Quantifications for multi-mode entanglement

TL;DR

This paper introduces two new measures for quantifying 3-mode and 4-mode entanglement, explores their conversion from single-mode nonclassicality in beam-splitters, and demonstrates their generalizability to other multi-mode entanglement scenarios.

Contribution

It presents two independent quantifications for multi-mode entanglement and shows their parallel behavior, with methods extendable to any multi-mode entanglement.

Findings

Parallel behavior of the two quantifications observed

Conversion of single-mode nonclassicality into multi-mode entanglement demonstrated

Methods can be generalized to other multi-mode entanglement quantifications

Abstract

We introduce two independent quantifications for 3-mode and 4-mode entanglement. We investigate the conversion of one type of nonclassicality, i.e. single-mode nonclassicality, into another type of nonclassicality, i.e. multi-mode entanglement, in beam-splitters. We observe parallel behavior of the two quantifications. The methods can be generalized to the quantification of any multi-mode entanglement.

Full text

Multi-mode continuous variable entanglement is a key element for quantum information processing. Quantum teleportation among multiple parties Furusawa and van Loock (2011), quantum computation with clusters Menicucci et al. (2006); Raussendorf and Briegel (2001), e.g. quantum networks, Armstrong et al. (2012) and quantum internet Kimble (2008), all, necessitate the presence of multi-mode entanglement. Multi-mode entanglement can be generated using several number of beam-splitters which converts the single-mode nonclassicality (SMNc), e.g. quadrature squeezing, of input beam(s) into multi-mode entanglement at the outputs of the beam splitters van Loock and Braunstein (2000). Hence, mechanism of the conversion of nonclassicality into multi-mode entanglement, via beam splitters, is important for distribution of the entanglement among multi-parties used in quantum information.

One can notice that quantum nonclassicality is like "energy". That is, it can be converted into different forms. For instance, single-mode nonclassicality (SMNc), two-mode entanglement (TME) and many-particle entanglement (MPE) are different forms nonclassicality (Nc) and can be converted to each other via linear interactions. 111Also the criteria witnessing them can be transformed into each other Tasgin (2017). A quadrature-squeezed light (SMNc) can transfer its squeezing to an ensemble of atoms Hald et al. (1999); Vitagliano et al. (2018); Vidal (2006) as spin-squeezing Kitagawa and Ueda (1993). This creates MPE Sørensen et al. (2001); Tasgin (2017) in the ensemble. Similarly, interaction of an ensemble with two entangled beams (TME) also creates MPE in the ensemble Regula et al. (2018); Taşgın and Meystre (2011). It is also possible to convert the nonclassicality of a single-mode, e.g. quadrature-squeezing, into two-mode entanglement at the output of a beam-splitter (BS) Kim et al. (2002); Asbóth et al. (2005). Generation of TME in a BS necessitates a nonclassical input light Kim et al. (2002).

Recent works Ge et al. (2015); Arkhipov et al. (2016a, b); Černoch et al. (2018) show that there appears a conservation-like relation between the generated TME and the remaining SMNc in a BS. A BS cannot convert all of the nonclassicality of the input beam into TME at the output. There still remains some SMNc in the two output modes Ge et al. (2015). However, the total SMNc decreases with respect to the input one. The form of the analytical expressions for the maximum TME extracted at the BS output, $E_{\cal N}={\rm max}\{0,-\frac{1}{2}\log_{2}(\lambda_{1,{\rm sm}}^{\rm(in)}\lambda_{2,{\rm sm}}^{\rm(in)})\}$ Tahira et al. (2009); Li et al. (2006), suggests us to quantify the SMNc in terms of a noise-area Ge et al. (2015), $\Omega=\lambda_{1,{\rm sm}}\lambda_{2,{\rm sm}}$. Here, $\lambda_{i,{\rm sm}}$ is the minimum noise of the $i$th beam input into the BS, where $\lambda_{i,{\rm sm}}<1$ implies the presence of squeezing in the $i$th beam. For mixing with vacuum or a coherent state, one of the modes becomes $\lambda_{1,{\rm sm}}=1$.

As an illuminating example: if a squeezed beam is mixed with a thermal noise Tahira et al. (2009) at the input channels of a BS, the output modes are entangled only if the noise-area $\Omega=\lambda_{\rm sqz}\>(1+2\bar{n})<1$, where $\lambda_{\rm sqz}<1$ is the reduced noise of the squeezed beam and $\bar{n}$ is the mean number of photons in the thermal noise. Also Ref. Li et al. (2006) shows that the maximum amount of entanglement extractable at the BS output is $E_{\cal N}=-\frac{1}{2}\log_{2}(2\nu_{-})=-\frac{1}{2}\log_{2}(\lambda_{1,{\rm sm}}^{\rm(in)}\lambda_{2,{\rm sm}}^{\rm(in)})\}$ if any two Gaussian states are mixed in the BS input. Here, one can observe that the smallest symplectic eigenvalue of an inseparable system Adesso et al. (2004); G. Vidal and R. F. Werner (2002) becomes $\mu=2\nu_{-}=\lambda_{1,{\rm sm}}^{\rm(in)}\lambda_{2,{\rm sm}}^{\rm(in)}$, which is actually the input noise-area. In Ref. Ge et al. (2015), we further show that the output TME is proportional to the change (increase) in the noise-area $S_{N}=\log_{2}(\lambda_{1,{\rm sm}}^{\rm(out)}\lambda_{2,{\rm sm}}^{\rm(out)})-\log_{2}(\lambda_{1,{\rm sm}}^{\rm(in)}\lambda_{2,{\rm sm}}^{\rm(in)})$ of the out beams with respect to the input beams. A geometric demonstration of this SMNc$\to$TME swap can be found in Fig. 1 of Ref. Tasgin (2019). More interestingly, well-known TME criteria, like Duan-Giedke-Cirac-Zoller Duan et al. (2000) and Hillery-Zubairy Hillery and Zubairy (2006), actually do search for a noise-area below unity in BS-like rotations Tasgin (2019).

In the present work, we investigate the swapping of single-mode nonclassicality into multi-mode entanglement (MME). In particular, we study three-mode entanglement (3ME) and four-mode entanglement (4ME) after two (Fig. 1) and three beam-splitters (Fig. 3), respectively. We define the remaining SM nonclassicalities in the 3 modes (or 4 modes) as a noise-volume (or a 4D noise-volume), $\Omega=\lambda_{1,{\rm sm}}^{\rm(out)}\>\lambda_{2,{\rm sm}}^{\rm(out)}\>\lambda_{3,{\rm sm}}^{\rm(out)}$. Here, $\tau_{i}^{\rm(out)}=1-\lambda_{i,{\rm sm}}^{\rm(out)}$ is the degree of the remaining SMNc, i.e. the nonclassical depth Lee (1991), of each mode after the BSs. It is calculated by wiping out the entanglement (correlations) between the modes Ge et al. (2015). We show that the SMNc decreases (noise-volume $\Omega$ increases) after the BSs while the the 3ME increases.

In the quantification of 3ME we use two different (independent) approaches. (i) We multiply the three noises associated with the symplectic eigenvalues ($\nu_{1}$,$\nu_{2}$,$\nu_{3}$) of the partial transposed system. $\nu_{1}$, for instance, is the smallest symplectic eigenvalue of the 3-mode system when the 1st mode is partial transposed Adesso et al. (2004); G. Vidal and R. F. Werner (2002). That is, smaller values of the eigenvalue compared to unity, i.e. $2\nu_{1}<1$, imply stronger entanglement of the 1st mode with the system composed of (2nd + 3rd) modes Plenio (2005). Similarly, $2\nu_{2}<1$ refers to the inseparability of the 2nd mode from the system of (1st + 3rd) modes. Hence, $\mu=2\nu_{1}\>2\nu_{2}\>2\nu_{3}$ refers to a kind of 3-mode entanglement strength, where $2\nu_{i}>1$ implies the absence of a genuine 3-mode entanglement.

(ii) Second, we use an alternative method given in Sec. II.3.2 of Ref. Tasgin (2019). The idea is very simple: Nonclassical depth $\tau$ quantifies the whole nonclassicality, i.e. SMNc + entanglement, in a multi-mode system Li et al. (2006). If we remove the unconverted (unused) SM nonclassicalities (after the BSs) from the noise-matrix, then the remaining nonclassicality is due to the entanglement only 222In Ref. Ge et al. (2015), we do the reverse. We remove the correlations in the noise matrix and examine the remaining SM nonclassicalitites.. Nonclassicality of a single-mode state can be determined by introducing a Gaussian filter function transformation on the Glauber-Sudarshan $P$-function, i.e. $P(\alpha,\tau)=\int d^{2}\alpha^{\prime}\>\exp\{-|\alpha-\alpha^{\prime}|^{2}/\tau\}\>P(\alpha^{\prime})\>/\tau\pi$. So that, the new $P$-function is non-negative Lee (1991). This corresponds to injecting noise which destroys the nonclassicality Kiesel and Vogel (2010). A similar method can be used also to determine the nonclassicality of a multi-mode system Li et al. (2006). Here, in difference to Ref. Li et al. (2006), we introduce different $\tau$’s for each mode. This way, we prevent the injection of unnecessary noise ($\tau$) by constraining $\tau_{1}=\tau_{2}=\tau_{3}=\tau$.

We emphasize that, our aim, in this short manuscript, is to introduce the basics of two possible quantifications for the multi-mode entanglement (MME). We do not aim to present a detailed analysis on MME, but we suffice with demonstrating that SMNc, quantified as a noise-volume, is converted into MME.

3-mode entanglement

We study the system depicted in Fig. 1. A single-mode (SM) nonclassical state $\hat{a}$ is mixed with vacuum noise in a BS with two output states $\hat{a}_{1}$ and $\hat{a}_{2}$. One of the output modes, $\hat{a}_{2}$, is input to a second BS, mixed with vacuum, resulting two new output modes $\hat{b}_{1}$ and $\hat{b}_{2}$. We examine the 3-mode entanglement (3ME) of $\hat{a}_{1}$, $\hat{b}_{1}$, and $\hat{b}_{2}$ modes.

In Fig. 2a and Fig. 2b, we examine the 3ME of the $\hat{a}_{1}$, $\hat{b}_{1}$, $\hat{b}_{2}$ modes using the two methods, (i) and (ii), respectively. We observe a similar behavior for the two quantifications. In Fig. 2c, one can observe that the SM nonclassicalities remaining in the $\hat{a}_{1}$, $\hat{b}_{1}$, $\hat{b}_{2}$ modes decreases (noise-volume increases), while the 3ME increases. In all Fig. 2a-2c, smaller $\mu$, $\eta$ and $\cal N$ imply stronger 3ME or SMNc. The first BS fed with a nonclassical light of squeezing parameter Scully and Zubairy (1997) $r=0.1$. We fix the angle of the first BS to $\theta_{{}_{\rm BS_{1}}}=\pi/4$ and vary the angle of the second BS $\theta_{{}_{\rm BS_{2}}}$.A similar behavior is obtained for varying $\theta_{{}_{\rm BS_{1}}}$ with a fixed $\theta_{{}_{\rm BS_{2}}}$.

$\mu$, $\eta_{\rm ent}$ and $\cal N$ are calculated from the 3 mode noise matrix as follows.

Calculation of $\mu$.— We compose the 6$\times$6 noise-matrix $V_{ij}=\langle\hat{\xi}_{i}\hat{\xi}_{j}+\hat{\xi}_{j}\hat{\xi}_{i}\rangle/2-\langle\hat{\xi}_{i}\rangle\langle\hat{\xi}_{j}\rangle$ for the 3 mode system in the real representation Simon et al. (1994) by introducing the operator $\hat{\xi}=[\hat{x}_{1}\>,\>\hat{p}_{1}\>,\>\hat{x}_{2}\>,\>\hat{p}_{2}\>,\>\hat{x}_{3}\>,\>\hat{p}_{3}]$, where $\hat{x}_{i}=(\hat{c}_{i}^{\dagger}+\hat{c}_{i})/\sqrt{2}$ and $\hat{p}_{i}=i(\hat{c}_{i}^{\dagger}-\hat{c}_{i})/\sqrt{2}$ are obtained using $\hat{c}_{i}=\hat{a}_{1}$,$\hat{b}_{1}$,$\hat{b}_{2}$, respectively. Partial transpose (PT) operation, e.g., on the 1st mode is equivalent to $\hat{p}_{1}\to-\hat{p}_{1}$ in the noise-matrix Simon (2000). If the 1st mode is separable from the system of (2+3) modes, symplectic eigenvalues of the partial transposed noise-matrix must be all larger than $1/2$ G. Vidal and R. F. Werner (2002); Adesso et al. (2004). (So, smallest eigenvalue $\nu_{1}$ also satisfies $2\nu_{1}>1$.) The symplectic eigenvalues of the partial transposed noise-matrix can alternatively be calculated as ${\rm eig}[i\chi_{1}V]$ where $\chi_{1}$ is the 6$\times$6 matrix $\chi_{1}=[-J\>,\>0_{2\times 2}\>,\>0_{2\times 2}\>;\>0_{2\times 2}\>,\>J\>,\>0_{2\times 2}\>;\>0_{2\times 2}\>,\>0_{2\times 2},\>J]$ with $0_{2\times 2}$ is 2$\times$2 matrix of zeros and $J=[0\>,\>1\>;\>-1\>,0]$. For $\chi_{2}$ and $\chi_{3}$ the "-" sing must be in the second and third $J$, respectively. ${\rm eig[i\chi_{1}V]}$ yields only a single eigenvalue with $2\nu_{1}<1$. Similarly, $\nu_{2}$ ($\nu_{3}$) is the only $2\nu_{2}<1$ ($2\nu_{3}<1$) eigenvalue from the partial transposition of the 2nd (3rd) mode.

We remind one more time that introduction of the noise-volume $\mu=2\nu_{1}\>2\nu_{2}\>2\nu_{3}$ via the symplectic eigenvalues follows from the observation $\log_{2}2\nu_{-}=\log_{2}(\lambda_{1,{\rm sm}}^{\rm(in)}\lambda_{2,{\rm sm}}^{\rm(in)})$ for a single BS Tahira et al. (2009); Li et al. (2006); Ge et al. (2015), where $2\nu_{-}=\lambda_{1,{\rm sm}}^{\rm(in)}\lambda_{2,{\rm sm}}^{\rm(in)}$ determines the maximum entanglement extractable from the input noise-area $\lambda_{1,{\rm sm}}^{\rm(in)}\lambda_{2,{\rm sm}}^{\rm(in)}$.

Calculation of $\eta_{\rm ent}$.— The nonclassical depth, associated only with the entanglement, is calculated as follows. We first transform the real noise-matrix $V$ into the complex representation Simon et al. (1994) $V^{\rm(c)}={\cal C}V{\cal C}^{\dagger}$, where ${\cal C}=[{\cal C}_{1},0_{2\times 2},0_{2\times 2};0_{2\times 2},{\cal C}_{1},0_{2\times 2};{\cal C}_{1},0_{2\times 2},0_{2\times 2}]$ with ${\cal C}_{1}=[1,i;1,-i]/\sqrt{2}$. In the 6$\times$6 complex noise-matrix $V^{\rm(c)}$, we wipe out the SM nonclassicalities of the 3 modes, e.g. in the 1st mode, by replacing the $[V_{11},V_{12};V_{2,1},V_{2,2}]$ with $[1/2,0;0,1/2]$, the noise-matrix for vacuum or a coherent state Tahira et al. (2009). We wipe out the SMNc of the other 2$\times$2 block-diagonals similarly. Then, we obtain the new noise-matrix $V_{\rm ent}^{\rm(c)}$, where the nonclassical depths $\tau_{1,2,3}$ accounts the entanglement only. For Gaussian states, we consider here, this can be performed by calculating the $\tau_{1}$, $\tau_{2}$, $\tau_{3}$ which makes all {\rm eig[V_{ent}^{\rm(c)}+\text{\boldmath\tau}]} positive where \tau$$={\rm diag[\tau_{1},\tau_{1},\tau_{2},\tau_{2},\tau_{3},\tau_{3}]}. Ref. Li et al. (2006) assigns a single $\tau_{1}=\tau_{2}=\tau_{3}=\tau$ for the $\tau$ matrix which certainly increases the injected noise. We quantify the nonclassicality of $V_{\rm ent}^{\rm(c)}$, or the entanglement, by choosing the minimum of $\tau_{\rm ent}=[\tau_{1}\tau_{2}\tau_{3}]_{\rm min}$, or $\eta_{\rm ent}=\left[(1-2\tau_{1})(1-2\tau_{2})(1-2\tau_{3})\right]_{\rm max}$ in terms in terms of the injected noise-volume. We note that, for a single-mode, $\lambda_{\rm sm}=(1-2\tau)$ corresponds to the reduced noise of that particular mode.

Calculation of $\cal N_{{}_{\rm SMNc}}$.— In the calculation of SMNc $\cal N_{{}_{\rm SMNc}}$, this time, we wipe out the correlations in the noise-matrix and left with the three 2$\times$2 block-diagonals. 2$\times$2 matrices give the SMNc associated with each mode Ge et al. (2015). Then, we introduce the SMNc noise-volume ${\cal N}_{{}_{\rm SMNc}}=(1-2\tau^{{}^{\rm(SMNc)}}_{1})(1-2\tau^{{}^{\rm(SMNc)}}_{2})(1-2\tau^{{}^{\rm(SMNc)}}_{3})$.

4-mode entanglement

We perform similar calculations also for a 4-mode system given in Fig. 3. We obtain the same behavior depicted in Fig. 4.

Summary

In summary, we introduce quantifications for 3-mode and 4-mode entanglement via two independent methods. We demonstrate how single-mode nonclasscality is converted into 3-mode and 4-mode entanglement. We quantify all nonclassicalities in terms of noise-volume. A smaller noise-volume implies a stronger nonclassicality or entanglement. The method we introduce here can be generalized to other multi-mode entanglement which has fundamental importance in quantum communication.

Acknowledgments

MET acknowledges support from TUBITAK Grant No: 1001-117F118, TUBA-GEBIP Award 2017, Hacettepe University BAP Grant No: FBI-2018-17423.