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.
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.
 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).
 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).
 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.
 K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Experimental realization of the quantum box problem, Phys. Lett. A 324, 125 (2004).
 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).
 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).
 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).
 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).
 N. Brunner and C. Simon, Measuring small longitudinal phase shifts: Weak measurements or standard interferometry? Phys. Rev. Lett. 105, 010405 (2010).
 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).
 M. F. Pusey, Anomalous weak values are proofs of contextuality, Phys. Rev. Lett. 113, 200401 (2014).
 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).
 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).
 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).
 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).
 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).
 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).
 A. Bednorz, K. Franke, and W. Belzig, Noninvasiveness and time symmetry of weak measurements, New J. Phys. 15, 023043 (2013).
 A. Bednorz and W. Belzig, Quasiprobabilistic interpretation of weak measurements in mesoscopic junctions, Phys. Rev. Lett. 105, 106803 (2010).
 J. S. Lundeen and C. Bamber, Procedure for direct measurement of general quantum states using weak measurement, Phys. Rev. Lett. 108, 070402 (2012).
 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).
 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).
 F. Costa and S. Shrapnel, Quantum causal modelling, New J. Phys. 18, 063032 (2016).
 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).
 M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, Witnessing causal nonseparability, New. J. Phys. 17, 102001 (2015).
 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).
 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).
 K. Franke, A. Bednorz, and W. Belzig, Time asymmetry in weak measurements, Phys. Scr. T151, 014013 (2012).
 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.
 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).
 Surya Narayan Sahoo, Sanchari Chakraborti, Arun K. Pati, and Urbasi Sinha, "Quantum State Interferography", Physical Review Letters 125 12, 123601 (2020).
 Eliahu Cohen, "Quantum measurements - yet another surprise", Quantum Views 3, 27 (2019).
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