Hardy nonlocality for entangled pairs in a four-particle system

TL;DR

This work extends Hardy-type nonlocality to a four-particle cyclic (ring) entanglement, where each particle couples to two neighbors and non-neighbors remain unentangled. Using a ring of four Toffoli gates and basis-transforming rotations, the authors construct three sets of conditions that yield a larger space of excluded states and correlations than in fully entangled configurations, increasing the number of paradox-contributing pathways. Quantum-circuit simulations validate the predicted state eliminations and correlation patterns, with a maximal nonlocal signal around $P_{ m{success}} \\approx 9\%$ at $ heta \\approx 0.439\pi$ rad; a thorough $ heta$-sweep confirms a near-optimal value. Experiments on IBM Brisbane show sizable deviations from simulations, highlighting practical challenges in hardware implementations, but overall the results demonstrate that cyclic entanglement can realize Hardy-type nonlocality with a rich structure of contradictions, albeit with a smaller success probability due to stricter state-selection rules.

Abstract

Nonlocality can be studied through different approaches, such as Bell's inequalities, and it can be found in numerous quantum states, including GHZ states or graph states. Hardy's paradox, or Hardy-type nonlocality, provides a way to investigate nonlocality for entangled states of particles without using inequalities. Previous studies of Hardy's nonlocality have mostly focused on the fully entangled systems, while other entanglement configurations remain less explored. In this work, the system under investigation consists of four particles arranged in a cyclic entanglement configuration, where each particle forms entangled pairs with two neighbors, while non-neighboring particles remain unentangled. We found that this entanglement structure offers a larger set of conditions that lead to the contradiction with the LHV model, compared to the fully entangled systems. This enhancement can be attributed to the presence of multiple excluded states and correlations, in which the measurement result of a particle only influences the result of its paired partners. We implement quantum circuits compatible with the cyclic entanglement structure, and through simulation, the correlation patterns and the states of interest are identified. We further execute the proposed circuits on IBM Brisbane, a practical backend; however, the results show considerable deviations from the simulation counterparts.

Figures (8)

• Figure 1: (a) Four Mach-Zehnder interferometers overlapping in pairs. Each particle travels along two possible paths labeled $v_i$ and $u_i$. The $u_i$ paths, highlighted in different colors, are trajectories where particles meet and annihilate each other at points $P_{ij}$. Specifically, $P_{12}$ corresponds to the intersection of $u_1$ and $u_2$; similarly, $P_{23}$ corresponds to $u_2$ and $u_3$, $P_{34}$ corresponds to $u_3$ and $u_4$, and $P_{14}$ corresponds to $u_1$ and $u_4$. Besides, fixed beam splitters are highlighted in different colors. (b) The quantum circuit setup of (a), four rotation gates $R_y(\theta)$ are highlighted to match the fixed beam splitters in the optical setup, which divide the particles' path into $u_i$ and $v_i$. Moreover, the line connecting two control qubits of Toffoli gates indicates that these gates entangle two qubits, analogous to the intersection of two particles' paths.
• Figure 2: (a) Quantum circuit that is used to examine the first set of conditions. Four qubits are initially prepared in {$\ket{c_k},\ket{d_k}$}. By applying $Ry(\theta=0.423\pi)$ to each of them, they are all transformed to {$\ket{u_k},\ket{v_k}$}. Through Toffoli gates, they are made to be entangled with their neighbors, and the post-selection process is carried out. In particular, the nine states listed in \ref{['e24']} are made to be undetectable, or have zero measurement probability. (b) Measurement result of states that can be detected from the circuit (2a).
• Figure 3: Quantum circuits and the corresponding measurement results when transforming each qubit's basis. The states highlighted in red are states of interest. (a, b) The quantum circuit and its measurement result for the first condition of the second set. The first qubit's basis is transformed back to {$\ket{c_k},\ket{d_k}$} while the remaining qubits are retained in {$\ket{u_k},\ket{v_k}$}. From the circuit's results, we have three states of interest: $\ket{1001}, \ket{1101}, \ket{1100}$ corresponding to $\ket{d_1v_2v_3u_4}, \ket{d_1u_2v_3u_4}, \ket{d_1v_2v_3u_4}$. (c, d) The quantum circuit and its results when changing the second qubit's basis. The three states of interest in this condition are $\ket{1110}, \ket{1100}, \ket{0110}$. (e, f) The quantum circuit and its results when changing the third qubit's basis. The three states of interest are $\ket{0110}, \ket{0111}, \ket{0011}$. (g, h) The quantum circuit and its results when changing the fourth qubit's basis. The three states of interest are $\ket{1011}, \ket{0011}, \ket{1001}$.
• Figure 4: (a,b) Quantum circuit and corresponding measurement's result for the third set of conditions. The states highlighted in red are states of interest. (a) All four qubits' basis are transformed back to {$\ket{c_k},\ket{d_k}$} by implementing the $R_y(-\theta)$ gates right after the Toffoli gates. (b) There are in total 16 states, matching the number of states described in Eq.\ref{['e34']}.
• Figure 5: $\theta$ sweep test. Using the circuit of the third set of conditions to examine. The range run from 0 to $\pi$ (rad) in increment of $\frac{\pi}{18}$ (rad). The maximum value of $P_{\text{success}}$ is approximately 9.09% with the corresponding $\theta\approx0.439\pi (rad)\approx79(^{\circ})$.