Separability of evolving W state in a noise environment
本站小编 Free考研考试/2022-01-02
Qian-Tong Men(门前同), Li-Zhen Jiang(蒋丽珍), Xiao-Yu Chen(陈小余),∗College of Information and Electronic Engineering, Zhejiang Gongshang University, Hangzhou 310018, Zhejiang, China
First author contact: Author to whom any correspondence should be addressed. Received:2020-09-20Revised:2020-12-7Accepted:2020-12-24Online:2021-02-16
Abstract Entanglement is an important resource for quantum information processing. We provide a new entanglement witness to detect the entanglement of an evolving W state. Our results show that the new entanglement witness matches the evolving W state better than other witnesses or methods. The new witness significantly improves the performance of entanglement detection for some three-qubit states. Keywords：entanglement witness;separable criterion;PPT criterion;W state
PDF (308KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Qian-Tong Men(门前同), Li-Zhen Jiang(蒋丽珍), Xiao-Yu Chen(陈小余). Separability of evolving W state in a noise environment. Communications in Theoretical Physics, 2021, 73(4): 045101- doi:10.1088/1572-9494/abd84b
1. Introduction
Quantum entanglement is one of the most fundamental concepts in physics and is a key resource for quantum information and computation [1–3]. Due to the important role of entanglement in many quantum technology protocols, how to optimally detect entanglement theoretically for experimentally prepared states has become a highly relevant topic [4]. A series of criteria have been proposed for bipartite systems, such as the positive partial transpose (PPT) criterion [5–7], the computable cross norm and realignment (CCNR) criterion [8] and so on [9–11].
Multipartite entanglement is the entanglement between three or more quantum systems. In contrast to bipartite entanglement, the structure of multipartite entanglement is far more intricate. In addition to particles in the system that are fully separable or fully entangled, there are also partially separable cases. The GHZ and W states are famous examples of multipartite entanglement.
The W state is a superposition of all states with exactly one qubit in an excited state ∣1⟩ while all the others are in a ground state ∣0⟩. The W states have applications in several protocols for quantum information processing, such as quantum teleportation, superdense coding, quantum key distribution and quantum games. Experimentally, W states have already been realized with photonic qubits [12, 13], Josephson flux qubits [14] and in an atomic system [15].
One of the challenges in quantum information is decoherence: the interaction of a qubit with its environment reduces the indispensable quantum coherence and entanglement of the quantum states. Decoherence is one of the main interferences in the evolution of all open systems, which seriously hinders the development of quantum applications that use coherence capabilities. The decoherence of multipartite entanglement has been studied with the generalized concurrence [16], geometric measure of entanglement [17], negativity [18] and global entanglement [19]. We will study the decoherence of the W state with an entanglement witness.
The entanglement witness [20–22] is a common method for detecting multipartite entanglement. An entanglement witness is a Hermitian operator, it has non-negative expectations for all separable states and has negative expectation for at least one entangled state. We will present a new entanglement witness, ${\hat{W}}_{1}$, which can be applied to the evolving W state. If we define the entanglement lifetime to be the time of a decayed state staying entangled according to some criterion or witness, then our entanglement witness, ${\hat{W}}_{1}$, gives a longest entanglement lifetime for the quantum state. We also compare the performance of entanglement detection using different witnesses.
This paper is arranged as follows: In section 2, we introduce some necessary definitions and give the form of entanglement witness ${\hat{W}}_{1}$. In section 3, we show some applications of our entanglement witness and present the necessary conditions of separability, while in section 4 conclusions are presented. We prove some of our results in the three appendices. In appendix A we provide the derivation process of witness ${\hat{W}}_{1}$. In appendix B we demonstrate the proof of the validity of entanglement witness ${\hat{W}}_{1}$. In appendix C we give the density matrix of the time evolution W state.
2. Entanglement witness
A fully separable three-qubit state can be written as [23]:$\begin{eqnarray}{\rho }_{s}=\displaystyle \sum _{i}{p}_{i}{\rho }_{A}^{(i)}\otimes {\rho }_{B}^{(i)}\otimes {\rho }_{C}^{(i)},\end{eqnarray}$where pi is the probability distribution. The entanglement witness is denoted as $\hat{W}$ [24]. For all separable quantum states ρs, there is $\mathrm{Tr}({\rho }_{s}\hat{W})\geqslant 0$, at least for one entangled state ρ, there is $\mathrm{Tr}(\rho \hat{W})\lt 0$. Therefore, if $\mathrm{Tr}(\rho \hat{W})\lt 0$ is measured, we know for sure that the state ρ is entangled and ρ is witnessed by $\hat{W}$. Let $\hat{W}={\rm{\Lambda }}I-\hat{M}$, where I is the identity operator and $\hat{M}$ is a Hermitian operator, then:$\begin{eqnarray}{\rm{\Lambda }}=\mathop{\max }\limits_{{\rho }_{s}}\mathrm{Tr}({\rho }_{s}\hat{M}).\end{eqnarray}$The witness is called the optimal entanglement witness [25].
We may replace the mixed separable states with the pure product states of the form ∣ψs⟩ = ∣ψA⟩∣ψB⟩∣ψC⟩. Therefore, formula (2) becomes:$\begin{eqnarray}{\rm{\Lambda }}=\mathop{\max }\limits_{| {\psi }_{s}\rangle }\langle {\psi }_{s}| \hat{M}| {\psi }_{s}\rangle .\end{eqnarray}$Giving a Hermitian operator $\hat{M}$, we can construct an entanglement witness. For a three-qubit state, we minimize the following expression by adjusting the Mi parameters of operator $\hat{M}$ (see appendix A):$\begin{eqnarray}{ \mathcal L }=\displaystyle \frac{{\rm{\Lambda }}}{\mathrm{Tr}(\rho \hat{M})}=\displaystyle \frac{{\rm{\Lambda }}}{{\boldsymbol{M}}\cdot {\boldsymbol{R}}},\end{eqnarray}$where Λ is positive, and M · R (the vectors M and R are defined in appendix A) is required to be positive [26]. So we get the following conclusion. If a quantum state is separable, it should satisfy:$\begin{eqnarray}{{ \mathcal L }}_{\min }\equiv \mathop{\min }\limits_{\hat{M}}{ \mathcal L }\geqslant 1.\end{eqnarray}$Violation of the above inequality implies entanglement.
In this paper, we present a new entanglement witness as follows:
$\hat{W}={\rm{\Lambda }}I-\hat{M}=2({M}_{1}+{M}_{6}){\hat{W}}_{1}$, with$\begin{eqnarray}\begin{array}{l}{\hat{W}}_{1}=\displaystyle \frac{1}{d}| 000\rangle \langle 000| +d(| 011\rangle \langle 011| \\ \qquad \ +\ | 101\rangle \langle 101| +| 110\rangle \langle 110| )\\ \qquad \ +\ \left(-\displaystyle \frac{1+\eta }{2}\right)[| 001\rangle (\langle 010| +\langle 100| )\\ \qquad \ +\ | 010\rangle (\langle 001| +\langle 100| )+| 100\rangle (\langle 001| +\langle 010| )]\\ \qquad \ +\ \left(-\displaystyle \frac{1-\eta }{2}\right)[| 000\rangle (\langle 011| +\langle 101| +\langle 110| )\\ \qquad \ +\ (| 011\rangle +| 101\rangle +| 110\rangle )\langle 000| ]\\ \qquad \ +\ \displaystyle \frac{d}{4}(1+3{\eta }^{2})[| 011\rangle (\langle 101| +\langle 110| )+| 101\rangle (\langle 011| \\ \qquad \ +\ \langle 110| )+| 110\rangle (\langle 011| +\langle 101| )]\\ \qquad \ +\ \displaystyle \frac{3d}{4}(1-{\eta }^{2})[| 111\rangle (\langle 001| +\langle 010| +\langle 100| )\\ \qquad \ +\ (| 001\rangle +| 010\rangle +| 100\rangle )\langle 111| ],\end{array}\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}d & = & \sqrt{\displaystyle \frac{1+{M}_{2}}{1-3{M}_{2}}},\\ \eta & = & \displaystyle \frac{{M}_{4}+{M}_{7}}{{M}_{1}+{M}_{6}}.\end{array}\end{eqnarray}$We have found a new entanglement witness, which is represented by the two variables η and d. We will show the applications of this entanglement witness ${\hat{W}}_{1}$ in section 3.3. The derivation process of witness ${\hat{W}}_{1}$ will be shown in appendix A. The validity of it as a witness will be shown in appendix B. The entanglement witness in articles [27, 28] is its special case of η = 0.
3. Application of the entanglement witness
3.1. Applicable quantum state models
It has been shown that, for an open system, the Lindblad master equation of the system affected by the environment influences and other factors can be written as [29, 30]:$\begin{eqnarray}\displaystyle \frac{\partial \rho }{\partial t}=-{\rm{i}}[{H}_{S},\rho ]+\displaystyle \sum _{i,\alpha }({L}_{i,\alpha }\rho {L}_{i,\alpha }^{\dagger }-\displaystyle \frac{1}{2}\{{L}_{i,\alpha }^{\dagger }{L}_{i,\alpha },\rho \}),\end{eqnarray}$where the Lindbald operator ${L}_{i,\alpha }\equiv \sqrt{\tfrac{{k}_{i,\alpha }}{2}}{\sigma }_{\alpha }^{(i)}$ acts on the ith qubit and is used to describe decoherence, and ${\sigma }_{\alpha }^{(i)}$ represents the Pauli matrix of the ith qubit, where α = x, y, z. The constant ki,α is approximately equals to the inverse of decoherence time, HS is the Hamiltonian of the system and the first term on the right-hand side of (8) can be removed when we work with the interaction picture [31]. The evolution state of the W state can be written in the following form:$\begin{eqnarray}{\rho }_{W}(t)=\left(\begin{array}{cccccccc}{\alpha }_{1} & 0 & 0 & {\beta }_{1} & 0 & {\beta }_{5} & {\beta }_{9} & 0\\ 0 & {\alpha }_{2} & {\beta }_{2} & 0 & {\beta }_{6} & 0 & 0 & {\beta }_{10}\\ 0 & {\beta }_{2} & {\alpha }_{3} & 0 & {\beta }_{11} & 0 & 0 & {\beta }_{7}\\ {\beta }_{1} & 0 & 0 & {\alpha }_{4} & 0 & {\beta }_{12} & {\beta }_{8} & 0\\ 0 & {\beta }_{6} & {\beta }_{11} & 0 & {\alpha }_{5} & 0 & 0 & {\beta }_{3}\\ {\beta }_{5} & 0 & 0 & {\beta }_{12} & 0 & {\alpha }_{6} & {\beta }_{4} & 0\\ {\beta }_{9} & 0 & 0 & {\beta }_{8} & 0 & {\beta }_{4} & {\alpha }_{7} & 0\\ 0 & {\beta }_{10} & {\beta }_{7} & 0 & {\beta }_{3} & 0 & 0 & {\alpha }_{8}\end{array}\right).\end{eqnarray}$The density matrix ρW(t) can be written as a summation of products of Pauli matrices, see appendix C. Next we build a quantum state model according the quantum state (9) as follows:$\begin{eqnarray}{\rho }_{{W}_{1}}=\left(\begin{array}{cccccccc}{\gamma }_{1}\, & 0\, & 0\, & {\delta }_{1}\, & 0\, & {\delta }_{1}\, & {\delta }_{1}\, & 0\\ 0 & {\gamma }_{2}\, & {\delta }_{2}\, & 0\, & {\delta }_{2}\, & 0\, & 0\, & {\delta }_{4}\\ 0 & {\delta }_{2}\, & {\gamma }_{2}\, & 0\, & {\delta }_{2}\, & 0\, & 0\, & {\delta }_{4}\\ {\delta }_{1} & \,0\, & 0\, & {\gamma }_{3}\, & 0\, & {\delta }_{3}\, & {\delta }_{3}\, & 0\\ 0 & {\delta }_{2}\, & {\delta }_{2}\, & 0\, & {\gamma }_{2}\, & 0\, & 0\, & {\delta }_{4}\\ {\delta }_{1} & 0\, & 0\, & {\delta }_{3}\, & 0\, & {\gamma }_{3}\, & {\delta }_{3}\, & 0\\ {\delta }_{1} & 0\, & 0\, & {\delta }_{3}\, & 0\, & {\delta }_{3}\, & {\gamma }_{3}\, & 0\\ 0 & {\delta }_{4}\, & {\delta }_{4}\, & 0\, & {\delta }_{4}\, & 0\, & 0\, & {\gamma }_{4}\end{array}\right).\end{eqnarray}$In order to satisfy the positive definite of the quantum state ${\rho }_{{W}_{1}}$, we require:$\begin{eqnarray}\begin{array}{l}{\gamma }_{2}\geqslant {\delta }_{2},\ {\gamma }_{3}\geqslant {\delta }_{3},\ {\gamma }_{1}({\delta }_{3}+2{\delta }_{3})\geqslant 3{\delta }_{1}^{2},\\ {\gamma }_{4}({\delta }_{2}+2{\delta }_{2})\geqslant 3{\delta }_{4}^{2}.\end{array}\end{eqnarray}$We set γ1 = γ2 = γ3 = γ4 = 1/8, and$\begin{eqnarray}{\delta }_{1}=\displaystyle \frac{1}{8}\sqrt{\displaystyle \frac{1}{3}(1+2x)},\ {\delta }_{2}=\displaystyle \frac{1}{8},\ {\delta }_{3}=\displaystyle \frac{1}{8}x,\ {\delta }_{4}=0.\end{eqnarray}$
3.2. Detecting entanglement of ${\rho }_{{W}_{1}}$
The mixture of quantum state ${\rho }_{{W}_{1}}$ and white noise is:$\begin{eqnarray}\rho =p{\rho }_{{W}_{1}}+\displaystyle \frac{1-p}{8}{I}_{8},\end{eqnarray}$where I8 is the 8 × 8 identity matrix and {p, 1 − p} is a probability distribution.
If we use a known witness [28] to detect the entanglement of the state (13), according to formula $\mathrm{Tr}(\rho \hat{W})\geqslant 0$, we obtain the necessary criterion of separability:$\begin{eqnarray}p\leqslant \displaystyle \frac{1}{3}(x+\sqrt{{x}^{2}+3}),\end{eqnarray}$the curve $p=\tfrac{1}{3}(x+\sqrt{{x}^{2}+3})$ is drawn in figure 1 as necessary condition curve of the witness in the literature [28]. It can be seen that the witness necessary condition curve coincides with the numerical necessary condition when 0 ≤ x ≤ 0.2.
Figure 1.
New window|Download| PPT slide Figure 1.Necessary conditions of separability for quantum state model (13). The numerical necessary conditions are obtained by minimizing ${ \mathcal L }$ with respect to randomly chosen parameter Mi.
In order to verify whether witness ${\hat{W}}_{1}$ matches the quantum state model ${\rho }_{{W}_{1}}$, we calculate the necessary condition due to the entanglement witness to see whether it coincides with the numerical necessary curve. Our entanglement witness is ${\hat{W}}_{1}$ and the density matrix is (13). According to formula $\mathrm{Tr}({\rho }_{{W}_{1}}\hat{{W}_{1}})\geqslant 0$ we have:$\begin{eqnarray}\begin{array}{l}\mathrm{Tr}({\rho }_{{W}_{1}}\hat{{W}_{1}})=\frac{3}{8}d+\frac{1}{8d}+3{\delta }_{1}p(\eta -1)-3{\delta }_{2}p(\eta +1)\\ \quad \ -\ \frac{9}{2}{\delta }_{4}{dp}({\eta }^{2}-1)+\frac{3}{2}{\delta }_{3}{dp}(3{\eta }^{2}+1)\geqslant 0.\end{array}\end{eqnarray}$Using mean value inequality to optimize variables η and d, then:$\begin{eqnarray}\begin{array}{l}\sqrt{\left\{\frac{3}{8}+\frac{9}{2}{\delta }_{4}p+\frac{3}{2}{\delta }_{3}p-18{p}^{2}{\left({\delta }_{1}+{\delta }_{2}\right)}^{2}\right\}\left\{\frac{9}{2}{\delta }_{3}p-\frac{9}{2}{\delta }_{4}p-18{p}^{2}{\left({\delta }_{2}-{\delta }_{1}\right)}^{2}\right\}}\\ \quad \ \geqslant \ 18{p}^{2}({\delta }_{2}^{2}-{\delta }_{1}^{2}).\end{array}\end{eqnarray}$By solving equation (16), we finally obtain the witness necessary condition curve of the state (we use a known witness). It can be seen from figure 1 that at 0 ≤ x ≤ 0.6 the witness necessary condition curve coincides with the numerical necessary curve. This shows that entanglement witness ${\hat{W}}_{1}$ is suitable for state (13).
At the same time, we calculate whether the given entanglement witness is suitable for the decayed states of state ${\rho }_{{W}_{1}}$. The decay factor is represented by τ, and its value range is from 0 to 1. We multiply the δ1∣000⟩(⟨011∣ + ⟨101∣ + ⟨110∣), δ1(∣011⟩ + ∣101⟩ + ∣110⟩)⟨000∣, δ4∣111⟩(⟨001∣ + ⟨010∣ + ⟨100∣), δ4(∣100⟩ + ∣001⟩ + ∣010⟩)⟨111∣ parts in the quantum state ${\rho }_{{W}_{1}}$ by the decay factors τ of 0.5 and 0.75, respectively. According to $\mathrm{Tr}({\rho }_{{W}_{1}}\hat{{W}_{1}})\geqslant 0$, we obtained the numerical necessary curve and the entanglement witness necessary condition curve for the decayed W states. The calculation shows that, when τ = 0.5, the witness necessary condition curve at 0 ≤ x ≤ 0.9 coincides with the numerical necessary curve. When τ = 0.75, the witness necessary condition curve at 0 ≤ x ≤ 0.8 coincides with the numerical necessary curve. So far our calculation results show that the necessary conditions of separability fit the numerical necessary condition well.
3.3. Detecting entanglement of ρW(t)
In order to detect the entanglement lifetime of quantum state ρW(t), we directly optimize the numerical random witness to evaluate the entanglement lifetime of quantum state ρW(t). By the following formula of separability:$\begin{eqnarray}{ \mathcal L }=\displaystyle \frac{{\rm{\Lambda }}}{\mathrm{Tr}({\rho }_{W}(t)\cdot \hat{M})}=\displaystyle \frac{{\rm{\Lambda }}}{{\boldsymbol{M}}\cdot {\boldsymbol{R}}}\geqslant 1.\end{eqnarray}$Numerical calculation shows that when t = 0.637 s, ${{ \mathcal L }}_{\min }\geqslant 1$. Thus, it can be seen that the entanglement lifetime for the state ρW(t) is not less than 0.637 s.
We use a known witness from [28] to detect the entanglement of the quantum state ρW(t), according to formula $\mathrm{Tr}({\rho }_{W}(t)\hat{W})\geqslant 0$, we get that the entanglement lifetime is t = 0.635 629 s.
Finally, we use witness ${\hat{W}}_{1}$ to calculate the entanglement lifetime of evolving W state ρW(t). According to the formula $\mathrm{Tr}({\rho }_{W}(t)\hat{{W}_{1}})\geqslant 0$ we have:$\begin{eqnarray}\begin{array}{l}\mathrm{Tr}({\rho }_{W}(t)\hat{{W}_{1}})=d\{{\alpha }_{4}+{\alpha }_{6}+{\alpha }_{7}+\displaystyle \frac{1+3{\eta }^{2}}{2}({\beta }_{4}+{\beta }_{8}+{\beta }_{12})\\ \quad \ +\ \displaystyle \frac{3(1-{\eta }^{2})}{2}({\beta }_{3}+{\beta }_{7}+{\beta }_{10})\}+\displaystyle \frac{{\alpha }_{1}}{d}\\ \quad \ +\ ({\beta }_{1}-{\beta }_{2}+{\beta }_{5}-{\beta }_{6}+{\beta }_{9}-{\beta }_{11})\eta \\ \quad \ -\ ({\beta }_{1}+{\beta }_{2}+{\beta }_{5}+{\beta }_{6}+{\beta }_{9}+{\beta }_{11})\\ \quad \ \geqslant \ 2\sqrt{{\alpha }_{1}\{{\alpha }_{4}+{\alpha }_{6}+{\alpha }_{7}+\displaystyle \frac{1+3{\eta }^{2}}{2}({\beta }_{4}+{\beta }_{8}+{\beta }_{12})+\displaystyle \frac{3(1-{\eta }^{2})}{2}({\beta }_{3}+{\beta }_{7}+{\beta }_{10})\}}\\ \quad \ +\ ({\beta }_{1}-{\beta }_{2}+{\beta }_{5}-{\beta }_{6}+{\beta }_{9}-{\beta }_{11})\eta \\ \quad \ -\ ({\beta }_{1}+{\beta }_{2}+{\beta }_{5}+{\beta }_{6}+{\beta }_{9}+{\beta }_{11})\geqslant 0.\end{array}\end{eqnarray}$Because 0 < η < 1, let S1 = β1 − β2 + β5 − β6 + β9 − β11, S2 = β1 + β2 + β5 + β6 + β9 + β11, S3 = 2α1[2(α4 + α6 + α7) + β4 + β8 + β12 + 3(β3 + β7 + β10)], S4 = 6α1(β4 + β8 + β12 − β3 − β7 − β10), so equation (18) can be further written as:$\begin{eqnarray}\sqrt{({S}_{4}-{S}_{1}^{2})({S}_{3}-{S}_{2}^{2})}+{S}_{1}{S}_{2}\geqslant 0.\end{eqnarray}$By solving equation (19), we can get that the entanglement lifetime of the quantum state ρW(t) under the action of entanglement witness ${\hat{W}}_{1}$ is t = 0.827 718 s. The results show that, compared with previous methods, witness ${\hat{W}}_{1}$ can detect entanglement for a longer lifetime, and witness ${\hat{W}}_{1}$ significantly improves the performance of entanglement detection for an evolving W state.
4. Conclusion
We have presented a new entanglement witness, ${\hat{W}}_{1}$, with two varying parameters. The parameters will vary to match the destination quantum state when detecting entanglement. We have shown that witness ${\hat{W}}_{1}$ perfectly detects the entanglement of our model quantum state ${\rho }_{{W}_{1}}$ and its decayed states in some parameter regions. The witness is also applicable to the evolving W state ρW(t). Our calculation shows that the entanglement lifetime of the evolving W state ρW(t) is t = 0.637 seconds when we directly optimize the numerical random witness. With the existing witness [28], the entanglement lifetime is t = 0.635 629 seconds. With witness ${\hat{W}}_{1}$, the entanglement lifetime is t = 0.827 718 seconds. This shows that for the evolving W state ρW(t), witness ${\hat{W}}_{1}$ is better at detecting entanglement than other known witnesses or methods.
Acknowledgments
Support from the National Natural Science Foundation of China (Grant No: 61 871 347) is gratefully acknowledged.
,∗College of Information and Electronic Engineering, 
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