Device-independent witness for the nonobjectivity of quantum dynamics

Quantum Darwinism offers an explanation for the emergence of classical objective features (those we are used to at macroscopic scales) from quantum properties at the microscopic level. The interaction of a quantum system with its surroundings redundantly proliferates information to many parts of the environment, turning it accessible and objective to different observers. However, given that one cannot probe the quantum system directly, only its environment, how to determine whether an unknown quantum property can be deemed objective? Here we propose a probabilistic framework to analyze this question and show that objectivity implies a Bell-like inequality. Among several other results, we show quantum violations of this inequality, a device-independent proof of the nonobjectivity of quantum correlations. We also implement a photonic experiment where the temporal degree of freedom of photons is the quantum system of interest, while their polarization acts as the environment. Employing a fully black-box approach, we achieve the violation of a Bell-like inequality, thus certifying the nonobjectivity of the underlying quantum dynamics in a fully device-independent framework.

Figure 1

General quantum Darwinism scenario. One central system $A$ interacts with the environment described by $n$ systems $B_1,...,B_n$. As a result of this interaction, part of the information contained in $A$ is transferred to the environment and replicated in each system $B_i$.

Figure 2

(a) and (b) Minimal values possible for $\delta$ as a function of the value of ${\mathrm{CHSH}}_{\delta ,\epsilon }$ corresponding to the observable distribution $p(b_1,b_2|x_1,x_2)$. Results were obtained using the third level of the Navascues-Pironio-Acin hierarchy [46]. Different curves correspond to different values for outcome agreement between observers, given by the constraint ${\sum }_{b_1}p(b_1=b_2|x_1=x^{*},x_2=x^{*})=1-\epsilon$ or equivalently $\langle B_1^0{B}_2^0\rangle =1-2\epsilon$. A change in the behavior for the maximal violation of the CHSH inequality can be seen between values (a) $\langle B_1^0{B}_2^0\rangle \le 1/\sqrt{2}$ and (b) $\langle B_1^0{B}_2^0\rangle \ge 1/\sqrt{2}$, where the former increases with increasing $\epsilon$ while the latter decreases with increasing $\epsilon$. At the same time, the restriction $\delta \ge \epsilon /2$ from Eq. (14) can be observed throughout the graphics, with saturation occurring in all cases for ${\mathrm{CHSH}}_{\delta ,\epsilon }=2$. It also can be seen that a sharp rise in $\delta$ occurs near maximal violation for each $\epsilon$, which leads to numerical instabilities in these regions. For this reason, no curve reaches $\delta =0.5$ and terminal points are different for each curve. (c) Optimal values for the violation of the CHSH inequality with the constraint $\langle B_1^0{B}_2^0\rangle =1-2\epsilon$, with explicit quantum realizations found numerically. The points achieve the self-testing criterion of [47], satisfying $\arcsin \left(\langle B_1^0{B}_2^0\rangle \right)+\arcsin \left(\langle B_1^0{B}_2^1\rangle \right)+\arcsin \left(\langle B_1^1{B}_2^0\rangle \right)-\arcsin \left(\langle B_1^1{B}_2^1\rangle \right)=\pi$, together with the constraints $\langle B_1^0{B}_2^1\rangle =\langle B_1^1{B}_2^0\rangle =-\langle B_1^1{B}_2^1\rangle$. The analytical curve is obtained by combining the equations, resulting in ${\mathrm{CHSH}}_{\delta ,\epsilon }=1-2\epsilon +3sin[\frac{\pi }{3}-\frac{1}{3}\arcsin (1-2\epsilon )]$.

Figure 3

Proof-of-principle experimental setup. (a) Conceptual scheme of the experiment, where the polarization of photons represents the environments while the temporal delay represents the quantum system of interest. (b) Experimental setup. A Sagnac-based polarization entangled photon source generates pairs of degenerate photons at $808\mathrm{nm}$. Inside the Sagnac interferometer, the interaction of the photons generated in the counterclockwise direction interacts with the birefringent plates which couple the polarizations with the temporal degree of freedom. The strength of the interaction is parametrized by $\mathrm{\Delta }$, indicating the overlap between the temporal wave functions of the horizontal and vertical polarizations, and is varied by changing the thickness of the birefringent plates. Finally, the photons are collected and detected by single-photon detectors, where M denotes mirror; PBS, polarizing beam splitter; HWP, half waveplate; QWP, quarter waveplate; DM, dichroic mirror; DHWP, dual-wavelength half waveplate; and APDs, avalanche photodiode detectors.

Figure 4

Experimental data. Optimal values of the experimental CHSH parameter as a function of the constraint $\langle B_1^0|B_2^0\rangle =1-2\epsilon$ for different values of temporal overlap $\mathrm{\Delta }$ between the wave functions of the two polarizations. The dashed lines represent the optimal values calculated from the theoretical model of the experimental state. Moreover, for $\mathrm{\Delta }=1$ and 0.91, there are also reported results obtained using the ab initio approach, which are indicated by squares. When no temporal overlap is present ($\mathrm{\Delta }=0$), we observe $S\approx 1.94<2$, due to an additional uncorrelated noise contribution, i.e., ${\rho }_{\text{dep}}\propto \mathbb{1}/4$, on the generated state. The error bars are calculated assuming Poissonian statistics.