A scheme for performing strong and weak sequential measurements of non-commuting observables. (English) Zbl 1386.81013
Summary: Quantum systems usually travel a multitude of different paths when evolving through time from an initial to a final state. In general, the possible paths will depend on the future and past boundary conditions, as well as the system’s dynamics. We present a gedanken experiment where a single system apparently follows mutually exclusive paths simultaneously, each with probability one, depending on which measurement was performed. This experiment involves the measurement of observables that do not correspond to Hermitian operators. Our main result is a scheme for measuring these operators. The scheme is based on the erasure protocol [the authors, “Nonlocal measurements via quantum erasure”, Phys. Rev. Lett. 116, No. 7, Article ID 070404, 6 p. (2016; doi:10.1103/PhysRevLett.116.070404)] and allows a wide range of sequential measurements at both the weak and strong limits. At the weak limit the back action of the measurement cannot be used to account for the surprising behaviour and the resulting weak values provide a consistent yet strange account of the system’s past.
MSC:
| 81P15 | Quantum measurement theory, state operations, state preparations |
| 16G10 | Representations of associative Artinian rings |
| 81P05 | General and philosophical questions in quantum theory |
References:
| [1] | Aharonov, Y., Albert, D.: Is the usual notion of time evolution adequate for quantum-mechanical systems? Phys. Rev. D 29, 223 (1984). doi:10.1103/PhysRevD.29.223. (ISSN 0556-2821) · doi:10.1103/PhysRevD.29.223 |
| [2] | Hofmann, H.F.: Sequential measurements of non-commuting observables with quantum controlled interactions. N. J. Phys. 16, 063056 (2014). doi:10.1088/1367-2630/16/6/063056. (ISSN 1367-2630) · Zbl 1451.81074 · doi:10.1088/1367-2630/16/6/063056 |
| [3] | Diosi, L.: Structural Features of Sequential Weak Measurements. (2015). arXiv:1511.03923 |
| [4] | Mermin, N.D.: Hidden variables and the two theorems of John Bell. Rev. Mod. Phys. 65, 803-815 (1993). doi:10.1103/RevModPhys.65.803. (ISSN 1539-0756) |
| [5] | Hardy, L.: Quantum Mechanics, Local Realistic Theories, and Lorentz-Invariant Realistic Theories. Phys. Rev. Lett. 68, 2981 (1992). doi:10.1103/PhysRevLett.68.2981. (ISSN 0031-9007) · Zbl 0969.81500 · doi:10.1103/PhysRevLett.68.2981 |
| [6] | Aharonov, Y., Botero, A., Popescu, S., Reznik, B., Tollaksen, J.: Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values. Phys. Lett. A 301, 130 (2002). http://www.sciencedirect.com/science/article/pii/S0375960102009866 · Zbl 0997.81011 |
| [7] | Guhne, O., Kleinmann, M., Cabello, A., Larsson, J.-A., Kirchmair, G., Zahringer, F., Gerritsma, R., Roos, C.F.: Compatibility and noncontextuality for sequential measurements. Phys. Rev. A 81 (2010). doi:10.1103/PhysRevA.81.022121. (ISSN 1094-1622) · Zbl 0969.81500 |
| [8] | Kochen, S., Specker, E.P.: The Problem of Hidden Variables in Quantum Mechanics. The Logico-Algebraic Approach to Quantum Mechanics, pp. 293-328. Springer, Dordrecht (1975). doi:10.1007/978-94-010-1795-4_17 |
| [9] | Pusey, M.F.: Anomalous Weak Values Are Proofs of Contextuality. Phys. Rev. Lett. 113 (2014).doi:10.1103/PhysRevLett.113.200401. (ISSN 1079-7114) |
| [10] | Leggett, A.J., Garg, A.: Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? Phys. Rev. Lett. 54, 857-860 (1985). doi:10.1103/PhysRevLett.54.857. (ISSN 0031-9007) |
| [11] | Williams, N.S., Jordan, A.N.: Weak Values and the Leggett-Garg Inequality in Solid-State Qubits. Phys. Rev. Lett. 100, 026804 (2008). http://prl.aps.org/abstract/PRL/v100/i2/e026804 |
| [12] | Marcovitch, S., Reznik, B.: Testing Bell inequalities with weak measurements. (2010). arXiv:1005.3236 · Zbl 1282.81055 |
| [13] | Goggin, M.E., Almeida, M.P., Barbieri, M., Lanyon, B.P., O’Brien, J.L., White, A.G., Pryde, G.J.: Violation of the Leggett-Garg inequality with weak measurements of photons. P. Natl. Acad. Sci. USA 108, 1256 (2011). http://www.pnas.org/content/108/4/1256.short |
| [14] | Dressel, J., Broadbent, C.J., Howell, J.C., Jordan, A.N.: Experimental Violation of Two-Party Leggett-Garg Inequalities with Semiweak Measurements. Phys. Rev. Lett. 106, 040402 (2011). http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.106.040402 |
| [15] | Suzuki, Y., Linuma, M., Hofmann, H.F.: Violation of Leggett-Garg inequalities in quantum measurements with variable resolution and back-action. New. N. J. Phys. 14, 103022 (2012). http://iopscience.iop.org/article/10.1088/1367-2630/14/10/103022/meta · Zbl 1448.81057 |
| [16] | Groen, J.P. et al.: Partial-Measurement Backaction and Nonclassical Weak Values in a Superconducting Circuit. Phys. Rev. Lett. 111, 090506 (2013). http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.090506 |
| [17] | White, T.C., Mutus, J.Y., Dressel, J., Kelly, J., Barends, R., Jeffrey, E., Sank, D., Megrant, A., Campbell, B., Chen, Y., et al.: Preserving entanglement during weak measurement demonstrated with a violation of the Bell-Leggett-Garg inequality. NPJ Quantum Inform. 2, bibinfopages15022 (2016). http://www.nature.com/articles/npjqi201522 |
| [18] | Katiyar, H., Brodutch, A., Lu, D., Laflamme, R.: Experimental violation of the Leggett-Garg inequality in a 3-level system. (2016). arXiv:1606.07151 |
| [19] | Nagali, E., Felicetti, S., de Assis, P.-L., D’Ambrosio, V., Filip, R., Sciarrino, F.: Testing sequential quantum measurements: how can maximal knowledge be extracted? Sci. Rep. 2 (2012). doi:10.1038/srep00443. (ISSN 2045-2322) |
| [20] | Aharonov, Y., Bergmann, P.G., Lebowitz, J.L.: Time Symmetry in the Quantum Process of Measurement. Phys. Rev. 134, B1410 (1964). http://prola.aps.org/abstract/PR/v134/i6B/pB1410_1 · Zbl 0127.43703 |
| [21] | Aharonov, Y., Vaidman, L.: The Two-State Vector Formalism: An Updated Review. Lect. Notes Phys. 734, 399-447 (2007). http://link.springer.com/content/pdf/10.1007/978-3-540-73473-4_13.pdf |
| [22] | Aharonov, Y., Rohrlich, D.: Quantum Paradoxes Quantum Theory for the Perplexed. Wiley-VCH, New York (2005). (ISBN 3527403914) · Zbl 1081.81001 |
| [23] | Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351 (1988). http://prl.aps.org/abstract/PRL/v60/i14/p1351_1 |
| [24] | Duck, I., Stevenson, P., Sudarshan, E.: The sense in which a “weak measurement” of a spin-1/2 particle’s spin component yields a value 100. Phys. Rev. D 40, 2112 (1989). http://prd.aps.org/abstract/PRD/v40/i6/p2112_1 |
| [25] | Ritchie, N., Story, J., Hulet, R.G.: Realization of a measurement of a “weak value”. Phys. Rev. Lett. 66, 1107 (1991). http://prl.aps.org/abstract/PRL/v66/i9/p1107_1 |
| [26] | Aharonov, Y., Cohen, E., Ben-Moshe, S.: Unusual Interactions of Pre-and-Post-Selected Particles. EPJ Web Conf. 70, 00053 (2014). http://www.epj-conferences.org/articles/epjconf/abs/2014/07/epjconf_icfp2012_00053/epjconf_icfp2012_00053.html |
| [27] | Mitchison, G., Jozsa, R., Popescu, S.: Sequential weak measurement. Phys. Rev. A 76, 062105 (2007). doi:10.1103/PhysRevA.76.062105 |
| [28] | Mitchison, G.: Weak measurement takes a simple form for cumulants. Phys. Rev. A 77, 052102 (2009). http://journals.aps.org/pra/abstract/10.1103/PhysRevA.77.052102 |
| [29] | Aberg, J., Mitchison, G.: Cumulants and the moment algebra: Tools for analyzing weak measurements. J. Math. Phys. 50, 042103 (2009). http://scitation.aip.org/content/aip/journal/jmp/50/4/10.1063/1.3093270 · Zbl 1214.81024 |
| [30] | Piacentini, F., et al.: Measuring incompatible observables of a single photon. (2015). arXiv:1508.03220 |
| [31] | Glick, J.R., Adami, C.: Quantum Mechanics of Consecutive Measurements. (2016). arXiv:0911.1142 |
| [32] | Brodutch, A., Cohen, E.: Nonlocal Measurements via Quantum Erasure. Phys. Rev. Lett. 116, 070404 (2016). http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.070404 |
| [33] | Kocsis, S., Braverman, B., Ravets, S., Stevens, M.J., Mirin, R.P., Shalm, L.K., Steinberg, A.M.: Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer. Science 332, 1170 (2011). http://www.sciencemag.org/content/332/6034/1170.short · Zbl 1355.81025 |
| [34] | Vaidman, L.: Past of a quantum particle. Phys. Rev. A 87, 052104 (2013). http://journals.aps.org/pra/abstract/10.1103/PhysRevA.87.052104 |
| [35] | Vaidman, L.: Tracing the past of a quantum particle. Phys. Rev. A 89, 024102 (2014). doi:10.1103/PhysRevA.89.024102. (ISSN 1094-1622) · Zbl 1304.81011 |
| [36] | Danan, A., Farfurnik, D., Bar-Ad, S., Vaidman, L.: Asking Photons Where They Have Been. Phys. Rev. Lett. 111, 240402 (2013). doi:10.1103/PhysRevLett.111.240402. (ISSN 1079-7114) |
| [37] | Li, Z.H., Al-Amri, M., Shuail, M.S.: Comment on “Past of a quantum particle”. Phys. Rev. A 88, 046102 (2013). http://link.aps.org/doi/10.1103/PhysRevA.88.046102 |
| [38] | Vaidman, L.: Reply to “Comment on ‘Past of a quantum particle”’. Phys. Rev. A 88, 046103 (2013). http://link.aps.org/doi/10.1103/PhysRevA.88.046103 |
| [39] | Saldanha, P.: Interpreting a nested Mach-Zehnder interferometer with classical optics. Phys. Rev. A 89, 033825 (2014). http://link.aps.org/doi/10.1103/PhysRevA.89.033825 |
| [40] | Bartkiewicz, K., Černoch, A., Javůrek, D., Lemr, K., Soubusta, J., Jiří, J.: One-state vector formalism for the evolution of a quantum state through nested Mach-Zehnder interferometers. Phys. Rev. A 91, 012103 (2015). http://link.aps.org/doi/10.1103/PhysRevA.91.012103 |
| [41] | Potoček, V., Ferenczi, G.: Which-way information in a nested Mach-Zehnder interferometer. Phys. Rev. A 92, 012103 (2015). http://link.aps.org/doi/10.1103/PhysRevA.92.023829 |
| [42] | Vaidman, L.: Comment on “One-state vector formalism for the evolution of a quantum state through nested Mach-Zehnder interferometers”. (2015). arXiv:1508.07614 |
| [43] | Vaidman, L.: Comment on “Which-way information in a nested Mach-Zehnder interferometer”. (2015). arXiv:1509.02388 |
| [44] | Aharonov, Y., Vaidman, L.: Complete description of a quantum system at a given time. J. Phys. A Math. Gen. 24, 2315 (1991). http://iopscience.iop.org/0305-4470/24/10/018 |
| [45] | Hu, M.J. Are observables necessarily Hermitian? (2016). arXiv:1601.04287 · Zbl 1387.81217 |
| [46] | Kedem, Y., Vaidman, L.: Modular Values and Weak Values of Quantum Observables. Phys. Rev. Lett. 105, 230401 (2010). http://prl.aps.org/abstract/PRL/v105/i23/e230401 |
| [47] | Silva, R., Guryanova, Y., Brunner, N., Linden, N., Short, A.J., Popescu, S.: Pre- and postselected quantum states: Density matrices, tomography, and Kraus operators. Phys. Rev. A 89, 012121 (2014). doi:10.1103/PhysRevA.89.012121. (ISSN 1094-1622) |
| [48] | Brun, T.A., Diósi, L., Strunz, W.T.: Test of weak measurement on a two- or three-qubit computer. Phys. Rev. A 77, 032101 (2008). http://pra.aps.org/abstract/PRA/v77/i3/e032101 |
| [49] | Wu, S., Molmer, K.: Weak measurements with a qubit meter. Phys. Lett. A 374, 34 (2009). doi:10.1016/j.physleta.2009.10.026. (ISSN 0375-9601) · Zbl 1235.81046 |
| [50] | Lu, D., Brodutch, A., Li, J., Li, H., Laflamme, R.: Experimental realization of post-selected weak measurements on an NMR quantum processor. New. N. J. Phys. 16, 053015 (2014). http://iopscience.iop.org/1367-2630/16/5/053015 |
| [51] | Resch, K., Steinberg, A.: Extracting Joint Weak Values with Local, Single-Particle Measurements. Phys. Rev. Lett. 92, 130402 (2004). http://dx.doi.org/10.1103/PhysRevLett.92.130402 |
| [52] | Brodutch, A., Vaidman, L.: Measurements of non local weak values. J. Phys. Conf. Ser. 174, 012004 (2009). http://iopscience.iop.org/1742-6596/174/1/012004 |
| [53] | Brodutch, A.: Weak measurements of non local variables. (2008). arXiv preprint arXiv:0811.1706 |
| [54] | Resch, K.J., Lundeen, J.S., Steinberg, A.M.: Experimental realization of the quantum box problem. Phys. Lett. A 324, 125 (2004). http://www.sciencedirect.com/science/article/pii/S0375960104002506 · Zbl 1123.81324 |
| [55] | George, R.E., Robledo, L.M., Maroney, O.J.E., Blok, M.S., Bernien, H., Markham, M.L., Twitchen, D.J., Morton, J.J.L., Briggs, G.A.D., Hanson, R.: Opening up three quantum boxes causes classically undetectable wavefunction collapse. P. Natl. Acad. Sci. USA 110, 3777-3781 (2013). doi:10.1073/pnas.1208374110. (ISSN 1091-6490) |
| [56] | Aharonov, Y., Popescu, S., Rohrlich, D., Skrzypczyk, P.: Quantum Cheshire Cats. New. J. Phys. 15, 113015 (2013). doi:10.1088/1367-2630/15/11/113015. (ISSN 1367-2630) · Zbl 1451.81018 |
| [57] | Denkmayr, T., Geppert, H., Sponar, S., Lemmel, H., Matzkin, A., Tollaksen, J., Hasegawa, Y.: Observation of a quantum Cheshire Cat in a matter-wave interferometer experiment. Nat. Commun. 5 (2014). doi:10.1038/ncomms5492. (ISSN 2041-1723) |
| [58] | Lundeen, J., Steinberg, A.: Experimental Joint Weak Measurement on a Photon Pair as a Probe of Hardy’s Paradox. Phys. Rev. Lett. 102, 020404 (2009). http://prl.aps.org/abstract/PRL/v102/i2/e020404 |
| [59] | Aharonov, Y., Nussinov, S., Popescu, S., Vaidman, L.: Peculiar features of entangled states with postselection. Phys. Rev. A 87, 014105 (2012). http://journals.aps.org/pra/abstract/10.1103/PhysRevA.87.014105 |
| [60] | Salvail, J.Z., Agnew, M., Johnson, A.S., Bolduc, E., Leach, J., Boyd, R.W.: Full characterization of polarization states of light via direct measurement. Nat. Photonics 7, 316-321 (2013). http://www.nature.com/nphoton/journal/v7/n4/abs/nphoton.2013.24.html |
| [61] | Dressel, J., Brun, T., Korotkov, A.N.: Implementing generalized measurements with superconducting qubits. Phys. Rev. A 90, 032302 (2014). http://journals.aps.org/pra/abstract/10.1103/PhysRevA.90.032302 · Zbl 0969.81500 |
| [62] | Schrödinger, E.: Discussion of Probability Relations between Separated Systems. Math. Proc. Camb. Phil. Soc. 31, 553 (1935). http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1737068&fileId=S0305004100013554 · JFM 61.1561.03 |
| [63] | Wiseman, H.M., Jones, S.J., Doherty, A.C.: Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox. Phys. Rev. Lett. 98, 140402 (2007). http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.98.140402 · Zbl 1228.81078 |
| [64] | Kent, A.: A no-summoning theorem in relativistic quantum theory. Quantum Inf. Process. 12, 1023-1032 (2013). http://link.springer.com/article/10.1007 · Zbl 1264.81079 |
| [65] | Hayden, P., May, A.: Summoning Information in Spacetime, or Where and When Can a Qubit Be? (2012). arXiv:1210.0913 · Zbl 1342.81071 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
