Towards an Experimental Device-Independent Verification of Indefinite Causal Order

Abstract

In classical physics, events follow a definite causal order: the past influences the future, but not the reverse. Quantum theory, however, permits superpositions of causal orders -- so-called indefinite causal orders -- which can provide operational advantages over classical scenarios. Verifying such phenomena has sparked significant interest, much like earlier efforts devoted to refuting local realism and confirming quantum entanglement. To date, demonstrations of indefinite causal order have all been based a process called the quantum switch and have relied on device-dependent or semi-device-independent protocols. Achieving a device-independent verification of indefinite causal order would imply that nature allows for correlations that do not respect causality, independent of any experimental assumptions or underlying theoretical description of the experiment. To this end, a recent theoretical development introduced a Bell-like inequality that allows for fully device-independent verification of indefinite causal order in a quantum switch. Here we implement this verification by experimentally violating this inequality. In particular, we measure a value of $1.8328 \pm 0.0045$, which is 18 standard deviations above the Definite Causal Order Bound of $1.75$. Our work presents the first implementation of a device-independent protocol to verify indefinite causal order, albeit in the presence of experimental loopholes. This represents an important step towards the device-independent verification of an indefinite causal order, and provides a context in which to identify loopholes specifically related to the verification of indefinite causal order.

Figures (4)

• Figure 1: Causal arrangement for the inequality test.a) Spacetime arrangement of the four parties, Alice 1 (A1), Alice 2 (A2), Bob (B), and Charlie (C), who will attempt to violate the inequality. Alice 1 and Alice 2 act before Charlie, while Bob (B) is space-like separated all other participants. b) Switch-based protocol to violate the inequality. A Bell state is shared between the control qubit of the quantum switch and Bob. Alice 1 and Alice 2 perform measurements on a target qubit in the quantum switch. Notice that our notation, which was introduced in van2023device, for Alice 1 and 2's settings and outcomes is not the standard Bell test notation. In particular, $a_{i}$ represents the Alices measurement outcomes as usual. However, they always measure in the computational basis. Thus no notation is used for their measurement settings. Instead, $x_{i}$ denote which state they prepare the post-measurement system in before it leaves their laboratory. Charlie and Bob's notation is more standard: Charlie measures the control qubit in basis determined by $z$ and receives outcome $c$, and Bob measurement basis is given by $y$ with outcome $b$.
• Figure 2: Experimental schematic: A type-0 spontaneous parametric downconversion (SPDC) source generates polarization entangled photon pairs in the $\ket{\phi^+}$ state. a) Photon 1 of the pair is sent to Bob’s measurement stage ($B$), represented by the orange-shaded area. Therein it is measured in a standard polarization measurement stage consisting of a quarter-waveplate (QWP), half-waveplate (HWP) and a polarizing beamsplitter (PBS). Its detection signal initiates the electrical trigger signal that allows the ultra-fast optical router (UFOR) to switch at the correct time.b) Photon 2 passes an optical circulator before entering the green area, where the polarization qubit is deterministically converted into a time-bin qubit using an imbalanced Mach-Zhender-like interferometer opening on a PBS and closing on an UFOR. For additional. To further refine the polarization, photon 2 passes through a PBS in the target-preparation stage. c) The time-bin qubit then serves as the control for the time-bin quantum switch indicated by the pink shaded region. Here Alice 1 and Alice 2 ($A_1, \ A_2$) perform measurement on the target qubit $a_1, a_2$ and re-prepare the state in the settings $x_1, x_2$ in the polarization DOF using a linear polariser and a HWP. After passing through the quantum switch, the photon travels in the opposite direction, ensuring that both time bins arrive simultaneously at the PBS. d)Depending on their spatial path and polarization, the photons are guided from here to one of Charlie’s measurement stages, $C_0$ or $C_1$ (shaded blue region). Charlie's two polarization measurement devices allow him to implement a complete set of measurements on the control and target qubits. * Two HWPs in $B$ allow the preparation of any Bell state as the input state, enabling additional measurements to test the effect of noise on the VBC inequality.
• Figure 3: Experimental violation of a local causal inequality.a) Density matrix of the two photon input state, as it is generated by our SPDC-Source with a purity of $0.98036 \pm 0.00034$. b) Maximally mixed input state with a purity of $0.25777 \pm 0.00015$ created by generating depolarising noise from experimental data via eq.\ref{['eq::depol']}. c) Effect of depolarising noise, applied on the two photon input state, on the values of the individual term probabilities $p_1, p_2, p_3$. d) Experimental violation of the VBC inequality, and simulation of the maximum possible VBC violation as a function of the purity of the input Bell states created by applying different amounts of depolarising noise. Here, the defined causal order (DCO) bound and the indefinite causal order (ICO) bound are marked in color, with the latter representing the maximum achievable value.
• Figure 4: The effect of dephasing noise on the probabilities $p_1$, $p_2$ and $p_3$ of the VBC-inequality Experimental data modelled with eq. \ref{['eq::depahse']} and simulated curves with a Switch process fidelity of $0.9816\pm0.0069$.