1. Introduction

Ever since Einstein-Podolsky-Rosen (EPR)’s argument against quantum mechanics was put forward [1], quantum entanglement has been a topic of great interest from a fundamental point of view. It can play a crucial role in manifesting striking differences between quantum and classical (e.g. local hidden-variable [2–4]) descriptions of nature. Furthermore, it has also drawn much attention of practical interest because quantum correlations can be employed to carry out information tasks to the extent far beyond their classical counterparts, e.g. quantum computing [5] and teleportation [6]. In a bipartite setting, the primitive entangled states for discrete variables are the so-called Bell states, the maximally entangled states of two qubits, e.g. singlet state. In the regime of continuous variables (CVs), the Bell state can be realized in the form of two-mode squeezed state (TMSS), which becomes maximally entangled in the limit of infinite squeezing. The TMSS has been mostly the target entangled resource to produce for various quantum information tasks [7] like CV quantum teleportation [8].

The TMSS belongs to the class of Gaussian states, which has been extensively studied both theoretically and experimentally for CV quantum informatics [9]. On the other hand, a great deal of attention has also been directed to the non-Gaussian regime (e.g. state engineering [10] and characterization [11, 12]), as the non-Gaussian entangled states can provide some practical merits [13–21] and even become an essential ingredient [22–25] for a number of quantum tasks. Furthermore, when the quantum information processing is performed under realistic conditions, the quantum correlation is inevitably degraded and it thus becomes an important question whether Gaussian or non-Gaussian entangled states can be more robust against decoherence [26,27]. It was recently demonstrated that there exists a broad parameter space in which non-Gaussian entanglement can survive longer than Gaussian entanglement under noisy environments [28–30] or quantum-limited amplifier [31]. With all these considered, it seems very desirable to have an experimental toolbox to generate a broad class of non-Gaussian entangled states in a controllable way.

In this paper we consider a class of CV entangled states in the photon-number entangled form ${\sum }_{n=0}^N{C}_n{|n〉}_a{|n〉}_b$, where |n〉 denotes a Fock state basis. One particular example is the TMSS with the coefficients C_n = λⁿ(1 − λ²)^1/2 (λ : squeezing parameter, N → ∞), which is the only Gaussian state among the photon-number entangled states. Another example is the pair-coherent state given by C_n ∼ ζⁿ/n! [32, 33] which can be useful for a number of applications including quantum teleportation [34], quantum metrology [35], and a Bell test [36]. In fact, a broad class of photon-number entangled states has been so far considered for the nonlocality test using homodyne detections [37–39]. Here we propose two experimental schemes to generate a finite-dimensional PNES with arbitrary coefficients, ${\sum }_{n=0}^N{C}_n{|n〉}_a{|n〉}_b$ ( ${\sum }_{n=0}^N{\left|C_n\right|}^2=1$), where the coefficients C_n can be controlled with beam splitting and squeezing parameters. Both of our proposed schemes make use of coherent superposition operations in single-photon interferometic settings that erase the which-path information on the realized photonic operations. The first scheme employs the second-order superposition operation tââ^† + râ^†â, which has been recently proposed and experimentally implemented in the context of proving bosonic commutation relation [â,â^†]= 1 [40–43], together with two-mode squeezing operation. We note that the coherent operation tââ^† + râ^†â was also discussed in the context of noiseless quantum amplifier [44]. On the other hand, the second scheme employs a sequence of nonlocal first-order coherent superpositions tâ + rb̂^†. Its single-mode version tâ + râ^† was recently proposed for a quantum state engineering [45] and also shown to be useful to enhance two-mode entanglement properties [20, 46].

We also address the usefulness of the finite-dimensional PNES for CV quantum teleportation [8] and nonlocality test [47, 48] compared with the two-mode squeezed state. Furthermore, it was very recently shown that the photon-number entangled states in finite dimension, e.g. C₀|0〉_a|0〉_b +C₁|1〉_a|1〉_b, can survive longer under noisy environments than the TMSS with the same degree of entanglement or energy [28, 29]. Therefore, our proposed schemes can be a useful tool not only for CV quantum applications but also for fundamental tests of quantum physics.

This paper is organized as follows. In Sec. 2, we first compare the entanglement properties of finite-dimensional PNES (non-Gaussian) and a two-mode squeezed state (Gaussian) in view of the degree of entanglement and the EPR correlation. In Sec. 3, we further investigate the usefulness of the PNES for CV quantum teleportation and nonlocality test. Then, we propose two experimental schemes to generate an arbitrary PNES, ${\sum }_{n=0}^N{C}_n{|n〉}_a{|n〉}_b$, in Sec. 4. We illustrate the feasibility of our schemes in Sec. 5 by investigating the generation of a PNES up to two-photon correlation, i.e. C₀|0〉_a|0〉_b +C₁|1〉_a|1〉_b +C₂|2〉_a|2〉_b, considering realistic experimental conditions. In Sec. 6, our results are summarized.

2. Entanglement and EPR correlation

First, we briefly compare the TMSS and the PNES in terms of entanglement properties in order to identify the practical relevance of the PNES for CV quantum informatics. For a pure two-mode state |Ψ〉_AB, the degree of entanglement can be quantified by the von Neumann entropy E(ρ_A)= −Tr_A[ρ_A log₂ ρ_A] for the reduced density operator ρ_A = Tr_B[|Ψ〉_AB〈Ψ|_AB]. For the class of photon-number entangled states ${\sum }_{n=0}^N{C}_n{|n〉}_a{|n〉}_b$, the von Neumann entropy becomes maximal when all the coefficients C_n are identical. Thus, for the case of TMSS with C_n = λⁿ(1 − λ²)^1/2, the state can have an infinite degree of entanglement with infinite squeezing, i.e. λ = 1, which is practically impossible to achieve. On the other hand, the finite-dimensional PNES can match or even surpass a finitely-squeezed TMSS as an entangled resource. In Fig. 1(a), we plot the degree of entanglement for the TMSS (blue solid) as a function of the squeezing parameter s = tanh⁻¹ λ. This is compared with the maximal possible entanglement for the PNES with equal coefficients (C₁ = C₂ = ⋯ = C_N) of dimensions N = 1 (red dotted), 2 (red dashed), and 10 (red dot-dashed). The degrees of entanglement for the PNESs are given by 1, 1.585 and 3.459, respectively. To achieve such degrees of entanglement, the squeezing of the TMSS should be s = 0.5185 (4.506 dB), s = 0.7335 (6.374 dB) and s = 1.391 (12.09 dB), respectively. In the pulsed-regime generation of squeezed states, the level of squeezing currently available from an optical parametric amplifier is s = 0.403 (3.5 dB) [49, 50] so that the PNES with N = 1 can already surpass the entanglement of the TMSS.

 

Fig. 1 (a) Degree of entanglement and (b) EPR correlation for the states: |TMSS〉 (blue solid) as a function of the squeezing parameter s, and ${\sum }_{n=0}^N{C}_n{|n〉}_a{|n〉}_b$ at N = 1 (red dotted), N = 2 (red dashed), N = 10 (red dot-dashed).

Download Full Size | PDF

We also look into another entanglement property, the EPR correlation, which is the total variance of a pair of EPR-like operators, EPR ≡ Δ²(x̂_A − x̂_B)+Δ²(p̂_A + p̂_B). Here ${\widehat{x}}_j=\frac{1}{\sqrt{2}}\left({\widehat{a}}_j+{\widehat{a}}_j^{†}\right)$ and ${\widehat{p}}_j=\frac{1}{i\sqrt{2}}\left({\widehat{a}}_j-{\widehat{a}}_j^{†}\right)$ (j = A,B) are the quadrature amplitudes of the field that can be measured in homodyne detection. The value of EPR below 2 represents the quantum correlation between the quadrature amplitudes of two modes. In Fig. 1(b), the EPR correlations of the PNESs for the dimensions N = 1, 2, and 10 are 1.172, 0.8315, and 0.2516, respectively. The corresponding levels of squeezing for the TMSS are given by s = 0.2674 (2.324 dB), s = 0.4388 (3.813 dB) and s = 1.037 (9.008 dB). Thus the PNES with N ≥ 2 can surpass the currently available TMSS (s = 0.403) in view of the EPR correlation.

3. Applications: CV teleportation and nonlocality test

In this section, we further investigate the usefulness of a finite-dimensional PNES particularly for continuous variable (CV) teleportation and nonlocality test. For this purpose, we evaluate the quality of CV teleportation by the average fidelity between an unknown input state and the teleported state [8], and investigate the nonlocality test by Banaszek and Wódkiewicz based on the phase-space distribution functions [47, 48].

(i) The teleportation fidelity in the Braunstein-Kimble (BK) scheme [8] can be evaluated in terms of the characteristic functions of an input state and its teleported state as

In Fig. 2(a), we compare the teleportation fidelity achieved via the PNES and the fidelity via the TMSS. The optimal fidelity via each PNES at N = 1, 2 and 3 corresponds to the fidelity via the TMSS with the squeezing parameters s = 0.320 (2.776 dB), s = 0.506 (4.397 dB), and s = 0.638 (5.548 dB), respectively. Thus, the PNES at N = 2 can surpass the fidelity via the TMSS with the currently available squeezing in the pulsed regime, i.e., s ≈ 0.403 (3.5 dB) [53]. As we show in Section 4, our proposed schemes do not require a high-level of squeezing to produce the optimal PNES (case of N = 2) for CV teleportation.

 

Fig. 2 (a) Average fidelity in teleporting a coherent state and (b) Bell parameter B_BW as a function of the squeezing parameter s for the |TMSS〉 (blue solid) and the PNES ${\sum }_{n=0}^N{C}_n{|n〉}_a{|n〉}_b$ at N = 1 (red dotted), N = 2 (red dashed) and N = 3 (red dot-dashed). The coefficients of the PNESs are optimized for each N.

Download Full Size | PDF

(ii) We next consider the nonlocality test by Banaszek and Wódkiewicz that is addressed in phase space using the two-mode Wigner function [47, 48]. This Bell inequality is given by

4. Experimental schemes

We now propose two optical schemes to generate an arbitrary PNES, ${\sum }_{n=0}^N{C}_n{|n〉}_a{|n〉}_b$. One scheme is based on a second-order coherent superposition operation with two-mode squeezing operations, and the other on a sequence of first-order coherent superposition operations.

(i) We first consider the operation tââ^† + râ^†â acting on a single-mode a, which is the coherent superposition of two product operations—photon addition followed by subtraction (ââ^†) and photon subtraction followed by addition (â^†â). This coherent operation was experimentally implemented to prove the bosonic commutation relation [â,â^†]= 1 [40, 41]. While such a commutator is addressed as an equal superposition of the two product operations,ââ^† −â^†â, we adopt an arbitrarily weighted superposition of the two operations, i.e., tââ^† + râ^†â. In particular, we show that the single-mode operation tââ^† + râ^†â together with two-mode squeezing operations Ŝ_ab(ξ)= exp(−ξâ^†b̂^† +ξ^*b̂â) can constitute an essential building block to generate an arbitrary PNES.

Suppose that a two-mode squeezing Ŝ_ab(ξ), the coherent operation tââ^† + râ^†â and the inverse squeezing ${\widehat{S}}_{ab}^{†}(\xi )$ are sequentially applied to an input state. That is, we apply a sequence of operations defined by

When a vacuum state |0〉_a|0〉_b is injected as an input, Ô₁ yields a superposition of number states as Ô₁|0〉_a|0〉_b = cosh² s₁[(t₁ + r₁ tanh² s₁)|0〉_a|0〉_b − exp(iφ₁)(t₁ + r₁)tanhs₁|1〉_a|1〉_b]. In principle, a succession of Ô_n applied on the vacuum states, ${\mathrm{\Pi }}_{n=1}^N{\widehat{O}}_n{|0〉}_a{|0〉}_b$, can yield an arbitrary superposition state ${\sum }_{n=0}^N{C}_n{|n〉}_a{|n〉}_b$ by properly choosing the parameters s_n, r_n, t_n and φ_n. For instance, the state Ô₁|0〉_a|0〉_b ∼ C₀|0〉_a|0〉_b + C₁|1〉_a|1〉_b can have a larger proportion of |1〉_a|1〉_b, i.e. |C₀| < |C₁|, under the condition r₁ tanhs₁ > t₁. For comparison, if one instead applies the original quantum scissor scheme on the TMSS that projects an input onto the subspace spanned by |0〉 and |1〉 [56], the output state becomes ∼ |0〉_a|0〉_b + λ|1〉_a|1〉_b. That is, the vacuum state |0〉_a|0〉_b is always more weighted than the single-photon state |1〉_a|1〉_b. On the other hand, a generalized scissor scheme proposed for the noiseless quantum amplifier [57] can be used to control each coefficient arbitrarily in the |0〉–|1〉 subspace, which will be briefly discussed in Sec. 5.

The elementary operation Ô_n can be experimentally realized as depicted in Fig. 3(a). An input state |ψ〉_ab is first injected into a nondegenerate parametric amplifier (NDPA) with coupling parameter ξ_n, and then into the beam splitter BS1 (transmittance: T₁ ≈ 1) with the other input mode c in a vacuum. This can be described by

 

Fig. 3 (a) Experimental scheme to implement the operation ${\widehat{S}}_{ab}^{†}(\xi )\left(t\widehat{a}{\widehat{a}}^{†}+r{\widehat{a}}^{†}\widehat{a}\right){\widehat{S}}_{ab}(\xi )$ on an arbitrary state. BS1, BS2, and BS3 are beam splitters with transmissivities T₁, T₂ and t_n, respectively. PD0, PD1 and PD2: photo detectors. The operation is successfully achieved under the detection of a single photon at only one of two detectors PD1 and PD2, with PD0 clicked. (b) For a vacuum input state, the sequence of operations Ô_n can yield a finite dimensional PNES, ${\sum }_{n=0}^N{C}_n{|n〉}_a{|n〉}_b$.

Download Full Size | PDF

(ii) Second, we show that the sequence of two first-order coherent superposition operations, (t_2nâ + r_2nb̂^†)(t_2n−1b̂ + r_2n−1â^†), can also yield an operation similar to Ô_n in Eq. (4). A similar type of coherent operation was previously investigated in a form acting on a single-mode, tâ + râ^†, which is the superposition of photon subtraction and addition [45]. Here we consider a nonlocal coherent superposition acting on two modes, tâ + rb̂^† (tb̂ + râ^†).

We define an operator

The operation Ô′_n can be implemented as depicted in Fig. 4. First, an arbitrary two-mode state |ψ〉_ab is injected into an NDPA with small coupling s₁ ≪ 1 and a BS1 with high transmissivity T₁ ≈ 1, with mode a (b) into NDPA (BS1). The other input modes to the NDPAs and the BSs are all in vacuum states. Then, the BS3 (transmissivity: t_2n−1) yielding the transformations ĉ^† → t_2n−1ĉ^† + r_2n−1d̂^† and d̂^† → t_2n−1d̂^† − r_2n−1ĉ^† gives the output

 

Fig. 4 Experimental scheme to implement the operation (t_2nâ + r_2nb̂^†)(t_2n−1b̂ + r_2n−1â^†) on an input state |ψ〉_ab. BS1, BS2, BS3 and BS4 are beam splitters with transmissivities T₁, T₂, t_2n−1, and t_2n, respectively. PD1, PD2, PD3 and PD4: photo detectors. The operation is successfully achieved under the detection of a single photon at only one of two detectors PD1 and PD2 and the detection of a single-photon at only one of two detectors PD3 and PD4.

Download Full Size | PDF

(iii) In Sec. 3, we have seen that the optimal PNES ${\sum }_{n=0}^N{C}_n{|n〉}_a{|n〉}_b$ with N = 2 for CV teleportation has the coefficients C₀ ≈ 0.765, C₁ ≈ 0.535, and C₂ ≈ 0.359. Under our first scheme, these coefficients can be obtained using the experimental parameters, e.g. s₁ = s₂ = 0.1, ϕ₁ = 0, ϕ₂ = π, r₁ ≈ 0.4589, and r₂ ≈ 0.9984, with $t_1={\left(1-r_1^2\right)}^{1/2}$ and $t_2=-{\left(1-r_2^2\right)}^{1/2}$. On the other hand, under the second scheme, the same coefficients can be obtained using the parameters r₂ ≈ 0.3, r₃ ≈ 0.3863 and r₄ ≈ 0.6193, with ${\text{t}}_2=-{\left(1-{\text{r}}_2^2\right)}^{1/2}$, ${\text{t}}_3=-{\left(1-{\text{r}}_3^2\right)}^{1/2}$, and ${\text{t}}_4=-{\left(1-{\text{r}}_4^2\right)}^{1/2}$. Note that in our first scheme, we generate such a PNES using the NDPAs in the weak squeezing regime, s = 0.1 (0.869 dB). Furthermore, in the second scheme, we can obtain the same PNES only by adjusting the beam-splitter parameters, therefore, a high-level of squeezing is not necessary in our schemes. This is also true for the case of nonlocality test shown in Sec. 3. Under the first scheme, the optimized coefficients of the PNES for the nonlocality test are obtained using the parameters s₁ = s₂ = 0.1, ϕ₁ = 0, ϕ₂ = π, r₁ ≈ 0.38 and r₂ ≈ 0.999, with $t_1={\left(1-r_1^2\right)}^{1/2}$ and $t_2={\left(1-r_2^2\right)}^{1/2}$. Under the second scheme, they are obtained using the parameters r₂ ≈ 0.3, r₃ ≈ 0.391 and r₄ ≈ 0.221, with ${\text{t}}_2={\left(1-{\text{r}}_2^2\right)}^{1/2}$, ${\text{t}}_3={\left(1-{\text{r}}_3^2\right)}^{1/2}$, and ${\text{t}}_4={\left(1-{\text{r}}_4^2\right)}^{1/2}$.

5. Experimental feasibility

In order to further illustrate the feasibility of our proposed schemes, we address realistic experimental conditions in producing a PNES up to one-photon correlation, C₀|0〉_a|0〉_b +C₁|1〉_a|1〉_b, or two-photon correlation, C₀|0〉_a|0〉_b + C₁|1〉_a|1〉_b + C₂|2〉_a|2〉_b, as examples. In the two schemes of Figs. 3 and 4, we particularly consider each photodetector as an on-off detector that only distingushes two events, detection and non-detection, with efficiency η. The photodetection can then be characterized by a positive operator-valued measure (POVM) [58–60] with two components ${\widehat{\mathrm{\Pi }}}_0={\sum }_n{\left(1-\eta \right)}^n|n〉〈n|$ (no click) and Π̂₁ =Î− Π̂₀ (click).

In the first scheme, an arbitrary input state goes through a sequence of operations—a two-mode squeezing, a second-order superposition tââ^† + râ^†â heralded by nonideal on-off detectors, and the inverse two-mode squeezing. This sequence yields an output state

 

Fig. 5 Fidelity between the ideal state C₀|0〉_a|0〉_b + C₁|1〉_a|1〉_b and the output state ρ_out obtained by applying ${\widehat{S}}_{ab}^{†}(\xi )\left(t\widehat{a}{\widehat{a}}^{†}+r{\widehat{a}}^{†}\widehat{a}\right){\widehat{S}}_{ab}(\xi )$ (blue circle) or (t₂â + r₂b̂^†)(t₁b̂ + r₁â^†) (red square), using on-off detectors with efficiency η to the input state ρ_in = |0〉_a|0〉_b as a function of |C₀|² for η = 0.66. Black triangle represents the output fidelity using the scissor scheme of [57], with the input two-mode squeezed state (s = 0.1) and the on-off detectors (η = 0.66).

Download Full Size | PDF

In Fig. 6, we further investigate the fidelity F₂ for the case of PNES up to two-photon correlation as a function of |C₁|² and |C₂|². With the same parameters (η = 0.66, s₁ = 0.1 and $T_1^2=T_2^2=0.99$) used in Fig. 5, the fidelity is achieved at least above 0.941 in the whole range of |C₁|² and |C₂|². For both of the cases, the fidelity slightly increases with the vacuum-state probability |C₀|², as the weak coupling (s₁ = 0.1) of the NDPA makes a low photon-number state better controlled.

 

Fig. 6 Fidelity between the ideal state C₀|0〉_a|0〉_b +C₁|1〉_a|1〉_b +C₂|2〉_a|2〉_b and the output state ρ_out obtained by applying twice (a) ${\widehat{S}}_{ab}^{†}\left({\xi }_2\right)\left(t_2\widehat{a}{\widehat{a}}^{†}+r_2{\widehat{a}}^{†}\widehat{a}\right){\widehat{S}}_{ab}\left({\xi }_2\right){\widehat{S}}_{ab}^{†}\left({\xi }_1\right)\left(t_1\widehat{a}{\widehat{a}}^{†}+r_1{\widehat{a}}^{†}\widehat{a}\right){\widehat{S}}_{ab}\left({\xi }_1\right)$ or (b) (t₄â+r₄b̂^†)(t₃b̂+r₃â^†)(t₂â+r₂b̂^†)(t₁b̂+r₁â^†), using on-off detectors with efficiency η to the input state ρ_in = |0〉_a|0〉_b as a function of |C₁|² and |C₂|² for η = 0.66.

Download Full Size | PDF

In the second scheme, two first-order superposition operations (t₂â + r₂b̂^†)(t₁b̂ + r₁â^†) heralded by nonideal on-off detectors are sequentially applied to an arbitrary input state. This yields an output state

We have also calculated the success probability numerically for the output states under each scheme. For the state C₀|0〉_a|0〉_b +C₁|1〉_a|1〉_b, the first scheme, with the condition of s₁ = 0.1, s = 0.1 and $T_1^2=T_2^2=0.99$, yields the success rate in the range of 2.4 × 10⁻⁶ (|C₀|² = 1/2) to 10⁻⁴, which increases with the coefficient |C₀|. On the other hand, the second scheme, with the condition of s₁ = s₂ = 0.1 and ${\text{T}}_1^2={\text{T}}_2^2=0.99$, yields the success rate ∼ 10⁻⁴. The success probability can of course be made larger by using higher-squeezing NDPAs in each scheme at the expense of output fidelity to some extent.

Other than nonideal detector efficiency, dark counts might potentially degrade the output fidelity. However, a recent experiment reported that a coincidence detection scheme recording only the synchronized events of laser pulse and a detector click in the pulsed regime can significantly eliminate dark count events [64]. Another experimental imperfection may also arise from the error in the transmissivity of beam splitter under our proposed schemes. In Fig. 7, we plot the output fidelity from each scheme by including the error Δt of beam-splitter transmissivity. Compared with Fig. 6, it turns out that a high output fidelity is still achievable and that the second scheme is particularly insensitive to the beam-splitter error.

 

Fig. 7 Fidelity between the ideal state C₀|0〉_a|0〉_b + C₁|1〉_a|1〉_b and the output state with the error Δt_i = ±0.01 of the beam-splitter transmissivity (i = 1,2). Other parameters are the same as those in Fig. 6.

Download Full Size | PDF

6. Summary

We have proposed two experimental schemes to generate a finite-dimensional photon number entangled state (PNES) with arbitrary coefficients, i.e., ${\sum }_{n=0}^N{C}_n{|n〉}_a{|n〉}_b$. One scheme is based on the second-order coherent superposition operation with two-mode squeezing operations, and the other on two first-order coherent superposition operations. We have shown that the coefficients of the PNES can be adjusted by the parameters of beam splitters and NDPAs in each scheme. In particular, our schemes do not require a high-level of squeezing for the nonlinear materials (NDPAs) and we further demonstrated that our schemes can generate the PNESs with high fidelity using realistic on-off photodetectors with nonideal efficiency. The class of PNES is useful for CV quantum informatics as we have considered its application to quantum teleportation and nonlocality test. We have shown that the PNES of finite dimension can surpass the performance of the TMSS with the level of squeezing currently available in the pulsed regime. Furthermore, the PNES includes a broad class of non-Gaussian entangled states together with the TMSS (a representative Gaussian entangled state), therefore, our schemes can also be used for fundamental tests of quantum physics, e.g. the robustness of Gaussian versus non-Gaussian entanglement under noisy environments [26, 28, 29].