Quantum Nonlocality Obtained from Local States by Entanglement Purification

We have applied an entanglement purification protocol to produce a single entangled pair of photons capable of violating a Clauser-Horne-Shimony-Holt Bell inequality from two pairs that individually could not. The initial poorly entangled photons were created by a controllable decoherence that introduced complex errors. All of the states were reconstructed using quantum state tomography which allowed for a quantitative description of the improvement of the state after purification.

Figure 1

Schematic for entanglement purification. A source produces two pairs of entangled photons, one photon from each pair travels to Alice and the others to Bob. In each quantum channel, decoherence degrades the entanglement and purity of the quantum states. Each party mixes their photons through a PBS oriented in the $H/V$ basis and measures the outputs $\mathrm{a}{1}^{\prime }$ and $\mathrm{b}{2}^{\prime }$ in the $|\pm ⟩=1/\sqrt{2}(|H⟩\pm |V⟩)$ basis. Given that the four photons each take a different PBS output and both Alice and Bob find the same measurement outcome (in our experiment $|+⟩$), the remaining photon pair in modes $\mathrm{a}{2}^{\prime }$ and $\mathrm{b}{1}^{\prime }$ has been purified.

Figure 2

The experimental setup for purifying mixed entangled states. Entangled photon pairs are created when an ultraviolet laser pulse makes two passes through a $\beta$-barium borate (BBO) crystal. Rotating the half-wave plate (HWP) in front of each compensating BBO crystal by less than $90°$ creates a tunable degree of decoherence (Decoherer). All four HWPs are left in to rotate $45°$ to enhance the efficiency of purification under our experimental noise conditions. Modes from the two different pairs (a1 and a2, and b1 and b2) are combined at the two PBSs which perform parity checks. In the output modes $\mathrm{a}{1}^{\prime }$ and $\mathrm{b}{2}^{\prime }$, projective measurements are made onto the state $|+⟩$ using single-mode fiber-coupled single-photon counting detectors behind 3-nm bandwidth interference filters. Tomographic and Bell inequality measurements are performed using the HWPs and quarter-wave plates (QWPs) in these output modes and two more fiber-coupled detectors also behind interference filters.

Figure 3

Tomographic reconstruction of the density matrices before and after purification. The real (left-hand side) and imaginary (right-hand side) parts of the density matrices of (a) the initial forward pair, (b) the initial backward pair, and (c) the purified pair. The maximum possible Bell parameter, $S_{\max }$, from the CHSH-Bell inequality for any set of measurements is $1.89\pm 0.02$ for the initial forward pair and $1.90\pm 0.014$ for the initial backward pair. For the purified state we measured the Bell parameter with the polarizer settings $-22.5°$ and $22.5°$ in mode a3 and $0°$ and $45°$ in mode b4 to be $S_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}=2.29\pm 0.13>2$—a $2.2$-$\sigma$ Bell inequality violation. The final purified state cannot be described by any local realistic theory.

Figure 4

Quantifying the state's improvement through purification. Quantum state purity and entanglement can be characterized by the linear entropy of the quantum state and by its tangle. The initial states (open circle and open square) are of high linear entropy and low tangle; i.e., they are highly mixed and have low entanglement. After purification, the new state (solid square) is of higher purity and higher entanglement. For reference, all physical states lie below the solid line [22].