1. Introduction

Schrödinger’s cat is one of the most famous concepts in quantum physics, which is a gedanken experiment formulated by Schrödinger in 1935 [1]. We first place a living cat and a radioactive atom inside a sealed box. An atomic decay triggers the death of the cat. Since the decay of the atoms occurs randomly, the time of death of the cat is unknown. In quantum physics, the situation of the cat inside the box is known as Schrödinger’s cat state, which is described by a coherent superposition of two classically distinguishable states, i. e., ‘alive’ and ‘dead’. In other words, the situation of the cat inside the box is simultaneously alive and dead, which seems to be weird or even counterintuitive for a cat in macroscopic world. However, such a mysterious Schrödinger’s cat state disappears once it is measured. If we open the box and observe the cat, Schrödinger’s cat state will be immediately projected onto a certain state, either alive or dead. Observation erases the quantum properties of Schrödinger’s cat, and transfers the superposition state of the microscopic quantum object into the macroscopic classical state.

To prepare such a Schrödinger’s cat state in the experiment, one should transpose Schrödinger’s cat to a more feasible physical system owning classical or quasi-classical properties. In quantum optics, coherent states $|\alpha \rangle$, nearly classical-like states, are used to prepare ‘Schrödinger’s cat’. An optical Schrödinger’s cat state (SCS) can be defined as a coherent superposition of two coherent states with opposite phase, i.e., $|\psi ^{\pm }_{\textrm {SCS}}\rangle =N_{\pm }(|\alpha \rangle \pm |-\alpha \rangle )$, where $N_{\pm }=(2\pm 2e^{-2\alpha ^{2}})^{-1/2}$ is the normalization factor. $|\psi ^{+}_{\textrm {SCS}}\rangle$ and $|\psi ^{-}_{\textrm {SCS}}\rangle$ are known as ‘even’ and ‘odd’ SCSs, because they contain only even and odd photon numbers, respectively. SCSs are of great interest in quantum optics, as they have remarkable properties that could allow them to have potential applications in various domains, such as fundamental tests of quantum theory [2–9], exploration of quantum-to-classical boundary [10–12], quantum information processing [13–17] and quantum metrology [18–20].

In recent decades, various schemes have been proposed to generate optical SCSs. All these schemes can be divided into two classes, according to choice of initial state and postselection scheme. The first one is to employ single-mode squeezed states as initial states, and use photon number measurement (PNM) in heralding the state preparation. In this endeavor, the first seminal scheme is subtracting one photon from a single-mode squeezed vacuum state [21–27]. It has been shown that such an 1-photon-subtracted squeezed state is extremely approximate to an ideal odd SCS with small size $|\alpha |\leq 1$. In order to achieve an approximate odd SCS with large size, it has been proposed to mix two single-mode squeezed states with opposite phase onto a linear beam splitter (BS) with an appropriate transimissivity [28,29]. By tapping a part of the light at one output port of the BS and detecting it with a photon number resolving detector (PNRD), a squeezed SCS with size $|\alpha |\sim 2$ can be heralded. The second class in generating optical SCSs is to employ photon number states (Fock states) as initial states, and use homodyne conditioning as the postselection scheme. For examples, it has been demonstrated that, by mixing a photon number state with vacuum onto a balanced BS and then using homodyne detection to measure the momentum quadrature $p$ in one mode, a squeezed SCS can be prepared under the condition that the measurement result $p$ is nearby zero [30]. In addition, ‘cat breeding’ scheme has also been proposed, which employs two 1-photon states to prepare a squeezed even SCS [8,31].

To date, most protocols for generating optical SCSs were based on remote state preparation (RSP). The RSP enables ones to create and manipulate quantum states remotely by using entanglement resources and a classical channel. RSP methods are of great significance in the application of quantum information processing due to its intrinsic security and high-efficiency [32]. It has been experimentally demonstrated that RSP methods can be used to prepare 1-photon state [33], single-mode photonic qubit [34], single-mode squeezed state [35] and hybrid entanglement between particle-like and wave-like optical qubits [36,37]. In the aspect of optical SCS generation, even-photon N00N state, also known as maximally path-entangled states of $N$ photons [38], has been proposed to remotely prepare an even SCS [39]. However, this scheme yields two severe practical problems. First, the parity of the generated SCS cannot be remotely manipulated. Second, high-N00N state is hard to be prepared in the experiment. Very recently, remote SCS preparation has also been experimentally realized via magnon-photon entanglement [40].

In this paper, we propose a ‘remote switch’ for SCS using Einstein-Podolsky-Rosen (EPR) entanglement. For an EPR entangled state shared by two modes $\mathcal {A}$ and $\mathcal {B}$, we demonstrate that a ‘hybrid projective measurement’ (HPM) performed by mode $\mathcal {A}$ is able to herald an approximate SCS generated at mode $\mathcal {B}$. Remarkably, the HPM can not only manipulate the size of the generated SCS but also control its parity. Here, the HPM consists of both PNM and homodyne conditioning. PNM acts as a photon creator or annihilator, which fulfills $m$-photon-addition ($m$-PA) or $m$-photon-subtraction ($m$-PS) on mode $\mathcal {A}$. Homodyne conditioning measures the momentum quadrature of mode $\mathcal {A}$. An approximate odd SCS can be heralded at party $\mathcal {B}$ when PNM is triggered for 1-PS (or 1-PA), while at the same time the measurement result of homodyne conditioning is nearby zero. Similarly, an approximate even SCS can be produced, using the same HPM scheme as that of the odd SCS preparation with the corresponding trigger condition of PNM varied from 1-PS (or 1-PA) to 2-PA. Resorting to nonlocal correlations, the HPM performed at mode $\mathcal {A}$ enables one to remotely prepare and fully manipulate a SCS at its entangled mode $\mathcal {B}$. Therefore, our scheme can be viewed as a remote switch for SCS.

2. Theoretical model for generating SCS using EPR entanglement

The diagrammatic sketch of remote switch for SCS is shown in Fig. 1. The EPR entangled state shared by two modes $\mathcal {A}$ and $\mathcal {B}$ can be expressed as $\hat {U}_{\textrm {S}}|0\rangle _{\mathcal {A}}\otimes |0\rangle _{\mathcal {B}}$ with two-mode squeezing operator $\hat {U}_{\textrm {S}}=\textrm{exp}[ {r}(\hat { {a}}\, \hat { {b}}-\hat { {a}}^{\dagger }\,\hat { {b}}^{\dagger })]$. Here, $|0\rangle _{\mathcal {A}}\otimes |0\rangle _{\mathcal {B}}$ represents the tensor product of two vacuum states, $r$ quantifies the two-mode squeezing [41], $\hat {a}$ ($\hat {a}^{\dagger }$) and $\hat {b}$ ($\hat {b}^{\dagger }$) are the annihilation (creation) operators of the corresponding modes. In momentum quadrature basis, the wavefunction of the EPR entangled state can be written as

 

Fig. 1. Diagrammatic sketch of remote switch for SCS using EPR entanglement. $|0\rangle _{\mathcal {A}}$ and $|0\rangle _{\mathcal {B}}$, vacuum states; $|\Psi _{\textrm {out}}\rangle _{\mathcal {B}}$, output state achieved at mode $\mathcal {B}$; $\hat {U}_{\textrm {S}}$, two-mode squeezing operation; $\hat {U}_{\textrm {PLO}}$, photon level operation; PNRD, photon number resolving detector; HD, homodyne detector; PSP, postselction protocol. The dashed boxes in the bottom denote three types of PLOs: (i), 1-photon-subtraction (1-PS); (ii), 1-photon-addition (1-PA); (iii), 2-photon-addition (2-PA). BS, beam splitter; PDCA, parametric down-conversion amplifier.

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After the preparation of the EPR entangled state, a HPM performed on mode $\mathcal {A}$ is able to herald an approximate SCS in mode $\mathcal {B}$. The HPM is a measurement consisting of both PNM and homodyne conditioning.

2.1 PLO protocol

Photon lever operation (PLO) can be viewed as a photon creator or annihilator, which adds or annihilates $m$-photon to/from mode $\mathcal {A}$. As shown by the dashed boxes in Fig. 1, three types of PLOs are considered: PLO$_{\textrm {i}}$, PLO$_{\textrm {ii}}$ and PLO$_{\textrm {iii}}$, which perform 1-PS, 1-PA and 2-PA on mode $\mathcal {A}$, respectively.

2.1.1. 1-PS scheme

PLO$_{\textrm {i}}$ annihilates 1-photon from mode $\mathcal {A}$. It can be performed by placing a BS on mode $\mathcal {A}$ with transmissivity $0\ll \tau <1$. Then the wavefunction of the state after the BS becomes

2.1.2. 1-PA and 2-PA schemes

PLO$_{\textrm {ii}}$ and PLO$_{\textrm {iii}}$ add 1-photon and 2-photon to mode $\mathcal {A}$, respectively. Such photon addition operations can be performed by parametric down-conversion amplifiers (PDCAs) [44]. Essentially, PDCA is a two-mode squeezer. When a nonlinear media, such as nonlinear crystal or atomic vapor cell, is pumped by a high-energy light, pairs of entangled photons are produced and emitted into symmetrically oriented directions, called signal and idler modes. In order to fulfill the photon-addition operation, mode $\mathcal {A}$ is seeded into the signal mode of the PDCA. Then the wavefunction of the state after the PDCA is given by

Similarly, the wavefunction $\Psi ^{\prime}_{\textrm {iii}}(p_{\mathcal {A}},p_{\mathcal {B}})$ of the state with 2-PA performed on mode $\mathcal {A}$ can be achieved by replacing $\Psi ^{\ast}_{1}(p_{\mu })$ with $\Psi ^{\ast}_{2}(p_{\mu })$ in Eq. (5), where $\Psi _{2}(p_{\mu })=2^{-3/2}\pi ^{-1/4}(2-4p^{2}_{\mu })\exp (-p^{2}_{\mu }/2)$ is the wavefunction of 2-photon Fock state.

2.2 Homodyne conditioning protocol

After the PNM, a homodyne detector (HD) is used to measure the momentum quadrature of mode $\mathcal {A}$. The corresponding measurement result is denoted as $p^{c}$. Such projective measurement makes the wavefunction of the output state becomes $\Psi ^{\prime}_{j}(p^{c}, p_{\mathcal {B}})$, where $j=\textrm {i}, \textrm {ii}, \textrm {iii}$. In the following, we show that an approximate SCS can be generated by a local success of conditional HPM$_{j}$. HPM$_{j}$ is fulfilled on the condition that PNM$_{j}$ is triggered, while at the same time the measurement result of the homodyne conditioning is $p^{c}=0$. In addition, it should be noted that exact measurement $p^{c}=0$ is unfeasible in the experiment, because it will lead to a zero success probability. Thus, a threshold $\Delta p^{c}$ has to be accepted in the practical implementation. The desired state is postselected for $|p^{c}|\le \Delta p^{c}\ll 1$. The selection of $\Delta p^{c}$ impacts not only the success probability of the state preparation, but also the fidelity between the output state and ideal SCS.

3. Results

In what follows, we show that an approximate odd SCS can be heralded at mode $\mathcal {B}$ when either HPM$_{\textrm {i}}$ or HPM$_{\textrm {ii}}$ is performed on mode $\mathcal {A}$. An approximate even SCS is heralded when HPM$_{\textrm {iii}}$ is performed. To obtain an interesting insight into the structure of the prepared state, it is easier to analyze the wavefunction in position quadrature basis. This can be achieved via a Fourier transform for the wavefunction $\Psi ^{\prime}_{j}(p^{c},p_{\mathcal {B}})$ with its variable transformed from $p_{\mathcal {B}}$ to $x_{\mathcal {B}}$, i.e.,

3.1 Odd SCS

The wavefunction of the output state from HPM$_{\textrm {i}}$ is given by

One can immediately notice that the output state in Eq. (7) has the same wavefunction of a squeezed 1-photon state. Similar results can also be found for the output state with HPM$_{\textrm {ii}}$ performed on mode $\mathcal {A}$. Its corresponding wavefunction in position quadrature basis is given by

The structure of the wavefunction $\Psi ^{\prime\prime}_{j}(0, x_{\mathcal {B}})$ as shown in Eqs. (7) and (9) has two peaks at $\pm \frac {1}{\sqrt {\xi _{j}}}$, where $j=\textrm {i}, \textrm {ii}$. On the other hand, the wavefunction of an ideal SCS in position quadrature basis has the mathematical form

 

Fig. 2. Fidelity $\mathcal {F}^{-}_{\textrm {i},\textrm {ii}}(\alpha, 0)$ versus coherent amplitude $\alpha$ of the ideal odd SCS. From $\alpha$ from 0 to 2, the blue dashed and red solid lines are the fidelity of the output states from HPM$_{\textrm {i}}$ and HPM$_{\textrm {ii}}$, respectivley. The inset shows $\tau _{\textrm {opt}}$ (with reference to the left-hand $y$ axis) and $r^{\prime}_{\textrm {opt}}$ (with reference to the right-hand $y$ axis) versus $\alpha$ for the corresponding HPM schemes. The three dashed boxes from top to bottom show the comparison of the Wigner function between the output states from HPM$_{\textrm {i}}$ (left-hand side) with $\tau =\tau _{\textrm {opt}}=0.37, 0.65, 0.85$, and HPM$_{\textrm {ii}}$ (middle) with $r^{\prime}=r^{\prime}_{\textrm {opt}}=1.08, 0.68, 0.41$, and the ideal odd SCS (right-hand side) with $\alpha =1, 1.5, 2$. The degree of the initial EPR entanglement is fixed at $r=1.5$ in plotting figure.

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In the above discussion, we have shown that an approximate odd SCS can be heralded on mode $\mathcal {B}$ if the homodyne conditioning on mode $\mathcal {A}$ with 1-photon added to (or 1-photon subtracted from) is $p^{c}=0$. However, exact measurement $p^{c}=0$ is impossible. In the practical implementation, one has to postselect the desired state within a window $p^{c}\in [-\Delta p^{c}, \Delta p^{c}]$, where $0<\Delta p^{c}\ll 1$ is the threshold. Then the fidelity in Eq. (12) is rewritten as

Based on numerical analysis, Fig. 3 shows the fidelity $\mathcal {F}^{-}_{\textrm {i}(\textrm {ii})}(\alpha, \Delta p^{c})$ between the postselected output state and an ideal odd SCS with $\alpha =1.2$, versus threshold $\Delta p^{c}$. The parameters $\tau$ and $r^{\prime}$ in plotting the dashed and solid lines in Fig. 3 are chosen as $0.49$ and $0.89$, which correspond to $\tau _{\textrm {opt}}$ and $r^{\prime}_{\textrm {opt}}$ as shown by the inset in Fig. 2. Our results show that an approximate odd SCS with high fidelity can be produced using experimentally realizable HPMs with finite thresholds. The corresponding success probability increases with the increasing of threshold. In addition, our numerical calculations show that HPM$_{\textrm {i}}$ is advantageous over HPM$_{\textrm {ii}}$, when the threshold is nonzero. Such advantages can be found not only in the fidelity but also in the success probability.

 

Fig. 3. Fidelity $\mathcal {F}^{-}_{\textrm {i},\textrm {ii}}(\alpha, \Delta p^{c})$ between the postselected output state and an ideal odd SCS with $\alpha =1.2$, versus threshold $\Delta p^{c}$. The red dashed (red solid) line with reference to the left-hand $y$ axis is the fidelity of the output state from HPM$_{\textrm {i}}$ (HPM$_{\textrm {ii}}$) with threshold considered, when the homodyne conditioning is $|p^{c}|\le \Delta p^{c}$. The blue dashed and blue solid lines with reference to the right-hand $y$ axis are the corresponding success probability. The parameters in plotting the dashed and solid lines are as follows: $\{r=1.5, \tau =0.49\}$ and $\{r=1.5, r^{\prime}=0.89\}$.

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3.2 Even SCS

We have shown that either HPM$_{\textrm {i}}$ or HPM$_{\textrm {ii}}$ performed at mode $\mathcal {A}$ can herald an approximate odd SCS at its entangled mode $\mathcal {B}$. In what follows, we show that our scheme can also produce an approximate even SCS. The corresponding scheme is HPM$_{\textrm {iii}}$, which adds 2-photon to mode $\mathcal {A}$ in PLO as shown by the third dashed box in Fig. 1. Then if the measurement result in the homodyne conditioning is $p^{c}=0$, the wavefunction of the output state is given by

 

Fig. 4. Fidelity $\mathcal {F}^{+}_{\textrm {iii}}(\alpha, 0)$ versus coherent amplitude $\alpha$ of the ideal even SCS. The line in the inset denotes the corresponding $r^{\prime}_{\textrm {opt}}$ maximizing the fidelity. The three dashed boxes from top to bottom show the comparison of the Wigner function between the output state from HPM$_{\textrm {iii}}$ (left-hand side) with $r^{\prime}=r^{\prime}_{\textrm {opt}}=1.12, 0.77, 0.55$, and the ideal even SCS (right-hand side) with $\alpha =1, 1.5, 2$. The degree of the initial EPR entanglement is fixed at $r=1.5$ in plotting figure.

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Figure 5 shows the fidelity $\mathcal {F}^{+}_{\textrm {iii}}(\alpha, \Delta p^{c})$ between the output state from HPM$_{\textrm {iii}}$ and an ideal even SCS with $\alpha =1.2$, when the threshold of the homodyne conditioning is nonzero. The parameter $r^{\prime}$ in plotting Fig. 5 is chosen as $r^{\prime}=r^{\prime}_{\textrm {opt}}=0.95$. It is clear that our scheme is feasible to produce an approximate even SCS for the nonzero postselection threshold.

 

Fig. 5. Fidelity $\mathcal {F}^{+}_{\textrm {iii}}(\alpha, \Delta p^{c})$ between the postselected output state from HPM$_{\textrm {iii}}$ and an ideal even SCS with $\alpha =1.2$, versus threshold $\Delta p^{c}$. The solid line with reference to the right-hand $y$ axis is the corresponding success probability. The parameters in plotting figure is as follows: $r=1.5, r^{\prime}=0.95$.

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4. Discussions

Our scheme enables one to realize the remote switch for SCS. In general, SCS preparation can be divided into two steps: initial state preparation and postselection. These two steps can also be viewed as degrees-of-freedoms to generate or manipulate SCS. Take two seminal works [28,30] as examples. In Ref. [30], it has been demonstrated that a SCS can be generated by mix a photon number state with vacuum onto a balanced BS and then use conditional momentum quadrature measurement in the postselection step. One can manipulate the initial photon number state to control both the size and parity of the generated SCS. In Ref. [28], one can generate a SCS by mixing two single-mode squeezed states with opposite phase onto a BS and use conditional PLO in the postselection step. One can manipulate the initial squeezing levels of the two single-mode squeezed states or the transmitivity of the BS to control the size of the generated SCS and vary the number of PLO to change its parity. Our present work is based on RSP, which transfers the degrees-of-freedom for SCS manipulation to a distant party with the help of nonlocality of EPR entanglement. We propose to perform HPM on one mode $\mathcal {A}$ of an EPR entangled state, and demonstrate that the an approximate SCS can be heralded at the other mode $\mathcal {B}$ when the probabilistic HPM is triggered, resorting to the non-local unfactorizable EPR entanglement. In our scheme, one can control the parity of the generated SCS, via varying the photon number manipulated in PLO. In addition, PLO provides another degrees-of-freedom to manipulate the size of the generated SCS. As shown in Fig. 2 and Fig. 4, the size of the approximate SCS is dependent on $\tau$ and $r^{\prime}$ in PLO. Such properties can also be explained by Eqs. (8) and (9). For an initial EPR entangled state with a given squeezing degree $r$, an appropriate $\tau$ (or $r^{\prime}$) can always be found, which maximizes the fidelity between the output state and ideal SCS with coherent amplitude $\alpha$. Thus, the HPM performed at mode $\mathcal {A}$ can fully manipulate both size and parity of the SCS generated at its entangled mode $\mathcal {B}$. Such properties also make our scheme advantageous over the method proposed in Ref. [39], which is limited to the parity manipulation.

Our scheme can be expandable to further enhance the size of the generated SCS state. In the above discussions, three types of HPMs performed at one mode of an ERP entangled state are considered to prepare an approximate SCS generated at the other one mode. In the next, we show that an approximate SCS with larger coherent amplitude can be prepared as the photon number manipulated in the photon number measurement increases. To see this intuitively, let us describe the output state in photon number basis. Take the HPM with $m$-photon-addition scheme performed in the PLO as an example. When the measurement result of the homodyne conditioning is $p^{c}=0$, then the corresponding output state is given by

5. Conclusion

In conclusion, we have demonstrated that, for an EPR entangled state shared between two modes $\mathcal {A}$ and $\mathcal {B}$, an approximate SCS can be heralded at mode $\mathcal {B}$ on the condition that a probabilistic HPM is performed on mode $\mathcal {A}$. The HPM consists of PNM and homodyne conditioning. We have shown that the HPM can fully manipulate the generated SCS, including both its size and parity. Different from the previous SCS protocols, our present scheme uses the nonlocal correlations of EPR entanglement to realize the remote preparation and manipulation of SCS. Such a remote switch for SCS may open up new ideas for remote non-Gaussian state manipulation.