Multipartite entanglement vs nonlocality for two families of $N$-qubit states

Introduction: Bell’s inequalities provide a practical method for testing whether correlations observed between spatially separated parts of a system are compatible with any local hidden variable (LHV) description [1]. Quantum entangled states can violate Bell-type inequalities, thereby confirming the fundamentally different nature of quantum correlations compared to those allowed by LHV models [2]. For the case of 2-qubit pure states, entanglement and nonlocality, as measured by Bell inequality violations, are directly related [3] [4]—the more entanglement there is, larger the observed violation. For multiqubit pure states, the much more complex relationship between $N$-qubit entanglement and nonlocality has not yet been explored in much detail. Such studies are important for characterizing and understanding the behaviour of many-body systems, shedding light on the subtle nature of quantum mechanics, and for designing large-scale, efficient information processing and communication protocols [5, 6, 7, 8].

In this letter, we generalize the well-known relationship between 2-qubit entanglement and violation of the Bell-CHSH inequality [2, 3, 4] to the case of $N$-qubit entanglement and violation of an N-qubit Bell inequality for two families of states. For two qubits, all pure states can be written in the form $\ket{\psi}=\alpha\ket{00}+\beta\ket{11}$ in some basis via the Schmidt decomposition. The maximum expected value of the Bell-CHSH operator with respect to these states is $\max[S_{2}]=2\sqrt{2}$ [9], which violates the LHV bound of $S_{2}\leq 2$ if $\ket{\psi}$ is an entangled state ($\alpha,\beta\neq 0$). Here, we analyse correlations in the $N-$qubit generalizations of two subsets of the GHZ class [10] of states: the $N-$qubit generalized GHZ (GGHZ) states $\ket{\psi_{g}}$ and the $N-$qubit maximal slice (MS) states $\ket{\psi_{s}}$ [11]:

$$\ket{\psi_{g}}=\cos\alpha\ket{0}^{\otimes N}+\sin\alpha\ket{1}^{\otimes N},$$

(1)

$\displaystyle\ket{\psi_{s}}=\frac{1}{\sqrt{2}}\left[\ket{0}^{\otimes N}+\ket{1}^{\otimes N-1}\left(\cos\alpha\ket{0}+\sin\alpha\ket{1}\right)\right].$

(2)

As the name suggests, the GGHZ states are generalizations of the well-known 3-qubit GHZ state $(N=3,\alpha=\pi/4)$, which is a useful entanglement resource for a variety of information processing protocols including dense coding, teleportation, etc [12, 13, 14, 15]. Tracing out any qubit in the GGHZ state leaves the remaining qubits in a mixed state, indicating genuine N-qubit entanglement. The maximal sliced state is bi-separable when $\sin\alpha=0$ and maximally entangled when $\sin\alpha=1$. The relation between entanglement and nonlocal properties of these states in terms of Bell inequality violations have been explored for $N=3$ [16, 17]. In order to examine the multiqubit nonlocal properties of these states, we consider the expectation value of the Svetlichny operator $S_{N}$ that is a generalization of the 2-qubit Bell-CHSH operator to the case of N qubits [18, 19, 20]. Violation of the inequality $|\left<S_{N}\right>|\leq 2^{N-1}$ implies genuine N-qubit non-separability as opposed to partial nonlocal correlations between less than N qubits. Finding the maximum of $|\left<S_{N}\right>|$ for arbitrary quantum states analytically or computationally is challenging as the number of terms in $S_{N}$ increases exponentially with $N$ [21]. Nevertheless, the symmetry of the GGHZ and MS states allows us to find explicit expressions for the maximum of $|\left<S_{N}\right>|$ for these states. We show that for the GGHZ states with $N>2$ the maximum of $S_{N}$, for odd $N$ is

$$S_{N\max}^{\pm}(\psi_{g})=\left\{\begin{array}[]{ll}2^{\frac{N+1}{2}}\sqrt{1-\tau(\psi_{g})},&\tau(\psi_{g})\leq\frac{1}{2^{N-2}+1}\\ \\ 2^{N-1}\sqrt{2\tau},&\tau(\psi_{g})\geq\frac{1}{2^{N-2}+1}\end{array}\right.$$

(3)

and for even $N$ is

$$S_{N\max}^{\pm}(\psi_{g})=\left\{\begin{array}[]{ll}2^{\frac{N}{2}},&\tau\leq 2^{1-N}\\ 2^{N-1}\sqrt{2\tau},&\tau\geq 2^{1-N},\end{array}\right.$$

(4)

where $\tau(\psi_{g})$ is the $n$-tangle [22, 23, 24] which quantifies the multipartite entanglement present in the GGHZ states. The above equations show that for the GGHZ states, the inequality $S_{N}\leq 2^{N-1}$ is not violated when $\tau(\psi_{g})\leq 1/2$, even though the states are genuinely $N$-qubit entangled. Thus, we show that the Svetlichny inequality is not sensitive to the $N$-qubit entanglement in these states. Interestingly, the critical value of $\tau(\psi_{g})\leq 1/2$ beyond which violation occurs, is the same for all values of $N$.

For the MS state, we find the maximum of $S_{N}$ is

$\displaystyle S_{N\max}^{\pm}(\psi_{s})=2^{N-1}\sqrt{1+\sin^{2}\alpha}.$

(5)

We see that the bound is violated for MS states as long as the state is not bi-separable, i.e $\sin\alpha\neq 0$. When $N=3$ and for even $N(>3)$, $\tau(\psi_{s})=\sin^{2}\alpha$ and hence for these cases the expression becomes

$\displaystyle S_{N\max}^{\pm}(\psi_{s})=2^{N-2}\sqrt{1+\tau(\psi_{s})},$

(6)

which matches previous results [16, 17].

Svetlichny Inequality: In order to derive the expressions above, we start with the $N$-qubit Svetlichny inequality constructed to test for genuine $N$-partite nonlocal correlations [19, 20]. Consider $N$ spatially separated particles and two dichotomic observables $A^{0}_{i},A^{1}_{i}$, $i=1,2\dots N$ for each particle. Then the following operators can be constructed:

$\displaystyle S_{N}^{\pm}=\sum_{\vec{x}}\nu^{\pm}(\vec{x})A(\vec{x})$

(7)

where $\{\vec{x}\}$ are bit strings of size $N$, $A(\vec{x})$ is an $N$-qubit operator of the form $A(\boldsymbol{x})=\bigotimes_{i=1}^{N}A_{i}^{x(i)}$ and $\nu^{\pm}(\boldsymbol{x})$ is a constant dependent on the Hamming weight $w(\boldsymbol{x})$ of the bit string

$\displaystyle\nu^{\pm}(\boldsymbol{x})=-1^{w(\boldsymbol{x})(w(\boldsymbol{x})\pm 1)/2}.$

(8)

For $N=2$, the above expression reduces to the standard CHSH form [2] and for $N=3$, they yield the operator in [18]. Svetlichny showed that if one allows nonlocal correlation between at most $N-1$ of the parties, then $|\braket{S_{N}^{\pm}}|\leq 2^{N-1}.$ Violation of this inequality confirms genuine $N$-partite nonlocality.

Maximum violation for GGHZ states: The two dichotomic observables $A^{0}_{i},A^{1}_{i}$, $i=1,2\dots N$ for each qubit can be written in terms of the Pauli operators $\sigma_{x},\sigma_{y}$ and $\sigma_{z}$ as $A^{x(i)}_{i}=\boldsymbol{v}^{x(i)}_{i}\cdot\vec{\sigma}$, where $\boldsymbol{v}^{x(i)}_{i}$ are unit vectors in $\mathbbm{R}^{3}$ and $\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$. Defining $\vec{v}^{x(i)}_{i}=(\sin\theta^{x(i)}_{i}\cos\phi^{x(i)}_{i},\sin\theta^{x(i)}_{i}\sin\phi^{x(i)}_{i},\cos\theta^{x(i)}_{i}),$ we can calculate the absolute value of the expectation value of the Svetlichny operator for GGHZ states as

	$\displaystyle|\braket{S_{N}^{\pm}}|=\Big{|}c_{1}\sum_{\boldsymbol{x}}\nu^{\pm}(\boldsymbol{x})\prod_{i}^{N}\cos\theta_{i}^{x(i)}$
	$\displaystyle+c_{2}\sum_{\boldsymbol{x}}\nu^{\pm}(\boldsymbol{x})\cos\left(\sum_{i}^{N}\phi_{i}^{x(i)}\right)\prod_{i}^{N}\sin\theta_{i}^{x(i)}\Big{|},$		(9)

which can be written in the form $|\braket{S_{N}^{\pm}}|=|c_{1}F_{N}^{\pm}+c_{2}G_{N}^{\pm}|,$ where $F_{N}$ and $G_{N}$ contain all the $\cos\theta$ and $\sin\theta$ terms of $S_{N}$ respectively and $c_{1}=\cos^{2}\alpha+(-1)^{N}\sin^{2}\alpha$ and $c_{2}=\sin 2\alpha$. We calculate the maximum value of $|\braket{S_{N}^{\pm}}|$ by considering the following cases:

When $\alpha=0$ or $\pi/2$, $c_{2}=0$. The state becomes a product state $\ket{0}^{\otimes N}$ or $\ket{1}^{\otimes N}$ and the maximum expectation value corresponds to the classical bound for local hidden variable models:

$\displaystyle|\braket{S_{N}^{\pm}}|_{\max}=\left|F_{N}^{\pm}\right|_{\max}$

$\displaystyle=\left|\sum_{k=0}^{N}(-1)^{k(k\pm 1)}{N\choose k}\right|.$

(10)

Evaluating the above sum, we get

$\displaystyle|\braket{S_{N}^{\pm}}|_{\max}=\left|F_{N}^{\pm}\right|_{\max}$

$\displaystyle=\left\{\begin{array}[]{ll}2^{\frac{N+1}{2}},&N\text{ is odd }\\ 2^{\frac{N}{2}},&N\text{ is even }\end{array}\right..$

(13)

This maxima is achieved by measuring all the qubits along the $Z$-axis (i.e, setting all $\theta_{i}^{x(i)}$ to $0$ or $\pi$).

When $\alpha=\pi/4$, we get the maximally entangled N-qubit GHZ state. Let us first consider the case of odd $N$. In this case the maximum expectation value is

	$\displaystyle|\braket{S_{N}^{\pm}}|_{\max}=|G^{\pm}_{N}|_{\max}=$
	$\displaystyle\max\left(\left|\sum_{\boldsymbol{x}}\nu^{\pm}(\boldsymbol{x})\cos\left(\sum_{i}^{N}\phi_{i}^{x(i)}\right)\prod_{i}^{N}\sin\theta_{i}^{x(i)}\right|\right).$		(14)

Note that the function $\nu^{\pm}(\boldsymbol{x})$ can be written as $\nu^{\pm}(\boldsymbol{x})=\sqrt{2}\cos\left(\pm\frac{\pi}{4}\pm w(\boldsymbol{x})\frac{\pi}{2}\right)$ [19]. Hence, if we fix $\phi_{1}^{0}=\phi_{1}^{1}=\pm\pi/4$, $\phi_{i}^{0}=0$ and $\phi_{i}^{1}=\pi/2$ for all $i\neq 1$, then we see that the cosine term in Eq.(Multipartite entanglement vs nonlocality for two families of $N$-qubit states) becomes equal to $\nu^{\pm}(\boldsymbol{x})$ and hence every term inside the summation becomes positive. Now setting all $\theta_{i}^{x(i)}=0$, we get the maxima

$\displaystyle|\braket{S_{N}^{\pm}}|_{\max}=|G^{\pm}_{N}|_{\max}=\sqrt{2}2^{N-1}$

(15)

which is in accordance with the quantum bound [20]. For the case of even $N$, the maximum expectation value is $|\braket{S_{N}^{\pm}}|_{\max}=|F_{N}^{\pm}+G_{N}^{\pm}|_{\max}$. Notice that for the choice of angles for which $|G^{\pm}_{N}|$ is maximum, $|F^{\pm}_{N}|=0$. Since this choice of angles achieves the quantum bound, the maximum for the case of even $N$ is the same as the odd $N$ case.

We can now derive the maxima for the general case of arbitrary value of $\alpha$ using the above two cases. When $F_{N}^{\pm}$ is maximum $G_{N}^{\pm}=0$ for the same choice of angles and vice-versa. Consider the derivative of $\left<S_{N}^{\pm}\right>$ with respect to $\theta_{k}^{l}$ ($l=0$ or 1):

			$\displaystyle\frac{\partial\left<S_{N}^{\pm}\right>}{\partial\theta_{k}^{l}}=-c_{1}\sum_{\boldsymbol{x}(k)=l}\nu^{\pm}(\boldsymbol{x})\sin\theta_{k}^{l}\prod_{i\neq k}^{N}\cos\theta_{i}^{x(i)}$
	$\displaystyle+$		$\displaystyle c_{2}\sum_{\boldsymbol{x}(k)=l}\nu^{\pm}(\boldsymbol{x})\cos\left(\sum_{i}^{N}\phi_{i}^{x(i)}\right)\cos\theta_{k}^{l}\prod_{i\neq k}^{N}\sin\theta_{i}^{x(i)}.$		(16)

We see that $\partial\left<S_{N}^{\pm}\right>/\partial\theta_{k}^{l}=0$ for the choices of $\{(\theta_{i},\phi_{i})\}$ where $F^{\pm}_{N}$ is maximum and for the choices where $G^{\pm}_{N}$ is maximum. It can be easily verified that the second derivative with respect to $\theta_{k}^{l}$ is negative at these points. Since Eq.(Multipartite entanglement vs nonlocality for two families of $N$-qubit states) is the form of the derivative with respect to all $\theta$, we can conclude that the points where are $F^{\pm}_{N}$ and $G^{\pm}_{N}$ are maximum are local maxima for $\left<S_{N}^{\pm}\right>$. As the first term $(c_{1}F^{\pm}_{N})$ is independent of $\phi$, checking the derivative with respect to $\phi$ here is not required. Considering the larger one of these two maxima, we get

	$\displaystyle|\braket{S_{N}^{\pm}}|^{\text{local}}_{\max}=\max\Big{[}$		$\displaystyle\cos^{2}\alpha+(-1)^{N}\sin^{2}\alpha|F_{N}^{\pm}|_{\max},$		(17)
			$\displaystyle\sin 2\alpha|G_{N}^{\pm}|_{\max}\Big{]}.$

We find that the maxima achieved with this approach is in fact the global maxima. The proof for this is provided in Appendix A. Hence the maximum violation in case of odd $N$ is

$$S_{N\max}^{\pm}(\psi_{g})=\left\{\begin{array}[]{ll}2^{\frac{N+1}{2}}\cos(2\alpha),&2^{\frac{N}{2}}|\tan(2\alpha)|\leq 2\\ \sqrt{2}2^{N-1}\sin(2\alpha),&2^{\frac{N}{2}}|\tan(2\alpha)|\geq 2\end{array}\right.$$

(18)

and for even $N$ is

$$S_{N\max}^{\pm}(\psi_{g})=\left\{\begin{array}[]{ll}2^{\frac{N}{2}},&2^{\frac{N}{2}}\sin(2\alpha)\leq\sqrt{2}\\ \sqrt{2}2^{N-1}\sin(2\alpha),&2^{\frac{N}{2}}\sin(2\alpha)\geq\sqrt{2}.\end{array}\right.$$

(19)

For the GGHZ state $\psi_{g}(\alpha)$, the $n$-tangle $\tau(\psi)$ is $\sin^{2}2\alpha$. Restating the above result in terms of $\tau(\psi)$ gives us the result in Eq.(3). We see that for N=3, our results match exactly with the bounds derived in [16, 17].

Maximum violation for MS states: To find the maximum of $S^{\pm}_{N}$ for the maximally sliced states, we define unit vectors $\boldsymbol{b}^{0}$ and $\boldsymbol{b}^{1}$ such that $\boldsymbol{v}_{N-1}^{0}+\boldsymbol{v}_{N-1}^{1}=2\cos\theta\boldsymbol{b}^{0}$ and $\boldsymbol{v}_{N-1}^{0}-\boldsymbol{v}_{N-1}^{1}=2\sin\theta\boldsymbol{b}^{1}$. Thus

	$\displaystyle\boldsymbol{b}^{0}\cdot\boldsymbol{b}^{1}=$		$\displaystyle\cos\theta_{b}^{0}\cos\theta_{b}^{1}$		(20)
			$\displaystyle+\sin\theta_{b}^{0}\sin\theta_{b}^{1}\cos\left(\phi_{b}^{0}-\phi_{b}^{1}\right)=0.$

By isolating the last two qubits, the Svetlichny operator can be written as

	$\displaystyle S^{\pm}_{N}=\sum_{\boldsymbol{\tilde{x}}}\nu^{\pm}(\boldsymbol{\tilde{x}})A(\boldsymbol{\tilde{x}})\left(A_{N-1}^{0}+(-1)^{w(\boldsymbol{\tilde{x}})}A_{N-1}^{1}\right)A_{N}^{0}$
	$\displaystyle+\sum_{\boldsymbol{\tilde{x}}}\nu^{\pm}(\boldsymbol{\tilde{x}})A(\boldsymbol{\tilde{x}})\left((-1)^{w(\boldsymbol{\tilde{x}})}A_{N-1}^{0}-A_{N-1}^{1}\right)A_{N}^{1}$		(21)

where $\boldsymbol{\tilde{x}}$ are now bit strings of size $N-2$. Defining $B_{N-1}^{k}=\boldsymbol{b^{k}}\cdot\boldsymbol{\sigma}$ and using the fact

$$x\cos\theta+y\sin\theta\leq(x^{2}+y^{2})^{\frac{1}{2}},$$

(22)

we get the inequality

	$\displaystyle\left<S^{\pm}_{N}\right>\leq 2\sum_{\boldsymbol{\tilde{x}_{e}}}\left[\left<A(\boldsymbol{\tilde{x}_{e}})B_{N-1}^{0}A_{N}^{0}\right>^{2}+\left<A(\boldsymbol{\tilde{x}_{e}})B_{N-1}^{1}A_{N}^{1}\right>^{2}\right]^{\frac{1}{2}}$
	$\displaystyle+2\sum_{\boldsymbol{\tilde{x}_{o}}}\left[\left<A(\boldsymbol{\tilde{x}_{o}})B_{N-1}^{1}A_{N}^{0}\right>^{2}+\left<A(\boldsymbol{\tilde{x}_{o}})B_{N-1}^{0}A_{N}^{1}\right>^{2}\right]^{\frac{1}{2}}.$		(23)

Where $\{\boldsymbol{\tilde{x}_{e}}\}$ are bit strings of size $N-2$ and even weight, and $\{\boldsymbol{\tilde{x}_{o}}\}$ are bit strings of size $N-2$ and odd weight. Consider the expectation value of the operator $A(\boldsymbol{\tilde{x}})B_{N-1}^{k}A_{N}^{l}$ with respect to $\psi_{s}$ for odd $N$:

	$\displaystyle\left<\psi_{s}|A(\boldsymbol{\tilde{x}})B_{N-1}^{k}A_{N}^{l}|\psi_{s}\right>$		$\displaystyle\leq\sum_{\boldsymbol{\tilde{x}}}\Big{(}\cos^{2}\alpha\prod_{i=1}^{N-2}\cos^{2}\theta_{i}^{\tilde{x}(i)}\left(\cos^{2}\alpha+\sin^{2}\alpha\cos^{2}(\phi_{N}^{l})\right)$		(24)
			$\displaystyle+\prod_{i=1}^{N-2}\sin^{2}\theta_{i}^{\tilde{x}(i)}\left(\cos^{2}\alpha\cos^{2}(\phi(\tilde{x})+\phi_{b}^{k})+\sin^{2}\alpha\cos^{2}(\phi(\tilde{x})+\phi_{b}^{k}+\phi_{N}^{l})\right)\Big{)}^{\frac{1}{2}}$

where $\phi(\tilde{x})=\sum_{i}^{N-2}\phi_{i}^{\tilde{x}(i)}$ and we have used Eq.(22) for the angles $\theta_{N}^{l}$ and $\theta_{b}^{k}$ to obtain the inequality. The right hand side of Eq.(24) can be maximized by setting all $\theta_{i}^{\tilde{x}(i)}=\pi/2$ . Hence we get

$\displaystyle\left<\psi_{s}|A(\boldsymbol{\tilde{x}})B_{N-1}^{k}A_{N}^{l}|\psi_{s}\right>\leq\big{(}\cos^{2}\alpha\cos^{2}(\phi(\tilde{x})+\phi_{b}^{k})+\sin^{2}\alpha\cos^{2}(\phi(\tilde{x})+\phi_{b}^{k}+\theta_{N}^{l})\big{)}^{\frac{1}{2}}.$

(25)

Inserting Eq.(25) into Eq.(Multipartite entanglement vs nonlocality for two families of $N$-qubit states) we obtain

	$\displaystyle\left<\psi_{s}|S_{N}^{\pm}|\psi_{s}\right>\leq 2\sum_{\boldsymbol{\tilde{y}}}\Big{[}\cos^{2}\alpha\cos^{2}(\phi(\tilde{x})+\phi_{b}^{0})+$		$\displaystyle\sin^{2}\alpha\cos^{2}(\phi(\tilde{x})+\phi_{b}^{0}+\theta_{N}^{0})$
			$\displaystyle+\cos^{2}\alpha\cos^{2}(\phi(\tilde{x})+\phi_{b}^{1})+\sin^{2}\alpha\cos^{2}(\phi(\tilde{x})+\phi_{b}^{1}+\theta_{N}^{1})\Big{]}^{\frac{1}{2}}$
	$\displaystyle+2\sum_{\boldsymbol{\tilde{z}}}\Big{[}\cos^{2}\alpha\cos^{2}(\phi(\tilde{x})+\phi_{b}^{1})+$		$\displaystyle\sin^{2}\alpha\cos^{2}(\phi(\tilde{x})+\phi_{b}^{1}+\theta_{N}^{0})$		(26)
			$\displaystyle+\cos^{2}\alpha\cos^{2}(\phi(\tilde{x})+\phi_{b}^{0})+\sin^{2}\alpha\cos^{2}(\phi(\tilde{x})+\phi_{b}^{0}+\theta_{N}^{1})\Big{]}^{\frac{1}{2}}$

In Eq.(Multipartite entanglement vs nonlocality for two families of $N$-qubit states) the right had side can now be maximized by setting $\cos^{2}(\phi(\tilde{x})+\phi_{b}^{k}+\phi_{N}^{l})=1$ $\forall\boldsymbol{\tilde{x}},k,l$. This has to be done by choosing angles such that $\phi_{b}^{0}-\phi_{b}^{1}=\pi/2$ to satisfy the constraint Eq.(20). Setting $\cos^{2}(\phi(\tilde{x})+\phi_{b}^{0})=\sin^{2}(\phi(\tilde{x})+\phi_{b}^{1})$ in Eq.(Multipartite entanglement vs nonlocality for two families of $N$-qubit states) we obtain

	$\displaystyle\left<\psi_{s}|S_{N}^{\pm}|\psi_{s}\right>$	$\displaystyle\leq$	$\displaystyle 2\sum_{\boldsymbol{\tilde{x}}}\left[\cos^{2}\alpha+2\sin^{2}\alpha\right]^{\frac{1}{2}}$		(27)
		$\displaystyle=$	$\displaystyle 2^{N-1}\sqrt{1+\sin^{2}\alpha}$

since there are $2^{N-2}$ such bit strings. A similar analysis follows for even $N$, giving the same bound.

Conclusion: In summary, we have analytically derived the maximum violation of the Svetlichny inequality for the GGHZ and MS states for any number of qubits. This gives us an analytical relationship between multipartite entanglement and nonlocality for these families of states. Our results show that for the GGHZ state, the relationship between the residual entanglement and nonlocality of the state as measured by the Svetlichny inequality is the same for any number of qubits. The range of $n$-tangle values for which the Svetlichny inequality is not violated $(\tau(\psi_{g})<1/2)$ is independent of the number of qubits. For the MS states, we have verified that for even number of qubits $(N>3)$, the analytical relationship between nonlocality and entanglement is the same as the case for three qubits. These findings shed light on the connection between nonlocality and the distribution of entanglement in multipartite systems. The non-violation of the Svetlichny inequality by some entangled $N-$qubit GGHZ states suggests that the hybrid local-nonlocal condition imposed by the Svetlichny inequality maybe too strong. The MS states violate the inequality for all values of parameter $\alpha$ for which $\ket{\psi_{s}}$ is entangled. This contrast between GGHZ and MS states raises practical questions like what the violation of this inequality actually certifies, and whether the states which violate this inequality are useful for particular information processing or communication protocols.