# Knowledge of Quantum Hidden Variables Enables Backwards-In-Time Signaling

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## Abstract

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## 1. Introduction

## 2. The Protocol

- Alice finds out the (contextual) hidden values of ${\sigma}_{1x}$, ${\sigma}_{2x}$, ${\sigma}_{1y},$ ${\sigma}_{2y}$, ${\sigma}_{1z}{\sigma}_{2z}$. Of course, quantum mechanics prevents us from knowing these values because they refer to non-commuting (complementary) properties. However, the explicit assumption here is that Alice can know the hidden variables. As proven by Peres, the hidden variables are contextual, but the context will be established by Alice in the future. Clearly, we suppose that the values of the hidden variables reflect the values that are (or were or will be) uncovered by the experiments that are (or were or will be) performed. As specified above, this is the minimal requirement for “hidden variables”. We emphasize the distinction between attaining knowledge of hidden variables and performing a quantum measurement. If one knew the hidden variable, one could predict the measurement outcome. Here we assume that one could somehow know the hidden variables without performing a quantum measurement. All interpretations of quantum mechanics prevent the knowledge of hidden variables (hence the name), but here we explore what would happen if it were possible to know them: if one assumes they exist, it is natural to ask what would happen if they could be known.
- Consider the products in (1): take the product of the hidden values of ${\sigma}_{1x}$, ${\sigma}_{2x}$, ${\sigma}_{1y},$ ${\sigma}_{2y}$ and ${\sigma}_{1z}{\sigma}_{2z}$. If this product is equal to $+1$, then set a bit $b=0$, if the product is equal to $-1$, then set $b=1$. The first choice is consistent with looking at the last line of the Peres–Mermin square, the second choice with looking at the last column.
- Then in the (distant) future, Alice can retroactively decide the value of b by deciding what to measure. If she measures ${\sigma}_{1x}{\sigma}_{2x}$ and ${\sigma}_{1y}{\sigma}_{2y}$ (which commute), then she sets $b=1$. Since the hidden variables, by assumption, reflect the preexisting values of the observables, this means that by setting $b=1$ she sends back to her past self $b=1$. Instead, if she measures ${\sigma}_{1x}{\sigma}_{2y}$ and ${\sigma}_{1y}{\sigma}_{2x}$ (which also commute), then she sends back to herself $b=0$. Note that quantum complementarity forces Alice to choose one of these two possibilities: she cannot perform the measurements connected to both choices since they do not commute.

## 3. Ways to Bypass the Argument?

## 4. Bohmian Mechanics

## 5. Causal Relations between Past and Future Events

- Undetermined choice. In the above narrative we supposed that in the future Alice will be “free” to choose whatever value of b she wants, namely that her choice is not determined by the value of b that was found in the past. This is tricky, since whatever her (free) choice, it will indeed be equal to the value of b that was found in the past: that value is what she will choose to send (her choice is already known before she makes it). This is the very meaning of communication to the past. It implies that, even though she is free to choose whatever b she wants, she cannot choose to send the value opposite to the one that was found in the past: whatever value is found in the past is the one she will (freely) choose in the future. While it is not strange that the value of her choice is known after she chose it, because of BITS, the value of her choice is known even before she chose it. Causality here is somewhat confusing because the temporal order of events is reversed with respect to usual expectations. Therefore, asking what Alice will feel if she decides to send back to the past the value opposite of b is meaningless, since she cannot decide to send back the opposite of what she actually decided to send. Sometimes what we called “undetermined choice” is termed “free will”, but the adherents to compatibilism have long argued that free will is not inconsistent with determinism (e.g., [31]), whereas here we simply require Alice’s choice to be not pre-determined. As a result that the “free will” and “undetermined choice” are quite slippery concepts, especially for those interpretations that consider quantum mechanics a deterministic theory, one would possibly want to avoid physical consequences being attached to this hypothesis, so we will list other three that can replace it. Namely, the following three hypotheses are compatible with a “no free will” condition according to which Alice is not “free” to choose the value of the bit, which is determined by some other process.
- Evolutionary principle. This principle states that “knowledge comes into existence only through evolutionary processes” [32]. This means that complex meaningful information (such as a Renaissance painting or a physics textbook) does not appear instantaneously from a random fluctuation, but it is the result of a lengthy evolutionary process or computation. For example, a Neolithic caveman could not have had the knowledge or the technique to paint Leonardo’s Mona Lisa painting, and Newton could not have had the knowledge to write a quantum mechanics textbook. This implies that by tying the transmitted bit to the result of a long evolutionary process, one is guaranteed that the value of the bit cannot have been already known in the past. For example, we could tie the value of b to some information that no one knows today, such as “will Apple stock shares increase in value in ten years?”. This guarantees that indeed the bit was transmitted, and not known beforehand. Communication with the past together with the evolutionary principle lead to a chronology paradox [32]: suppose that Alice sends to Leonardo a picture of his painting and he painted it by copying her picture, then the painting would be generated spontaneously since Alice obtained it from Leonardo and Leonardo from Alice.
- Relativistic causality (strictly enforcing non-superluminal communication of at least one event). A completely independent way of ensuring that b is sent to the past is to assume relativity and causality (as defined above) and tie the value of b to the unknown value of some degree of freedom that is spacelike separated from the observer at step 2 of the above protocol, but is accessible at step 3 of the protocol when one must “choose” the value of b. For example: “has the star Betelgeuse turned into a supernova?” (Betelgeuse, in Orion, is a supernova candidate [33]). The supernova event is spacelike separated from step 2 and hence inaccessible if relativity plus causality is assumed, but it will become accessible at step 3, when it enters in step 3’s past lightcone. A caveat is in order here. As discussed above, relativity + causality is incompatible with superluminal communication, but the knowledge of hidden variables immediately implies superluminal communication. Therefore, one can assume relativity + causality for the event that causes the value of b (e.g., the supernova explosion) only if one is sure that the event in question is not communicated superluminally through some quantum hidden variables.
- Ontic randomness for some hidden variable. This entails that, at least in some cases, the outcomes of quantum measurements are intrinsically random. This is what the Copenhagen interpretation of quantum mechanics suggests, but an interpretation that uses hidden variables could say that randomness only arises because of lack of knowledge of the hidden variables. Indeed, hidden variables are based on the idea that a definite, pre-determined value existed prior to the measurement. Therefore, to be really intrinsically random, the hidden variables themselves must possess (at least in some cases) an ontic intrinsic randomness. Under this hypothesis, Alice may perform a ${\sigma}_{z}$ measurement of a qubit in an eigenstate of ${\sigma}_{x}$ using an ancillary qubit whose hidden value is not predetermined. She then sends back the outcome as the value of b, which could not have been determined in the far past under this hypothesis.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Genovese, M. Research on hidden variable theories: A review of recent progresses. Phys. Rep.
**2005**, 413, 319–396. [Google Scholar] [CrossRef] [Green Version] - Mermin, N.D. Hidden variables and the two theorems of John Bell. Rev. Mod. Phys.
**1993**, 65, 803. [Google Scholar] [CrossRef] - Belinfante, F.J. A Survey of Hidden-Variables Theories; Pergamon Press: Oxford, UK, 1973. [Google Scholar]
- Gödel, K. An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation. Rev. Mod. Phys.
**1949**, 21, 447. [Google Scholar] [CrossRef] [Green Version] - Kochen, S.; Specker, E.P. The problem of hidden variables in quantum mechanics. J. Math. Mech.
**1967**, 17, 59. [Google Scholar] [CrossRef] - Peres, A. Incompatible results of quantum measurements. Phys. Lett. A
**1990**, 151, 107. [Google Scholar] [CrossRef] - Bell, J.S. On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys.
**1966**, 38, 447. [Google Scholar] [CrossRef] - Wharton, K.B.; Argaman, N. Colloquium: Bell’s theorem and locally mediated reformulations of quantum mechanics. Rev. Mod. Phys.
**2020**, 92, 021002. [Google Scholar] [CrossRef] - Brunner, N.; Cavalcanti, D.; Pironio, S.; Scarani, V.; Wehner, S. Bell nonlocality. Rev. Mod. Phys.
**2014**, 86, 419. [Google Scholar] [CrossRef] [Green Version] - Werner, R.F. Quantum information theory—An invitation. In Quantum Information—An Introduction to Basic Theoretical Concepts and Experiments; Springer Tracts in Modern Physics; Springer: Berlin/Heidelberg, Germany, 2001. [Google Scholar]
- Liberati, S.; Sonego, S.; Visser, M. Faster-than-c Signals, Special Relativity, and Causality. Ann. Phys.
**2002**, 298, 167. [Google Scholar] [CrossRef] [Green Version] - Sutherland, R.I. Causally symmetric Bohm model. Stud. Hist. Philos. Mod. Phys.
**2008**, 39, 782–805. [Google Scholar] [CrossRef] [Green Version] - Price, H. Toy models for retrocausality. Stud. Hist. Philos. Mod. Phys.
**2008**, 39, 752–761. [Google Scholar] [CrossRef] [Green Version] - Price, H. Does Time-Symmetry Imply Retrocausality? How the Quantum World Says “Maybe”. Stud. Hist. Philos. Mod. Phys.
**2012**, 43, 75. [Google Scholar] [CrossRef] [Green Version] - Price, H.; Wharton, K. Disentangling the quantum world. Entropy
**2015**, 17, 7752–7767. [Google Scholar] [CrossRef] [Green Version] - Leifer, M.S.; Pusey, M.F. Is a time symmetric interpretation of quantum theory possible without retrocausality? Proc. R. Soc. A
**2017**, 473, 20160607. [Google Scholar] [CrossRef] - Wharton, K. A New Class of Retrocausal Models. Entropy
**2018**, 20, 410. [Google Scholar] [CrossRef] [Green Version] - Guryanova, Y.; Silva, R.; Short, A.J.; Skrzypczyk, P.; Brunner, N.; Popescu, S. Exploring the limits of no backward in time signalling. Quantum
**2019**, 3, 211. [Google Scholar] [CrossRef] - Aharonov, Y.; Cohen, E.; Landsberger, T. The two-time interpretation and macroscopic time-reversibility. Entropy
**2017**, 19, 111. [Google Scholar] [CrossRef] [Green Version] - Bohm, D.; Hiley, B.J. The Undivided Universe; Routledge: London, UK, 1993. [Google Scholar]
- Holland, P.R. The Quantum Theory of Motion; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Dürr, D.; Teufel, S. Bohmian Mechanics; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Dürr, D.; Goldstein, S.; Tumulka, R.; Zanghí, N. Bohmian Mechanics. In Compendium of Quantum Physics; Greenberger, D., Hentschel, K., Weinert, F., Eds.; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Mermin, N.D. Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett.
**1990**, 65, 3373. [Google Scholar] [CrossRef] - Vaidman, L. Lorentz-invariant “elements of reality” and the joint measurability of commuting observables. Phys. Rev. Lett.
**1993**, 70, 3369. [Google Scholar] [CrossRef] [Green Version] - Popescu, S.; Rohrlich, D. Generic quantum nonlocality. Phys. Lett. A
**1992**, 166, 293. [Google Scholar] [CrossRef] - Xu, X.-Y.; Pan, W.-W.; Wang, Q.-Q.; Dziewior, J.; Knips, L.; Kedem, Y.; Sun, K.; Xu, J.-S.; Han, Y.-J.; Li, C.-F.; et al. Measurements of nonlocal variables and demonstration of the failure of the product rule for a pre- and postselected pair of photons. Phys. Rev. Lett.
**2019**, 122, 100405. [Google Scholar] [CrossRef] [Green Version] - Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. Phys. Rev.
**1952**, 85, 166. [Google Scholar] [CrossRef] - Margalit, Y.; Zhou, Z.; Dobkowski, O.; Japha, Y.; Rohrlich, D.; Moukouri, S.; Folman, R. Realization of a complete Stern-Gerlach interferometer. arXiv
**2018**, arXiv:1801.02708. [Google Scholar] - Daumer, M.; Dürr, D.; Goldstein, S.; Zanghì, N. Naive realism about operators. In Proceedings of the International Conference “Probability, Dynamics and Causality”, Beijing, China, 14–17 October 1996. [Google Scholar]
- Lloyd, S. A Turing test for free will. Philos. Trans. Roy. Soc. A
**2012**, 28, 3597. [Google Scholar] [CrossRef] - Deutsch, D. Quantum mechanics near closed timelike lines. Phys. Rev. D
**1991**, 44, 3197. [Google Scholar] [CrossRef] [Green Version] - Firestone, R.B. Observation of 23 supernovae that exploded <300 pc from Earth during the past 300 kyr. Astrophys. J.
**2014**, 789, 29. [Google Scholar] [CrossRef] [Green Version] - Nikolić, H. Bohmian mechanics for instrumentalists. Int. J. Quantum Inf.
**2019**, 17, 1950029. [Google Scholar] [CrossRef] - Fankauser, J. Taming the delayed choice quantum eraser. arXiv
**2017**, arXiv:1707.07884v2. [Google Scholar] - Aharonov, Y.; Cohen, E.; Elitzur, A.C. Can a future choice affect a past measurement’s outcome? Ann. Phys.
**2015**, 355, 258. [Google Scholar] [CrossRef] [Green Version] - Aharonov, Y.; Cohen, E.; Colombo, F.; Landsberger, T.; Sabadini, I.; Struppa, D.C.; Tollaksen, J. Finally making sense of the double-slit experiment. Proc. Natl. Acad. Sci. USA
**2017**, 114, 6480. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Carmi, A.; Cohen, E. Relativistic independence bounds nonlocality. Sci. Adv.
**2019**, 5, eaav8370. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Peres, A. Quantum Theory: Concepts and Methods; Kluwer ac. Publ.: Dordrecht, The Netherlands, 1993. [Google Scholar]
- Peres, A. Unperformed experiments have no results. Am. J. Phys.
**1978**, 46, 745. [Google Scholar] [CrossRef]

**Figure 1.**Depiction of the protocol. First, Alice finds out the values of the (contextual) hidden variables and uses them to calculate the value of a bit b. Later she establishes the context by choosing which set of observables to measure. Her later choice determines the value of b, sending it back in time.

**Figure 2.**FULOs change the hidden variables. A full-loop Stern–Gerlach apparatus (FULO) maps the spin hidden variables onto a particle trajectory. However, the FULO does not preserve the values of the hidden variables as shown by the conceptual experiment depicted here, where a FULO oriented along the x axis is followed by a FULO oriented opposite to the x axis. If the initial state of the spin is an equal superposition of “up” and “down” along the x axis, namely $|+\rangle =\left(\right|a\rangle +|b\rangle )/\sqrt{2}$, then the trajectory is as depicted: if the trajectory in the first FULO is in the a-arm, it is in the b-arm in the second (which is flipped by 180 degrees) and vice versa. A possible interpretation is that the FULO has changed the spin hidden variable for x, even though it is an identity transformation that does not change the quantum state.

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**MDPI and ACS Style**

Carmi, A.; Cohen, E.; Maccone, L.; Nikolić, H.
Knowledge of Quantum Hidden Variables Enables Backwards-In-Time Signaling. *Appl. Sci.* **2021**, *11*, 4477.
https://doi.org/10.3390/app11104477

**AMA Style**

Carmi A, Cohen E, Maccone L, Nikolić H.
Knowledge of Quantum Hidden Variables Enables Backwards-In-Time Signaling. *Applied Sciences*. 2021; 11(10):4477.
https://doi.org/10.3390/app11104477

**Chicago/Turabian Style**

Carmi, Avishy, Eliahu Cohen, Lorenzo Maccone, and Hrvoje Nikolić.
2021. "Knowledge of Quantum Hidden Variables Enables Backwards-In-Time Signaling" *Applied Sciences* 11, no. 10: 4477.
https://doi.org/10.3390/app11104477