Experimental Demonstration of Input-Output Indefiniteness in a Single Quantum Device

Quantum theory allows information to flow through a single device in a coherent superposition of two opposite directions, resulting into situations where the input-output direction is indefinite. Here we introduce a theoretical method to witness input-output indefiniteness in a single quantum device, and we experimentally demonstrate it by constructing a photonic setup that exhibits input-output indefiniteness with a statistical significance exceeding 69 standard deviations. Our results provide a way to characterize input-output indefiniteness as a resource for quantum information and photonic quantum technologies and enable tabletop simulations of hypothetical scenarios exhibiting quantum indefiniteness in the direction of time.

Figure 1

Input-output indefiniteness in a bidirectional quantum device. A bidirectional device with ports $A$ and $B$ can be traversed in two opposite directions: from $A$ to $B$ (a) or from $B$ to $A$ (b). When these two configurations take place in a quantum superposition (c), the direction of the information flow between $A$ and $B$ becomes indefinite. To generate the superposition, we introduce a control qubit that coherently controls the direction, with basis states $|0⟩$ and $|1⟩$ corresponding to directions $A\to B$ and $B\to A$, respectively. Our setup (d) sets the control qubit in the state $|+{⟩}_{\mathrm{c}}=(|0⟩+|1⟩)/\sqrt{2}$ and witnesses input-output indefiniteness by performing local measurements on the target and control system, with the target initialized in a quantum state ${\rho }_{\mathrm{t}}$.

Figure 2

Experimental setup. A 2.5 mW continuous wave violet laser at 404 nm pumps a type-II cut ppKTP crystal, effectively working as a heralded single photon source when the idler photons trigger a single photon detector (SPD). The single photon’s polarization serves as the target qubit and is initialized with a fiber polarizer controller, a half-wave plate (HWP), and a quarter-wave plate (QWP). Spatial modes of the photon serve as the control qubit, and BS1 is used to coherently control the input-output direction. Measure-and-reprepare operations on the polarization are implemented by two HWPs, two QWPs, and a PBS (dotted rectangle), while measurements on spatial modes are implemented by two liquid crystal variable retardes (LCs) and BS2. A trombone-arm delay line (DL) and a piezoelectric transducer are used to set the path length and the relative phases of the interferometer. [reflection mirror (RM), fiber coupler (FC)].

Figure 3

Experimental data for the optimal witness. The figure shows the outcome probabilities of different measurements on the control and target qubits, with the control initialized in the state $|+⟩$ and the target in one of the states $|0⟩$ (red), $|1⟩$ (cyan), $|+⟩$ (yellow), and $(|0⟩+i|1⟩)/\sqrt{2}$ (green). The measurement outcomes are labeled by numbers 0, 1, 2, 3, corresponding to projections on the states $|0⟩,|1⟩,|+⟩,(|0⟩+i|1⟩)/\sqrt{2}$, respectively. For example, “03” labels the outcome that projects the control qubit onto $|0⟩$ and the target qubit onto $(|0⟩+i|1⟩)/\sqrt{2}$. The bars show the theoretical predictions, while the blue diamonds show the experimental data. We omit the experimental data for outcomes that are irrelevant to the evaluation of the optimal witness. All the data in this figure refer to the setting where the device inside our setup implements a measure-and-reprepare process. Specifically, they refer to the event where the target is measured on the $X$ eigenstate $|+⟩$ and reprepared in the $Z$ eigenstate $|0⟩$. The experimental data for the remaining settings are shown in the Supplemental Material [48].