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Abstract
The entanglement improvement is theoretically investigated when applying a single-side quantum scissors (SSQS) with a local squeezing operation and two-asymmetrical beam splitters (BSs) to one mode of an input two-mode squeezed vacuum state (TMSV). It is found that the gain factor can be significantly enhanced with the increasing of local squeezing parameter at the expense of the success probability. The entanglement can also be further improved adjusting the local-squeezing or the transmissivity of BSs in a small initial squeezing region. In addition, our scheme is robust against the photon loss in TMSV. The improved effect becomes more obvious due to the presence of local squeezing. However, the case is not true for a more realistic SSQS. In both cases, the asymmetric BSs play a positive role for the entanglement improvement. These results suggest that the squeezing-based SSQS at single-photon level is beneficial to effectively improve the entanglement, which may have potential applications in quantum communication.
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1. Introduction
Preparation and manipulation of the entangled resource is the prerequisite for both theoretical and experimental implementations of quantum information tasks, such as quantum metrology [1–3] and quantum communication [4–9]. The two-mode squeezed vacuum state (TMSV), which is seen as the most useful Gaussian resource, is often applied to various tasks [1,4,9]. For instance, using TMSV as input of optical interferometer, the so-called Heisenberg limit can be beaten [1]. Besides, Braunstein et al. proposed a scheme for teleporting a coherent state with the high fidelity by using TMSV as an entangled resource [4]. Interestingly, compared to the prepare-and-measure scheme in continuous-variable quantum key distribution (CV-QKD), the entanglement-based scheme using TMSV is beneficial to security analysis [10–13]. However, it is still a challenging task to experimentally prepare TMSV with a large squeezing. So far, the maximum value of two-mode squeezing is only about $10$ dB [14]. In order to realize various quantum tasks, it is usually required to prepare a higher entanglement or squeezing resource. Thus, it is a key issue how to enhance the entanglement of TMSV especially in the low squeezing levels.
Based on the restrictions of the no-go theorem [15–17], it is shown that the entanglement cannot be enhanced by using local Gaussian unitary operations. To solve this problem, lots of efforts have been made to focus on non-Gaussian operations [18–28], such as photon addition (PA), photon subtraction (PS) and the coherent superposition of both PA and PS. Due to the easier implementation with current technologies, these typical non-Gaussian operations not only have potential applications in the quantum state engineering, but also play an important role in QKD [29,30], quantum metrology [31,32] and quantum illumination [33]. For example, by performing PS on both modes of TMSV, the entanglement between Gaussian states can be improved distinctly, and the corresponding teleportation fidelity under the Braustein-Kimble scheme can also be enhanced using the generated non-Gaussian state as entangled resource [34]. However, the success probability for implementing single PS is less than 0.25, which might lead to the limited improvement of CV-QKD [10]. To avoid this problem, a kind of non-Gaussian operation, quantum catalysis first proposed by Lvovsky [35], has been successfully used to enhance the nonclassicality and entanglement properties of quantum states, improving the quantum coherence and expanding the transmission distance of QKD. In Ref. [25], Hu et al. pointed out that performing quantum catalysis on TMSV, although no photon is added or subtracted in this process, can effectively improve the entanglement and quantum-teleportation fidelity.
In addition to the aforementioned non-Gaussian operations, another interesting non-Gaussian operation, quantum scissors (QS) first suggested by Pegg et al. [36], is also a feasible strategy to generate entangled states. It is shown that, for an input coherent state, the resulting state can be truncated into the superposition of zero- and single-photon states. The QS can not only realize teleportation of single-mode optical states [37], but also can be used to improve the entanglement even in a realistic environment [38]. It is found that the entanglement improvement can be still achieved by using single-side (SS) QS instead of two-side QS even when both modes of TMSV pass through the lossy channels. In addition, the QS can achieve an ideal noiseless linear amplification (NLA) [39]. Inspired by this work, Yang et al. suggested a QS scheme including local-quadrature squeezing operation and applied this QS on coherent state to realize the improvement of NLA [40]. A problem arises naturally: can this squeezing-based QS realize the improvement of the TMSV entanglement. In this paper, we pay our attention to the entanglement improvement by applying the squeezing-based SSQS on one mode of TMSV rather than two-mode case. The single-mode case presents much higher success probability than two-mode case. In addition, two asymmetrical beam splitters (BSs) are involved in our scheme, different from that in Ref. [40]. The simulation results show that the entanglement (involving Shchukin–Vogel (SV) criteria, logarithmic negativity and entropy of entanglement) can be effectively improved with the increase of local squeezing parameter.
This paper is arranged as follows. In Sec. 2, we describe the theoretical scheme to generate non-Gaussian entangled states with squeezing-based SSQS. In Sec. 3, we discuss the entanglement degree with SV criteria, logarithmic negativity and entropy of entanglement and then make some comparisons. In Sec. 4, we further consider the ideal SSQS to the realistic TMSV and the realistic SSQS to the ideal TMSV. Finally, our main results are concluded in Sec. 5.
2. Generation of non-Gaussian entangled states with squeezing-based SSQS
In this section, we first give detailed descriptions of the squeezing-based SSQS, and derive its equivalent operator. Subsequently, we give the schematic generation of non-Gaussian entangled states by applying the SSQS with quadrature squeezing on one mode of TMSV, and its success probability is analyzed in detail.
2.1 Squeezing-based SSQS and its equivalent operator
In Fig. 1, we show the model of the squeezing-based SSQS comprised of two BSs (denoted as $B_{1}$ and $B_{2}$), two ideal photon-counting detectors (denoted as $D_{1}$ and $D_{2}$) and a local squeezing (denoted as $S_{c}\left ( \zeta \right )$ with squeezing parameter $\zeta$) in the transferring path between $B_{1}$ and $B_{2}$. $T_{1}$ and $T_{2}$ are the transmissivities of $B_{1}$ and $B_{2}$, respectively. When the assistant Fock states $\left \vert 1\right \rangle _{a}$ and $\left \vert 0\right \rangle _{c}$ on modes $a$ and $c$ are respectively injected into the first BS $B_{1}$, after undergoing the local squeezing on mode $c$ and interfering with arbitrary input states in the second BS $B_{2}$, the detectors $D_{2}$ and $D_{1}$ on modes $b$ and $c$ register only $\left \vert 1\right \rangle _{b}$ and $\left \vert 0\right \rangle _{c}$, respectively. Actually, the whole process can be viewed as an equivalent operator, i.e.,
Fig. 1. The squeezing-based SSQS for any input state. The effect will convert any input state into an output state in a Fock space spanned by both a vacuum and single photon. $B_{1}$ and $B_{2}$ are asymmetrical beam splitters with transmissivity $T_{1}$ and $T_{2}$. $D_{1}$ and $D_{2}$ are two ideal photon-counting detectors. $S_{c}\left ( \zeta \right )$ is a single-mode squeezing operator. The relation between input and output can be described by the equivalent operator $\widehat {O}\left ( \zeta,T_{1},T_{2}\right ) .$
From Eq. (1), one can see that the equivalent operator $\hat {O}\left ( \zeta,T_{1},T_{2}\right )$ consists of a superposition of vacuum and single-photon projection operators. Hence, after an arbitrary input state on mode $b$ goes through the squeezing-based SSQS, the resulting state can be truncated into the superposition of vacuum state and single-photon state. For instance, when inputting a weak coherent state $\left \vert \alpha \right \rangle =e^{-\left \vert \alpha \right \vert ^{2}/2}\sum _{n=0}^{\infty }\alpha ^{n}/\sqrt {n!}\left \vert n\right \rangle _{b},$ the resulting state becomes
Fig. 2. The gain factor $g$ of the generated state as a function of the BSs transmissivity for several different values of $\zeta =0,0.5,1$. (a) $T_{1}=T,$ $T_{2}=0.5$ and (b) $T_{1}=T_{2}=T$, respectively.
2.2 Generation of non-Gaussian entangled states
Now, let us begin with the schematic descriptions of generating non-Gaussian entangled states by performing the squeezing-based SSQS on one mode of TMSV, shown in Fig. 3. According to Eq. (1), the generated states can be given by
From Eq. (5), it is clear that when the SSQS is applied to one-mode of TMSV, the output state is truncated to be a kind of Bell-like state. Actually, for a small squeezing $r$, the TMSV is approximated as ($\left \vert 00\right \rangle +\tanh r\left \vert 11\right \rangle ).$ Thus, after the SSQS operation, the TMSV with small squeezing can be realized an ideal NLA with a gain factor $g$ defined in Eq. (4). This indicates that the NLA for TMSV can be further improved due to the presence of both two asymmetrical BSs and squeezing $\zeta$.
Fig. 3. Scheme of the squeezing-based SSQS device applied to one-mode of TMSV. The relation between input and output can be described by the operator $\widehat {O}\left ( \zeta,T_{1},T_{2}\right ) .$
To examine the effects of local squeezing parameter $\zeta$ and transmissivity on success probability, in Fig. 4, we show the success probability as a function of $T$ and $r$ for several different values of $\zeta =0,0.5,1.$ Different from the gain factor described in Fig. 2, it is clearly seen from Fig. 4 that the success probability decreases with the increase of $\zeta =0,0.5,1$, and the high probability distribution is in the low transmissivity range. In addition, at a fixed value $\zeta,$ we can see that the success probability for $\left ( T_{1},T_{2}\right ) =\left ( T,0.5\right )$ performs worse than that for $\left ( T_{1},T_{2}\right ) =\left ( T,T\right )$ (see Fig. 4). These results indicate that the embedded local squeezing operation is beneficial to enhance the gain factor at the expense of the success probability, while two identical-asymmetrical BSs can achieve amplification with a higher success probability compared to the former. The reason for the decrease of success probability with squeezing parameter $\zeta$ can be explained as follows. From the first equation in Eq. (1), for ideal zero/single photon input/detection, it is clear that the equivalent operator $\propto$ $-\left \vert 0\right \rangle _{ab}\left \langle 0\right \vert \left [ _{c}\left \langle 1\right \vert S_{c}\left \vert 1\right \rangle _{c}\right ] +\left \vert 1\right \rangle _{ab}\left \langle 1\right \vert \left [ _{c}\left \langle 0\right \vert S_{c}\left \vert 0\right \rangle _{c}\right ]$, where we take $T_{1}=T_{2}=0.5$ for simplicity. This indicates that the introduction of squeezing operator $S_{c}$ can adjust the weights of single and zero photons in the output state. With the increasing $\zeta$, the probability of zero/single photon state will be smaller in the squeezed vacuum/single photon state. This point may be clear by noting $\left [ _{c}\left \langle 0\right \vert S_{c}\left \vert 0\right \rangle _{c}\right ] =\sqrt {\textrm{sech}\zeta }$ and $\left [ _{c}\left \langle 1\right \vert S_{c}\left \vert 1\right \rangle _{c}\right ] =$ sech$^{3/2}\zeta.$ This case is true for TMSV as input. In addition, for TMSV as input, the entanglement is originated from single-photon and zero-photon components. For unideal case, this case is not true.
Fig. 4. The success probability $P_{d}$ of achieving the generated state as a function of $r$ and $T$ with several different values of $\zeta =0$ (green surface), $0.5$ (red surface), $1$ (blue surface). (a) $T_{1}=T,$ $T_{2}=0.5$ and (b) $T_{1}=T_{2}=T$, respectively.
3. Entanglement properties of the generated non-Gaussian states
Most of quantum-communication protocols, such as quantum teleportation [5–7], dense coding [41,42] and entanglement-based CV-QKD [12,13,43], require the faithful transmission of entanglement, which reveals that how to measure the degree of correlation is a key element. Fortunately, in Refs. [44,45], there are several common methods of quantifying the entanglement properties, such as Einstein-Podolsky-Rosen correlation, Shchukin-Vogel (SV) criteria, logarithmic negativity, entropy of entanglement and concurrence. Different entanglement measures may show different properties of entanglement. Among them, the logarithmic negativity is an easily computable measure especially for mixed states, and the entropy of entanglement is one of the standard and simplest measures. To achieve a combination of qualitative and quantitative descriptions, in this section, we shall adopt SV criteria, logarithmic negativity and entropy of entanglement to investigate the entanglement properties of the generated non-Gaussian states.
3.1 SV criteria
On the basis of SV criteria, the acceptable and sufficient condition of inseparability (or entanglement) is given by
Generally speaking, the smaller the SV value, the higher the entanglement degree. According to Eq. (5), $\left \langle a^{\dagger }b^{\dagger }\right \rangle$ and $\left \langle ab\right \rangle$ can be easily calculated as
First, let’s examine the improved region of entanglement for the generated states. As shown in Fig. 5, we contour the difference SV between the generated states and the TMSV as a function of $r$ and $T$ for several different values of $\zeta =0,0.5,1$. For the case of $\left ( T_{1},T_{2}\right ) =\left ( T,0.5\right ) ,$ it is shown that the improved area of entanglement is distributed within small squeezing $r$ and high transmissivity $T$, which also increases with the increase of squeezing parameter $\zeta =0,0.5,1.$ This implies that the embedded local-squeezing operation is beneficial to broadening the improved entanglement area spanned by $r$ and $T$. This case is also true for the case of $T_{1}=T_{2}=T.$ Different from the former, however, the improved area of the latter is relatively smaller under the same accessible parameters.
Fig. 5. The difference of SV between the generated states and the TMSV as a function of $r$ and $T$ for several different $\zeta =0,0.5,1$ (from left to right). (a)-(c) $T_{1}=T,$ $T_{2}=0.5$ and (d)-(f) $T_{1}=T_{2}=T$, respectively. Note that the color area represents the improvement of SV.
Second, we consider the optimal SV changing with $r$ for several different cases of $\left ( T_{1},T_{2}\right ) =\left ( 0.2,0.2\right )$, $\left ( 0.5,0.5\right )$, $\left ( 0.7,0.7\right )$ in Fig. 6(a) when optimized over $\zeta.$ To show the advantage of our scheme, we also consider the case without local squeezing ($\zeta =0$, black solid line). Here, we should point out that, considering the fact that the experimentally available squeezing parameter is small and corresponding to the above discussion in Fig. 5, the optimized parameter $\zeta$ is fixed at $\left ( 0,1\right )$. For the TMSV, from Fig. 6(a) it is clear that only when the squeezing parameter $r$ exceeds a certain threshold value ($\simeq 0.35$), the condition of $SV<0$ can be achieved. For low transmissivity case with $\left ( T_{1},T_{2}\right ) =\left ( 0.2,0.2\right )$, the condition of $SV<0$ can not be realized in the whole region of $r$ (see red lines). However, for high transmissivity cases with $\left ( T_{1},T_{2}\right ) =\left ( 0.5,0.5\right ) ,\left ( 0.7,0.7\right )$, the condition of $SV<0$ can be achieved, and the threshold values of $r$ decrease with the increase of $T_{1}=T_{2}=0.5,0.7$. For instance, the threshold values are about 0.25 and 0.12 for $T_{1}=T_{2}=0.5,0.7,$ respectively, which are smaller than that (0.35) for TMSV. In addition, compared to SV for $\left ( T_{1},T_{2}\right ) =\left ( 0.2,0.2\right )$, $\left ( 0.5,0.5\right )$, $\left ( 0.7,0.7\right )$ and $\zeta =0$ (see Fig. 6(a)), it is shown that although the condition of $SV<0$ can also be improved by the SQSS without the local squeezing, the improved effect is much obvious by the SQSS with local squeezing. These results indicate that the local squeezing operation contributes to the existence and improvement of entanglement in a small squeezing region (about $r<0.48$) and high transmissivity region. In order to see clearly whether the SV can be improved by $\zeta$ which is not larger than $r$, we further consider the SV as a function of $\zeta$ for given $r$ in Fig. 6(b). From Fig. 6(b), it is clear that the SV can be further improved by $\zeta$ smaller than $r$.
Fig. 6. (a) The optimal SV as a function of squeezing parameter for several different $T_{1}=T_{2}=0.2,0.5,0.7$ when optimized over $\zeta$. For comparison, we plot the SV with $T_{1}=T_{2}=0.2,0.5,0.7$ corresponding to $\zeta =0$ and the original TMSV in the solid line. (b) The SV as a function of $\zeta$ for given $r=0.3$ and $T_{1}=T_{2}=0.7.$ For comparison, we plot SV for the TMSV and the TMSV with traditional scissor ($\zeta =0$).
3.2 Logarithmic negativity and entropy of entanglement
In this subsection, we further quantify the entanglement degree of the generated states by means of logarithmic negativity and entropy of entanglement. The entropy of entanglement is one of the standard and simplest entanglement measures, whereas the logarithmic negativity is an easily computable measure of entanglement in contrast to other methods, such as SV criteria and logarithmic negativity. These two measures, i.e., logarithmic negativity and entropy of entanglement, are respectively defined as [46,47]
where $\left \Vert \rho ^{T_{A}}\right \Vert =$Tr$\left \vert \rho ^{T_{A}}\right \vert$ is the trace norm ($\rho ^{T_{A}}$ representing the partial transpose of density operator $\rho$ with respect to party $A$) and $S\left [ \rho \right ] =-$Tr$\left ( \rho \log _{2}\rho \right )$ is the von Neumann entropy of the density operator $\rho.$ For an arbitrary pure state in the Schmidt form expressed as $\left \vert \Theta \right \rangle _{ab}=\sum _{l=0}c_{l}\left \vert \Theta _{l}\right \rangle _{a}\left \vert \Theta _{l}\right \rangle _{b}$ ($c_{l}$: normalized factor) with the orthonormal states $\left \vert \Theta _{l}\right \rangle _{a}$ and $\left \vert \Theta _{l}\right \rangle _{b},$ the explicit forms of both logarithmic negativity and entropy of entanglement are respectively given byThus, based on Eqs. (12) and (13), for the generated states, the explicit expressions of both logarithmic negativity and entropy of entanglement are respectively calculated as
For a comparison, after substituting Eq. (6) into Eqs. (12) and ( 13) and direct calculation, the logarithmic negativity and entanglement entropy for TMSV can be derived as
Next, we first examine the entanglement improvement for the generated states, compared to TMSV. Combining Eqs. (14) and ( 15) with Eqs. (17) and (18), we plot the difference of logarithmic negativity (entropy of entanglement) between the generated states and the TMSV as a function of $r$ and $T$ for several different values of $\zeta =0,0.5,1,$ involving two special cases $\left ( T_{1},T_{2}\right ) =\left ( T,0.5\right ) ,\left ( T,T\right ) ,$ as shown in Fig. 7 (Fig. 8). It is evident that, under the same parameters, for both the logarithmic negativity and the entropy of entanglement, the region of entanglement-enhanced can be further expanded with the increase of $\zeta =0,0.5,1,$ which is similar to the SV case (see Fig. 6). The improved area of the entanglement is limited at the range of both small squeezing $r$ and high transmissivity $T$. In addition, comparing Figs. 7 with 8, it is found that the improved area for $\left ( T_{1},T_{2}\right ) =\left ( T,0.5\right )$ is larger than that for $\left ( T_{1},T_{2}\right ) =\left ( T,T\right ) .$ This indicates that, according to the sizes of the improved area, utilizing the logarithmic negativity to quantify the entanglement improvement seems more rigorous than using the entropy of entanglement, due to the fact that the improvement of the former does not mean that of the latter, but the opposite is true.
Fig. 7. The difference $\Delta E_{N}=E_{N}\left ( \left \vert \psi \right \rangle _{out}\right ) -E_{N}\left ( \left \vert \psi \right \rangle _{in}\right )$ as a function of $r$ and $T$ for several different $\zeta =0,0.5,1$ (from left to right). (a)-(c) $T_{1}=T,$ $T_{2}=0.5$ and (d)-(f) $T_{1}=T_{2}=T$, respectively. Note that the color area represents the improvement of logarithmic negativity.
Fig. 8. The difference $\Delta E_{V}=E_{V}\left ( \left \vert \psi \right \rangle _{out}\right ) -E_{V}\left ( \left \vert \psi \right \rangle _{in}\right )$ as a function of $r$ and $T$ for several different $\zeta =0,0.5,1$ (from left to right). (a)-(c) $T_{1}=T,$ $T_{2}=0.5$ and (d)-(f) $T_{1}=T_{2}=T$, respectively. Note that the color area represents the improvement of entropy of entanglement.
Similarly, in Figs. 9(a) and 9(b), we illustrate the optimal logarithmic negativity and entropy of entanglement as a function of $r$ for several different cases with $\left ( T_{1},T_{2}\right ) =\left ( 0.2,0.2\right )$, $\left ( 0.5,0.5\right )$, $\left ( 0.7,0.7\right ) ,$ when optimized over $\zeta \in.\left ( 0,1\right ) .$ As a comparison, the black line is the entanglement for TMSV. From Figs. 9(a) and (b), it is clear that, for low transmissivity case with $\left ( T_{1},T_{2}\right ) =\left ( 0.2,0.2\right )$, the entanglement can not be improved, even in the presence of local squeezing. While for the high-transmittance cases with $\left ( T_{1},T_{2}\right ) =\left ( 0.5,0.5\right ) ,\left ( 0.7,0.7\right )$, the entanglement can not only be enhanced in a small squeezing region, but also the enhanced effect becomes more obvious especially for the existing local squeezing $\left ( \zeta \neq 0\right )$. This indicates that the embedded local-squeezing operation is conducive to increasing the entanglement well. In addition, compared Figs. 9(a) to 9(b), it is interesting to notice that the entanglement-enhanced are limited at the regions of $r<0.35$ and $r<0.48$ for measures of $E_{N}$ and $E_{V}$, respectively. This case of $E_{V}$ is similar to SV, thus the case of $E_{N}$ can be seen as a subspace for both of them to describe the entanglement. In order to see clearly whether the $E_{N}$ and $E_{V}$ can be improved by $\zeta$ which is not larger than $r$, we further consider $E_{N}$ and $E_{V}$ as a function of $\zeta$ for given $r$ in Figs. 9(c) and (d), from which it is clear that $E_{N}$ and $E_{V}$ can be further improved by $\zeta$ smaller than $r$.
Fig. 9. The optimal $E_{N}$ (a) and $E_{V}$ (b) as a function of $r$ for several different $T_{1}=T_{2}=0.2,0.5,0.7$ when optimized over $\zeta \in (0,1)$. For comparison, we plot the logarithmic negativity and the entropy of entanglement with $T_{1}=T_{2}=0.2,0.5,0.7$ corresponding $\zeta =0$ and the original TMSV in the solid line. $E_{N}$ (c) and $E_{V}$ (d) as as a function of $\zeta$ for given $r=0.3$ and $T_{1}=T_{2}=0.7.$ For comparison, we plot $E_{N}$ and $E_{V}$ for the TMSV and the TMSV with traditional scissor ($\zeta =0$).
4. Effects of quantum scissors in realistic cases
In this section, we further examine the ideal SSQS to the realistic TMSV and the realistic SSQS to the ideal TMSV, respectively.
4.1 Realistic quantum scissors to TMSV
When the two-mode of TMSV go through two symmetrical loss channels (characteristic of transmission coefficient $\eta _{t}=1-\eta _{r}$), the output state after losses can be given by [38]
After the SSQS is applied to mode $b$, then the output state is given by
where $P=$Tr$\left [ \hat {O}\rho _{loss}\hat {O}^{\dagger}\right ]$. Substituting Eq. (1) into (21) and using Eq. (19), it is ready to have the expansion of $\rho _{out}$ in Fock space, not shown here for simplicity (For more details, see Appendix A).In Fig. 10, we plot the entanglement as a function of squeezing parameter $r$ for given $T_{1},T_{2},\zeta$ and $\eta _{t}$. As a comparison, the entanglement of ideal TMSV is also included (see black line). For given $\zeta =1.0$ and $\eta _{t}=0.9$, Fig. 10(a) shows the effects of different values of $T_{1},T_{2}$ on the entanglement. It is shown that, in a small squeezing region of $r,$ (i) although $T_{1}=T_{2}=0.5$ has no contribution for NLA, the local squeezing $\zeta$ can still achieve the entanglement improvement. (ii) For the case of $T_{2}=0.5$, both the squeezing region and degree of improved entanglement become more obvious as $T_{1}$ increases. (iii) For the case of $T_{1}=T_{2}$, the degree of improved entanglement becomes larger but in a smaller squeezing region.
Fig. 10. The logarithmic negativity is plotted as a function of squeezing parameter $r$ for given $T_{1},T_{2}, \zeta$ and $\eta _{t}$. As a comparison, the entanglement of ideal TMSV is also included (see black-dashed line)
To clearly see the effects of different local squeezing $\zeta$ on entanglement, in Fig. 10(b) shows the entanglement as a function of squeezing parameter $r$ for given $T_{1}=T_{2}=0.7,$ and $\eta _{t}=0.9$. It is found that the degree of improved entanglement becomes larger but in a smaller squeezing region as the increasing $\zeta$. This is similar to the case of $T_{1}=T_{2}$. These results indicate that, even in a loss environment, the SSQS can still improve the entanglement in the small squeezing region, which implies that the proposed SSQS can be used to combat against photon losses of TMSV.
4.2 Realistic quantum scissors to TMSV
Next we examine the case of realistic quantum scissors to TMSV. Here we replace single-photon detector $D_{2}=\left \vert 1\right \rangle _{bb}\left \langle 1\right \vert$ and single-photon resource $\left \vert 1\right \rangle _{aa}\left \langle 1\right \vert$ with “on-off” detector $\Pi _{b}=1-\left \vert 0\right \rangle _{bb}\left \langle 0\right \vert$ and the heralded single-photon state, respectively. The heralded single-photon state is prepared using the “on-off” detector $\Pi =1-\left \vert 0\right \rangle \left \langle 0\right \vert$ to one mode of TMSV $\left \vert \Psi \left ( \lambda \right ) \right \rangle$ with a squeezing parameter $\lambda$, shown in Fig. 11.
Fig. 11. Realistic scheme of the SSQS for any inpt state. The heralded single photon is generated from the TMSV with squeezing parameter $\lambda$. Here all detectors are "on-off" detectors.
After the modified SSQS device is applied to $b$ mode of TMSV ($\rho _{bb_{1}}$), the output state $\rho _{out}$ is derived analytically (for more details, see Appendix B). In Fig. 12 we plot the entanglement as a function of squeezing parameter $r$ for given $\lambda,$ $T_{1}=T_{2},$ and $\zeta$. As a comparison, the entanglement of ideal TMSV is also plotted (see black line). From Fig. 12(a), it is clear that for given $\lambda$ and $\zeta$, the degree of improved entanglement becomes larger but in a smaller squeezing region. This is similar to the case above. However, from Fig. 12(b) it is interesting that the improved effect of entanglement becomes worse or even absent as $\zeta$ increases. This is in sharp contrast to this situation that increasing $\zeta$ can improve the entanglement of realistic TMSV. These imply that, for the realistic SSQS, the entanglement can be improved by adjusting transmission factor $T_{1},T_{2}$ rather than local squeezing $\zeta$.
Fig. 12. The logarithmic negativity is plotted as a function of squeezing parameter $r$ for given $T_{1},T_{2}, \zeta$ and $\lambda$. As a comparison, the entanglement of ideal TMSV is also included (see black-dashed line)
5. Conclusion
In this paper, we propose an improving scheme of the entanglement properties of the generated states by using squeezing-based SSQS and two asymmetrical BSs on one mode of TMSV. In the SSQS system, the embedded local squeezing operation and the asymmetrical BSs can be used to further enhance the gain factor when inputting a weak coherent state or TMSV. In addition, compared to the case without the local squeezing or two asymmetrical BSs, our scheme also has a better performance at the expense of the success probability. After that, the entanglement properties of the generated states have been investigated and discussed by means of SV criteria, logarithmic negativity and entropy of entanglement. Our numerical simulations show that the entanglement with respect to these three measurements can be further improved with the increase of local squeezing parameter in an initial small squeezing region, which means that the use of local squeezing-based SSQS is beneficial to significantly enhance the entanglement.
In addition, some realistic cases are considered, including the decohered TMSV and the more realistic SSQS modified by replacing single-photon detector and single-photon resource with “on-off” detector and the heralded single-photon state, respectively. For the decohered TMSV, it is found that our scheme presents robust against the photon-loss in the decohered TMSV. The robustness becomes obvious with the increasing of local squeezing and transmissivity. However, for the more realistic SSQS, the improvement of entanglement can be kept by adjusting transmissivity rather than local squeezing in initial small squeezing region. These results suggest that the use of squeezing-based SSQS at single-photon level is helpful to effectively improving the entanglement, thereby making it more potential to the quantum information processing.
Actually, the quantum scissors have lots of applications. For example, the usual quantum scissors can be used to realize the long distance of CV-QKD [48], and continuous-variable quantum repeater [49]. In addition, a generalized quantum scissors is proposed to truncate and amplify quantum states up to higher order without distorting the amplified Fock coefficients [50]. Thus it will be our future work to expand our scheme to these aspects.
Appendix A: The output state for ideal quantum scissors to the realistic TMSV
Using Eq. (9) and the coherent state representation of Fock state, i.e., $\left \vert m\right \rangle =\left. 1/\sqrt {m!}\frac {\partial ^{m}}{ \partial t^{m}}\left \Vert t\right \rangle \right \vert _{t=0}$ where $\left \Vert t\right \rangle =e^{tb^{\dagger }}\left \vert 0\right \rangle$ is un-normalized coherent state (satisfying $\left \langle \nu \right. \left \Vert t\right \rangle =e^{\nu t}$), the matrix elements of $\rho _{loss}$ in Fock space can be rewritten as
When the SSQS is applied to mode $b$, then the output state is given by
After substituting Eqs. (1) into (A3), it is ready to have
Using Eq. (A4) one can numerically calculate the degree entanglement.
Appendix B: The output state in realistic quantum scissors device with local squeezing
For simplicity, we first consider any single mode state $\rho _{b}$ as inputs. In realistic case, the output state $\rho _{out}$ can be derived as
The coherent state representation of thermal state will be used in the following calculation.
Substituting Eqs. (B2) into (B1), it is ready to have
Then let’s derive $O_{j}(j=1,2,3,4)$. Using the normal order forms of beam splitter operators and single-mode squeezing operator [38], i.e.,
In the whole deriviation, the IWOP technique [51] is fully used.
Substituting Eqs. (B21)–(B22) into (B7)–(B8), one can get
Further using Eq. (B3) and the IWOP technique, $O_{5}$ can be put into the normal ordering form, i.e.,
Next, we consider the case that the quantum scissors is applied to single mode ($b$) of two-mode squeezed vacuum (TMSV) $\rho _{bb_{1}}$. At this time, we only replace $\rho _{b}$ with the TMSV denoted as $\rho _{bb_{1}}$ in Eq. (5). For simplicity of calculation, here we present the normal ordering form of the TMSV,
Substituting Eq. (B30) into (B5)–(B6), and using the IWOP technique and completeness relation of coherent state, one can finally obtain
In a similar way, one can derive
whereThus the final output state is obtained as
In Eq. (B45) we dropped the normalized factor, which has no effect on the entanglement. From Eqs. (B45)–(B47), it is clear that the output state is non-Gaussian mixed state, composed of two-mode entangled state $\hat {K}_{1}$, squeezed thermal and vacuum states.
In order to evaluate the degree of entanglement in realistic case, one needs to expand $\hat {K}_{1}\left ( a^{\dagger},a,b_{1}^{\dagger},b_{1}\right ) ,\hat {K} _{2}\left ( b_{1}^{\dagger},b_{1}\right )$ and $\hat {K}_{3}\left ( a^{\dagger},a\right )$ in Fock space to get the factors
Using Eqs. (B45)–(B53), one can numerically calculate the degree of entanglement in realistic case.
Funding
National Natural Science Foundation of China (11664017, 11964013); Major Discipline Academic and Technical Leaders Training Program of Jiangxi Province (20204BCJL22053).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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