Violation of local realism with freedom of choice

Abstract

Bell's theorem shows that local realistic theories place strong restrictions on observable correlations between different systems, giving rise to Bell's inequality which can be violated in experiments using entangled quantum states. Bell's theorem is based on the assumptions of realism, locality, and the freedom to choose between measurement settings. In experimental tests, "loopholes" arise which allow observed violations to still be explained by local realistic theories. Violating Bell's inequality while simultaneously closing all such loopholes is one of the most significant still open challenges in fundamental physics today. In this paper, we present an experiment that violates Bell's inequality while simultaneously closing the locality loophole and addressing the freedom-of-choice loophole, also closing the latter within a reasonable set of assumptions. We also explain that the locality and freedom-of-choice loopholes can be closed only within nondeterminism, i.e., in the context of stochastic local realism.

Fig. 1.

Experimental setup. The Bell experiment was carried out between the islands of La Palma and Tenerife at an altitude of 2,400 m. (La Palma) A 405-nm laser diode (LD) pumped a periodically poled potassium titanyl phosphate (ppKTP) crystal in a polarization-based Sagnac interferometer, to generate entangled photon pairs in the ψ^- singlet state. One photon per pair was sent through a 6-km-long, coiled optical single-mode fiber (SMF) to Alice (located next to the source). Alice’s polarization analyzer consisted of half- and quarter-wave plates (HWP, QWP), an electro-optical modulator (EOM), a polarizing beam splitter (PBS) and two photodetectors (D_T, D_R). A quantum random number generator (30) (QRNG_A) located at a distance of 1.2 km, consisting of a light-emitting diode (LED), a 50/50 beam splitter (BS), and two photomultipliers (PMs), generated random bits which were sent to Alice via a 2.4 GHz radio link. The random bits were used to switch the EOM, determining if the incoming photon was measured in the 22.5°/112.5° or 67.5°/157.5° linear polarization basis. A time-tagging unit (TTU), locked to the global positioning system (GPS) time standard and compensated (31) for small drifts up to 10 ns, recorded every detection event (arrival time, detector channel, and setting information) onto a local hard disk. The other photon was guided to a transmitter telescope and sent through a 144-km optical free-space link to Bob on Tenerife. (Tenerife) The incoming photon was received by the 1-m optical ground station (OGS) telescope of the European Space Agency. At Bob’s polarization analyzer (triggered by an equal but independent quantum random number generator QRNG_B), the photons were measured in either the horizontal (0°)/vertical (90°), or the 45°/135° linear polarization basis. Bob’s data acquisition was equivalent to Alice’s. (See also Materials and Methods for details.) (Geographic pictures taken from Google Earth, ©2008 Google, Map Data ©2008 Tele Atlas.)

Fig. 2.

Space-time diagrams. (A) Source reference frame. The forward (backward) light cone of the photon-pair emission event E, shaded in gray, contains all space-time events which can be causally influenced by E (can causally influence E). Alice’s random setting choices (indicated by small green dots in the zoomed part of A), each applied for a 1-μs interval, were transmitted over a 1.2-km classical link (green line), which took 4.5 μs (3.9 μs classical rf link, 0.6-μs electronics). This signal was electronically delayed by 24.6 μs, so that the choice event a, corresponding to a given measurement A, occurred simultaneously within a time window of ± 0.5 μs with the emission event E, i.e., E occurred on average in the middle of the 1-μs setting interval. The choice and emission events were therefore space-like separated. The same electronic delay (24.6 μs) was applied to Bob’s choice b, so that it was also space-like separated from the source. (B) Moving reference frame. From the perspective of an observer moving at a speed of 0.938·c parallel to the direction from La Palma (Alice) to Tenerife (Bob), the measurement events, A and B, occur simultaneously with the emission event approximately in the middle of the two. The locality and the freedom-of-choice loopholes are closed in the source reference frame, and because space-like separation is invariant under Lorentz transformations, they are closed in all reference frames. In the diagrams above, the total uncertainty of the event times is below the size of the illustrated points (see Materials and Methods).

Fig. 3.

State tomography. Reconstructed density matrix ρ for Alice’s and Bob’s nonlocal two-photon state, with tangle (36, 37) T = 0.68 ± 0.04, confirming the entanglement of the widely separated photons, with linear entropy (37) 0.21 ± 0.03, and an optimal fidelity with a maximally entangled state F_opt = 0.91 ± 0.01. The measured state predicts a Bell parameter of S^tomo = 2.41 ± 0.06, which agrees with the directly measured value, and an optimal violation of S^opt = 2.54 ± 0.06 for a rotated set of polarization measurements. The nonzero imaginary components are mainly due to polarization rotations resulting from imperfections in the alignment of Alice’s and Bob’s shared reference frame.