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Project GHZ Experiment Project GHZ experiment Reference 2: “Experimental Test of Quantum Nonlocality in Three-Photon GHZ Entanglement” From the journal Nature February 2000. Comments on notation etc. This paper is a almost a direct copy of the whole problem assigned. There are small notational differences to note in reading this article. 1) They begin with the GHZ state which is slightly different than we have (although note that in reference 1 the GHZ state is the same as ours). They start with |H>|H>|H> + |V>|V>|V>. The difference is only in the phases of the terms you get for the quantum kets. See eqn 4. This should be identical to the state you found in part C2 (the yyx state), except for the phases (they have all plus signs). These phases do not effect the argument of the article. 2) You probably immediately noticed that eqn 4 doesn’t agree with yours. If you examine equations 2 and 3, you will see what we call |P> and |M>, they call |H’> and |V’> respectively. Thus whenever you see a H’ in the article, think P and V’ think M. It all works. 3) There is some stuff on experimental details, you can gloss over these. A paragraph by paragraph comparison goes as follows: Paragraph 1: Just states the GHZ state in the zzz basis (HV basis). This you derived in step B1 (though note difference above. Paragraph 2: This defines the basis as you did problem A. See that they agree. Paragraph 3: This corresponds to problem C2. Paragraph 4: Says you can get others (C3, and C4) by permutation. You derived them explicitly. Paragraph 5: Start real local hidden variable model. Reasoning for instruction sets. Paragraph 6: Defines what we call parity. Paragraph 7: Just includes the result from problem C1. Paragraph 8: Argues what xxx state instruction set should be. This is a different argument than we pursued. This argument states that the instruction set should be the opposite of our xxx ket. What we tried to do is to make our instruction set compatible with the C1,C3, and C4 quantum kets and found that we got differences with the C2 (yyx) result then. A slightly different approach. They argue that (given parity of yyx, yxy, and xyy) the instruction set should have parity –1. Paragraph 9: States what xxx state is (problem C1 again). States also that these terms do not agree with what the instruction set gives. Paragraph 10: GHZ only needs 1 measurement (ideally) to discriminate between QM and Classical. Paragraph 11: Discussion of experimental set up. Paragraph 12: Differences with older Bell Ineq. Violations experiments. Paragraph 13: Explains exactly how you observe a violation. Paragraph 14: Starts to discuss experiment error (details). Paragraph 15: Possible counter-arguments since measurements are not perfect. *Paragraph 16: The results for one possible case (yyx) and what data look like. Figure 3 is what you want to examine. Compare these results (especially the top one) with the ket you obtained in C2 AND your instruction sets (which are opposite. You should find that they agree with your ket and not your set. Paragraph 17: Explains some experimental error and results. See fig 4. This is based on xxx state as argued in paragraph 8. Paragraph 18: Discusses the experimental results discussed in previous paragraph. Paragraph 19: Some details (don’t worry). Paragraph 20: Extensions they would like to do. letters to nature electrons could be employed to probe atoms and molecules ................................................................. remotely while minimizing the perturbing in¯uences of a nearby Experimental test of quantum local probeÐthese unwanted interactions might be chemical in nature or result from electric or magnetic ®elds emanating from an nonlocality in three-photon STM tip or other probe device. Our results also suggest altered geometries such as ellipsoids con®ning bulk electrons, or specially Greenberger±Horne±Zeilinger shaped electron mirrors (such as parabolic re¯ectors) electronically coupling two or more points. One might additionally envision entanglement performing other types of `spectroscopy-at-a-distance' beyond electronic structure measurements, for example, detecting Jian-Wei Pan*, Dik Bouwmeester², Matthew Daniell*, vibrational16 or magnetic excitations. Harald Weinfurter³ & Anton Zeilinger* We conclude with some remaining questions. Our calculations show that ellipses have a class of eigenmodes that possess strongly * Institut fuÈr Experimentalphysik, UniversitaÈt Wien, Boltzmanngasse 5, peaked probability amplitude near the classical foci (Fig. 3e and f are 1090 Wien, Austria good examples). Our experiments reveal that the strongest mirages ² Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK ³ Sektion Physik, Ludwig-Maximilians-UniversitaÈtofMuÈnchen, occur when one of these eigenmodes is at EF and therefore at the energy of the Kondo resonance. When perturbed by a focal atom, Schellingstrasse 4/III, D-80799 MuÈnchen, Germany this eigenmode still has substantial density surrounding both foci; it .............................................................................................................................................. is therefore plausible that this quantum state `samples' the Kondo Bell's theorem1 states that certain statistical correlations predicted resonance on the real atom and transmits that signal to the other by quantum physics for measurements on two-particle systems focus. The physics behind this process is not completely under- cannot be understood within a realistic picture based on local stood. Also, given that a Kondo signature is detected at the empty properties of each individual particleÐeven if the two particles focus, does this imply that a measurement there is simply providing are separated by large distances. Einstein, Podolsky and Rosen us with a remote probe of the Co atom with the intervening two- ®rst recognized2 the fundamental signi®cance of these quantum dimensional electrons acting essentially like a wire? Or, as the Kondo correlations (termed `entanglement' by SchroÈdinger3) and the effect on the Co atom requires a net spin polarization of the surrounding electron gas, does the projection of the Kondo reso- D nance imply a modi®ed spin polarization at the empty focus? We D1 2 speculate that both interpretations are actually correct. Full answers to these questions, however, await further experimental and theo- F F retical efforts. λ /4 D POL POL 3 Received 12 October; accepted 14 December 1999. T 1. Spector, J., Stormer, H. L., Baldwin, K. W., Pfeiffer, L. N. & West, K. W. Electron focusing in two- F F dimensional systems by means of an electrostatic lens. Appl. Phys. Lett. 56, 1290±1292 (1990). PBS 2. Crommie, M. F., Lutz, C. P. & Eigler, D. M. Con®nement of electrons to quantum corrals on a metal λ/4 surface. Science 262, 218±220 (1993). λ/2 POL 3. Heremans, J. J., von MolnaÂr, S., Awschalom, D. D. & Gossard, A. C. Ballistic electron focusing by elliptic re¯ecting barriers. Appl. Phys. Lett. 74, 1281±1283 (1999). 4. Kondo, J. Resistance minimum in dilute magnetic alloys. Prog. Theor. Phys. 32, 37±49 (1964). BS 5. Li, J., Schneider, W.-D., Berndt, R. & Delley, B. Kondo scattering observed at a single magnetic PBS impurity. Phys. Rev. Lett. 80, 2893±2896 (1998). 6. Madhavan, V., Chen, W., Jamneala, T., Crommie, M. F. & Wingreen, N. S. Tunnelling into a single ab magnetic atom: Spectroscopic evidence of the Kondo resonance. Science 280, 567±569 (1998). 7. Kittel, C. Quantum Theory of Solids (Wiley, New York, 1963). BBO 8. Hewson, A. C. The Kondo Problem to Heavy Fermions (Cambridge Univ. Press, Cambridge, 1997). 9. Fano, U. Effects of con®guration interaction on intensities and phase shifts. Phys. Rev. 124, 1866±1878 (1961). Pulse 10. Crommie, M. F., Lutz, C. P.& Eigler, D. M. Imaging standing waves in a two-dimensional electron gas. Nature 363, 524±527 (1993). 11. Hasegawa, Y. & Avouris, P. Direct observation of standing wave formation at surface steps using scanning tunneling spectroscopy. Phys. Rev. Lett. 71, 1071±1074 (1993). 12. Eigler, D. M. & Schweizer, E. K. Positioning single atoms with a scanning tunnelling microscope. Nature 344, 524±526 (1990). Figure 1 Experimental set-up for Greenberger±Horne±Zeilinger (GHZ) tests of quantum 13. Stroscio, J. A. & Eigler, D. M. Atomic and molecular manipulation with the scanning tunneling nonlocality. Pairs of polarization-entangled photons28 (one photon H polarized and the microscope. Science 254, 1319±1326 (1991). 14. Tomsovic, S. & Heller, E. J. Semiclassical construction of chaotic eigenstates. Phys. Rev. Lett. 70, 1405± other V ) are generated by a short pulse of ultraviolet light (, 200 fs, l = 394 nm). 1408 (1993). Observation of the desired GHZ correlations requires fourfold coincidence and therefore 15. Chan, Y. S. & Heller, E. J. Scanning tunnel microscopy surface state electron scattering: Two-tip results two pairs29. The photon registered at T is always H and thus its partner in b must be V. The from one-tip data. Phys. Rev. Lett. 78, 2570±2572 (1997). 16. Stipe, B. C., Rezaei, M. A. & Ho, W. Single-molecule vibrational spectroscopy and microscopy. Science photon re¯ected at the polarizing beam-splitter (PBS) in arm a is always V, being turned 280, 1732±1735 (1998). into equal superposition of V and H by the l/2 plate, and its partner in arm b must be H. 17. Lang, N. D. Spectroscopy of single atoms in the scanning tunneling microscope. Phys. Rev. B 34, 5947± Thus if all four detectors register at the same time, the two photons in D1 and D2 must 5950 (1986). either both have been VVand re¯ected by the last PBS or HH and transmitted. The photon 18. Everson, M. P., Jaklevic, R. C. & Shen, W. Measurement of the local density of states on a metal surface: Scanning tunneling spectroscopic imaging of Au(111). J. Vac. Sci. Technol. A 8, 3662±3665 (1990). at D3 was therefore H or V, respectively. Both possibilities are made indistinguishable by 19.
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