Analysing nonlocality robustness in multiqubit systems under noisy conditions and weak measurements

Abstract

We analyse robustness of nonlocal correlation in multiqubit entangled states—three- and four-qubit GHZ class and three-qubit W class—useful for quantum information and computation, under noisy conditions and weak measurements. For this, we use a Bell-type inequality whose violation is considered as a signature for confirming the presence of genuine nonlocal correlations between the qubits. In order to demonstrate the effects of noise and weak measurements, an analytical relation is established between the maximum expectation value of three and four-qubit Svetlichny operators for the systems under study, noise parameter and strengths of weak measurements. Our results show that for a set of three- and four-qubit GHZ class states, maximal nonlocality does not coincide with maximum entanglement for a given noise parameter and a certain range of weak measurement parameter. Our analysis further shows an excellent agreement between the analytical and numerical results.

Appendix

1.1 Maximization of the expectation value of the three-qubit Svetlichny operator for W states

For maximizing the value of Svetlichny operator in Eq. (34) described in Sec. V, we assume $\phi _{i}=0$ [128], and then add the first four terms in Eq. (8) to get

$$\begin{aligned} \left\langle M \right\rangle= & {} \frac{1}{4}\left[ \left( - \Delta -C_{12}-C_{23}-C_{31}\right) \left\{ \cos \left( \theta _{a}+\theta _{b}+\theta _{c'}\right) \right. \right. \nonumber \\&\left. \left. +\,\cos \left( \theta _{a'}+\theta _{b}+\theta _{c}\right) +\cos \left( \theta _{a}+\theta _{b'}+\theta _{c}\right) -\cos \left( \theta _{a'}+\theta _{b'}+\theta _{c'}\right) \right\} \right. \nonumber \\&\left. +\left( -\, \Delta +C_{12}-C_{23}+C_{31}\right) \left\{ \cos \left( \theta _{a}+\theta _{b}-\theta _{c'}\right) \right. \right. \nonumber \\&\left. \left. +\,\cos \left( \theta _{a'}+\theta _{b}-\theta _{c}\right) +\cos \left( \theta _{a}+\theta _{b'}-\theta _{c}\right) -\cos \left( \theta _{a'}+\theta _{b'}-\theta _{c'}\right) \right\} \right. \nonumber \\&\left. + \left( -\,\Delta +C_{12}+C_{23}-C_{31}\right) \left\{ \cos \left( \theta _{a}-\theta _{b}+\theta _{c'}\right) \right. \right. \nonumber \\&\left. \left. +\,\cos \left( \theta _{a'}-\theta _{b}+\theta _{c}\right) +\cos \left( \theta _{a}-\theta _{b'}+\theta _{c}\right) -\cos \left( \theta _{a'}-\theta _{b'}+\theta _{c'}\right) \right\} \right. \nonumber \\&\left. +\, \left( -\, \Delta -C_{12}+C_{23}+C_{31}\right) \left\{ \cos \left( \theta _{a}-\theta _{b}-\theta _{c'}\right) \right. \right. \nonumber \\&\left. \left. +\,\cos \left( \theta _{a'}-\theta _{b}-\theta _{c}\right) +\cos \left( \theta _{a}-\theta _{b'}-\theta _{c}\right) -\cos \left( \theta _{a'}-\theta _{b'}-\theta _{c'}\right) \right\} \right] \nonumber \\ \end{aligned}$$

(A.1)

The expression for $M'$ can be written in a similar fashion. For simplicity and mathematical convenience, let us define $\Sigma =\left( \theta '_{a}+\theta '_{b}+\theta '_{c}\right)$ , $\Sigma _{k}=\Sigma -2\theta '_{k}$, $\tilde{\theta _{k}}=(\theta _{k}+\theta _{k'})/2$, and $\theta '_{k}=(\theta _{k'}-\theta _{k})/2$ where $k\epsilon \left\{ a,b,c\right\}$ such that

$$\begin{aligned} S_{v}\left( \rho ^{wk}_{W}\right) _\mathrm{opt}= & {} \frac{1}{2}\left[ \left( - \Delta -C_{12}-C_{23}-C_{31}\right) \right. \nonumber \\&\left. \times \sin \left( \tilde{\theta }_{a}+\tilde{\theta }_{b}+\tilde{\theta }_{c}\right) \left\{ K-2\sin \left( \theta '_{a}-\theta '_{b}-\theta '_{c}\right) \right\} \right. \nonumber \\&+ \left. \left( - \Delta +C_{12}-C_{23}+C_{31}\right) \right. \nonumber \\&\left. \times \sin \left( \tilde{\theta }_{a}+\tilde{\theta }_{b}-\tilde{\theta }_{c}\right) \left\{ K-2\sin \left( \theta '_{a}-\theta '_{b}+\theta '_{c}\right) \right\} \right. \nonumber \\&+ \left. \left( - \Delta +C_{12}+C_{23}-C_{31}\right) \right. \nonumber \\&\left. \times \sin \left( \tilde{\theta }_{a}-\tilde{\theta }_{b}+\tilde{\theta }_{c}\right) \left\{ K-2\sin \left( \theta '_{a}+\theta '_{b}-\theta '_{c}\right) \right\} \right. \nonumber \\&+ \left. \left( - \Delta -C_{12}+C_{23}+C_{31}\right) \right. \nonumber \\&\left. \times \sin \left( \tilde{\theta }_{a}-\tilde{\theta }_{b}-\tilde{\theta }_{c}\right) \left\{ K-2\sin \left( \theta '_{a}+\theta '_{b}+\theta '_{c}\right) \right\} \right] \nonumber \\= & {} \Delta \left\{ \sin \Sigma _{a}+\sin \Sigma _{b}+\sin \Sigma _{c}-\sin \Sigma \right\} \nonumber \\&+ C_{12}\left\{ \sin \Sigma -\sin \Sigma _{a}+\sin \Sigma _{b}+\sin \Sigma _{c}\right\} \nonumber \\&+ C_{23}\left\{ \sin \Sigma +\sin \Sigma _{a}+\sin \Sigma _{b}-\sin \Sigma _{c}\right\} \nonumber \\&+ C_{31}\left\{ \sin \Sigma +\sin \Sigma _{a}-\sin \Sigma _{b}+\sin \Sigma _{c}\right\} \end{aligned}$$

(A.2)

$$\begin{aligned}\equiv & {} 4\left( s_{1} \Delta + s_{2} C_{12} +s_{3} C_{23} +s_{4} C_{31}\right) \end{aligned}$$

(A.3)

where

$$\begin{aligned} K= & {} \left\{ \sin \left( \theta '_{a}+\theta '_{b}+\theta '_{c}\right) +\sin \left( \theta '_{a}-\theta '_{b}+\theta '_{c}\right) \right. \nonumber \\&+ \left. \sin \left( \theta '_{a}+\theta '_{b}-\theta '_{c}\right) \sin \left( \theta '_{a}-\theta '_{b}-\theta '_{c}\right) \right\} \end{aligned}$$

(A.4)

and the equality in Eq. (A.2) can be achieved by considering $\tilde{\theta }_{a}=\tilde{\theta }_{b}=\tilde{\theta }_{c}=\pi /2$.

1.2 Derivation for the relationship between $t_{1}$ and $t_{2}$

For evaluating the relationship between $t_{1}$ and $t_{2}$, we further consider two unit vectors p and $p'$ where $\vec {b}+\vec {b'}=2\vec {p}\cos \theta _{1}$ and $\vec {b}-\vec {b'}=2\vec {p'}\sin \theta _{1}$ such that

$$\begin{aligned} \vec {p}.\vec {p'}=\cos \theta _{p}\cos \theta _{p'}+\sin \theta _{p}\sin \theta _{p'}\cos (\phi _{p}-\phi _{p'})=0 \end{aligned}$$

(B.1)

Therefore, $t_{1}$ and $t_{2}$ can be re-expressed as

$$\begin{aligned} \vert t_{1}\vert= & {} \vert l_{ap'cd}\sin \theta _{1}-l_{apcd'}\cos \theta _{1}-l_{apc'd}\cos \theta _{1} \nonumber \\&-\, l_{ap'c'd'}\sin \theta _{1} + l_{a'pc'd'}\cos \theta _{1}\nonumber \\&-\,l_{a'p'c'd}\sin \theta _{1}-l_{a'pcd}\cos \theta _{1}-l_{a'p'cd'}\sin \theta _{1}\vert \end{aligned}$$

(B.2)

and

$$\begin{aligned} \vert t_{2}\vert= & {} \vert s_{ap'cd}\sin \theta _{1}-s_{apcd'}\cos \theta _{1}-s_{apc'd}\cos \theta _{1} \nonumber \\&-\, s_{ap'c'd'}\sin \theta _{1} +s_{a'pc'd'}\cos \theta _{1} \nonumber \\&-\, s_{a'p'c'd}\sin \theta _{1}-s_{a'pcd}\cos \theta _{1}-s_{a'p'cd'}\sin \theta _{1}\vert \end{aligned}$$

(B.3)

where

$$\begin{aligned} l_{ap'cd}= & {} 2\cos \theta _{a}\cos \theta _{p'}\cos \theta _{c}\cos \theta _{d} \end{aligned}$$

(B.4)

$$\begin{aligned} s_{ap'cd}= & {} 2\sin \theta _{a}\sin \theta _{p'}\sin \theta _{c}\sin \theta _{d}\cos \phi _{ap'cd} \end{aligned}$$

(B.5)

The other coefficients $l_{apcd'}$, $s_{apcd'}$ etc. can be defined in a similar fashion with prime on different angles. In order to simplify and optimize the expressions further, we assume $\theta _{c}=\theta _{c'}$, and define two unit vectors q and $q'$ such that $\vec {d}+\vec {d'}=2\vec {q}\cos \theta _{2}$ and $\vec {d}-\vec {d'}=2\vec {q'}\sin \theta _{2}$, i.e.,

$$\begin{aligned} \vec {q}\cdot \vec {q'}=\cos \theta _{q}\cos \theta _{q'}+\sin \theta _{q}\sin \theta _{q'}\cos (\phi _{q}-\phi _{q'})=0 \end{aligned}$$

(B.6)

This allows us to re-express Eqs. (B.2) and (B.3) as

$$\begin{aligned} \vert t_{1}\vert= & {} \vert l'_{ap'cq'}\sin \theta _{1}\sin \theta _{2}-l'_{apcq}\cos \theta _{1}\cos \theta _{2}\nonumber \\&-\, l'_{a'pcq'}\cos \theta _{1}\sin \theta _{2}- l'_{a'p'cq}\sin \theta _{1}\cos \theta _{2} \vert \end{aligned}$$

(B.7)

and

$$\begin{aligned} \vert t_{2}\vert= & {} \vert s'_{ap'cq'}\sin \theta _{1}\sin \theta _{2}-s'_{apcq}\cos \theta _{1}\cos \theta _{2}\nonumber \\&-\, s'_{a'pcq'}\cos \theta _{1}\sin \theta _{2}- s'_{a'p'cq}\sin \theta _{1}\cos \theta _{2} \vert \end{aligned}$$

(B.8)

where

$$\begin{aligned} l'_{apcq}= & {} 4\cos \theta _{a}\cos \theta _{p}\cos \theta _{c}\cos \theta _{q} \end{aligned}$$

(B.9)

$$\begin{aligned} s'_{apcq}= & {} 4\sin \theta _{a}\sin \theta _{p}\sin \theta _{c}\sin \theta _{q}\cos \phi _{apcq} \end{aligned}$$

(B.10)

From Eqs. (B.7) and (B.8), one can get

$$\begin{aligned} \vert t_{1}\vert\le & {} \vert l'_{ap'cq'}\vert \vert \sin \theta _{1}\vert \vert \sin \theta _{2}\vert +\vert l'_{apcq}\vert \vert \cos \theta _{1}\vert \vert \cos \theta _{2}\vert \nonumber \\&+\, \vert l'_{a'pcq'}\vert \vert \cos \theta _{1}\vert \vert \sin \theta _{2}\vert +\vert l'_{a'p'cq}\vert \vert \sin \theta _{1}\vert \vert \cos \theta _{2}\vert \end{aligned}$$

(B.11)

$$\begin{aligned} \vert t_{2}\vert\le & {} \vert s'_{ap'cq'}\vert \vert \sin \theta _{1}\vert \vert \sin \theta _{2}\vert +\vert s'_{apcq}\vert \vert \cos \theta _{1}\vert \vert \cos \theta _{2}\vert \nonumber \\&+\, \vert s'_{a'pcq'}\vert \vert \cos \theta _{1}\vert \vert \sin \theta _{2}\vert +\vert s'_{a'p'cq}\vert \vert \sin \theta _{1}\vert \vert \cos \theta _{2}\vert \end{aligned}$$

(B.12)

Using these inequalities, the iterative maximization of Eq. (39) can be summarized below as

$$\begin{aligned} 2\sqrt{2}\kappa \vert t_{1}\vert +\vert t_{2} \vert\le & {} \left[ \left\{ (2\sqrt{2} \kappa \vert l'_{apcq}\vert + \vert s'_{apcq}\vert ) \vert \cos \theta _{1}\vert \right. \right. \nonumber \\&\left. \left. +\,(2\sqrt{2} \kappa \vert l'_{a'p'cq}\vert + \vert s'_{a'p'cq}\vert ) \vert \sin \theta _{1}\vert \right\} \vert \cos \theta _{2}\vert \right. \nonumber \\&\left. +\, \left\{ (2\sqrt{2} \kappa \vert l'_{a'pcq'}\vert + \vert s'_{a'pcq'}\vert ) \vert \cos \theta _{1}\vert \right. \right. \nonumber \\&\left. \left. +\,(2\sqrt{2} \kappa \vert l'_{ap'cq'}\vert + \vert s'_{ap'cq'}\vert ) \vert \sin \theta _{1}\vert \right\} \vert \sin \theta _{2}\vert \right] \end{aligned}$$

(B.13)

$$\begin{aligned} 2\sqrt{2}\kappa \vert t_{1}\vert +\vert t_{2} \vert\le & {} \left[ \left\{ (2\sqrt{2} \kappa \vert l'_{apcq}\vert + \vert s'_{apcq}\vert ) \vert \cos \theta _{1}\vert \right. \right. \nonumber \\&\left. \left. +\,(2\sqrt{2} \kappa \vert l'_{a'p'cq}\vert + \vert s'_{a'p'cq}\vert ) \vert \sin \theta _{1}\vert \right\} ^{2} \right. \nonumber \\&\left. +\, \left\{ (2\sqrt{2} \kappa \vert l'_{a'pcq'}\vert + \vert s'_{a'pcq'}\vert ) \vert \cos \theta _{1}\vert \right. \right. \nonumber \\&\left. \left. +\,(2\sqrt{2} \kappa \vert l'_{ap'cq'}\vert + \vert s'_{ap'cq'}\vert ) \vert \sin \theta _{1}\vert \right\} ^{2}\right] ^{\frac{1}{2}} \end{aligned}$$

(B.14)

$$\begin{aligned} 2\sqrt{2}\kappa \vert t_{1}\vert +\vert t_{2} \vert\le & {} \left[ (2\sqrt{2} \kappa \vert l'_{apcq}\vert + \vert s'_{apcq}\vert )^{2}+(2\sqrt{2} \kappa \vert l'_{a'p'cq}\vert + \vert s'_{a'p'cq}\vert )^{2} \right. \nonumber \\&\left. +\, (2\sqrt{2} \kappa \vert l'_{a'pcq'}\vert + \vert s'_{a'pcq'}\vert )^{2}+(2\sqrt{2} \kappa \vert l'_{ap'cq'}\vert \right. \nonumber \\&\left. +\, \vert s'_{ap'cq'}\vert )^{2}\right] ^{\frac{1}{2}} \end{aligned}$$

(B.15)

where Eq. (B.13) is maximized with respect to $\theta _{2}$, and the first and second terms in Eq. (B.14) are maximized separately with respect to $\theta _{1}$. To simplify and optimize Eq. (B.15), we use Eqs. (B.9) and (B.10), such that

$$\begin{aligned} (2\sqrt{2} \kappa \vert l'_{apcq}\vert + \vert s'_{apcq}\vert )= & {} 8\sqrt{2}\kappa \vert \cos \theta _{a}\cos \theta _{p}\cos \theta _{c}\cos \theta _{q}\vert \nonumber \\&+\, 4\vert \sin \theta _{a}\sin \theta _{p}\sin \theta _{c}\sin \theta _{q}\cos \phi _{apcq}\vert \end{aligned}$$

(B.16)

Equation (B.16) when maximized with respect to $\theta _{a}$, where we have used the inequality (15), gives

$$\begin{aligned} (2\sqrt{2} \kappa \vert l'_{apcq}\vert + \vert s'_{apcq}\vert )\le & {} 4\left[ 8\kappa ^{2}\cos ^{2}\theta _{c}\cos ^{2}\theta _{p}\cos ^{2}\theta _{q}\right. \nonumber \\&\left. +\,\sin ^{2}\theta _{c}\sin ^{2}\theta _{p}\sin ^{2}\theta _{q}\cos ^{2}\phi _{apcq}\right] ^{\frac{1}{2}} \end{aligned}$$

(B.17)

Similarly, the other terms in Eq. (B-15) can be evaluated as

$$\begin{aligned} (2\sqrt{2} \kappa \vert l'_{ap'cq'}\vert + \vert s'_{ap'cq'}\vert )\le & {} 4\left[ 8\kappa ^{2}\cos ^{2}\theta _{c}\cos ^{2}\theta _{p'}\cos ^{2}\theta _{q'}\right. \nonumber \\&\left. +\,\sin ^{2}\theta _{c}\sin ^{2}\theta _{p'}\sin ^{2}\theta _{q'}\cos ^{2}\phi _{ap'cq'}\right] ^{\frac{1}{2}} \end{aligned}$$

(B.18)

$$\begin{aligned} (2\sqrt{2} \kappa \vert l'_{a'p'cq}\vert + \vert s'_{a'p'cq}\vert )\le & {} 4\left[ 8\kappa ^{2}\cos ^{2}\theta _{c}\cos ^{2}\theta _{p'}\cos ^{2}\theta _{q}\right. \nonumber \\&\left. +\,\sin ^{2}\theta _{c}\sin ^{2}\theta _{p'}\sin ^{2}\theta _{q}\cos ^{2}\phi _{a'p'cq}\right] ^{\frac{1}{2}} \end{aligned}$$

(B.19)

$$\begin{aligned} (2\sqrt{2} \kappa \vert l'_{a'pcq'}\vert + \vert s'_{a'pcq'}\vert )\le & {} 4\left[ 8\kappa ^{2}\cos ^{2}\theta _{c}\cos ^{2}\theta _{p}\cos ^{2}\theta _{q'}\right. \nonumber \\&\left. +\,\sin ^{2}\theta _{c}\sin ^{2}\theta _{p}\sin ^{2}\theta _{q'}\cos ^{2}\phi _{a'pcq'}\right] ^{\frac{1}{2}} \end{aligned}$$

(B.20)

where, for optimization, we consider $\cos ^{2}\phi _{apcq}=\cos ^{2}\phi _{ap'cq'}=\cos ^{2}\phi _{a'p'cq}=\cos ^{2}\phi _{a'pcq'}=1$. Therefore, using Eqs. (B.17–B.20), Eq. (B.15) can be re-expressed as

$$\begin{aligned} 2\sqrt{2}\kappa \vert t_{2}\vert +\vert t_{1} \vert\le & {} 4\left[ 8\kappa ^{2}\cos ^{2}\theta _{c}\cos ^{2}\theta _{q}(\cos ^{2}\theta _{p}+\cos ^{2}\theta _{p'})\right. \nonumber \\&\left. +\,\sin ^{2}\theta _{c}\sin ^{2}\theta _{q}(\sin ^{2}\theta _{p}+\sin ^{2}\theta _{p'})\right. \nonumber \\&\left. +\, 8\kappa ^{2}\cos ^{2}\theta _{c}\cos ^{2}\theta _{q'}(\cos ^{2}\theta _{p}+\cos ^{2}\theta _{p'})\right. \nonumber \\&\left. +\,\sin ^{2}\theta _{c}\sin ^{2}\theta _{q'}(\sin ^{2}\theta _{p}+\sin ^{2}\theta _{p'}) \right] ^{\frac{1}{2}} \end{aligned}$$

(B.21)

Considering the orthogonality of unit vectors $\vec {p}$ and $\vec {p'}$, the maximum value of $(\sin ^{2}\theta _{p}+\sin ^{2}\theta _{p'})$ is 2 and maximum value of $(\cos ^{2}\theta _{p}+\cos ^{2}\theta _{p'})$ is 1, i.e.,

$$\begin{aligned} 2\sqrt{2}\kappa \vert t_{1}\vert +\vert t_{2} \vert\le & {} 4\left[ 8\kappa ^{2}\cos ^{2}\theta _{c}(\cos ^{2}\theta _{q}+\cos ^{2}\theta _{q'})\right. \nonumber \\&\left. +\,2\sin ^{2}\theta _{c}(\sin ^{2}\theta _{q}+\sin ^{2}\theta _{q'})\right] ^{\frac{1}{2}} \end{aligned}$$

(B.22)

Similarly from the orthogonality of unit vectors $\vec {q}$ and $\vec {q'}$, Eq. (B.22) can be further optimized as

$$\begin{aligned} 2\sqrt{2}\kappa \vert t_{1}\vert +\vert t_{2} \vert \le 4\left[ 8\kappa ^{2}\cos ^{2}\theta _{c}+4\sin ^{2}\theta _{c}\right] ^{\frac{1}{2}} \end{aligned}$$

(B.23)

A further maximization on the parameter $\kappa$ gives

$$\begin{aligned} 2\sqrt{2}\vert t_{1}\vert +\vert t_{2} \vert\le & {} 8\left[ 1+\cos ^{2}\theta _{c}\right] ^{\frac{1}{2}} \le 8\sqrt{2} \end{aligned}$$

(B.24)

Therefore, the relationship between $t_{1}$ and $t_{2}$ can be defined as

$$\begin{aligned} \vert t_{2} \vert \le 8\sqrt{2}-2\sqrt{2}\vert t_{1} \vert \end{aligned}$$

(B.25)