Investigation of quantum-entanglement simulation in random-variable theories augmented by either classical communication or nonlocal effects

Bell's theorem states that quantum mechanics is not a locally causal theory. This state is often interpreted as nonlocality in quantum mechanics. Toner and Bacon [Phys. Rev. Lett. 91, 187904 (2003)] have shown that a shared random-variable theory augmented by one bit of classical communication exactly simulates the Bell correlation in a singlet state. In this paper, we show that in Toner and Bacon protocol, one of the parties (Bob) can deduce another party's (Alice) measurement outputs, if she only informs Bob of one of her own outputs. Afterwards, we suggest a nonlocal version of Toner and Bacon protocol wherein classical communications is replaced by nonlocal effects, so that Alice's measurements cause instantaneous effects on Bob's outputs. In the nonlocal version of Toner and Bacon's protocol, we get the same result again. We also demonstrate that the same approach is applicable to Svozil's protocol.

Figure 1

The random unit vectors ${\widehat{\lambda }}_1$ and ${\widehat{\lambda }}_2$ divide the Poincare sphere into four parts. If Alice's measurement setting $\widehat{a}$ lies in the shaded region, she sends $c=-1$ and if her measurement axis lies in the unshaded region, she sends $c=+1$ to Bob. Toner and Bacon deduce that Bob obtains no information about Alice's output from the communication.

Figure 2

(a) Subsets of shared random variables lie in the $xy$ plan. The blue zone corresponds to random variables $({\widehat{\lambda }}_{1,l}^{xy},{\widehat{\lambda }}_{2,l+1}^{xy})$ with $c_l^{xy}=-1$ (Fig. 1). According to the $c$ definition, Alice's measurement setting $\widehat{a}$ must lie in a plane with unit vector ${\widehat{\lambda }}_{1,l}^{xy}$ (blue zone in the $l\mathrm{th}$ round of experiment). The azimuthal angle $\left(\widehat{a}\right)$ is equal to $l\delta \pm \pi /2$. (b) Subsets of shared random variables lie in the $xz$ plane. The red zone corresponds to one set of random variables ${\widehat{\lambda }}_{1,p}^{xz},{\widehat{\lambda }}_{2,p+1}^{xz}$ with $c_p^{xz}=-1$ (Fig. 1). Alice's measurement setting $\widehat{a}$ must lie in a plane with unit vector ${\widehat{\lambda }}_{1,p}^{xz}$ (red zone in the $p\mathrm{th}$ round of experiment). In fact, the thin strips sweep the surface of the unit sphere so that $c=-1$ for $l\mathrm{th}$ and $p\mathrm{th}$ strips and $c=+1$ elsewhere. (c) These two strips cross each other at two small areas and consequently, Alice's measurement setting $\widehat{a}$ is in the same (or in the opposite) direction of the unit vector which connects the origin of the Poincare sphere to the cross points.

Figure 3

Svozil's protocol for the case of $\omega =\frac{\pi }{2}$. Here ${\widehat{\lambda }}_k$ is given by rotating ${\widehat{\lambda }}_{k-1}(\forall k)$ clockwise around the origin by $\delta$, and ${\widehat{\lambda }}_{k,\pm }={\widehat{\lambda }}_k\pm {\widehat{\mathrm{\Delta }}}_k$. According to the $c_k=\mathrm{sgn}(\widehat{a}·{\widehat{\lambda }}_k)\mathrm{sgn}(\widehat{a}·{\widehat{\mathrm{\Delta }}}_k)$ definition, in the $l+1\mathrm{th}$ round of the protocol the sign of the communicated bits change and consequently $\widehat{a}$ becomes parallel (or antiparallel) to ${\widehat{\lambda }}_l$.

Figure 4

NTB protocol as an elementary resource for simulating physical systems, where parties shared two sets of independent random variables ${\widehat{\lambda }}_1$ and ${\widehat{\lambda }}_2$; Alice and Bob inputs are $\widehat{a}$ and $\widehat{b}$, respectively. NTB protocol outputs are $\alpha =-\mathrm{sgn}(\widehat{a}.{\widehat{\lambda }}_1)$ and $\beta =\mathrm{sgn}[\widehat{b}.({\widehat{\lambda }}_1+c{\widehat{\lambda }}_2)]$, where $c=\mathrm{sgn}(\widehat{a}·{\widehat{\lambda }}_1)\mathrm{sgn}(\widehat{a}·{\widehat{\lambda }}_2)$.

Figure 5

(a) Subsets of shared random variables lie in the $xy$ plan. The blue zone defines a plane with unit vectors ${\widehat{\lambda }}_{2,l}^{xy}$. According to the definition of $c$, Alice's measurement setting $\widehat{a}$ must lie in the blue zone. (b) Subsets of shared random variables lie in the $xz$ plane. The red zone defines a plane with unit vectors ${\widehat{\lambda }}_{2,p}^{xz}$, so that Alice's measurement setting $\widehat{a}$ must lie in the red zone. In fact, these thin strips sweep the surface of the unit sphere.