Anomalous Weak Values Without Post-Selection

Alastair A. Abbott1, Ralph Silva2,3, Julian Wechs1, Nicolas Brunner2, and Cyril Branciard1

1Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
2Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland
3Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

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A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observable's spectrum. This effect, usually referred to as an ``anomalous weak value'', is generally believed to be possible only when a non-trivial post-selection is performed, i.e., when only a particular subset of the data is considered. Here we show, however, that this is not the case in general: in scenarios in which several weak measurements are sequentially performed, an anomalous weak value can be obtained without post-selection, i.e., without discarding any data. We discuss several questions that this raises about the subtle relation between weak values and pointer positions for sequential weak measurements. Finally, we consider some implications of our results for the problem of distinguishing different causal structures.

A hallmark of quantum mechanics is the “no information without disturbance” principle: to learn about a quantum system one must measure it in some way, but doing so generally perturbs it irreversibly. As a result, it is often useful to perform “weak measurements”, in which one only disturbs the system of interest very weakly but, in turn, only learns a small amount of information about it. Such weak measurements can lead to surprising phenomena, such as “anomalous weak values”: when a weak-measurement is followed by a “post-selection” on certain final conditions, the average result of the measurement can be far outside the range of expected values (for example, one may find that the average score of a “quantum dice” is -100). This paradoxical effect has even found applications in quantum metrology and beyond.

The need to post-select on certain outcomes, thereby discarding the unselected data, is traditionally seen as an essential prerequisite to obtaining these anomalous weak-values. In this paper we show that, in some situations, this popular belief is in fact incorrect. In situations in which one performs several weak measurements sequentially on a system, one can obtain anomalous outcomes without any post-selection. We analyse this unexpected phenomenon in detail, showing how the final measurement can act as an “effective post-selection” but, crucially, without the need to discard any data. These results and insights shed new light on the phenomenon of anomalous weak-values, and raise questions about the relation between “weak values” – the formal property used to calculate the measurement outcomes in such scenarios – and the actual reading shown by the measurement device.

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[1] Y. Aharonov, D. Z. Albert, and L. Vaidman, 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).

[2] H. M. Wiseman, Weak values, quantum trajectories, and the cavity-qed experiment on wave-particle correlation, Phys. Rev. A 65, 032111 (2002).

[3] N. Brunner, A. Acín, D. Collins, N. Gisin, and V. Scarani, Optical telecom networks as weak quantum measurements with postselection, Phys. Rev. Lett. 91, 180402 (2003).

[4] L. Diósi, Structural features of sequential weak measurements, Phys. Rev. A 94, 010103(R) (2016).

[5] Y. Aharonov and L. Vaidman, Properties of a quantum system during the time interval between two measurements, Phys. Rev. A 41, 11 (1990).

[6] Y. Aharonov and L. Vaidman, The two-state vector formalism: An updated review, in Time in Quantum Mechanics, Lecture Notes in Physics, Vol. 734, edited by R. S. Mayato, J. G. Muga, and Í. Egusquiza (Springer Verlag, Berlin, 2008) Chap. 13, pp. 339–447.

[7] N. W. M. Ritchie, J. G. Story, and R. G. Hulet, Realization of a measurement of a ``weak value'', Phys. Rev. Lett. 66, 1107 (1991).

[8] K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Experimental realization of the quantum box problem, Phys. Lett. A 324, 125 (2004).

[9] G. J. Pryde, J. L. O'Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, Measurement of quantum weak values of photon polarization, Phys. Rev. Lett. 94, 220405 (2005).

[10] A. M. Steinberg, How much time does a tunneling particle spend in the barrier region? Phys. Rev. Lett. 74, 2405 (1995).

[11] N. Brunner, V. Scarani, M. Wegmüller, M. Legré, and N. Gisin, Direct measurement of superluminal group velocity and signal velocity in an optical fiber, Phys. Rev. Lett. 93, 203902 (2004).

[12] D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, Fast light, slow light, and phase singularities: A connection to generalized weak values, Phys. Rev. Lett. 92, 043601 (2004).

[13] O. Hosten and P. Kwiat, Observation of the spin hall effect of light via weak measurements, Science 319, 787 (2008).

[14] P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Ultrasensitive beam deflection measurement via interferometric weak value amplification, Phys. Rev. Lett. 102, 173601 (2009).

[15] N. Brunner and C. Simon, Measuring small longitudinal phase shifts: Weak measurements or standard interferometry? Phys. Rev. Lett. 105, 010405 (2010).

[16] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, Understanding quantum weak values: Basics and applications, Rev. Mod. Phys. 86, 307 (2014).

[17] M. F. Pusey, Anomalous weak values are proofs of contextuality, Phys. Rev. Lett. 113, 200401 (2014).

[18] Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, Revisiting hardy's paradox: counterfactual statements, real measurements, entanglement and weak values, Phys. Lett. A 301, 130 (2002).

[19] J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Direct measurement of the quantum wavefunction, Nature 474, 188 (2011).

[20] S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Observing the average trajectories of single photons in a two-slit interferometer, Science 332, 1170 (2011).

[21] G. Mitchison, R. Jozsa, and S. Popescu, Sequential weak measurement, Phys. Rev. A 76, 062105 (2007).

[22] F. Piacentini, A. Avella, M. Levi, M. Gramegna, G. Brida, I. Degiovanni, E. Cohen, R. Lussana, F. Villa, A. Tosi, F. Zappa, and M. Genovese, Measuring incompatible observables by exploiting sequential weak values, Phys. Rev. Lett. 117, 170402 (2016).

[23] Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, Direct quantum process tomography via measuring sequential weak values of incompatible observables, Nat. Commun. 9, 192 (2018).

[24] J.-S. Chen, M.-J. Hu, X.-M. Hu, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, and Y.-S. Zhang, Experimental realization of sequential weak measurements of arbitrary non-commuting pauli observables, Opt. Express 27, 6089 (2019).

[25] D. Curic, M. C. Richardson, G. S. Thekkadath, J. Flórez, L. Giner, and J. S. Lundeen, Experimental investigation of measurement-induced disturbance and time symmetry in quantum physics, Phys. Rev. A 97, 042128 (2018).

[26] D. Georgiev and E. Cohen, Probing finite coarse-grained virtual Feynman histories with sequential weak values, Phys. Rev. A 97, 052102 (2018).

[27] K. J. Resch and A. M. Steinberg, Extracting joint weak values with local, single-particle measurements, Phys. Rev. Lett. 92, 130402 (2004).

[28] A. Bednorz, K. Franke, and W. Belzig, Noninvasiveness and time symmetry of weak measurements, New J. Phys. 15, 023043 (2013).

[29] A. Bednorz and W. Belzig, Quasiprobabilistic interpretation of weak measurements in mesoscopic junctions, Phys. Rev. Lett. 105, 106803 (2010).

[30] J. S. Lundeen and C. Bamber, Procedure for direct measurement of general quantum states using weak measurement, Phys. Rev. Lett. 108, 070402 (2012).

[31] G. Thekkadath, L. Giner, Y. Chalich, M. Horton, J. Banker, and J. Lundeen, Direct measurement of the density matrix of a quantum system, Phys. Rev. Lett. 117, 120401 (2016).

[32] L. Vaidman, Weak-measurement elements of reality, Found. Phys. 26, 895 (1996).

[33] L. Vaidman, A. Ben-Israel, J. Dziewior, L. Knips, M. Weißl, J. Meinecke, C. Schwemmer, R. Ber, and H. Weinfurter, Weak value beyond conditional expectation value of the pointer readings, Phys. Rev. A 96, 032114 (2017).

[34] J. F. Fitzsimons, J. A. Jones, and V. Vedral, Quantum correlations which imply causation, Sci. Rep. 5, 18281 (2015).

[35] K. Ried, M. Agnew, L. Vermeyden, D. Janzing, R. W. Spekkens, and K. J. Resch, A quantum advantage for inferring causal structure, Nat. Phys. 11, 414 (2015).

[36] F. Costa and S. Shrapnel, Quantum causal modelling, New J. Phys. 18, 063032 (2016).

[37] J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee, and R. W. Spekkens, Quantum common causes and quantum causal models, Phys. Rev. X 7, 031021 (2017).

[38] C. Giarmatzi and F. Costa, A quantum causal discovery algorithm, npj Quantum Inf. 4, 17 (2018).

[39] O. Oreshkov, F. Costa, and Č. Brukner, Quantum correlations with no causal order, Nat. Commun. 3, 1092 (2012).

[40] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, Witnessing causal nonseparability, New. J. Phys. 17, 102001 (2015).

[41] C. Branciard, Witnesses of causal nonseparability: an introduction and a few case studies, Sci. Rep. 6, 26018 (2016).

[42] G. Rubino, L. A. Rozema, A. Feix, M. Araújo, J. M. Zeuner, L. M. Procopio, Č. Brukner, and P. Walther, Experimental verification of an indefinite causal order, Sci. Adv. 3, e1602589 (2017).

[43] K. Goswami, C. Giarmatzi, M. Kewming, F. Costa, C. Branciard, J. Romero, and A. G. White, Indefinite causal order in a quantum switch, Phys. Rev. Lett. 121, 090503 (2018).

[44] K. Franke, A. Bednorz, and W. Belzig, Time asymmetry in weak measurements, Phys. Scr. T151, 014013 (2012).

[45] Y. Aharonov and L. Vaidman, The two-state vector formalism of quantum mechanics, in Time in Quantum Mechanics, edited by J. G. Muga, R. Sala Mayato, and I. L. Egusquiza (Springer-Verlag, Berlin Heidelberg, 2002) Chap. 13, pp. 369–412.

Cited by

[1] Mu Yang, Qiang Li, Zheng-Hao Liu, Ze-Yan Hao, Chang-Liang Ren, Jin-Shi Xu, Chuan-Feng Li, and Guang-Can Guo, "Experimental observation of an anomalous weak value without post-selection", Photonics Research 8 9, 1468 (2020).

[2] Surya Narayan Sahoo, Sanchari Chakraborti, Arun K. Pati, and Urbasi Sinha, "Quantum State Interferography", Physical Review Letters 125 12, 123601 (2020).

[3] Eliahu Cohen, "Quantum measurements - yet another surprise", Quantum Views 3, 27 (2019).

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