2}}(|2\rangle+|4\rangle)+e^{i\phi_{1}}\frac{1}{\sqrt{2}}(|2\rangle-|4\rangle)\right]$
$\displaystyle=$ $\displaystyle
e^{i\frac{\phi_{1}}{2}}\left(\cos\frac{\phi_{1}}{2}|2\rangle-i\sin\frac{\phi_{1}}{2}|4\rangle\right)\overset{\sigma^{1234}}{\longrightarrow}$
$\displaystyle
e^{i\frac{\phi_{1}}{2}}\left(\cos\frac{\phi_{1}}{2}|1\rangle-i\sin\frac{\phi_{1}}{2}|3\rangle\right)=|W\rangle.$
(20)

The same happens for primed paths with
$|n^{\prime}\rangle\in\left(|1^{\prime}\rangle,|2^{\prime}\rangle,|3^{\prime}\rangle,|4^{\prime}\rangle\right)$.
Thus, after the photons in $L_{1}$ and $R_{2}$ arms simultaneously go through
beam splitters $BS_{1}$ and $BS_{2}$ followed by the $\sigma^{1234}$
operators, our post-selected state is given by

$|\psi_{f^{\prime}}\rangle=\frac{1}{\sqrt{2}}\left(|R_{1}L_{2}\rangle+|L_{1}R_{2}\rangle\right)|WW^{\prime}\rangle.$
(21)

Next, by adjusting the beam-splitters $BS_{3}$ and $BS_{4}$, respectively, we
make the states $|R_{1}L_{2}\rangle$ and $|L_{1}R_{2}\rangle$ go through
$X_{3}$ and $X_{4}$, such that the detectors $D_{1}$ and $D_{2}$ will not
click. The $X_{3}=X_{4}=|W\rangle\langle W|$ operators transmit only the wave
state, and this action guarantees that detectors $D_{3}$ and $D_{4}$ will not
click. Finally, by adjusting the beam splitter $BS_{5}$, the state
$|R_{1}L_{2}\rangle+|L_{1}R_{2}\rangle$ reaches the detector $D_{5}$. Overall,
we conclude that if and only if the detector $D_{5}$ clicks we have made a
successful post-selection. Otherwise, if any other detector clicks post-
selection fails, and we fail to separate the wave and particle attributes of
the two entangled photons.

The operators that measure whether the wave and particle attributes of two
entangled photons are present in the left and right arms is defined as

$\Pi_{\nu}^{\mu_{i}}=|\mu_{i}\rangle\langle\mu_{i}|\otimes|\nu\rangle\langle\nu|,$
(22)

with $\nu\in\\{W,W^{\prime},P,P^{\prime}\\}$, $\mu_{i}\in\\{L_{i},R_{i}\\}$
and $i\in\\{1,2\\}$. After preparing the pre-selected state, we choose to
post-select by $|\psi_{f}\rangle$ and perform weak measurement for observables
$\Pi_{\nu}^{\mu_{i}}$. By substituting the above-defined pre- and post-
selected states into Eq. (3), we find the corresponding weak values of the
observables $\Pi_{\nu}^{\mu_{i}}$, i.e.,

$\displaystyle\langle\Pi_{W}^{\mu_{i}}\rangle$
$\displaystyle=\frac{\cos\alpha}{\cos\alpha+\sin\alpha}\delta_{1i}\delta_{R_{i}\mu_{i}},$
(23a) $\displaystyle\langle\Pi_{P}^{\mu_{i}}\rangle$
$\displaystyle=\frac{\sin\alpha}{\cos\alpha+\sin\alpha}\delta_{1i}\delta_{L_{i}\mu_{i}},$
(23b) $\displaystyle\langle\Pi_{W^{\prime}}^{\mu_{i}}\rangle$
$\displaystyle=\frac{\cos\alpha}{\cos\alpha+\sin\alpha}\delta_{2i}\delta_{L_{i}\mu_{i}},$
(23c) $\displaystyle\langle\Pi_{P^{\prime}}^{\mu_{i}}\rangle$
$\displaystyle=\frac{\sin\alpha}{\cos\alpha+\sin\alpha}\delta_{2i}\delta_{R_{i}\mu_{i}}.$
(23d)

If $\alpha\neq 0,\frac{\pi}{2}$, these weak values indicated that the wave and
particle attributes of two entangled photons are perfectly separated, and in
each one of the four arms we only have one attribute (wave or particle ) of
the two photons. In the particular case, $\alpha=\frac{\pi}{4},$ the weak
values for the wave and particle attributes may be written as
$\langle\Pi_{P}^{L_{1}}\rangle=\langle\Pi_{W}^{R_{1}}\rangle=\frac{1}{2}$,
$\langle\Pi_{P^{\prime}}^{R_{2}}\rangle=\langle\Pi_{W^{\prime}}^{L_{2}}\rangle=\frac{1}{2}$.
This means that for $\alpha=\frac{\pi}{4}$ case, half of the particle
attributes of the two entangled photons occurrs in $L_{1}$ and $R_{2}$ arms
and half of the wave attributes are present in $R_{1}$ and $L_{2}$ arms of the
interferometer, respectively. Furthermore, it is clear that these weak values
cannot take arbitrary large values, and always obey the Bohr’s complementarity
principle, i.e.,

$\displaystyle\langle\Pi_{P}^{L_{1}}\rangle+\langle\Pi_{W}^{R_{1}}\rangle$
$\displaystyle=1,$ (24)
$\displaystyle\langle\Pi_{P^{\prime}}^{R_{2}}\rangle+\langle\Pi_{W^{\prime}}^{L_{2}}\rangle$
$\displaystyle=1.$ (25)

##  IV conclusion

In this paper, we have investigated the possibility of spatially separating
the wave and particle features of two entangled photons. Our work is based on
measuring the weak values of two-level quantum systems by selecting
appropriate pre- and post-selection states in order to generate polarization-
path like quantum Cheshire cats.

We found that properly pre- and post-selecting the two-photon state, the wave
and particle attributes of two entangled photons can be spatially separated,
while keeping the validity of Bohr’s complementarity principle. We also
proposed a feasible setup for the implementation of our thought experiment on
a quantum optical platform.

The quantum entanglement is a cornerstone of quantum informatics and it has an
irreplaceable role in quantum precision measurements. In particular, in
quantum metrology one would generally require that the measurement of a
physical quantity on an entangled state does non disturb to other variable.
Our results may foster the exploration of schemes to separate entangled
observables and enhance precision of quantum metrological protocols. At the
same time, in the emerging field of counterfactual communication (Salih  _et
al._ , 2013; Li  _et al._ , 2015), information may be exchanged even without
the transmission of quantum state. In this framework, entanglement also plays
a vital role, but the basic principles of communication is still unclear. Also
in this case, our proposal may promote the understanding of the fundamental
mechanism behind counterfactual communication.

Finally, we anticipate that our theoretical proposal is feasible with current
technology and may be implemented in quantum optical labs in the near future.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(No. 12365005).

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