Direct Measurement of a Nonlocal Entangled Quantum State

Entanglement and the wave function description are two of the core concepts that make quantum mechanics such a unique theory. A method to directly measure the wave function, using weak values, was demonstrated by Lundeen et al. [Nature 474, 188 (2011)]. However, it is not applicable to a scenario of two disjoint systems, where nonlocal entanglement can be a crucial element, since that requires obtaining weak values of nonlocal observables. Here, for the first time, we propose a method to directly measure a nonlocal wave function of a bipartite system, using modular values. The method is experimentally implemented for a photon pair in a hyperentangled state, i.e., entangled both in polarization and momentum degrees of freedom.

Figure 1

Measurement of a pre- and postselected system, using qubit meters. (a) A logic circuit diagram shows a single-qubit meter prepared in a state $|{\uparrow }_x⟩=(|0⟩+|1⟩)/\sqrt{2}$ and a single system prepared in a state $|\psi ⟩$. Then, a controlled gate modifies the state of the meter, depending on the state of the system and, finally, if the system is found in a state $|\varphi ⟩$, the meter is readout. (b) Experimental scheme to realize the circuit in (a) using the path of a photon as a meter and its polarization as the system [half-wave plate (HWP), quarter-wave plate (QWP), polarized beam splitter (PBS)]. (c) The logic circuit diagram for the case of a bipartite system (two spatiality separated qubits) is composed from similar components to the single-qubit case shown in (a) and (b). The main difference is that the system can be initially entangled and thus the meter might have to be entangled as well.

Figure 2

Experimental setup. (Inset) A $\beta \text{−}{\mathrm{BaB}}_2{\mathrm{O}}_4$ (BBO) crystal pumped by a vertically polarized ultraviolet laser. A spherical mirror reflects the photon pairs, which are emitted due to a degenerate spontaneous parametric down-conversion, and the pump beam to a second pass through the crystal, after passing a QWP (or $\lambda /4$) with its optical axis orienting at 45°. Choosing four points in the entanglement ring yields a photon pair entangled both in path and polarization, which enters the main setup shown in the main panel. The up path (green) stands for the first photon (labeled $A$), while the bottom path (red) represents the second one (labeled $B$); they are vertically separated around 5 mm. For each photon, they have two spatially separated path modes (left and right with around one meter from each other) and two polarization modes (horizontal and vertical); the former is taken as the meter and the latter as the system. For preparing the initial meter state, we use two neutral optical attenuators (ATTs) to precisely reduce the relative intensity of relevant modes. Half- and quarter-wave plates are used to manipulate the polarization of each path independently and glass compensators (COs) are used to offset the phase difference. The two paths of each photon are combined in one unpolarized beam splitter (BS). Glass plates (GPs) are used to introduce a controllable phase so different path states can be measured. The PBS postselects the final polarization state. After spectrum filters, the photons are collected by four single-mode fibers and guided to four avalanche single-photon detectors (D-1,2,3,4).

Figure 3

Demonstration of the method for the state $1/\sqrt{2}(|HH⟩+e^{i\theta }|VV⟩)$. (a) Modular values (upper, for real part, and lower, for imaginary part): To avoid clutter, we only show the corresponding values for the projector $P_V^A$, $P_V^B$, and $P_V^A+P_V^B$. (b) The normalized probability amplitude for the corresponding component ${\mathrm{\Psi }}_{VV}$ of the state. While all the real parts of modular values in (a) exhibit constant value, the oscillations in both the real and imaginary part of ${\mathrm{\Psi }}_{VV}$ emerge when performing the normalization to the whole state. In both (a) and (b), solid lines are values derived from a perfectly prepared state, dashed lines are predicted results taking into account the higher order of $\epsilon$ ($=0.2$ in our case), and markers are experimental results. The error bars (calculated through Monte Carlo simulations considering the counting noise) are smaller than the point size.

Figure 4

Experimentally measured probability amplitudes. The upper part of each panel shows the directly measured probability amplitudes (Exp). The theoretical values (Th) refer to the state in the case of an ideal preparation, which in each panel is given by (a) $|\mathrm{\Psi }⟩=(|HH⟩+|VV⟩)/\sqrt{2}$, (b) $|\mathrm{\Psi }⟩=(|HH⟩+i|VV⟩)/\sqrt{2}$, (c) $|\mathrm{\Psi }⟩=(|H⟩+|V⟩)(|H⟩-i|V⟩)/2$, and (d) $[0.8|HH⟩-0.6i|HV⟩-0.8|VH⟩-0.6i|VV⟩]/\sqrt{2}$. The lower part of each panel shows the reconstructed density matrix (colored bars) for each state, obtained by standard tomography. Errors are estimated through Monte Carlo simulations considering the counting noise.