Robust violation of a multipartite Bell inequality from the perspective of a single-system game ¹¹1Mod. Phys. Lett. A 37, 2250082(2022)

Quantum entanglement Schrödinger (1935); Amico et al. (2008); Horodecki et al. (2009); Hill and Wootters (1997); Wootters (1998); Torun and Yildiz (2019), brought about by the superpositionprinciple, puzzled many physicists in the early days of quantum theory. In the original work for Einstein-Podolsky-Rosen (EPR) paradox in entangled states Einstein et al. (1935), Einstein and his collaborators proposed that quantum mechanics provides probabilistic results because of its incompleteness. In 1964, Bell proposed an inequality Bell (1964) to solve the EPR paradox under the assumptions of local reality and hidden variable theory. Such inequality and its generalized versions Das et al. (2017, 2018) revealed nonlocality Brunner et al. (2014); Scarani (2019); Gisin and Bechmann-Pasquinucci (1998); Ren et al. (2019); Reid et al. (2009); Mermin (1990); Ardehali (1992) in entangled states. The Clauser-Horne-Shimony-Holt inequality Clauser et al. (1969) is the most widely studied Bell inequality for two-qubit systems, which is written as

$$\langle I_{CHSH}^{2}\rangle=\langle A_{0}B_{0}\rangle+\langle A_{0}B_{1}\rangle+\langle A_{1}B_{0}\rangle-\langle A_{1}B_{1}\rangle\leq 2,$$

(1)

with $A_{a},B_{b}$ ($a,b=0,1$) being measurement settings. It can be violated by all the two-qubit entangled pure states.

Researchers have tried to understand the nonlocality from the perspective of game theory van Dam (2000); Ji et al. (2008); Henaut et al. (2018). The author of Ref. van Dam (2000) setted up the so-called CHSH game to show the advantage of quantum strategies. There are two players, Alice and Bob, in the game who cannot communicate with each other. They share a two-qubit system and have measurement operators $A_{a}$ and $B_{b}$ respectively. Here, $a$ and $b$ $=0,1$ are two input values. Let $x$ and $y$ $=0,1$ represent the outcomes of Alice and Bob. When $x\oplus y=ab$, the players win the game, with $\oplus$ denoting modulo $2$ addition. It is directly to find the linear relationship between their success probability

$$\frac{1}{4}\sum_{a,b}\mbox{Prob}\left(x\oplus y=ab|a,b\right)$$

(2)

and the expected value of $\langle I_{CHSH}^{2}\rangle$.

Recently, Henaut et al. Henaut et al. (2018) introduced a single-player CHSH* game with two inputs $a$ and $b$. Any strategy in the CHSH* game can be mapped to a strategy in the CHSH game with two-qubit maximally entangled states. Without loss of generality, let Alice and Bob share one of the Bell states, $|\psi_{+}\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle+|11\rangle\right)$. The player of the CHSH* game, Carol, has a qubit in state $|+\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle+|1\rangle\right)$. Alice and Bob apply arbitrary local unitary transformations $\mathcal{A}_{a}^{T}$ and $\mathcal{B}_{b}$ to their qubits, and then measure the Pauli operator $\sigma_{x}$ on their qubits respectively. This is equivalent to the local measurement of ${A}_{a}{B}_{b}=\mathcal{A}_{a}^{\ast}\sigma_{x}\mathcal{A}_{a}^{T}\otimes\mathcal{B}_{b}^{\dagger}\sigma_{x}\mathcal{B}_{b}$ on $|\psi_{+}\rangle$. Carol applies $\mathcal{A}_{a}$ and $\mathcal{B}_{b}$ on the state $|+\rangle$ and measures $\sigma_{x}$ on her qubit. Similarly, this represents the measurement of $C_{ab}=\mathcal{A}_{a}^{\dagger}\mathcal{B}_{b}^{\dagger}\sigma_{x}\mathcal{B}_{b}\mathcal{A}_{a}$ on $|+\rangle$. She wins the game when her outcome $c=ab\ (\!\!\!\mod 2)$. The success probabilities of the two games are equal; i.e.,

$$\frac{1}{4}\sum_{a,b}\mbox{Prob}\left(x\oplus y=ab|a,b\right)=\frac{1}{4}\sum_{a,b}\mbox{Prob}\left(c=ab|a,b\right),$$

(3)

which arises from the expected values

$$\langle I_{CHSH}^{2}\rangle=\langle I_{CHSH}^{1}\rangle,$$

(4)

with $\langle I_{CHSH}^{2}\rangle=\sum_{a,b}(-1)^{ab}\langle\psi_{+}|\mathcal{A}_{a}^{\ast}\sigma_{x}\mathcal{A}_{a}^{T}\otimes\mathcal{B}_{b}^{\dagger}\sigma_{x}\mathcal{B}_{b}|\psi_{+}\rangle$ and $\langle I_{CHSH}^{1}\rangle=\sum_{a,b}(-1)^{ab}\langle+|\mathcal{A}_{a}^{\dagger}\mathcal{B}_{b}^{\dagger}\sigma_{x}\mathcal{B}_{b}\mathcal{A}_{a}|+\rangle$.

On the other hand, to distinguish $N$-qubit Greenberger-Horne-Zeilinger (GHZ) states, Fan et al. presented a generalized CHSH inequality in their very recent work Fan et al. (2021) . It is expressed as

$$\langle I_{CHSH}^{N}\rangle=\langle\mathbb{A}_{0}\mathbb{B}_{0}\rangle+\langle\mathbb{A}_{0}\mathbb{B}_{1}\rangle+\langle\mathbb{A}_{1}\mathbb{B}_{0}\rangle-\langle\mathbb{A}_{1}\mathbb{B}_{1}\rangle\leq 2.$$

(5)

$\mathbb{A}_{0}$ and $\mathbb{A}_{1}$ denote the tensor products of local observables for the first $N-1$ qubit. $\mathbb{B}_{0}$ and $\mathbb{B}_{1}$ represent two observables of the $N$th qubit. The $N$-qubit GHZ states can be identified by the maximal violations of the inequality. Besides, they found an interesting quantum phenomenon of robust violations of the generalized CHSH inequality, in which the maximal violation can be robust under some specific noises. Such phenomenon originates from the degeneracy of the largest eigenvalue of Bell-function $I_{CHSH}^{N}$.

In this work, we show the mapping between the CHSH* and CHSH games proposed by Henaut et al. Henaut et al. (2018) can be extended to the generalized CHSH inequality for $N$-qubit case. The relations among the inequalities (and the CHSH* game) provide an explanation for the robust violations and give the degeneracy of $I_{CHSH}^{N}$. Namely, we map the $N$-qubit Bell function $I_{CHSH}^{N}$ to $I_{CHSH}^{2}$ for the two-qubit case, and consequently to $I_{CHSH}^{1}$ for the one-qubit system. For the $N$-qubit GHZ states ($|\psi_{+}\rangle$ and $|+\rangle$for $N=2$ and $1$), the expected values satisfy $|\langle I^{1}_{CHSH}\rangle|=|\langle I^{2}_{CHSH}\rangle|\geq|\langle I^{N}_{CHSH}\rangle|$ when $|\langle I^{N}_{CHSH}\rangle|\geq 2$. The equality holds when the generalized CHSH inequality achieves the Tsirelson’s bound, i.e.

$$\langle I^{1}_{CHSH}\rangle=\langle I^{2}_{CHSH}\rangle=\langle I^{N}_{CHSH}\rangle=\pm 2\sqrt{2}.$$

(6)

This equation is invariable under specific local unitary transformations on the two-qubit and $N$-qubit Bell-functions, which corresponds to the identity operation on the single-qubit system. By using these invariance, we exactly demonstrate the degeneracy of $\langle I^{N}_{CHSH}\rangle$, which causes the robust violations.

We first introduce the mappings from $\langle I_{CHSH}^{N}\rangle$ to $\langle I_{CHSH}^{2}\rangle$, where the measured quantum states are the GHZ states

$$|G\rangle=\frac{1}{\sqrt{2}}(|00...0\rangle+|11...1\rangle)$$

(7)

and the Bell state $|\psi_{+}\rangle$. Unless explicitly stated otherwise, all the expected values in this paper are of the states $|G\rangle$, $|\psi_{+}\rangle$ and $|+\rangle$ corresponding to $N$-, two- and one-qubit system. The local measurement operators in $N$-qubit system can be written as

$$X_{j}=\vec{n}_{j}\cdot\vec{\sigma},\quad X_{j}^{\prime}=\vec{n}_{j}^{\prime}\cdot\vec{\sigma}\quad(j=1,2\cdots N),$$

(8)

where $\vec{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the vector of Pauli matrices, $\vec{n}_{j}=(\sin\alpha_{j}\cos\varphi_{j},\sin\alpha_{j}\sin\varphi_{j},\cos\alpha_{j})$ and $\vec{n}_{j}^{\prime}=(\sin\alpha_{j}^{\prime}\cos\varphi_{j}^{\prime},\sin\alpha_{j}^{\prime}\sin\varphi_{j}^{\prime},\cos\alpha_{j}^{\prime})$ denote the measurement direction of the $j$th qubit. Fan et al. Fan et al. (2021) defined the measurement operators of Alice and Bob in (5) as

$$\mathbb{A}_{0}=\bigotimes_{j=1}^{N-1}X_{j},\mathbb{A}_{1}=\bigotimes_{j=1}^{N-1}X_{j}^{\prime},\mathbb{B}_{0}=X_{N},\mathbb{B}_{1}=X_{N}^{\prime}.$$

(9)

One can derive the first term of $\langle I_{CHSH}^{N}\rangle$ as

$\displaystyle\begin{split}\langle\mathbb{A}_{0}\otimes\mathbb{B}_{0}\rangle=\frac{1}{2}[1+(-1)^{N}]\prod_{j=1}^{N}\cos\alpha_{j}+\cos(\sum_{j=1}^{N}\varphi_{j})\prod_{j=1}^{N}\sin\alpha_{j}.\end{split}$

(10)

Obviously, only the terms $\prod_{j=1}^{N}\cos\alpha_{j}$, $\cos(\sum_{j=1}^{N}\varphi_{j})$ and $\prod_{j=1}^{N}\sin\alpha_{j}$ are contributed by the projections of the measurement operators in the subspace of $\{|00...0\rangle,|11...1\rangle\}$.

For brevity, we ignore the cases: (i) ${{(\prod_{j=1}^{N-1}\cos\alpha_{j})^{2}+(\prod_{j=1}^{N-1}\sin\alpha_{j})^{2}}}=0$; (ii) ${{(\prod_{j=1}^{N-1}\cos\alpha_{j}^{\prime})^{2}+(\prod_{j=1}^{N-1}\sin\alpha_{j}^{\prime})^{2}}}=0$; (iii) $\sin\alpha_{N}=0$ when $N$ is odd; (iv) $\sin\alpha_{N}^{\prime}=0$ when $N$ is odd. These cases compose a zero measure set, and do not violate the generalized CHSH inequality (5). To connect the expected value to the two-qubit system in the state $|\psi_{+}\rangle$, we define the following two mappings. The first one uniquely leads to a single-qubit observable, for a given $(N-1)$-qubit operator in (9), as

$$\Gamma[\mathbb{A}_{a}]:=\vec{\mathrm{n}}_{a}\cdot\vec{\sigma}.$$

(11)

Take $\Gamma[\mathbb{A}_{0}]:=\vec{\mathrm{n}}_{0}\cdot\vec{\sigma}$ as an example. The Bloch vector $\vec{\mathrm{n}}_{0}=(\sin\gamma\cos\beta,\sin\gamma\sin\beta,\cos\gamma)$, with $\sin\gamma={\prod_{j=1}^{N-1}\sin\alpha_{j}}/{\sqrt{(\prod_{j=1}^{N-1}\cos\alpha_{j})^{2}+(\prod_{j=1}^{N-1}\sin\alpha_{j})^{2}}}$ and $\beta=\sum_{j=1}^{N-1}\varphi_{j}$. The other vector $\vec{\mathrm{n}}_{1}$ is in the similar form. The second mapping projects the single-qubit observable in (9) onto the equator of Bloch sphere; i.e.,

$$\Theta[\mathbb{B}_{b}]:=\vec{r}_{b}\cdot\vec{\sigma},$$

(12)

with $\vec{r}_{0}=(\cos\varphi_{N},\sin\varphi_{N},0)$ and $\vec{r}_{1}=(\cos\varphi_{N}^{\prime},\sin\varphi_{N}^{\prime},0)$. When $N$ is even, one can choose

$$A_{0}=\Gamma\left[\mathbb{A}_{0}\right],A_{1}=\Gamma\left[\mathbb{A}_{1}\right],B_{0}=\mathbb{B}_{0},B_{1}=\mathbb{B}_{1},$$

(13)

while

$$A_{0}=\Gamma\left[\mathbb{A}_{0}\right],A_{1}=\Gamma\left[\mathbb{A}_{1}\right],B_{0}=\Theta[\mathbb{B}_{0}],{B}_{1}=\Theta[\mathbb{B}_{1}]$$

(14)

when $N$ is odd, and obtain the two-qubit Bell function

$$\langle I_{CHSH}^{2}\rangle=\langle A_{0}B_{0}\rangle+\langle A_{0}B_{1}\rangle+\langle A_{1}B_{0}\rangle-\langle A_{1}B_{1}\rangle,$$

(15)

corresponding to $\langle I_{CHSH}^{N}\rangle$ in (5).

The above measurement operators $A_{a}$ and $B_{b}$ can always be expressed as ${A}_{a}=\mathcal{A}_{a}^{\ast}\sigma_{x}\mathcal{A}_{a}^{T}$ and ${B}_{b}=\mathcal{B}_{b}^{\dagger}\sigma_{x}\mathcal{B}_{b}$, with $\mathcal{A}_{a}^{T}$ and $\mathcal{B}_{b}$ being two local unitary transformations. That is, any measurement of the Bell function $\langle I_{CHSH}^{N}\rangle$ in (5) can be mapped to a strategy in the CHSH game, and consequently to the CHSH* game according to the relation $C_{ab}=\mathcal{A}_{a}^{\dagger}\mathcal{B}_{b}^{\dagger}\sigma_{x}\mathcal{B}_{b}\mathcal{A}_{a}$ given by Henaut et al. Henaut et al. (2018). The two maps are constructed based on the fact that, the expected value of $\mathbb{A}_{a}\mathbb{B}_{b}$ on the state $|G\rangle$ is equivalent to the one of $\bar{A}_{a}\bar{B}_{b}$ on the state $|\psi_{+}\rangle$. The two single-qubit operators, $\bar{A}_{a}$ and $\bar{B}_{b}$, are in the form of (8), but the Bloch vector of $\bar{A}_{a}$ is in the unit ball in general. Therfore, we introduce a normalization coeffieient in (11). Under these mappings, we have the two following theorems.

There are two corollaries of Theorem 1 as follows. (i) The maximal violations of the GHZ state cannot exceed the Tsirelson’s bound $\pm 2\sqrt{2}$, which has been found in Ref. Fan et al. (2021). (ii) When the expected value for the GHZ state $\langle I^{N}_{CHSH}\rangle=\pm 2\sqrt{2}$ , the corresponding $\langle I^{2}_{CHSH}\rangle=\langle I^{N}_{CHSH}\rangle$.