# When Photons Are Lying about Where They Have Been

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## Abstract

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## 1. Introduction

## 2. Alonso and Jordan Modified Interferometer

## 3. The Trace Analysis

## 4. Do the Photons Have Any Presence in E?

“The photons themselves tell us where they have been. And the story they tell is surprising. The photons do not always follow continuous trajectories. Some of them have been inside the nested interferometer (otherwise, they could not have known the frequencies ${f}_{A},\phantom{\rule{0.166667em}{0ex}}{f}_{B}$), but they never entered and never left the nested interferometer, since otherwise they could not avoid the imprints of frequencies ${f}_{E}$ and ${f}_{F}$ of mirrors E and F leading photons into and out of the interferometer.”

## 5. Quantifying the Presence of Photons

## 6. Bohmian Trajectory

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Danan, A.; Farfurnik, D.; Bar-Ad, S.; Vaidman, L. Asking Photons Where They Have Been. Phys. Rev. Lett.
**2013**, 111, 240402. [Google Scholar] [CrossRef] [PubMed] - Alonso, M.; Jordan, A. Can a Dove prism change the past of a single photon? Quantum Stud. Math. Found.
**2015**, 2, 255–261. [Google Scholar] [CrossRef] [Green Version] - Vaidman, L. Past of a quantum particle. Phys. Rev. A
**2013**, 87, 052104. [Google Scholar] [CrossRef] - Vaidman, L.; Ben-Israel, A.; Dziewior, A.J.; Knips, L.; Weißl, M.; Meinecke, J.; Schwemmer, C.; Ber, R.; Weinfurter, H. Weak value beyond conditional expectation value of the pointer readings. Phys. Rev. A
**2017**, 96, 032114. [Google Scholar] [CrossRef] - Piacentini, F.; Avella, A.; Rebufello, E.; Lussana, R.; Villa, F.; Tosi, A.; Gramegna, M.; Brida, G.; Cohen, E.; Vaidman, L.; et al. Determining the quantum expectation value by measuring a single photon. Nat. Phys.
**2017**, 13, 1191–1194. [Google Scholar] [CrossRef] [Green Version] - Wheeler, J.A. The ‘Past’ and the ‘Delayed-Choice Double-Slit Experiment’. In Mathematical Foundations of Quantum Theory; Marlow, A.R., Ed.; Academic Press: New York, NY, USA, 1978; pp. 9–48. [Google Scholar]
- Aharonov, Y.; Vaidman, L. Properties of a quantum system during the time interval between two measurements. Phys. Rev. A
**1990**, 41, 11. [Google Scholar] [CrossRef] [PubMed] - Li, Z.H.; Al-Amri, M.; Zubairy, M.S. Comment on “Past of a Quantum Particle”. Phys. Rev. A
**2013**, 88, 046102. [Google Scholar] [CrossRef] - Vaidman, L. Reply to the “Comment on ‘Past of a Quantum Particle’”. Phys. Rev. A
**2013**, 88, 046103. [Google Scholar] [CrossRef] - Vaidman, L. Tracing the past of a quantum particle. Phys. Rev. A
**2014**, 89, 024102. [Google Scholar] [CrossRef] - Saldanha, P.L. Interpreting a nested Mach–Zehnder interferometer with classical optics. Phys. Rev. A
**2014**, 89, 033825. [Google Scholar] [CrossRef] - Li, F.; Hashmi, F.A.; Zhang, J.-X.; Zhu, S.-Y. An Ideal Experiment to Determine the ‘Past of a Particle’ in the Nested Mach–Zehnder Interferometer. Chin. Phys. Lett.
**2015**, 32, 050303. [Google Scholar] [CrossRef] - Ben-Israel, A.; Knips, L.; Dziewior, J.; Meinecke, J.; Danan, A.; Weinfurter, H.; Vaidman, L. An Improved Experiment to Determine the ‘Past of a Particle’ in the Nested Mach–Zehnder Interferometer. Chin. Phys. Lett.
**2017**, 34, 020301. [Google Scholar] [CrossRef] [Green Version] - Potoček, V.; Ferenczi, G. Which-way information in a nested Mach–Zehnder interferometer. Phys. Rev. A
**2015**, 92, 023829. [Google Scholar] [CrossRef] - Vaidman, L. Comment on “Which-way information in a nested Mach–Zehnder interferometer”. Phys. Rev. A
**2016**, 93, 017801. [Google Scholar] [CrossRef] - Salih, H. Commentary: “Asking photons where they have been”—Without telling them what to say. Front. Phys.
**2015**, 3, 47. [Google Scholar] [CrossRef] - Vaidman, L.; Danan, A.; Farfurnik, D.; Bar-Ad, S. Response: Commentary: “Asking photons where they have been”—Without telling them what to say. Front. Phys.
**2015**, 3, 48. [Google Scholar] - Bartkiewicz, K.; Černoch, A.; Javůrek, D.; Lemr, K.; Soubusta, J.; Svozilík, J. One-state vector formalism for the evolution of a quantum state through nested Mach–Zehnder interferometers. Phys. Rev. A
**2015**, 91, 012103. [Google Scholar] [CrossRef] - Vaidman, L. Comment on ‘One-state vector formalism for the evolution of a quantum state through nested Mach–Zehnder interferometers’. Phys. Rev. A
**2016**, 93, 036103. [Google Scholar] [CrossRef] - Bartkiewicz, K.; Černoch, A.; Javůrek, D.; Lemr, K.; Soubusta, J.; Svozilík, J. Reply to “Comment on ‘One-state vector formalism for the evolution of a quantum state through nested Mach–Zehnder interferometers’”. Phys. Rev. A
**2016**, 93, 036104. [Google Scholar] [CrossRef] - Bartkiewicz, K.; Černoch, A.; Javůrek, D.; Lemr, K.; Soubusta, J.; Svozilík, J. Measuring evolution of a photon in an interferometer with spectrally resolved modes. Phys. Rev. A
**2016**, 94, 052106. [Google Scholar] - Hashmi, F.; Li, F.; Zhu, S.-Y.; Zubairy, M. Two-state vector formalism and quantum interference. J. Phys. A Math. Theor.
**2016**, 49, 345302. [Google Scholar] [CrossRef] - Vaidman, L. Comment on ‘Two-state vector formalism and quantum interference’. J. Phys. A Math. Theor.
**2018**, 51, 068002. [Google Scholar] [CrossRef] [Green Version] - Hashmi, F.; Li, F.; Zhu, S.-Y.; Zubairy, M. Reply to the comment on ‘Two-state vector formalism and quantum interference’. J. Phys. A Math. Theor.
**2018**, 51, 068001. [Google Scholar] [CrossRef] [Green Version] - Wu, Z.-Q.; Cao, H.; Huang, J.-H.; Hu, L.; Xu, X.-X.; Zhang, H.-L.; Zhu, S.-Y. Tracing the trajectory of photons through Fourier spectrum. Opt. Exp.
**2015**, 23, 10032–10039. [Google Scholar] [CrossRef] [PubMed] - Griffiths, R. Particle path through a nested Mach–Zehnder interferometer. Phys. Rev. A
**2016**, 94, 032115. [Google Scholar] [CrossRef] - Vaidman, L. Comment on “Particle path through a nested Mach–Zehnder interferometer”. Phys. Rev. A
**2017**, 95, 066101. [Google Scholar] [CrossRef] - Griffiths, R. Reply to “Comment on ‘Particle path through a nested Mach–Zehnder interferometer’”. Phys. Rev. A
**2017**, 95, 066102. [Google Scholar] [CrossRef] - Svensson, B. Non-representative Quantum Mechanical Weak Values. Found. Phys.
**2015**, 45, 1645–1656. [Google Scholar] [CrossRef] [Green Version] - Ben-Israel, A.; Vaidman, L. Comment on ‘Non-representative Quantum Mechanical Weak Values’. Found. Phys.
**2017**, 47, 467–470. [Google Scholar] [CrossRef] - Svensson, B. Response to Comment on ‘Non-representative Quantum Mechanical Weak Values’. Found. Phys.
**2017**, 47, 1258–1260. [Google Scholar] [CrossRef] - Zho, Z.-Q.; Liu, X.; Kedem, Y.; Cui, J.-M.; Li, Z.-F.; Hua, Y.-L.; Li, C.-F.; Guo, G.-C. Experimental observation of anomalous trajectories of single photons. Phys. Rev. A
**2017**, 95, 042121. [Google Scholar] [CrossRef] - Sokolovski, D. Asking photons where they have been in plain language. Phys. Lett.
**2017**, A381, 227–232. [Google Scholar] [CrossRef] - Vaidman, L. A Comment on “Asking photons where they have been in plain language”. arXiv, 2017; arXiv:1703.03615. [Google Scholar]
- Nikolaev, G. Paradox of photons disconnected trajectories being located by means of “weak measurements” in the nested Mach–Zehnder interferometer. JETP Lett.
**2017**, 105, 152–157. [Google Scholar] [CrossRef] [Green Version] - Vaidman, L. A comment on “Paradox of photons disconnected trajectories being located by means of ‘weak measurements’ in the nested Mach–Zehnder interferometer”. JETP Lett.
**2017**, 105, 473–474. [Google Scholar] [CrossRef] - Nikolaev, G. Response to the comment on “Paradox of photons disconnected trajectories being located by means of ‘weak measurements’ in the nested Mach–Zehnder interferometer”. JETP Lett.
**2017**, 105, 475. [Google Scholar] [CrossRef] - Duprey, Q.; Matzkin, A. Null weak values and the past of a quantum particle. Phys. Rev. A
**2017**, 95, 032110. [Google Scholar] [CrossRef] - Sokolovski, D. Comment on “Null weak values and the past of a quantum particle”. Phys. Rev. A
**2018**, 97, 046102. [Google Scholar] [CrossRef] [Green Version] - Duprey, Q.; Matzkin, A. Reply to Comment on “Null weak values and the past of a quantum particle”. Phys. Rev. A
**2018**, 97, 046103. [Google Scholar] [CrossRef] - Englert, B.; Horia, K.; Dai, J.; Len, Y.; Ng, H. Past of a quantum particle revisited. Phys. Rev. A
**2017**, 96, 022126. [Google Scholar] [CrossRef] - Peleg, U.; Vaidman, L. Comment on “Past of a quantum particle revisited”. arXiv, 2018; arXiv:1805.12171. [Google Scholar]
- Bernardo, B.; Canabarro, A.; Azevedo, S. How a single particle simultaneously modifies the physical reality of two distant others: A quantum nonlocality and weak value study. Sci. Rep.
**2017**, 7, 39767. [Google Scholar] [CrossRef] [PubMed] - Paneru, D.; Cohen, E. Past of a particle in an entangled state. Int. J. Quantum Inf.
**2017**, 15, 1740019. [Google Scholar] [CrossRef] [Green Version] - Aharonov, Y.; Cohen, E.; Landau, A.; Elitzur, A. The Case of the Disappearing (and Re-Appearing) Particle. Sci. Rep.
**2017**, 7, 531. [Google Scholar] [CrossRef] [PubMed] - Geppert-Kleinrath, H.; Denkmayr, T.; Sponar, S.; Lemmel, H.; Jenke, T.; Hasegawa, Y. Multifold paths of neutrons in the three-beam interferometer detected by a tiny energy kick. Phys. Rev. A
**2018**, 97, 052111. [Google Scholar] [CrossRef] [Green Version] - Hallaji, M.; Feizpour, A.; Dmochowski, G.; Sinclair, J.; Steinberg, A.M. Weak-value amplification of the nonlinear effect of a single photon. Nat. Phys.
**2017**, 13, 540–544. [Google Scholar] [CrossRef] - Dziewior, J.; Knips, L.; Farfurnik, D.; Senkalla, K.; Benshalom, N.; Efroni, J.; Meinecke, J.; Bar-Ad, S.; Weinfurter, H.; Vaidman, L. Universality property of local weak interactions and its application for interferometric alignment. arXiv, 2018; arXiv:1804.05400. [Google Scholar]
- Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. I, II. Phys. Rev.
**1952**, 85, 166. [Google Scholar] [CrossRef] - Vaidman, L. Surrealistic trajectories. In Quantum Paths: Festschrift in Honor of Berge Englert on His 60th Birthday; Ng, H.K., Han, R., Eds.; Worlds Scientific: Hackensack, NJ, USA, 2015; pp. 182–186. [Google Scholar]
- Englert, B.G.; Scully, M.O.; Sussmann, G.; Walther, H. Surrealistic Bohm trajectories. Z. Naturforsch. A
**1992**, 47, 1175–1186. [Google Scholar] [CrossRef] - Naaman-Marom, G.; Erez, N.; Vaidman, L. Position Measurements in the de Broglie-Bohm Interpretation of Quantum Mechanics. Ann. Phys.
**2012**, 327, 2522–2542. [Google Scholar] [CrossRef]

**Figure 1.**Nested Mach–Zehnder interferometer with inner interferometer tuned to destructive interference towards F. Although our ‘common sense’ suggests that the only possible path for the photon detected in D is path C, the trace was found also inside the inner interferometer supporting the TSVF proposal according to which the particle was present in the places where forward (red continuous line) and backward (green dashed line) evolving wavefunctions overlap. The latter is demonstrated by the results of the measurement by Danan et al. [1].

**Figure 2.**Nested Mach–Zehnder interferometer with a Dove prism inside the inner interferometer as suggested by Alonso and Jordan [2]. The region of the overlap of the forward and the backward evolving states remains the same, but predicted results of an experiment similar to [1] include a signal from mirror E where the photon was not supposed to be.

**Figure 3.**Nested Mach–Zehnder interferometer tuned to destructive interference towards F when a single photon is detected in D. The dashed line represents a common sense proposal by Wheeler, the thick gray line describes the past according to Vaidman’s proposal as places where the particle leaves a weak trace, the continuous line represents the Bohmian trajectory.

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**MDPI and ACS Style**

Vaidman, L.; Tsutsui, I.
When Photons Are Lying about Where They Have Been. *Entropy* **2018**, *20*, 538.
https://doi.org/10.3390/e20070538

**AMA Style**

Vaidman L, Tsutsui I.
When Photons Are Lying about Where They Have Been. *Entropy*. 2018; 20(7):538.
https://doi.org/10.3390/e20070538

**Chicago/Turabian Style**

Vaidman, Lev, and Izumi Tsutsui.
2018. "When Photons Are Lying about Where They Have Been" *Entropy* 20, no. 7: 538.
https://doi.org/10.3390/e20070538