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Abstract
Recently, the strong single-mode squeezed vacuum state of light with $-15$ dB of classic noise has been created [Phys. Rev. Lett. 117, 110801 (2016) [CrossRef] ]. This is an important resource in continuous variable (CV) quantum communication and fault-tolerant quantum computing. Nevertheless, strong squeezing means a non-negligible population in high-photon-number subspace and may pose a strong challenge to photon-based technology, for example, entanglement distillation. Entanglement distillation is an efficient method for retrieving a higher quality of entanglement from a weakly entangled state. Up till now, almost all the schemes for entanglement distillation are restricted to the low-squeezing regime. The distillation of strong-squeezing induced entanglement is an interesting but open topic in the near future. Here, we take the single-mode squeezed entangled state (SMSE) as an example and show that conventional photon subtraction based distillation fails to generate a higher quality of entanglement with experimentally feasible optical beam splitters. To this point, we show that the superposition of photon annihilation and creation could be an effective method. A practical scheme for implementing such a superposition with on-line squeezing is suggested. Our method is verified numerically by the calculation of a 25 dB SMSE state with an ideal photon detector and for distillation of 20.5 dB SMSE state with practical dichotic on-off detectors. The analysis of the distillation of the strong squeezing effect could be extended straightforwardly to other squeezing-based quantum information processing, such as quantum metrology and quantum illumination.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Squeezed state is one of the most important resource in continuous variable (CV) quantum information processing. Strongly squeezed entanglement state is beneficial to improving the fidelity of teleportation [1], precision of gravitational wave detector [2] and fault-tolerance of CV quantum computing [3]. At the same time, strong squeezing means non-negligible population in high-photon-number subspace, which may offer a challenge to photon-number based quantum information processing. In this paper, we are choosing photon subtraction as an example and investigating the effect of strong squeezing on practical entanglement distillation schemes.
Generally, the study of continuous variable entanglement distillation starts in the end of 1990s [4,5]. As is well known, entanglement is a fragile quantum resource and it will be easily disturbed by environment-induced decoherence. To this point, entanglement distillation is proposed to produce a small set of stronger entanglements from a large amounts of weakly entangled ones. According to the dimension of the Hilbert space where an entangled quantum state resides, research on entanglement distillation can be generally divided into discrete variable (DV) [6–9] and continuous variable (CV) entanglement distillation. The distillation of CV quantum entanglement is more complicated since it was then noticed that non-Gaussian operation was a necessary element in distillation of commonly used Gaussian entanglement [10–12]. There are two methods to generate non-Gaussian elements in optical CV entanglement state. (1) The first method is non-Gaussian operation on single copy entanglement state, such as photon subtraction [5], photon addition [13], photon catalysis [14,15] and noiseless amplification [16]. These schemes can be implemented with beam splitters and photon detections. (2) The second one is non-Gaussian post-selection. Typical examples can be found in entanglement distillation with two-copy and multi-copy inputs [17], where two pairs of entanglement are interfered by two $50:50$ beam splitters and then biside homodyne measurement is performed on two of the outgoing modes. Non-Gaussianity is introduced by post-selecting the homodyne measurement records. Both single-copy and two-copy distillation have been demonstrated in state-of-the-art experiments [18–20].
Almost all these results on continuous variable entanglement distillation assume that the two-mode entanglement state are in weak-squeezing regime. More recently, strong single-mode squeezing at $15$ dB has been successfully generated in experiment [21]. Enlightened by this, in this manuscript, we investigate the distillation of single-mode-squeezed entangled state in case of strong squeezing. Our results show that, remarkably, schemes with perfect single-side and biside photon annihilation can still be used for distilling single-mode-squeezed entangled states, even in strong squeezing regime. But the situations for schemes with practical photon annihilation ( photon subtraction, PS) are quite different. It is shown that strong squeezing results in a huge population in high-photon-number subspace and only practical PS-based schemes with beam splitter of extremely high transmittance $T\approx 1$ is competent for distillation. Generally, a beam splitter with $T\approx 1$ is not always available in most laboratories and we further propose an alternative scheme with superposition of photon annihilation and addition to overcome such a unit-$T$ limit.
The paper is organized as follows. First, in Sec. II, we show the infeasibility of entanglement distillation scheme with conventional photon subtractions. Actually, to distill two-mode entanglement corresponding to beam-splitting of 25 dB single mode squeezed state, a beam splitter with extremely high transmittance ($T>0.9921$) is required for effective distillation. This poses a challenge to state-of-art distillation scheme. Thus to relieve such a burden, in Sec. III, we demonstrate that the scheme of coherent superposition of photon annihilation and creation could be employed to implement effective distillation. Such a superposition can be implemented with a weak on-line squeezing and numerical calculation is performed for distillation of SMSE upto 25 dB input. Sec. IV is devoted to the analysis of nonunit effect of on-off photon detectors in the distillation scheme. Finally, we conclude with a discussion in Sec.5.
2. Infeasibility of distillation with conventional photon subtractions
We start with the single-mode-squeezed entangled state. It is generated by mixing a single mode squeezed state $|\xi \rangle = S(\xi )|0\rangle =\exp [\frac {1}{2}\xi \hat {a}^{\dagger 2}-\frac {1}{2}\xi ^* \hat {a}^2]|0\rangle$ with vacuum state $|0\rangle$ on a balanced beam splitter (See Fig. 1(a)(b)). This is a rather simple method for generating continuous variable two-mode entanglement [22]. In photon number space, the single-mode-squeezed entanglement can be written as
Fig. 1. Schematic diagram of entanglement distillation with single-side (a) and biside (b) photon annihilation operation. $|\xi \rangle$ and $|0\rangle$ represents single-mode-squeezed state and vacuum state. $BS$ denotes the optical beam splitter with transmittance $T=0.50$.
Fig. 2. (a)Average photon number of $|\xi \rangle$ as a function of squeezing $S_{\mathrm {dB}}$. (b) Entanglement (measured with logarithmic negativity) and (c) variance (measured with $\log _{10}\textrm {Var}(P_1+P_2)$ )before and after single-side photon annihilation (lines with circles ) and biside photon annihilation (lines with asterisks). Photon number is truncated at $\overline {D}=1999$.
2.1 Ideal photon subtraction: photon annihilation
We first show that distillation of single-mode-squeezed entanglement can be implemented by single-side (Fig. 1(a)) or biside photon annihilation(Fig. 1(b)), even for strong-squeezing inputs. The state after single-side and biside photon annihilation can be given by
Theorem 1. Let $|\psi \rangle =\sum _{n,m=0}^{\overline {D}-1} c_{nm}|n\rangle |m\rangle$ and $C=[c_{nm}]$ be $\overline {D}\times \overline {D}$ matrix. The logarithmic negativity of $|\psi \rangle$ can be given by the singular values of $C$.
The proof of Theorem 1 is given in Appendix A. Theorem 1 helps to reduce the computing complexity from $\overline {D}^4$ to $\overline {D}^2$. In Fig. 2(b) we truncate each optical mode at $\overline {D}=1999$ and show the logarithmic negativity as a function of squeezing $S_{\mathrm {dB}}$ . Our results show that the entanglement of single-mode-squeezed entanglement can always be further improved both by single-side photon annihilation and biside photon annihilation, even in case of strong squeezing inputs. In terms of logarithmic negativity, single-side photon annihilation performs better than biside photon annihilation in weak squeezing input, which coincides very well with the results in Ref.[25,26]. For strong squeezing $S_{\mathrm {dB}}>4.0$ dB, biside photon annihilation could generate a higher entanglement due to the symmetry property of distilled entangled state. However, biside photon annihilation does not show any substantial improvement (about only $3.23\%$ improvement in logarithmic negativity) upon that after single-side annihilation, even though more resources will be used in biside settings. Thus, for a practical point of view, single-side annihilation is more convenient for distillation.
Logarithmic negativity is a relatively easily-computable measure for two mode continuous variable entanglement [27,28] and is widely adopted in theory and experimental investigation of CV entanglement distillation [18]. However, such a figure of merit is not completely equivalent to the squeezing of quadrature. The SMSE state in Eq.(1) is symmetric with respect to A and B mode. It is squeezed in the quadrature $P_1+P_2$, with $P_1=\frac {\hat {a}_A-\hat {a}_A^\dagger }{\sqrt {2}i}$ and $P_2=\frac {\hat {a}_B-\hat {a}_B^\dagger }{\sqrt {2}i}$. As shown in the black solid line in Fig. 2(c), the variance of operator $P_1+P_2$ ($\textrm {Var}(P_1+P_2)\equiv \langle (P_1+P_2)^2\rangle -\langle P_1+P_2\rangle ^2$) decreases exponentially with $S_{\mathrm {dB}}$. After distillation, the entanglement (measured with logarithmic negativity ) increases but the squeezing does not always .The variance of operator $P_1+P_2$ is even bigger than that before distillation. The SMSE state after single-side annihilated performs even worse due to the asymmetric property in the distillation scheme. That is to say, single-side annihilation and bi-side annihilation result cannot be translated directly to the squeezing of $P_1+P_2$. On the other hand, photon subtraction or photon addition has been applied to generate single mode squeezing in literature [29]. However, the squeezing generated is quite weak. The probabilistic scheme for generating two-mode squeezing would be an interesting topic and we believe it requires a more sophisticated procedure to implement a strong squeezing ($>10$ dB) with weak squeezing and probabilistic photon subtraction.
2.2 Practical photon subtraction: with beam splitters and photon detectors
Perfect photon annihilation operator is actually non-physical and non-unitary and it is convenient to approximate it with physical available operation such as beam splitter and photon detection, which is always termed as photon subtraction [5]. This is actually what is always adopted in state-of-art laboratories [30–32]. A scheme with beam splitter and ideal single photon detector is shown in Fig. 3(a)(b). It should be noted that such practical scheme of PS has been intensively investigated in literature. For example, Ref[33,34] have investigated the characteristic of the photon subtracted single mode squeezed vacuum state $S(\xi )|0\rangle$ and the distillation of two mode squeezed vacuum in noisy environment. However, their state to be distilled is different from the SMSE state in experiment [18] and their consideration is more focused in low squeezing regime. For simplicity, we assume that the beam splitters for biside photon subtraction are of the same transmittance $T$. $C$, $D$ modes, initialized in pure vacuum state, are assistant optical modes. Successful photon subtractions are heralded when all detectors register a single photon state.
Fig. 3. Practical single-side and biside distillation with optical beam splitters with transmittance $T$ and photon detectors. Successful photon subtractions are heralded when all photon detectors in each scheme register a single photon state. The input of $C$ and $D$ are pure vacuum states.
Now we want to see why the practical scheme with beam splitters and photon detectors fails in distillation in strong squeezing regime. Let us write the pure two mode state in $A^\prime - B^\prime$ with $|\psi \rangle =\sum _{n,m=0}^\infty c_{nm}|n\rangle |m\rangle$. Here we take the single-side photon subtraction as an example and similar result also applies to biside subtraction scenarios. The coupling between mode $A^\prime$ and $C$ can be formulated by
As is shown in Fig. 4(a), to distill the single-mode-squeezed entangled (SMSE) state with 15 dB, a beam splitter with transmittance $T<0.9158$ will be unable to generate a higher entanglement. Whereas to distill SMSE with 25 dB, the transmittance $T$ of up to $0.9921$ (Fig. 5 (a)) is highly in need. Fig. 4 (c)(d) and Fig. 5(c)(d) show a similar phenomenon for biside photon subtraction. It is interesting to note that the probability of single-side and biside photon subtraction get a pronounced increment due to strong population in high-photon-number subspace. They can even surpass easily the upper bound of $1-T$ (single-side PS) and $(1-T)^2$(biside PS) in weak-squeezing scenarios.
Fig. 4. Distillation SMSE state generated with 15 dB single mode $|\xi \rangle$. Photon number is truncated at $\overline {D}=199$.
Fig. 5. Distillation SMSE state generated with 25 dB single mode $|\xi \rangle$. Photon number is truncated at $\overline {D}=1199$.
The results in Fig. 4 and Fig. 5 also show the infeasibility of distillation with conventional beam splitter and photon detectors. Going ahead with this method, a beam splitter with $T=0.9985$ is highly demanding in distilling SMSE state generated by 30 dB single mode squeezed vacuum. This on the other hand will be a technique challenge for most quantum information laboratories. In the follwing, we will show an alternative method with photon subtraction and on-line weak squeezing.
3. Distillation of strong SMSE with on-line weak squeezing
Continuous variable entanglement distillation with photon subtraction and on-line squeezing has been proposed by one of us and Peter van Loock in 2011 [35] where it was shown that on-line local squeezing can be used to enhance continuous variable entanglement distillation scheme. Examples were shown for distilling two-mode squeezed vacuum state in weak squeezing regimes. Here, we extend our analysis to distillation of beam splitted single-mode squeezed vacuum. Already-known experiment by A. Furusawa’s group [18] was carried for $5.4$ dB squeezing. Here, with the Theorem 1 we developed, we could extend the related analysis to strong squeezing regime (upto 25 dB squeezing).
The method of on-line local squeezing is in essence equivalent to preparing a coherent superposition of the photon annihilation and creation operations. This is actually a single-mode non-Gaussian operation and can be given by $f(\hat {a},\hat {a}^\dagger )=\mu \hat {a}+\nu \hat {a}^\dagger$, with $\mu , \nu \in \mathbb {C}$ being the complex coefficient [36]. Since single-side distillation has the advantage of high probability of success, in what follows we will mainly focus on distillation schemes with single-side operation. The distilled state (after single-side operation on mode $A^\prime$) can be given by
The coherent superposition $f(\hat {a},\hat {a}^\dagger )$ is non-Gaussian operation and can be implemented by cascade of single-mode squeezing and photon annihilation [35], single photon interference [36], cascade of quadrature displacement and photon annihilation [37]. This provides a powerful tool for continuous variable entanglement distillation and quantum state engineering [38–44]. Here, we consider the implementation scheme with local squeezing and photon annihilation.
Fig. 6(a) and (b) shows the single-side distillation with a series of single-mode squeezing $S(r^\prime )$ and ideal and practical photon annihilation. The effect is local unitarily equivalent to the coherent superposition of photon annihilation and addition operation. This can be seen as follows:
Fig. 6. Squeezing operation assisted single-side distillation of SMSE state with ideal (a) and practical (b) photon subtraction. Success in practical single-side distillation (b) is heralded when photon detector registers the single photon state.
Local squeezing $S(r^\prime )$ provides an additional tool for improving the entanglement in the distilled state. To show how squeezing parameter $r^\prime$ influences the distillation scheme, the entanglement of the distilled state is studied as a function of $r^\prime$. Results for $-1\le r^\prime \le 1$ is plotted in Fig. 7. It is of note that, the weights of $\hat {a}_{A^\prime }$ and $\hat {a}_{A^\prime }^\dagger$ is easily modulated when the absolute value of squeezing parameter $r^\prime$ is small ( e.g. $|r^\prime | \lesssim 1$). For $|r^\prime | \gg 1$, $\cosh (r^\prime )\approx |\sinh (r^\prime )|, f({a}_{A^\prime },{a}_{A^\prime }^\dagger )\propto \hat {a}_{A^\prime }+\hat {a}_{A^\prime }^\dagger$ or $\hat {a}_{A^\prime }-\hat {a}_{A^\prime }^\dagger$ and the weights of $\hat {a}_{A^\prime }$ and $\hat {a}_{A^\prime }^\dagger$ will not change with $r^\prime$ any longer.
Fig. 7. Logarithmic negativity and probability of success for distilling $|\psi \rangle _{\mathrm {{SMS}}}$ with $S_{\mathrm {dB}}=15$ dB (a)(b) and $S_{\mathrm {dB}}=25$ dB (c)(d). Photon number is truncated at $\overline {D}=199$ for (a)(b) and at $\overline {D}=1999$ for (c)(d).
The result in Fig. 7 confirms that the coherent superposition brought by local squeezing operation could be used to enhance the distillation. The black solid lines in (a) (c) indicate the entanglement before distillation. The circled lines represent the entanglement of the output state in Fig. 1(a). The squared lines represent the entanglement of output state in Fig. 6(a). It is clearly shown that the entanglement can be adjusted by tuning the additional squeezing parameter $r^\prime$. Moreover, for $r^\prime <0$, the enhancement of logarithmic negativity occurs and this gives the chance of relieving the burden of high transmittance beam splitters in practical photon annihilation. We use the lines marked with asterisks, upward and downward triangles to represent the performance of entanglement distillation with practical beam splitters of different transmittance. As shown in Fig. 7(a), for distilling $|\psi \rangle _{\mathrm {{SMSE}}}$ with $S_{\mathrm {dB}}=15$ dB (a)(b), the beam splitters with $T=0.9158$ fail to distill higher entanglement without local squeezing ($r^\prime =0$). However, the scenarios may change if local squeezing is applied. For example, when $r^\prime =-0.80$, all beam splitters with $T>0.80$ could help to distill the $|\psi \rangle _{\mathrm {{SMSE}}}$ with $S_{\mathrm {dB}}=15$ dB. Similar results hold for even higher entanglement with $S_{\mathrm {dB}}=25$ dB, as shown in Fig. 7(c). The success probability is also tuned by the squeezing $r^\prime$ due to the the change of photon number population induced by squeezing $S(r^\prime )$.
The result in Fig. 8 shows the dependence of entanglement $E_N$ on beam splitter $T$. We choose $r^\prime =-0.1151, -0.3453, -0.5756$, corresponding to $1$ dB, $3$ dB and $5$ dB on-line squeezing. As shown in Fig. 8(a), to distill $15$ dB SMSE, the local squeezing-enhanced distillation even outperforms the ideal single-side photon annihilation for $r^\prime = -0.5756, T>0.95$. However, to distill $25$ dB SMSE, $5$ dB online squeezing is not sufficient and the distilled state is less entangled than that of SMSE before distillation. This suggests that it requires a stronger online squeezing to distill a stronger SMSE. Namely, there exists a lower bound $r_L^\prime$, representing the minimal value of on-line squeezing $r^\prime$ that still allows to generate a stronger entanglement(compared with that before distillation). The bound $r_L^\prime$ is dependent on the value $T$ used for beam splitter and in Fig. 9, we show the bound $r_L^\prime$ as a function of squeezing $S_{\mathrm {dB}}$.
Fig. 8. Logarithmic negativity and probability of success for distilling $|\psi \rangle _{\mathrm {{SMS}}}$ versus beam splitter transmittance $T$. Other parameters: (a)(b) $S_{\mathrm {dB}}=15$ dB, $\overline {D}=199$ and (c)(d) $S_{\mathrm {dB}}=25$ dB, $\overline {D}=1999$.
Fig. 9. Lower bound $|r_L^\prime | (r_L^\prime \, <\, 0)$ as a function of squeezing $S_{\mathrm {dB}}$. The letters “U,N” represents the regime in which local squeezing is unnecessary and necessary for successful distillation and the letter “F” denote the regime where even local squeezing-enhanced distillation fails to improve the entanglement. Other parameters: (a)$T=0.80$,(b)$T=0.85$,(c)$T=0.90$,(d)$T=0.95$.
For example, for $T=0.80$ and $S_{\mathrm {dB}} < 11.62$dB, only ideal single-side photon annihilation (with no local squeezing ) is sufficient for generating a entanglement higher than that before distillation. This regime is termed as “Unnecessary (U)”. For $11.62$dB $< S_{\mathrm {dB}} <18.94$dB, we obtain a non-trivial lower bound of $r_L^\prime$ (expressed in terms of the correspong squeezing in dB) below which the distillation scheme of single-side annihilation with local squeezing will never generate a higher entanglement. We plot $(20 |r_L ^\prime | log_{10}e )$ dB as a function of $S_{\mathrm {dB}}$ for fair comparison. This regime is termed as “Necessary (N)”. Finally, for $S_{\mathrm {dB}} >18.94$dB, we find that even with a stronger local squeezing of $25$ dB it is still not possible to increase the squeezing. This regime is termed as “Fail (F)” since it costs more squeezing in distilling itself than the original squeezed state. For other values of $T=0. 85, 0.90$ and $0.95$, the regime is also numerically evaluated in Fig. 9 (b)(c)(d).
Finally, before ending this section, we want to emphasize the budget of our distillation scheme. By using additional local online squeezing, the mean number of photon that enter the photon detectors could be extremely high. This will blind the gated single photon detectors (in Fig. 6(b)). Similar phenomenon also happens in quantum eavesdropping on quantum key distribution systems with bright illumination [45]. Thus, the evaluation of mean photon number is quite important. After straightforward algebraic calculations, we obtain the mean photon number
Fig. 10. Mean photon number that enters the single photon detector in Fig. 6(b). Other parameters: $T=0.90, r^\prime <0$.
Fig. 11. Photon detection with non-unit on-off detector. A beam splitter (transmittance $\eta _D$ ) and ancillary mode D (initialized in vacuum) is used to simulate the quantum efficiency $\eta _D$. The trash box denotes the discarding of the ancillar optical mode. We refer to a successful distillation event when the on-off detector registers ‘on’ result.
4. Discussion: distillation with inefficient on-off photon detectors
In all the scheme above, photon detectors are assumed to be ideal single photon detector and successful distillations are heralded when all the detectors register exact a single photon state. In laboratories, on-off photon detectors with non-unit efficiency are more often employed. Photon loss and the accompanying decoherence may happen even in the detector itself. It is quite interesting to identify the regime when realistic on-off detectors can be applied to distill strong squeezing entanglement state.
The scheme for entanglement distillation is the same as Fig. 6 (b), except that the ideal single photon detector is replaced with non-unit on-off detector (see Fig. 11 for more information). Such a kind of detector can be simulated with a beam splitter of transmittance $\eta _{_D}$ and a 100% efficiency on-off detector. The on-off detector can only register two results. ‘Off’ means no photons are detected and ‘on’ means one or more number of photons are detected. In this case, successful distillation happens when the detector register ‘on’ result. The effect of non-unit efficiency means that photons to be detected may be lost during its interaction with the beam splitter $\eta _{_D}$.
The derivation of state evolution in Sec.3. still applies except by using the measurement operator of on-off detector. In photon number space, the measurement operator corresponding to the detection of ‘on’ and ‘off’ can be given by positive-definite operators $\{ \hat {\Pi }_{\mathrm {on}}=\mathop{\sum}\limits _{k=1}^\infty [1-(1-\eta _{_D})^k] |k\rangle \langle k|, \hat {\Pi }_{\mathrm {off}}=\mathop{\sum}\limits _{k=0}^\infty (1-\eta _{_D})^k |k\rangle \langle k| \}$ [46].
The state after distillation will be a mixed state whose entanglement can only be evaluated with all its density matrix entries. However, the number of matrix elements of the distilled two-mode entanglement states scales as $\overline {D}^4$, which quickly consumes all the computing resource in our computers (See Appendix B ). For convenience, we truncate our state in each optical mode such that the entanglement $|E_N(\overline {D}+10)-E_N(\overline {D})|<0.004 E_N(\overline {D})$. For a given entangled state $|\psi \rangle _{\mathrm {{SMSE}}}$ with $S_{\mathrm {dB}}=15$ dB, we truncate each optical mode at $\overline {D}=140$ and plot the performance of logarithmic negativity as well as its contour plot in Fig. 12, showing the regime in which the combination of $(\eta _D, r^\prime )$ enables to output a higher entanglement. Similar result for $20.5$ dB case is shown in Fig. 13, with $T=0.95$ and photons in each optical mode truncated at $\overline {D}=189$.
Fig. 12. Performance of practical single-side distillation of SMSE with $S_{\mathrm {dB}}=15$ dB. Entanglement and its contour plot is given for different combination of $(\eta _{_D},r^\prime )$. All optical modes are truncated at $\overline {D}=139$. Other parameter: $T=0.90$.
Fig. 13. Performance of practical single-side distillation of SMSE with $S_{\mathrm {dB}}=20.5$ dB. Entanglement and its contour plot is given for different combination of $(\eta _{_D},r^\prime )$. All optical modes are truncated at $\overline {D}=189$. Other paramter: $T=0.95$.
Once again in Fig. 12 and Fig. 13, it should be noted that the entanglement after successful distillation is almost immune to the detection efficiency $\eta _D$. Similarly phenomenon is also observed in two-copy distillation [25]. The reason is that the detector efficiency affects the detection rates only, but not the essence of heralding of photon’s existance. Thus once the detector clicks, although with a lower efficiency, the photon will be subtracted from the incoming optical mode and the desired non-Gaussian operation is implemented. Of course, the detection loss will take effect on the projection to vacuum subspace. It will mistake a non-zero photon for zero photon due to efficiency loss $1-\eta _D$. Fortunately, this will correspond to a failed entanglement distillation and the distilled state is not used any longer.
5. Conclusions
In conclusion, strong squeezing is what can be expected in the forthcoming continuous variable quantum information processing. Motivated by the recent rapid progress in preparation of strongly squeezed state, we here investigate the distillation problem of single-mode squeezed entangled state. We demonstrate that ideal single-side and biside photon annihilation are sufficient for distilling single-mode-squeezed entangled state, even in case of extreme strong squeezing of $25$ dB and $30$dB. However, the performance of distillation decays rapidly if ideal photon annihilation is replaced with beam splitter and photon detectors. The transmittance of beam splitter should approach $0.9921$ for distilling $|\psi \rangle _{\mathrm {SMSE}}$ with single mode squeezing of $25$ dB, which is a big challenge in state-of-art laboratories. Then, we show the superposition of photon annihilation and photon addition could help to relieve such a technical burden. Practical scheme with local squeezing are suggested for generating such a superposition operation. This shows that local squeezing operations not only provide a method for improving the photon-subtraction based distillation for weak entanglement [35], but also provide the only feasible way (to our knowledge) to distill the SMSE entanglement generated by extremely strong squeezed state. Finally, our investigations are extended to realistic on-off photon detectors of non-unit efficiency. The regime in which the realistic on-off photon detectors can be applied for distillation is identified for experimentally feasible beam splitters.
Finally, it should be noted that the implementation of local squeezing is becoming state of art. Actually, it is a fundamental building block for continuous variable quantum computation. It can implemented by employing an offline-generated auxiliary squeezed vacuum state combined with interference, homodyne measurement and feedforward [47]. In Ref.[48], A. Furusawa’s group provided an experimental demonstration of single mode squeezing operation of 3dB, 6dB and even 10dB. The local squeezing-enhanced scheme is a complement to the previous results for continuous variable entanglement distillation with photon subtraction [18] . We hope our investigation in strongly-squeezed entanglement distillation can be experimentally checked in the near future.
A. Find $E_N$ for two-mode pure state
To perform the numeric calculation, we can use a two-dimensional array to represent $|\psi _1\rangle$ and $|\psi _2\rangle$. The calculation of entanglement $E_N$ for $|\psi _1\rangle$ and $|\psi _2\rangle$ can be formed straightforwardly by the definition [24]
Proof of Theorem 1: Let $U$ and $V$ be the unitary matrix corresponding to the singular value decomposition of $C$: $C=U\Lambda V$, $\Lambda$ being the diagonal matrix. Then $c_{nm}=(U\Lambda V)_{nm}=\langle n |U\Lambda V|m\rangle$ and $|\psi \rangle =\sum _{nm}\langle n| U\Lambda V |m\rangle |n\rangle |m\rangle$. Since $\Lambda$ is diagonal matrix $\Lambda =\sum _{p} \Lambda _p |p\rangle \langle p|$, we can rewrite $|\psi \rangle$ as
With Theorem 1, one can reduce the storage complexity of $|\psi _i\rangle$ $(i=1,2)$ from $\overline {D}^4$ to $\overline {D}^2$ and perform the numeric calculation with photon truncation upto $\overline {D}=1999$ in each optical mode.
B. Find $E_N$ for two-mode mixed state
For the distillation with realistic on-off photon detectors, the distilled state is two-mode mixed state. The number of density entries scales as $\overline {D}^4$ and no efficient methods exist to storage the density matrix without any further approximation. To faithfully show the performance of distillation, we make no approximation and evaluate all the $\overline {D}^4$ entries in Fig. 12 and Fig. 13. We calculate the up-right triangle of density matrix elements (due to its hermiticity) in a parallel way and finally evaluate its eigenvalues with a multi-core computer. In Table 1, we show the computer memory used by a single density matrix $\rho _{A^\prime B^\prime }{(\overline {D})}$ as well as the time used (in unit of second) to find the eigenvalues of $\rho _{A^\prime B^\prime } (\overline {D})$. The computer server is equipped with Intel Xeon E5-2682 2.5 Ghz CPU (32 core) and 256GB memory. In Fig. 14, the entanglement distillation of SMSE with $20.5$ dB single mode squeezing is evaluated with photon subspace spanned by $\{|0\rangle , |1\rangle ,\ldots , |\overline {D}-1\rangle \}$ of increasing size $\overline {D}$. The figure of merit we use is logarithmic negativity which quickly converges as $\overline {D}$ increases.
Fig. 14. Numerical convergence for calculation of entanglement distill with on-off photon detectors within photon number space of increasing size $\overline {D}$. Other paramter: $T=0.95,\eta _D=0.9990, r^\prime = -1.8824$.
Table 1. Computer memory (in GB) and Computing time (in second) in storage and evaluating eigenvalues of distilled state.
Funding
National Natural Science Foundation of China (NSFC) (11574400); Beijing Institute of Technology (BIT); NSFC-ICTP proposal (1191101236).
Acknowledgement
The authors acknowledge support from the National Natural Science Foundation of China (Grants No.11574400 and No. 11204379) , Beijing Institute of Technology Research Fund Program for Young Scholars, NSFC-ICTP proposal( Grants No.1191101236 ).
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