Space Isotropy and Homogeneity Principles Determine the Maximum Nonlocality of Nature

Abstract

One of the fundamental questions in physics concerns the relation between spacetime and quantum entanglement. The spacetime is usually considered as a fixed background physical space, and the quantum entanglement is usually manifested as a ``spooky action at a distance" or the existence of ``nonlocality" in nature. Here, we propose the flat-space isotropy and homogeneity principles as the fundamental criteria for determining the maximum degree of nonlocality of nature. More specifically, we consider abstract and deterministic nonlocal-box models which have stronger correlations than in quantum mechanics, whereas therein instantaneous communication remains impossible. We impose space-symmetry group structures on these models and derive a measure for the degree of space symmetries. Surprisingly, there is a tradeoff or inconsistency between the degree of space symmetries and the degree of nonlocality, where this inconsistency is exactly lifted at the Tsirelson bound, as predicted by quantum physics and also predicted in the experiments. Moreover, we prove this result in the general framework of deterministic nonlocal models and conclude that the probabilistic interpretation of the nonlocal box models is an emergent property of the flat-space symmetries.

Figures (6)

• Figure 1: Schematic correspondence between a physical experiment (performed in the real space) and an NL-box mathematical model. In the Bell-like experiments, Alice and Bob hold one part of a shared system and each party chooses one arbitrary measurement direction ($\mathbf{a}$ and $\mathbf{b}$) to apply the action of a physical apparatus (e.g., magnetic fields) on their systems and detect the state of the binary measurement outcomes $A$ and $B$, respectively. The parties repeat the process many times to estimate the probability distribution $P_{\mathrm{exp}}(A, B|\mathbf{a},\mathbf{b})$ by using the measurement inputs and recorded outputs. Any experiment can be viewed an abstract black box, which is more convenient for mathematical analyzes. As pointed out in one of the EPR axioms EPR (and its reverse in the no-restriction hypothesis Alm1), a mathematical model is complete if every observable quantity in the experiment has a counterpart in the physical model (and viceversa). To make the NL-box model represent a complete physical model, we propose correspondence relations among the measurement inputs/outputs in the real space $(\mathbf{a},\mathbf{b})/(A, B)$ and the inputs/outputs of corresponding the NL-box models $(\mathbf{x},\mathbf{y})/(\alpha, \beta)$, where $A=(-1)^{\alpha}$ and $B=(-1)^{\beta}$. For details, see the main text.
• Figure 2: The probability distributions for the imperfect version of the NL-box models. We show later in appendices that probability for $\mathcal{W}$ to vanish is $\mathcal{Q}(\mathcal{W}=0)=(1/4)[1+3E^{4}]$ and probability for $\mathcal{W}$ to be unity is given by $\mathcal{P}(\mathcal{W}=1)=(1/4)[1+E^{2}]^{2}$. This figure depicts $\mathcal{Q}(\mathcal{W}=0)$ (brown), $\mathcal{P}(\mathcal{W}=1)$ (red) and $\mathcal{Q}(\mathcal{W}=0)+\mathcal{P}(\mathcal{W}=1)$ (green) as function of the correlation $E$, which shows the symmetry condition and nonlocality are mutually consistent in the interval $-\frac{\sqrt{2}}{2}\leqslant E\leqslant\frac{\sqrt{2}}{2}$ (blue interval). It exactly coincides with the Tsirelson bound as the threshold for internal consistency of NL-box models.
• Figure 3: Schematic presentation of the rotation symmetry transformations in the real space. After receiving the qudit to laboratory, internal party (Alice) chooses an arbitrary direction $\mathbf{a}\in\mathcal{R}^{3}$ and obtains one of outcomes $A\in\left\{1,\cdots,d_{A}\right\}$. The external (Bob) can text the isotropy/homogeneity of space by rotation/translation of all elements of each experimental setup, including the source (S), qudit, the polarizer (P), detector (D), and the hole lab by the same amount and asks from Alice about detecting any impact on the value of observable quantities. We represent arbitrary rotation/translation direction of the lab by $\mathbf{n}$ and the value $\theta$.
• Figure 4: Alice and Bob select $n$-tuple of inputs $\mathbf{x}$ and $\mathbf{y}$ from input spaces $\{\mathcal{X}\}$ and $\{\mathcal{Y}\}$, respectively, where $\mathbf{x}=(x_{1},...,x_{n}), \mathbf{y}=(y_{1},...,y_{n}), x_{i},y_{j}\in\{0,1\}$. The parties outputs are represented by ($\alpha,\beta$), where $\alpha,\beta\in\{0,1\}$. These boxes are characterized by input-output correlation $\alpha\oplus\beta=\mathbf{x}\cdot \mathbf{y}=x_{1}y_{1}\oplus...\oplus x_{n}y_{n},\space(\text{mod}\space 2)$. The NL-box models are reference frame independent which means that it is sufficient to apply transformation matrices $F$ on the parties measurement inputs without refer to interior structure of NL-boxes.
• Figure 5: ($a$) A general representation of the probability distribution $\mathcal{Q}(\mathcal{W})$ for correlation function in interval $-1\leqslant E \leqslant1$. ($b$) Similar representation of the probability distributions $\mathcal{P}(\mathcal{W})$ for an arbitrary $1\leqslant E \leqslant1$. Here, we have used different probability symbols $\mathcal{P}$ and $\mathcal{Q}$ to stress that they are derived from independent methods and have different distributions, the former from the symmetry condition and the latter the imperfect NL-box correlation function. ($c$) In order to have consistent NL-box models, it is sufficient that $\mathcal{Q}(\mathcal{W}=0)$ be a subset of $\mathcal{P}(\mathcal{W}=0)$ in which the two incompatible properties $\mathcal{W}=0$ and $\mathcal{W}=1$ cannot be detected simultaneously. In other words, $\mathcal{Q}(\mathcal{W}=0)+\mathcal{P}(\mathcal{W}=1)\leqslant 1$ which is equivalent to $-\sqrt{2}/2\leqslant E\leqslant\sqrt{2}/2$. ($d$) The Tsirelson bound $E=\sqrt{2}/2$ is given as $\mathcal{Q}(\mathcal{W}=0)+\mathcal{P}(\mathcal{W}=1)=1$.