Creation and manipulation of Schrödinger cat states based on semiclassical predictions

Abstract

We consider the generation of Schrödinger cat states using a quantum measurement-induced logical gate where entanglement between the input state of the target oscillator and the Fock state of the ancillary system produced by the quantum non-demolition entangling {\hat{C}}_Z operation is combined with the homodyne measurement. We utilize the semiclassical approach to construct both the input-output mapping of the field variables in the phase space and the wave function of the output state. This approach is found to predict that the state at the gate output can be represented by a minimally disturbed cat-like state which is a superposition of two copies of the initial state symmetrically displaced by momentum variable. For the target oscillator prepared in the coherent state, we show that the fidelity between the exact solution for the gate output state and the “perfect” Schrödinger cat reconstructed from the semiclassical theory can reach high values exceeding 0.99.

Graphical abstract

Keywords

Schrödinger cat state / homodyne measurement / measurement-induced logical gate / semiclassical approach

1 Introduction

Continuous-variable (CV) optical systems are a promising platform for large-scale quantum information processing. The properties of the CV schemes with embedded non-Gaussian gates are being actively investigated nowadays [1–3]. Along with the large-scale Gaussian operations enabled by the cluster states [4, 5], quantum non-Gaussian gates are essential components [6, 7] for a variety of practical tasks in optical quantum technology and advanced quantum information processing including quantum communication, quantum computation, quantum algorithms, and quantum control. It has been found that non-Gaussianity [8–11] in the form of non-Gaussian quantum states [12–21] and non-Gaussian operations [2, 22–27] is crucial, due to the limited capability of Gaussian states and operations, for various CV quantum information protocols required for quantum teleportation [28–30], entanglement distillation [31, 32], error correction [33, 34], fault-tolerant universal quantum computing [2, 35–38], loophole-free test of quantum non-locality [39, 40], and quantum simulations [41, 42].

Optical Schrödinger cat states have been considered as a substantial non-Gaussian resource for practical employment in quantum science since the early 2000s [43, 44]. They play a considerable role in up-to-date quantum technologies [44–55] and CV quantum information processing [44, 54, 56, 57], including quantum metrology [52, 58–60], quantum teleportation [61, 62], quantum communication and quantum repeaters [48–51], and error correction schemes for fault-tolerant quantum computing [44, 45, 46, 63–67]. From a basic research point of view, the Schrödinger cat states have been of great interest both for testing the foundations of quantum mechanics and determining the limits of its validity by exploring the quantum-to-classical transition.

The main challenge for most applications in quantum information technologies is to produce Schrödinger cat-like states whose “size”, namely the distance in phase space between the two coherent states, is sufficiently large to enable good-quality operations [43, 45, 68–70]. Such large-amplitude coherent-state superpositions exhibiting unique non-classical attributes such as sub-Planck phase-space structures [71] and non-Gaussian interference features [72–74] are commonly used as a base for preparing qubits in CV quantum computing [43, 75], and a resource for quantum error-correcting codes [35, 76–78].

Currently, cat-like coherent superpositions are being successfully simulated in various physical systems [79–83], but developing realistic schemes to produce optical Schrödinger cat states with a large number of photons and controlled quantum properties remains a challenging task. To achieve this goal, a variety of well-established conditional generation approaches have been proposed, including schemes based on such non-Gaussian operations as photon-number measurement and subtraction [47, 84–90] and other cat-states generation methods [12, 69, 70, 86, 91–93], some of which have been successfully implemented [57, 69, 88, 89, 91–93]. In nondeterministic schemes which create the target state only under predetermined conditions, the auxiliary channel can be prepared in the Fock state [68, 69, 91, 93, 94], or even in complex superpositions which arise in the iterative cat breeding schemes [69, 70, 95, 96]. In particular, a low-frequency regime of optical Schrödinger cat state generation using the Fock state as a resource, a beam splitter as an entangling element, and homodyne detection was experimentally demonstrated [69, 91].

In this paper, we consider the conditional generation of a Schrödinger cat state from an arbitrary coherent state employing a two-node non-Gaussian gate (“cat gate”) based on the Fock state of the ancillary oscillator as an elementary non-Gaussian resource, the quantum non-demolition (QND) entangling operation {\hat{C}}_Z, and the projective homodyne measurement. In the context of the above discussed conditional-generation platforms, this Fock-state–based {\hat{C}}_Z gate that not only achieves the fidelities comparable to those reported in leading homodyne-postselected schemes such as [91] but also delivers substantially higher heralding rates and flexible tuning of non-Gaussian resources through its measurement-dependent operation.

The controlled-Z (CZ) gate {\hat{C}}_Z =e^{i{\hat{q}}_1{\hat{q}}_2} is a canonical two-mode operation and CV analog of the two-qubit CPHASE gate, recognized as a fundamental CV quantum gate. Despite the cross-Kerr nonlinear interactions provide a conceptual pathway for generating photon-number-dependent phase correlations between optical modes [97, 98], experimental implementations of the CZ gate using nonlinear media remain challenging due to weak nonlinearities and noise in materials like crystals/fibers [99]. Alternative experimental approaches include teleportation-based realizations via squeezed cluster states [100–102]. While finite squeezing constrains the fidelity in cluster-state implementations [101], these approaches leverage deterministic generation and multiplexing capabilities of optical systems that enable large-scale entanglement for quantum applications [103]. We show that a CV quantum circuit with such measurement-induced two-node non-Gaussian element may generate a cat-like quantum superposition and, under optimal conditions, the output state is close to the “perfect” Schrödinger cat state which is a superposition of the two (in general, more) symmetrically displaced undistorted copies of the input state. In parallel with the exact theoretical description of the gate operation, we introduce a clear visual interpretation of the output state based on the semiclassical mapping of the input field variables and construct the semiclassical wave function of the undisturbed cat state closest to the exact output state.

This study builds upon and significantly extends the work [104], where the generation of Schrödinger cat states was examined for input wavefunctions localized near the origin. Here, we generalize this approach to arbitrary coherent states, thereby broadening the applicability of the method. In addition, we introduce a geometric semiclassical mapping that provides an intuitive visualization of the transformation induced by the non-Gaussian gate. By establishing a quantitative correspondence between the exact output state and its semiclassical counterpart, we demonstrate that the fidelity remains high across a wide range of input parameters. Furthermore, we analyze the inherent constraints of the prior method and propose refinements that enhance the predictive accuracy of the semiclassical description. In contrast to Ref. [104] which is focused on making a comparison between the cubic phase and Fock resource states, this manuscript provides a dedicated investigation of Fock-state-based gates, establishing their advantages in fidelity and success probability through both exact and semiclassical frameworks. Note that the results of a complementary study dealing with the cubic phase resource states has been recently reported in the preprint [105].

The scheme can produce optical Schrödinger cat states of any desired size with high fidelity exceeding 0.99 and we find the conditions under which the gate generates high-quality cat-like states by computing the fidelity between the exact output state and the superposition of two symmetrically displaced undistorted copies of the input coherent state. We also demonstrate the output state quantum statistics in terms of the Wigner function.

The paper is organized as follows. In Section 2, we provide a semiclassical description of the creation of Schrödinger cat states using a measurement-induced two-node non-Gaussian logic gate. In Section 3, we derive the target oscillator’s exact output state following the non-Gaussian gate’s action, providing an analytic expression for the output wave function. Section 4 focuses on generating cat-like states from a coherent state. Section 5 concludes the paper. Technical details on the method used to evaluate the Wigner functions are relegated to Appendix A.

2 Semiclassical description of Schrödinger cat state creation

Following to Ref. [104], we consider the measurement-induced two-node non-Gaussian logic gate shown in Fig.1. It can be seen that this gate uses the photon number (Fock) state of the ancilla as an elementary non-Gaussian resource, the {\hat{C}}_Z operation which entangles an input signal with the ancilla, and the projective homodyne measurement. An essential feature of the scheme is that the ancilla measurement outcome provides multivalued information about the target oscillator output momentum that gives rise to a cat-like output state. This peculiarity can be easily interpreted [106] in terms of a clear visual representation of the quadrature amplitudes transformations in the scheme which demonstrates that two-component (or multi-component) Schrödinger cat state arises when the measurement outcome is compatible with not one but with multiple (two or more) values of the target oscillator variables. Such a pictorial description might also be useful for the analysis of measurement-induced schemes based on more complicated non-Gaussian resource states where a closed-form (analytic) expression for the output state is not available.

According to Ref. [104], for the photon number state-based gate, a cat-like superposition of two “copies” of the target state closest to the exact output state can be effectively evaluated using the semiclassical approximation. To this end, we assume the Heisenberg representation and consider how the canonical variables, coordinate and momentum operators, of both oscillators are transformed under the action of the entangling operation. Then the relations imposed on the canonical variables are treated as c-numerical and the measurement of the ancilla momentum is also described semiclassically by substituting momentum with its observed value giving an explicit expression for the input−output mapping between the target oscillator variables.

More specifically, let us introduce the coordinate \hat{q} and momentum \hat{p} operators for each of the oscillators in conventional way as \hat{a}=(\hat{q}+\mathrm{i}\hat{p})/ \sqrt{2}, where [ \hat{q},\hat{p}]= \mathrm{i}. Next, the two-mode entangling unitary evolution operator {\hat{C}}_Z=\exp (\mathrm{i}\hat{q}{\hat{q}}_a) is applied to the initial state of the oscillators. In the Heisenberg picture, we have the relations

(1)\begin{array}{rl}& {\hat{q}}^{(\mathrm{o}\mathrm{u}\mathrm{t})}={\hat{q}}^{(\mathrm{i}\mathrm{n})},{\hat{p}}^{(\mathrm{o}\mathrm{u}\mathrm{t})}={\hat{p}}^{(\mathrm{i}\mathrm{n})}+{\hat{q}}_a^{(\mathrm{i}\mathrm{n})},\\ & {\hat{q}}_a^{(\mathrm{o}\mathrm{u}\mathrm{t})}={\hat{q}}_a^{(\mathrm{i}\mathrm{n})},{\hat{p}}_a^{(\mathrm{o}\mathrm{u}\mathrm{t})}={\hat{p}}_a^{(\mathrm{i}\mathrm{n})}+{\hat{q}}^{(\mathrm{i}\mathrm{n})},\end{array}

where the index a marks the ancilla variables. In what follows, the variables of the subsystems that enter Eq. (1) will be interpreted as c-numbers. For an ancilla initially prepared in the Fock resource state | n⟩, n is the photon number, the semiclassical amplitudes q_a and p_a are related as follows

(2)q_a^{(\mathrm{i}\mathrm{n})2}+p_a^{(\mathrm{i}\mathrm{n})2}=2n+ 1.

After applying the operation {\hat{C}}_Z, the resource state is described by the resource curve equation

(3)q_a^{(\mathrm{o}\mathrm{u}\mathrm{t})2}+(p_a^{(\mathrm{o}\mathrm{u}\mathrm{t})}-q^{(\mathrm{i}\mathrm{n})}{)}^2=2n+1,

which is a circle of radius \sqrt{2n+1} vertically displaced by q^{(\mathrm{i}\mathrm{n})} (see Fig.2).

Finally, a homodyne measurement of the ancillary oscillator momentum p_a^{(\mathrm{o}\mathrm{u}\mathrm{t})} with outcome y_m is described by the change p_a^{(\mathrm{o}\mathrm{u}\mathrm{t})}\to y_m giving two values of the ancilla coordinate: q_a^{(\mathrm{o}\mathrm{u}\mathrm{t})}=x_a^{(\pm )}\equiv \pm \sqrt{2n+1-(y_m-q^{(\mathrm{i}\mathrm{n})}{)}^2}. As is shown in Fig.2, these values are the coordinates of the points of intersection of the horizontal line p_a^{(\mathrm{o}\mathrm{u}\mathrm{t})}=y_m with the circle (3). Fig.2 presents a clear geometrical description based on the semiclassical mapping that directly indicates the number, position, and offset of these points depending on the target oscillator coordinate.

Owing the quantum correlation of the target and ancillary subsystems, the ambiguity of the solution for the ancilla coordinate arising from results in the possibility of multiple values of the target oscillator momentum (hereafter x\equiv q denotes the target oscillator coordinate) that enter the semiclassical mapping

(4)(q^{(\mathrm{o}\mathrm{u}\mathrm{t})},p^{(\mathrm{o}\mathrm{u}\mathrm{t})})=(x,p^{(\mathrm{i}\mathrm{n})}\pm \sqrt{2n+1-(y_m-x{)}^2}),

and, under additional conditions discussed below, to the emergence of a cat-like state which is a coherent superposition of two macroscopically distinguishable coherent states.

Note that Eq. (4) implies distortion of the copies of the initial state: at any fixed y_m the region of phase space where x\approx y_m experiences the greatest displacement that decreases with the magnitude of the difference | y_m-x| leading to deformation of the cat components. The distortion of copies is minimal if, for a given measurement outcome y_m, the resource curve at its intersection points is x_a^{(\pm )} of the horizontal line p_a^{(\mathrm{o}\mathrm{u}\mathrm{t})}=y_m is close to vertical (Fig.2, the top row of (b) column). In this case, measurement-induced splitting of the ancilla coordinate (indicated by dashed horizontal arrows) is almost independent of the target oscillator coordinate x (i.e., from the resource curve’s vertical shift). In general, the ideal case is when the coordinates of the intersection points x_a^{(\pm )} do not change when shifting the position x of the initial point, chosen within the support region of the target oscillator in the phase space. For the resource Fock state, as can be seen from Fig.2, the deformation of the copies is minimal for the phase space points belonging to the region x\approx y_m—for them the horizontal dashed arrows lie on the diameter of the Fock circle.

In order to illustrate these effects, let us assume that the target oscillator is prepared in the coherent state |\alpha ⟩ with the amplitude \alpha = (x_0+\mathrm{i}{p}_0)/\sqrt{2} and the uncertainty region (x- x_0{)}^2+(p-p_0{)}^2\le 1 indicated in Fig.3 as a blue colored circle. Fig.3 shows what happens to the uncertainty circle under the semiclassical mapping (x,p )\mapsto (x,p \pm \sqrt{2n+1-(y_m-x{)}^2}) (images of the circle are shown as the orange colored regions) at different values of y_m. In Fig.3 the area liable to minimal deformation is: (a) x\approx 3 (the center of the uncertainty circle); (b) x\approx 2 (the left edge of the uncertainty circle); (c) x\approx 1 (the set of points outside the uncertainty region of the initial coherent state).

The obtained semiclassical relations can be used to reconstruct the target oscillator wave function when the ancilla oscillator in the Fock state. Indeed, in the semiclassical treatment, as is shown in Ref. [104], the input-output mapping performed by the gate under consideration

(5){\psi }_{\mathrm{o}\mathrm{u}\mathrm{t}}(x )={\psi }_{\mathrm{i}\mathrm{n}}(x ){\phi }_{\mathrm{s}\mathrm{c}\mathrm{l}}(x) ,

can be described by multiplying the input wave function of the target oscillator {\psi }_{\mathrm{i}\mathrm{n}}(x) by the factor

(6)\begin{array}{rl}{\phi }_{\mathrm{s}\mathrm{c}\mathrm{l}}(x)\sim & \sqrt{P^{(+)}(x,y_m)}\exp \left[\mathrm{i}\int \mathrm{d}x\delta p(x)\right]\\ & +\sqrt{P^{(-)}(x,y_m)}\exp \left[-\mathrm{i}\int \mathrm{d}x\delta p(x)\right],\end{array}

which is the sum of the added factors coming from two intersection points of the resource curve with the horizontal line p_a^{(\mathrm{o}\mathrm{u}\mathrm{t})}=y_m. It depends only on the state of the ancilla, the target oscillator coordinate x, and the ancilla momentum measurement outcome y_m. Equation (5) reflects universality of the gate implying that its action is independent of the target oscillator state at the input of the scheme.

From Eq. (4), we have

(7)\delta p(x)\equiv p^{(\mathrm{o}\mathrm{u}\mathrm{t})}-p^{(\mathrm{i}\mathrm{n})}=\pm \sqrt{2n+1-(y_m-x{)}^2},

and, according to [104], the quantities P^{(\pm )} given by

(8)P^{(+)}(x,y_m)=P^{(-)}(x,y_m)\sim {\displaystyle \frac{1}{|\delta p(x)|}}

can be interpreted as the classical probability of obtaining the measurement outcome y_m at the value of the ancilla coordinate in the vicinity of the overlap points x_a^{(\pm )} \equiv \pm \sqrt{2n+1-(y_m-x{)}^2} of the horizontal line p_a^{(\mathrm{o}\mathrm{u}\mathrm{t})}=y_m with the resource curve (p_a^{(\mathrm{o}\mathrm{u}\mathrm{t})}-x{)}^2+ x_a^2=2n+1 (see Fig.2(b)).

The exponential factors in Eq. (6) provide the symmetrical displacement of the output state components in phase space along the momentum axis by \delta p(x). Substituting Eq. (7) into the momentum produced by the gate in the semiclassical added factor (6) gives the factor in the following explicit form:

(9){\phi }_{\mathrm{s}\mathrm{c}\mathrm{l}}(n,z )\sim {\displaystyle \frac{1}{(1-z^2{)}^{1/4}}}[{\mathrm{e}}^{\mathrm{i}\varphi (n,z)}+(- 1{)}^n{\mathrm{e}}^{-\mathrm{i}\varphi (n,z)}],

where

(10)\varphi (n, z)\equiv {\displaystyle \frac{1}{2}}(2n+1)\left(z\sqrt{1-z^2}+\arcsin z\right),

(11)z\equiv {\displaystyle \frac{x-y_m}{\sqrt{2n+1}}}.

3 Exact output state

In the Schrödinger representation, the action of the non-Gaussian gate schematically depicted in Fig.1 can be described based on the von Neumann reduction postulate. This consideration yields an exact analytic expression for the output wave function of the target oscillator initially prepared in a quantum state that, in the coordinate representation, is given by

(12)| {\psi }_t^{(\mathrm{i}\mathrm{n})}⟩={\displaystyle \int \mathrm{d}x{\psi }^{(\mathrm{i}\mathrm{n})}(x)|x⟩,}

where {\psi }^{(\mathrm{i}\mathrm{n})}(x) is the wave function of the input state. The ancillary oscillator is prepared in the Fock state with n photons

(13)| {\psi }_a⟩={\displaystyle \int \mathrm{d}{x}_1{\psi }^{(n)}(x_1)|x_1⟩}

with the wave function

(14){\psi }^{(n)}(x_1)={\displaystyle \frac{1}{{\pi }^{1/4}\sqrt{2^{n}n!}}} H_n(x_1){\mathrm{e}}^{-x_1^2/2},

where H_n(x_1) is the Hermite polynomial.

A two-mode entangling QND operation {\hat{C}}_Z applied to the composite system state |{\psi }_t^{(\mathrm{i}\mathrm{n})} ⟩\otimes | {\psi }_a⟩ leads to the state given by

(15)\begin{array}{rl}{\hat{C}}_Z|{\psi }_t^{(\mathrm{i}\mathrm{n})}⟩\otimes |{\psi }_a⟩=& \int {\psi }^{(\mathrm{i}\mathrm{n})}(x){\psi }^{(n)}\left(x_1\right)\\ & \times {\mathrm{e}}^{\mathrm{i}x{x}_1}|x⟩\otimes |x_1⟩\mathrm{d}x\mathrm{d}{x}_1.\end{array}

A subsequent projective ancilla momentum measurement performed on the state (15) with the outcome p_a^{(\mathrm{o}\mathrm{u}\mathrm{t})}=y_m and the corresponding momentum eigenstate

(16)|y_m⟩={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle \int {\mathrm{e}}^{\mathrm{i}{y}_m{x}_1}|x_1⟩\mathrm{d}{x}_1}

results in a reduction of the total state that can be conveniently described in terms of the unnormalized output state of the target oscillator state

(17)\begin{array}{rl}|{\tilde{\psi }}_{\mathrm{o}\mathrm{u}\mathrm{t}}⟩=& ⟨y_m|{\hat{C}}_Z|{\psi }_t^{(\mathrm{i}\mathrm{n})}⟩\otimes |{\psi }_a⟩\\ =& \int {\psi }^{(\mathrm{i}\mathrm{n})}\left(x\right)\mathcal{F}[{\psi }^{(n)}](x-y_m)|x⟩\mathrm{d}x,\end{array}

where we have used the relation ⟨y_m|x_1⟩={\mathrm{e}}^{-\mathrm{i}{y}_m{x}_1}/\sqrt{2\pi } and \mathcal{F}[{\psi }^{(n)}] is the Fourier transform of {\psi }^{(n)} given by

(18)\mathcal{F}[{\psi }^{(n)}](p )={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle \int {\mathrm{e}}^{\mathrm{i}p{x}_1}{\psi }^{(n)}(x_1)\mathrm{d}{x}_1={\mathrm{i}}^n{\psi }^{(n)}(p).}

So, the gate induced factor is the Fourier transform of the resource state coordinate wave function corresponding to the value of momentum x- y_m and the closed-form expression for the wave function of the output state reads

(19){\psi }^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x,y_m)={\displaystyle \frac{1}{\sqrt{N}}}{\tilde{\psi }}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x, y_m),

(20)\begin{array}{rl}{\tilde{\psi }}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x,y_m)=& {\psi }^{(\mathrm{i}\mathrm{n})}(x)\\ & \times \frac{{\mathrm{i}}^n}{{\pi }^{1/4}\sqrt{2^{n}n!}}{H}_n(x-y_m){\mathrm{e}}^{-(x-y_m{)}^2/2},\end{array}

where {\tilde{\psi }}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x, y_m) is the unnormalized output wave function of the target oscillator and N is the normalization factor. Equations (19) and (20) define the state that will be referred to as the exact output state.

Note that, similar to the semiclassical Eq. (5), the output wave function is obtained by multiplying the input wave function by the factor representing the gate action. This factor is solely determined by the initial state of the ancilla and the difference between the variables x and y_m while remaining independent of the initial state of the target oscillator. In the geometric representation shown in Fig.2, the latter is consistent with the upward shift of x in the circle representing the Fock resource state on the phase plane [see top row of (b) column in Fig.2].

In the next section we demonstrate that, when the input state is a Glauber coherent state and the value of the measurement outcome is optimal, the exact output state (19) will be close to the Schrödinger cat state with high fidelity regardless of the photon number of the resource Fock state.

4 Generation of cat-like states from coherent state

In this section, we consider the conditional generation of the Schrödinger cat state via the “cat gate” described above and concentrate on the special case where the initial state of the target oscillator is the Glauber coherent state |{\alpha }_0⟩ with the amplitude {\alpha }_0=( x_0+\mathrm{i}{p}_0)/\sqrt{2}, whereas the ancillary one is still assumed to be prepared in the n-photon Fock state. In the qp phase space, the coherent state |{\alpha }_0⟩ can be represented by the uncertainty (localization) region bounded by the circle of the radius \delta x \sim 1/\sqrt{2} centered at the point (x_0 ,p_0). In this case, the wave function of the input state is

(21)\begin{array}{rl}{\psi }^{(\mathrm{i}\mathrm{n})}(x)=& ⟨x|{\alpha }_0⟩={\psi }_{vac}(x-x_0)\exp \{\mathrm{i}{p}_{0}x-\mathrm{i}{p}_0{x}_0/2\}\\ =& {\pi }^{-1/4}\exp \{-(x-x_0{)}^2/2+\mathrm{i}{p}_{0}x-\mathrm{i}{p}_0{x}_0/2\},\end{array}

where {\psi }_{vac} is the vacuum wave function in the coordinate representation, and Fig.4 schematically shows the semiclassical input-output mapping of the quadrature amplitudes of the target and auxiliary oscillators predicting an emergence of the output state in the form of a cat-like superposition of two “copies” of the input target state.

In Fig.5, we compare the Wigner function of the exact output state (19)

(22)\begin{array}{rl}W(x,p)=& \frac{1}{\pi }\int \mathrm{d}z{\psi }^{(\mathrm{o}\mathrm{u}\mathrm{t})\ast }(x+z,y_m)\\ & \times {\psi }^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x-z,y_m){\mathrm{e}}^{2\mathrm{i}pz}\end{array}

computed for the input coherent state (21) with the appropriate semiclassical mapping of the phase space region corresponding to the Gaussian function (21) performed by the gate according to Eq. (4). Calculations were performed at y_m=0 for the photon number n=10 with x_0\in \{ 0,1,2 \} assuming that the uncertainty region radius of the mapped coherent state is unity. For calculating the Wigner functions, we have developed the fast and efficient method which is based on the technique of generating functions and, thus, avoids performing time consuming numerical integration. Details on this method are relegated to Appendix A.

Referring to Fig.5, results for the Wigner functions and the semiclassical mapping are in good agreement. Thus, a simple geometric representation in the phase space based on the semiclassical description reproduces the output state with high accuracy and also predicts the appearance of a minimum disturbed cat-like superposition at the gate output (see the left column in Fig.5).

Given the input state, from Eqs. (5) and (9), considering that the input state in the phase space occupies a limited range of the coordinate near the point x_0 and the weighting factor in Eq. (9) can be assumed constant, we can construct the semiclassical wave function at the gate output

(23){\psi }_{\mathrm{s}\mathrm{c}\mathrm{l}}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x,y_m )\sim {\psi }^{(\mathrm{i}\mathrm{n})}(x)[{\mathrm{e}}^{\mathrm{i}\varphi (n,z)}+(-1{)}^n{\mathrm{e}}^{-\mathrm{i}\varphi (n,z)}],

where \varphi (n,z ) is the phase function given by Eq. (10), and calculate the fidelity F_{\mathrm{s}\mathrm{c}\mathrm{l}} between the semiclassical and exact output states

(24)F_{\mathrm{s}\mathrm{c}\mathrm{l}}(y_m,x_0,n)={\left|{\displaystyle \int \mathrm{d}x{\psi }^{(\mathrm{o}\mathrm{u}\mathrm{t})\ast }(x,y_m){\psi }_{\mathrm{s}\mathrm{c}\mathrm{l}}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x,y_m)}\right|}^2.

Note that, for the input wave function (21), this fidelity is independent of p_0.

The plots of F_{\mathrm{s}\mathrm{c}\mathrm{l}} computed in relation to the photon number n of the resource state at y_m=0 and different values canonical variable x_0 are shown in Fig.6. It is seen that the exact output state and the output state recovered from the semiclassical theory are in perfect agreement at small x_0, namely, with fidelity F_{\mathrm{s}\mathrm{c}\mathrm{l}}> 0.9970 at x_0=0 for any n, F_{\mathrm{s}\mathrm{c}\mathrm{l}}>0.9733 at x_0=1 for any n, F_{\mathrm{s}\mathrm{c}\mathrm{l}}>0.9056 at x_0=2 for n\ge 2.

In subsequent sections, we shall use Eq. (23) to deduce the expression for the semiclassical wave function of a Schrödinger cat-like superposition of two “copies” of the initial target state representing the state closest to the exact output state (19) for an arbitrary coherent input state.

4.1 Schrödinger cat-like state when the input wave function is localized near zero

We begin with the case of cat state generation by the Fock state-based gate when the input state in the phase space occupies a limited range of the coordinate near the point x_0=0, i.e., x\approx x_0\ll \sqrt{2n+1}. Under this assumption, the function \varphi (n,z) can be decomposed into a Taylor series in powers of x, which converges at provided that |y_m|\ll \sqrt{2n+1}. For x\ll \sqrt{2n+1}, we can limit ourselves to the first few terms of the Taylor series for phase \varphi. So, up to the second order in the coordinate x, we have

(25)\varphi (n, z)\approx \theta +p^{(+)}x+\delta p^{(+)}{x}^2,

where

(26)\begin{array}{rl}& \theta \equiv \varphi \left(n,-y_m/\sqrt{2n+1}\right),p^{(+)}\equiv \sqrt{2n+1-y_m^2},\\ & \delta p^{(+)}\equiv \frac{y_m}{2\sqrt{2n+1-y_m^2}}.\end{array}

When the measurement result y_m=0 the Taylor series (25) can be written up to a linear term in the coordinate x. This approximation yields the gate output state (23) in the form

(27){\psi }_{\mathrm{c}\mathrm{a}\mathrm{t}}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x)\sim {\psi }^{(\mathrm{i}\mathrm{n})}(x) [{\mathrm{e}}^{\mathrm{i}(\theta +p^{(+)}x)}+(-1{)}^n{\mathrm{e}}^{-\mathrm{i}(\theta +p^{(+)}x)}],

which corresponds to the “perfect” Schrödinger cat state, namely a superposition of two undistorted copies of the target oscillator initial state symmetrically shifted in the phase q- p space along the momentum axis by \pm p^{(+)}=\pm \sqrt{2n+1}, with the phase \theta =0. Such semiclassical “perfect” cat state may be represented in terms of a superposition of the Glauber coherent states | {\alpha }_{+}⟩ and |{\alpha }_{-}⟩,

(28)| {\psi }_{\mathrm{c}\mathrm{a}\mathrm{t}}^{(\mathrm{o}\mathrm{u}\mathrm{t})}⟩={\displaystyle \frac{1}{\sqrt{\mathcal{N}}}}[{\mathrm{e}}^{\mathrm{i}\theta }|{\alpha }_{+} ⟩+(-1{)}^n{\mathrm{e}}^{-\mathrm{i}\theta }|{\alpha }_{-}⟩],

where {\alpha }_{\pm }=[x_0+\mathrm{i}(p_0\pm p^{(+)})]/ \sqrt{2} and \mathcal{N} is the normalization factor.

The terms of the second order and higher concerning x in the Taylor series of the added factor phase \varphi (n,z), indicate the dependence of the momentum transmitted by the gate on the target oscillator coordinate x and lead to a distortion of the shape of the region on the phase plane where the Wigner function component is localized. In the geometrical description (see Fig.2 and Fig.4), this dependence follows from the fact that, for a given measurement outcome y_m, the intersection points x_a^{(\pm )} are displaced when the resource curve is shifted along the vertical axis in changing the coordinate x of the target oscillator if the resource curve is not a vertical line at the intersection with the horizontal line p_a=y_m. This results in the copies of the input state of the target oscillator undergo shear deformation of the opposite sign due to the nearly linear dependence of the displacement on x for large enough measurement outcomes y_m, which is described by the quadratic term in the Taylor expand of \varphi (n,z ) and characterized by the measure of the linear shear deformation being y_m/(2\sqrt{2n+1-y_m^2}), as follows from Eq. (26).

In order to evaluate the “proximity” of the exact output state to the Schrödinger cat state, we consider the fidelity

(29)F_{\mathrm{c}\mathrm{a}\mathrm{t}}(y_m,x_0,n)={\left|{\displaystyle \int \mathrm{d}x{\psi }^{(\mathrm{o}\mathrm{u}\mathrm{t})\ast }(x,y_m){\psi }_{\mathrm{c}\mathrm{a}\mathrm{t}}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x)}\right|}^2

between the state (19) and the “perfect” cat state (27) recovered from the semiclassical picture.

From the explicit expressions for the wave functions {\psi }^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x, y_m) and {\psi }_{\mathrm{c}\mathrm{a}\mathrm{t}}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x), it is not difficult to conclude that F_{\mathrm{c}\mathrm{a}\mathrm{t}} does not depend on the coordinate y_0. In Fig.7, F_{\mathrm{c}\mathrm{a}\mathrm{t}} is plotted against x_0 at y_m=0 for various values of the photon number. It is shown that the fidelity reaches values close to unity at x_0=0 for any value of n. So, F_{\mathrm{c}\mathrm{a}\mathrm{t}} remains close to unity only for such coherent states where x_0 lies in the vicinity of the origin. It can also be seen that the fidelity rapidly drops with x_0.

The Wigner functions of the exact solution (19) at y_m=0 for the input coherent state with x_0\in \{0, 1.5,2\} pictured in the top row Fig.5 demonstrate that, for the resource Fock state with n=10, the output state is close to the undisturbed Schrödinger-cat-like superposition (27) at x_0=0 in agreement with the curves for F_{\mathrm{c}\mathrm{a}\mathrm{t}} (29) plotted in Fig.7 and Fig.8. Note that, for the columns shown in Fig.5, the values of the fidelity F_{\mathrm{c}\mathrm{a}\mathrm{t}} are: 0.9974 (left column), 0.8926 (central column), and 0.704 (right column).

4.2 Schrödinger cat-like state when the input wave function is localized away from zero

From Fig.7 and Fig.8, at the measurement outcome y_m=0, the fidelity F_{\mathrm{c}\mathrm{a}\mathrm{t}} between the exact solution (19) and the “perfect” semiclassical cat (27) for the input state (21) is close to unity when the x-quadrature x_0 of the input coherent state of amplitude {\alpha }_0 is in the immediate vicinity of zero. It is expected that the size of the vicinity is of the order of vacuum fluctuations level \delta x =1/\sqrt{2}, i.e., |x_0|\le \delta x (the maximum magnitude for x_0 can be chosen based on the requirements for the value of F_{\mathrm{c}\mathrm{a}\mathrm{t}}).

However, the fidelity F_{\mathrm{c}\mathrm{a}\mathrm{t}} rapidly declines with x_0 and the main reason for this significant reduction is that the wave function (27) of the semiclassical cat state obtained in the previous subsection fails to give a good approximation for the output state (19) provided that the magnitude of x_0 exceeds \delta x.

In this section, we give a prescription for constructing the Schrödinger cat state closest to the exact output state for the general case when the coherent state (21) with x_0\ne 0 is sent to the input of the Fock state-based gate. In other words, the support of the input wave function {\psi }^{(\mathrm{i}\mathrm{n})}(x) is now localized in the vicinity of the point x_0 which is well separated from the origin of the phase space.

To construct a cat-like superposition state giving a high fidelity approximation of the exact output state, we expand the phase (10) in the added factor {\phi }_{\mathrm{s}\mathrm{c}\mathrm{l}} in a Taylor series in the localization region of the Gaussian {\psi }^{(\mathrm{i}\mathrm{n})}(x) as was performed in the previous subsection, i.e., now in the vicinity of the point x=x_0. By analogy with (25), in the region under the condition |x-y_m|\ll \sqrt{2n+1} we expand the phase function \varphi (n,z) into the Taylor series in powers of x-x_0. In the sufficiently small neighborhood of x_0 with | x-x_0|\ll \sqrt{2n+1}, the expansion of the function \varphi (n,z) can be truncated up to the second order terms quadratic in x- x_0. So, we have

(30){\phi }_{\mathrm{s}\mathrm{c}\mathrm{l}}\sim {\mathrm{e}}^{\mathrm{i}\varphi (n,z)}+(- 1{)}^n{\mathrm{e}}^{-\mathrm{i}\varphi (n,z)},

(31)\varphi (n, z)\approx {\theta }_0+ p_0^{(+)}\cdot (x-x_0)+\delta p_0^{(+)}\cdot (x-x_0{)}^2,

where

(32){\theta }_0=\varphi (n, (x_0-y_m)/\sqrt{2n+1}),

(33)p_0^{(+)}=\sqrt{2n+1-(x_0-y_m{)}^2} ,

(34)\delta p_0^{(+)}=-{\displaystyle \frac{x_0-y_m}{2\sqrt{2n+1-(x_0-y_m{)}^2}}}.

At \delta p_0^{(+)}=0 when the measurement result of the ancillary oscillator is y_m=x_0, the approximate gate output state (5) with the semiclassical factor determined by Eqs. (30)–(34) will be the undistorted semiclassical Schrödinger cat given by

(35){\psi }_{\mathrm{c}\mathrm{a}\mathrm{t}}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x,x_0)\sim {\psi }^{(\mathrm{i}\mathrm{n})}(x)[{\mathrm{e}}^{\mathrm{i}({\varphi }_0+p_0^{(+)}x)}+(-1{)}^n{\mathrm{e}}^{-\mathrm{i}({\varphi }_0+p_0^{(+)}x)}],

where

(36){\varphi }_0\equiv -p_0^{(+)}{x}_0,p_0^{(+)}=\sqrt{2n+1}.

The “perfect” Schrödinger cat can also be written as a superposition of Glauber coherent states of the form:

(37)| {\psi }_{\mathrm{c}\mathrm{a}\mathrm{t}}^{(\mathrm{o}\mathrm{u}\mathrm{t})}⟩\sim {\mathrm{e}}^{\mathrm{i}{\varphi }_0}|{\tilde{\alpha }}_{+} ⟩+(-1{)}^n{\mathrm{e}}^{-\mathrm{i}{\varphi }_0}|{\tilde{\alpha }}_{-}⟩,

where {\tilde{\alpha }}_{\pm }\equiv [x_0+i(p_0\pm p_0^{(+)})]/\sqrt{2}.

From relations (19), (20), and (35), the fidelity between {\psi }^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x, y_m) and {\psi }_{\mathrm{c}\mathrm{a}\mathrm{t}}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x,x_0 ) at y_m=x_0

(38)F_{\mathrm{c}\mathrm{a}\mathrm{t}}(n )={\left|{\displaystyle \int \mathrm{d}x{\psi }^{(\mathrm{o}\mathrm{u}\mathrm{t})\ast }(x,x_0){\psi }_{\mathrm{c}\mathrm{a}\mathrm{t}}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x,x_0)}\right|}_{y_m=x_0}^2

is independent of both p_0 and x_0 (since the integrand depends only on the difference x- x_0 and the integral is taken over an unbounded interval) and, thus, is determined solely by the photon number n.

Note that from the semiclassical description it can be inferred the output state created by the scheme under consideration from the coherent state |{\alpha }_0⟩ with the amplitude {\alpha }_0=(x_0+i p_0)/ \sqrt{2}, at y_m=x_0, will be close to the “perfect” cat (35) with high fidelity for any photon number of the resource state because target state is localized in proximity of x_0 with the characteristic length \delta x \sim 1/\sqrt{2}. Therefore, for all x belonging to the input function support, the semiclassical relation | x-x_0|\ll \sqrt{2n+1} will be fulfilled, which guarantees the generation of a minimally distorted cat state (see Fig.9, top row).

The fact that the quality of the semiclassical approximation given by Eq. (37) improves with increasing photon number n is illustrated in Fig.9 where, for y_m=x_0 =3 and n\in \{1, 5,15\}, the Wigner functions of the exact output state (19) are compared with their semiclassical counterparts computed for the undeformed cat (35).

Statistics of the homodyne measurements of the ancilla momentum is described by the probability density to observe the outcome y_m given by the norm of the unnormalized output wave function (20)

(39)P(y_m)= ⟨{\tilde{\psi }}^{(\mathrm{o}\mathrm{u}\mathrm{t})}|{\tilde{\psi }}^{(\mathrm{o}\mathrm{u}\mathrm{t})}⟩={\displaystyle \int \mathrm{d}x|{\tilde{\psi }}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x,y_m){|}^2,}

so that the wave function (19) of the output state can be written as

(40){\psi }^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x,y_m)={\displaystyle \frac{{\tilde{\psi }}^{(\mathrm{o}\mathrm{u}\mathrm{t})}(x,y_m)}{\sqrt{P(y_m)}}}.

For the input coherent state (21) of the target oscillator and the Fock resource state (13) with n photons, the probability density takes the explicit form

(41)\begin{array}{rl}P(y_m,x_0)=& \frac{1}{\pi 2^{n}n!}\int H_n^2(x-y_m){\mathrm{e}}^{-(x-y_m{)}^2}{\mathrm{e}}^{-(x-x_0{)}^2}\mathrm{d}x\\ =& P(y_m-x_0,0)=P(x_0-y_m,0),\end{array}

where the probability density P(y_m,0) corresponds to the vacuum state of the target oscillator [104]. Note that P(y_m,0) is even in y_m due to the parity in \xi of the absolute value of the Hermite polynomials |H_n(\xi )|. In Fig.10, the probability density P(y_m- x_0,0 ) is plotted as a function of | y_m- x_0| at different values of the photon number. Each curve shows a local maximum that decreases with n while shifting outward, reflecting a fundamental quantum trade-off: higher photon numbers n increase the cat-state displacement \sqrt{2n+1} but broaden the ancilla’s wavefunction, reducing peak success probabilities. Conversely, smaller n yields higher P but produces smaller cats with reduced macroscopic distinguishability. The optimal regime, characterized by simultaneously moderate P\sim 0.2–0.4 and high fidelity F_{\mathrm{c}\mathrm{a}\mathrm{t}}>0.95, is achieved with small photon numbers (n\le 5) and minor deviations of y_m from x_0, specifically | x_0- y_m|\lesssim 1/\sqrt{2} (as illustrated for n=1 in Fig.7 and Fig.10). Within this regime, the displacement offset \mathrm{\Delta } y_m=|y_m-x_0| can be adjusted to prioritize either P or F_{\mathrm{c}\mathrm{a}\mathrm{t}}, allowing for a tunable balance between success rate and output state quality tailored to application-specific needs.

Since, in real-world experiments, the measurement outcome can only be identified with a certain precision, it is instructive to consider a mixed output state that emerges when the observed ancilla momentum falls within an acceptance interval ranged from -d /2 to d/ 2 of the width d centered at y_m=0 provided x_0=0. In this case, the weighted fidelity between the “perfect” semiclassical cat state (27) and the mixed state at the gate output is evaluated as follows

(42)F^{\mathrm{m}\mathrm{i}\mathrm{x}}(d) ={\displaystyle \frac{1}{P^{\mathrm{m}\mathrm{i}\mathrm{x}}(d)}}{\displaystyle {\int }_{-d/2}^{+d/2}\mathrm{d}{y}_{m}P(y_m,0)F_{\mathrm{c}\mathrm{a}\mathrm{t}}(y_m),}

where P^{\mathrm{m}\mathrm{i}\mathrm{x}}( d)\equiv {\int }_{-d/2}^{+d/2}\mathrm{d}{y}_{m}P (y_m,0) is the probability that the measurement outcome is within the acceptance interval.

When x_0\ne 0, we choose the acceptance interval to be centered at y_m=x_0 and insert the ideal cat (35) into the expression for F_{\mathrm{c}\mathrm{a}\mathrm{t}}(y_m ). After change of the variable: x\to x-x_0, the expression for F^{\mathrm{m}\mathrm{i}\mathrm{x}}(d) is reduced to Eq. (42).

Fig.11 presents the results for the fidelity F^{\mathrm{m}\mathrm{i}\mathrm{x}}(d) evaluated as a function of the acceptance interval length, d, at different values of the photon number, n\in \{1, 5,10\}. From Fig.11 it is essential to use sufficiently narrow acceptance intervals in order to prepare a mixed state with high fidelity F^{\mathrm{m}\mathrm{i}\mathrm{x}}(d) to the undistorted semiclassical cat state (27).

It is useful to note that, for small intervals with d\ll \sqrt{2n+1}, the probability that the measurement outcome lies in the acceptance interval is proportional to d and can be estimated as P^{\mathrm{m}\mathrm{i}\mathrm{x}}(d) \approx P(0,0) d. This behavior comes from the weak dependence of the probability density on y_m at low y_m (see Fig.10).

5 Discussion and conclusion

We have studied the CV measurement-induced Fock state-based “cat gate” which can conditionally generate a two-component Schrödinger cat-like superposition from the input coherent state. It is shown that the geometric semiclassical mapping in the phase space can be used to evaluate such important characteristics as the number of components of the cat-like superposition, their positions, and distortions produced by the logical element. We have identified the gate operation regime in which the output state is closest to the “perfect” semiclassical Schrödinger cat state represented by the superposition of two undeformed copies of the input state symmetrically displaced in the phase space. The “size” of the generated Schrödinger cat at the gate output − the “distance” between the copies − can be made as large as required by an appropriate choice of the gate parameters. A key feature of the gate under consideration is that the Schrödinger cat state emerges when the ancillary oscillator measurement is compatible with multiple values of the target oscillator physical variables.

We have performed a detailed analysis of the fidelity between the gate output state and “perfect” Schrödinger cat state derived from semiclassical theory. We have found the criteria for the gate operation with high fidelity values exceeding 0.99 for a coherent state at the gate input and have illustrated the qualitative features of the output cat states in terms of their Wigner functions depending on on the gate parameters and measurement outcome. A clear interpretation of the output state quantum statistics in terms of the Wigner function in dependence on the gate parameters and measurement outcome was presented.

For small photon numbers (n\le 5) and minor deviations of the measurement outcome from the input coherent state’s coordinate (| y_m- x_0|\lesssim 1), the protocol simultaneously achieves high fidelity F_{\mathrm{c}\mathrm{a}\mathrm{t}}>0.95 and moderate P\sim 0.2-0.4. This demonstrates practical viability, as multiple experimental runs can compensate for the success rate while maintaining state quality.

In analogy to some other non-Gaussian CV schemes, the studied “cat gate” uses the same key elements such as the Fock resource state, the entangling {\hat{C}}_Z operation, and the homodyne measurement. A key feature of the regime where the cat-like superposition emerges is that the projective measurement provides multivalued information about the target system’s physical variables. From our analysis, it might be concluded that this feature may arise in CV quantum networks with embedded non-Gaussian gates of a general kind using more complex resource states and other types of measurements that may cause measurement-induced evolution leading to the generation of the multi-component Schrödinger cat states which can be used as the logical qubit basis for error correction codes. The cat-breeding transformations of an arbitrary input state may be combined with standard Gaussian operations such as displacement, rotation, squeezing, and shear deformation, and can be successfully applied in complex non-Gaussian quantum networks.

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