Self-guided method to search maximal Bell violations for unknown quantum states

In recent decades, a great variety of research and applications concerning Bell nonlocality have been developed with the advent of quantum information science. Providing that Bell nonlocality can be revealed by the violation of a family of Bell inequalities, finding maximal Bell violation (MBV) for unknown quantum states becomes an important and inevitable task during Bell experiments. In this paper we introduce a self-guided method to find MBVs for unknown states using a stochastic gradient ascent algorithm (SGA), by parametrizing the corresponding Bell operators. For three investigated systems (two qubit, three qubit, and two qutrit), this method can ascertain the MBV of general two-setting inequalities within 100 iterations. Furthermore, we prove SGA is also feasible when facing more complex Bell scenarios, e.g., $d$-setting $d$-outcome Bell inequality. Moreover, compared to other possible methods, SGA exhibits significant superiority in efficiency, robustness, and versatility.

Figure 1

Searching process for MBVs of (a) two-qubit, (b) three-qubit, and (c) two-qutrit systems when SGA is applied. For each system, three different states, including pure maximally entangled state and partially mixed state represented by three colors [black, orange (gray), green (light gray)], are studied and the solid lines denote the calculated true values of MBVs. The searched values of MBVs with certain iterations are listed in the inset table on each figure, together with the calculated true values.

Figure 2

When utilizing $d$-setting $d$-outcome Bell inequality, the searching process for MBVs when (a) $d=3$ and (b) $d=5$. The red solid lines indicate the true values. The final searched values by SGA as well as the calculated true values are shown in the inset table on each figure.

Figure 3

Searched MBVs of CHSH inequality for a two-qubit singlet with different numbers of photon pairs used per measurement (i.e., different level of Poisson noise). Points denote the average of ten repetitions and the error bar gives the standard deviation. The blue solid line denotes the true value of MBV to be $\approx 2.828$.

Figure 4

With identical amount of resources, CVT and SGA are applied to the singlet state considering experimental errors. Three levels of errors are studied, representing the wave-plate uncertainty to be $1^{\circ }$ (black), $3^{\circ }$ [orange (gray)], and $5^{\circ }$ [green (light gray)], respectively. The lines show the results of CVT while the points show the process of SGA. SGA can outperform CVT giving a higher MBV as shown in the inset table.

Figure 5

Searching process for MBVs of two-qubit CHSH inequality when the target states are not uniform. Three different levels of standard deviation are studied. The red solid lines represent the weighted average over the true MBV values of all the states.

Figure 6

Simulations of SGA with uncalibrated wave plates for a two-qubit singlet. The initial estimate of operators are randomly selected and eventually they can converge at $\approx 2.828$ through different routes. The purple solid line denotes the true value of 2.828 and the searched MBV from five different initial estimates of operators are shown in the inset table.