So far, all experimental tests of Bell inequalities which must be satisfied by all local realistic hidden-variable theories and are violated by quantum mechanical predictions have left at least one loophole open. We propose a feasible setup allowing for a loophole-free test of the Bell inequalities. Two electron spin qubits of donors<sup>31</sup>P in a nanoscale silicon host in different cavities 300 m apart are entangled through a bright coherent light and postselections using homodyne measurements. The electron spins are then read out randomly and independently by Alice and Bob, respectively, with unity efficiency in less than 0.7 µs by using optically induced spin to charge transduction detected by radio-frequency single electron transistor. A violation of Bell inequality larger than 37% and 18% is achievable provided that the detection accuracy is 0.99 and 0.95, respectively.
Most working scientists hold fast to the concepts of “realism” according to which an external reality exists independent of observation and “locality” which means that local events cannot be affected by actions in space-like separated regions [
Several schemes were proposed closing these loopholes based on entangled photon pairs [14,17], Hg atoms [
A Bell measurement of inequality of Clauser, Horne, Shimony, and Holt (CHSH) [
where and denote the numbers of measurements in which the measured results are the same or different, respectively. The CHSH form of Bell inequalities states that the correlations resulting from local realistic theories must satisfy:
where and (and) are specific values of (). For a Bell measurement based on electron spins, we have
where with (i = A, B, and) being the Pauli matrices, and () are unit vectors. The CHSH inequality (2) is maximally violated by quantum mechanics at certain sets of and, one such set is that both of the () are in the xy-plane, and the polar angles of () are, for, and, for. For these phase angles and state
, the quantum mechanics gives
The architecture of the basic phosphorus 31P donor electron spin qubit in silicon with control gates and a resonant readout mechanism are shown in
To generate entanglement between donor qubits at Alice’s and Bob’s sites 300 m apart, a bright coherent pulse sequentially interacts with the qubits, entangled qubit pairs will then be postselected conditioned upon the results of probe homodyne measurements. For a sufficient dispersive interaction between the donor electron and the light, the system should be placed in a cavity resonant with the light. For the cavity, weak coupling is
sufficient, but a high value of is required, where Q is the quality and V is the mode-volume of the cavity [
The donor electron spin system in the cavity is treated as a system with two stable and metastable ground states and, and an excited state provided by the bound-exciton state. For the coherent pulses, the transition between and is suppressed due to a prohibitive selection rule and only and participate in the interaction with the cavity mode [
The probe beam is then sent to the cavity at the Bob’s site and interacts with the qubit donor in the same way. Applying a further linear phase shift of θ to the pulse after it leaves the cavity will yield the total state
where with the conventional denotation and hereafter. In the presence of channel loss, we may model the photon loss by considering a beam splitter in the channel that transmits only a part of the probe pulse with transmission [
where
Here is the decoherence factor arising from the dispersive light-matter interaction in the cavities,
, and an extra phase
can be set to be naught, since it is independent of the measurement results and can be locally removed via static phase shifters.
With the balanced homodyne detection [
where, , and is the selection window of the homodyne measurements. The desired entangled output state is, so the average fidelity after postselection has the form [
For the state obtained through the postselection with the configuration for () aforesaid, the violation of the CHSH inequalities reads
After the pulse leaves the cavity at the Bob’s site, Alice and Bob randomly and dependently manipulate the electron spins of the SET 31P from the initial state to the state corresponding to or the state and to state
or the state
, respectively. These manipulations on the SET 31P electron spins equivalent to the actions on the qubit donor spins can be finished in 0.1 µs [
Assuming the telecom fiber and wavelength where losses are about 1 dB/km [
Building the setup shown in
versus the detection accuracy with. See the text for the values of other parameters.
As a summary, we present a scheme for the loopholefree test of the Bell inequalities. The detection efficiency of donor electron spins is unity using the optically induced spin to charge transfer detected by a rf-SET, and the fair sampling assumption is not required, thereby the detection loophole in this scheme is closed. The two qubit donors are 300 m apart, and the time of the random and independent measurement of the two qubits by Alice and Bob, respectively, is within 0.7 µs, thus the lightcone loophole may be closed too. The experimental realization of this scheme is within the reach of the current technology. Large violation of the CHSH inequality for the detection accuracy is achievable. Even if the detection accuracy is so low that, we may still have. This scheme may open a promising avenue towards a complete experimental Bell test which has a profound significance far beyond science.
This work was supported by the National Nature Science Foundation of China (Grant No.10874071, 50672088, 11072218, 11005031, and 60571029), by Zhejiang Provincial Natural Science Foundation of China (Grant No. Y6110314), by Scientific Research Fund of Zhejiang Provincial Education Department (Grant No.Y200909693), and by Science Foundation of Zhejiang Sci-Tech University (ZSTU) under Grant No.0913842-Y.