Photonic Simulation of Beyond-Quantum Nonlocal Correlations (e.g. Popescu-Rohrlich Box) with Non-Signaling Quantum Resources

TL;DR

This work investigates how no-signaling beyond-quantum correlations (PR-box) can be realized within quantum circuits while preserving non-signaling. It introduces a four-qubit photonic implementation that uses restricted gates and intrinsic randomness to dynamically couple distinct physical systems, enabling PR-like correlations without violating no-signaling. Experimentally, CHSH values up to $S \approx 3.91$ are observed with entangled resources, and PR behavior is demonstrated across multiple bases; the protocol also works with a classical oracle for PR in the computational basis. The approach extends to multipartite scenarios and provides a versatile platform for studying foundational aspects and potential applications in computation and secure communication, including links to the one-time-pad model of PR correlations and broader nonlocality-resources in quantum networks.

Abstract

Bell nonlocality exemplifies the most profound departure of quantum theory from classical realism. Yet, the extent of nonlocality in quantum theory is intrinsically bounded, falling short of the correlations permitted by the relativistic causality (the no-signaling) principle. A paradigmatic example is the Popescu-Rohrlich correlation: two distant parties sharing arbitrary entanglement cannot achieve this correlation, though it can be simulated with classical communication between them. Here we show how such post-quantum correlations can instead be simulated using intrinsically non-signaling physical resources, and implement the proposed scheme using a quantum circuit on a four-qubit photonic platform. Unlike the conventional approaches, our method exploits dynamical correlations between distinct physical systems, with intrinsic randomness suppressing any signaling capacity. This enables the realization of post-quantum correlations both with and without entanglement. We also analyze how the simulation scheme extends to beyond quantum nonlocal correlations in multipartite systems. Our experimental demonstration using a photonic system establishes a versatile framework for exploring post-quantum correlations in both foundational settings and as a resource for computation and security applications.

Figures (7)

• Figure 1:
• Figure 2: [Left]: Four-qubit quantum oracle (blue shaded box) -- qubits $A,A'$ correspond to Alice’s inputs and $B,B'$ to Bob’s. When the primed qubits are initialized in $\ket{0}$ and $A$ and $B$ are respectively prepared in $\ket{x}$ and $\ket{y}$ with $x,y\in \{0,1\}$, the output state of the primed systems after the oracle action becomes $\ket{\phi^{xy}}$ (see Eq. \ref{['qpr']}). [Right]: Corresponding classical circuit, obtained by replacing qubits and quantum gates with their classical analogues. In this case, after the action of the classical oracle (green shaded box), the state of the primed systems becomes $\rho^{xy}_{B'A'}=\tfrac{1}{2}[(x y)_{B'}(0)_{A'}+(x y\oplus1)_{B'}(1)_{A'}]$.
• Figure 3: (a) Experimentally obtained CHSH values $\mathcal{S}$ when Alice and Bob perform local measurements on the entangled state $\ket{\phi^{xy}}$ in the bases listed along the x-axis. Computational basis measurement $\{\ket{H},\ket{V}\}$ and circular basis measurement $\{\ket{L},\ket{R}\}$ approach the algebraic maximum $\mathcal{S}=4$ (PR correlations), whereas diagonal basis $\{\ket{A},\ket{D}\}$ attains at most $\mathcal{S}=2$. (b) CHSH values for local measurement bases $\{\ket{\psi}:=\cos(\theta/2)\ket{0}+e^{i\phi}\sin(\theta/2)\ket{1},\ket{\psi^\perp}\}$ parameterized by the polar angle $\theta$, with $\phi=0$ within the Bloch sphere. Experimentally obtained CHSH values (red dots) for the entangled state $\ket{\phi^{xy}}$ agree with the theoretical expectations (solid violet curve) within the experimental errors (teal bars). (c) Numerical simulation of CHSH values for the unentangled mixed state $\rho^{xy}$. The measurement bases are parameterized by $\theta$ as above; the final bar confirms that, unlike the entangled state $\ket{\phi^{xy}}$, the CHSH value $\mathcal{S}$ for $\rho^{xy}$ is independent of the azimuthal parameter $\phi$.
• Figure 4: Experimental setup. (a) State Preparation: A polarization-entangled state $\ket{\Psi^{+}}_{B'A'} = \left(\ket{H}_{B'}\ket{V}_{A'} + \ket{V}_{B'}\ket{H}_{A'}\right)/\footnotesize{\sqrt{2}}$ is prepared through SPDC process using a type-II PPKTP crystal in a Sagnac interferometer configuration. The photon pairs are then coupled to two single mode fibers and two polarization fiber controllers are used to maintain entanglement during the fiber transmission. (b) The oracle: The state $\ket{\Psi^{+}}_{B'A'}$ is converted to $\ket{\Phi^{+}}_{B'A'}$ using a half-wave plate (HWP) as $\sigma_{x}$ operator in the path of one of the photons. A photon from the polarization-entangled pair is then sent through one of the four distinct paths represented by $\ket{x}_{A}\ket{y}_{B}$, with $x,y \in \{0,1\}$. Followed by the path selection, a Toffoli gate is applied to the path-path-polarization encoded state with the path d.o.f. in $A$ and $B$ as the controls and polarization d.o.f. in $B'$ as the target, resulting in the state $\ket{x}_{A}\ket{y}_{B}\ket{\phi^{xy}}_{B'A'}$. (c) Measurement unit: The entangled state $\ket{\phi^{xy}}_{B'A'}$ is measured in different polarization basis using QWP, HWP and PBS combinations, and recording the coincidences between the photon in one of the four paths and the other photon.
• Figure S1: PR bases for quantum state $\ket{\phi^{xy}}$ on Bloch sphere.