# A Macroscopic Quantum Three-Box Paradox: Finding Consistency with Weak Macroscopic Realism

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Three-Box Paradox

#### Experimental Realization

## 3. Mesoscopic Paradox

#### Number States

## 4. Macroscopic “Three-box” Paradox with Cat States

#### 4.1. Coherent-State Model: $k>2$

#### Experimental Realization

#### 4.2. Coherent-State Model: $k=2$

## 5. Finding Consistency with Weak Macroscopic Realism

#### 5.1. The Weak Macroscopic Realism (wMR) Model

- (1)
- A real property for the pointer measurement:

- (2)
- A weak form of locality:

#### 5.2. Finding Consistency with wMR

#### 5.2.1. Three-Box Mesoscopic Paradox

#### 5.2.2. Cat-State Paradox

#### 5.2.3. Summary

## 6. Leggett–Garg Test of Macro-Realism

#### 6.1. Weak Macroscopic Realism (wMR) and the Leggett–Garg Assumptions

#### 6.2. Original Three-Box Paradox and the Mesoscopic Realization

#### 6.3. Macroscopic Cat-State Realization of “Three-Box” Paradox

#### 6.4. Finding Consistency with Weak Macroscopic Realism

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Aharonov, Y.; Vaidman, L. Complete description of a quantum system at a given time. J. Phys. A Math. Gen.
**1991**, 24, 2315. [Google Scholar] [CrossRef] - Aharonov, Y.; Bergmann, P.G.; Lebowitz, J.L. Time Symmetry in the Quantum Process of Measurement. Phys. Rev. B
**1964**, 134, 1410. [Google Scholar] [CrossRef] - Kirkpatrick, K.A. Classical three-box ‘paradox’. J. Phys. A
**2003**, 36, 4891. [Google Scholar] [CrossRef] - Leifer, M.S.; Spekkens, R.W. Logical Pre- and Post-Selection Paradoxes, Measurement-Disturbance and Contextuality. Int. J. Theor. Phys.
**2005**, 44, 1977. [Google Scholar] [CrossRef] - Leifer, M.S.; Spekkens, R.W. Pre- and Post-Selection Paradoxes and Contextuality in Quantum Mechanics. Phys. Rev. Lett.
**2005**, 95, 200405. [Google Scholar] [CrossRef] - Finkelstein, J. What is paradoxical about the “Three-box paradox”? arXiv
**2006**, arXiv:quant-ph/0606218. [Google Scholar] - Ravon, T.; Vaidman, L. The three-box paradox revisited. J. Phys. A Math. Theor.
**2007**, 40, 2873. [Google Scholar] [CrossRef] - Kirkpatrick, K.A. Reply to ’The three-box paradox revisited’ by T Ravon and L Vaidman. J. Phys. A
**2007**, 40, 2883. [Google Scholar] [CrossRef] - Maroney, O.J.E. Measurements, disturbance and the three-box paradox. Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Mod. Phys.
**2017**, 58, 41. [Google Scholar] [CrossRef] - Kastner, R.E. The Three-box “Paradox” and Other Reasons to Reject the Counterfactual Usage of the ABL Rule. Found. Phys.
**1999**, 29, 851. [Google Scholar] [CrossRef] - Vaidman, L. The Meaning of Elements of Reality and Quantum Counterfactuals: Reply to Kastner. Found. Phys.
**1999**, 29, 865. [Google Scholar] [CrossRef] - Boyle, C.; Schafir, R. The N-box paradox in orthodox quantum mechanics. arXiv
**2001**, arXiv:quant-ph/0108113. [Google Scholar] - Aharonov, Y.; Cohen, E.; Landau, A.; Elitzur, A.C. The Case of the Disappearing (and Re-Appearing) Particle. Sci. Rep.
**2017**, 7, 531. [Google Scholar] [CrossRef] [PubMed] - Blasiak, P.; Borsuk, E. Causal reappraisal of the quantum three-box paradox. Phys. Rev. A
**2021**, 105, 012207. [Google Scholar] [CrossRef] - Kolenderski, P.; Sinha, U.; Youning, L.; Zhao, T.; Volpini, M.; Cabello, A.; Laflamme, R.; Jennewein, T. Playing the Aharon-Vaidman quantum game with a Young type photonic qutrit. arXiv
**2011**, arXiv:1107.5828. [Google Scholar] - Resch, K.J.; Lundeen, J.S.; Steinberg, A.M. Experimental Realization of the Quantum box Problem. Phys. Lett. A
**2004**, 324, 125. [Google Scholar] [CrossRef] - 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. Proc. Natl. Acad. Sci. USA
**2013**, 110, 3777. [Google Scholar] [CrossRef] - Kirchmair, G.; Vlastakis, B.; Leghtas, Z.; Nigg, S.E.; Paik, H.; Ginossar, E.; Mirrahimi, M.; Frunzio, L.; Girvin, S.M.; Schoelkopf, R.J. Observation of the quantum state collapse and revival due to a single-photon Kerr effect. Nature
**2013**, 495, 205. [Google Scholar] [CrossRef] - Vlastakis, B.; Kirchmair, G.; Leghtas, Z.; Nigg, S.E.; Frunzio, L.; Girvin, S.M.; Mirrahimi, M.; Devoret, M.H.; Schoelkopf, R.J. Deterministically encoding quantum information using 100-photon schrödinger cat states. Science
**2013**, 342, 607. [Google Scholar] [CrossRef] - Thenabadu, M.; Reid, M.D. Bipartite Leggett–Garg and macroscopic Bell inequality violations using cat states: Distinguishing weak and deterministic macroscopic realism. Phys. Rev. A
**2022**, 105, 052207. [Google Scholar] [CrossRef] - Fulton, J.; Teh, R.Y.; Reid, M.D. Argument for the incompleteness of quantum mechanics based on macroscopic and contextual realism: GHZ and Bohm-EPR paradoxes with cat states. arXiv
**2022**, arXiv:2208.01225. [Google Scholar] - Fulton, J.; Thenabadu, M.; Teh, R.; Reid, M.D. Weak versus deterministic macroscopic realism, and elements of reality. arXiv
**2021**, arXiv:2101.09476. [Google Scholar] - Joseph, R.R.; Thenabadu, M.; Hatharasinghe, C.; Fulton, J.; Teh, R.-Y.; Drummond, P.D.; Reid, M.D. Wigner’s Friend paradoxes: Consistency with weak-contextual and weak-macroscopic realism models. arXiv
**2022**, arXiv:2211.02877. [Google Scholar] - Schrödinger, E. The Present Status of Quantum Mechanics. Die Naturwissenschaften
**1935**, 23, 807. [Google Scholar] - Leggett, A.; Garg, A. Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? Phys. Rev. Lett.
**1985**, 54, 857. [Google Scholar] [CrossRef] [PubMed] - Thenabadu, M.; Cheng, G.-L.; Pham, T.L.H.; Drummond, L.V.; Rosales-Zárate, L.; Reid, M.D. Testing macroscopic local realism using local nonlinear dynamics and time settings. Phys. Rev. A
**2020**, 102, 022202. [Google Scholar] [CrossRef] - Thenabadu, M.; Reid, M.D. Leggett–Garg tests of macrorealism for dynamical cat states evolving in a nonlinear medium. Phys. Rev. A
**2019**, 99, 032125. [Google Scholar] - Thenabadu, M.; Reid, M.D. Macroscopic delayed-choice and retrocausality: Quantum eraser, Leggett–Garg and dimension witness tests with cat states. Phys. Rev. A
**2022**, 105, 062209. [Google Scholar] [CrossRef] - Dowling, J.P. Quantum optical metrology–the lowdown on high-n00n states. Contemp. Phys.
**2008**, 49, 125. [Google Scholar] [CrossRef] - Lipkin, H.J.; Meshkov, N.; Glick, A.J. Validity of many-body approximation methods for a solvable model: Exact solutions and perturbation theory. Nucl. Phys.
**1965**, 62, 188. [Google Scholar] [CrossRef] - Steel, M.; Collett, M.J. Quantum state of two trapped Bose-Einstein condensates with a Josephson coupling. Phys. Rev. A
**1998**, 57, 2920. [Google Scholar] [CrossRef] - Carr, L.D.; Dounas-Frazer, D.; Garcia-March, M.A. Dynamical realization of macroscopic superposition states of cold bosons in a tilted double well. Europhys. Lett.
**2010**, 90, 10005. [Google Scholar] - Fröwis, F.; Sekatski, P.; Dür, W.; Gisin, N.; Sangouard, N. Macroscopic quantum states: Measures, fragility, and implementations. Rev. Mod. Phys.
**2018**, 90, 025004. [Google Scholar] - Yurke, B.; Stoler, D. Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion. Phys. Rev. Lett.
**1986**, 57, 13. [Google Scholar] - Wineland, D.J. Nobel Lecture: Superposition, entanglement, and raising Schrödinger’s cat. Rev. Mod. Phys.
**2013**, 85, 1103. [Google Scholar] [CrossRef] - Leghtas, Z.; Kirchmair, G.; Vlastakis, B.; Devoret, M.H.; Schoelkopf, R.J.; Mirrahimi, M. Deterministic protocol for mapping a qubit to coherent state superpositions in a cavity. Phys. Rev. A
**2013**, 87, 042315. [Google Scholar] - Ourjoumtsev, A.; Jeong, H.; Tualle-Brouri, R.; Grangier, P. Generation of optical ‘Schrödinger cats’ from photon number states. Nature
**2007**, 448, 784. [Google Scholar] - Wang, C.; Gao, Y.Y.; Reinhold, P.; Heeres, R.W.; Ofek, N.; Chou, K.; Axline, C.; Reagor, M.; Blumoff, J.; Sliwa, K.M.; et al. A Schrödinger cat living in two boxes. Science
**2016**, 352, 1087. [Google Scholar] [CrossRef] - Wolinsky, M.; Carmichael, H.J. Quantum noise in the parametric oscillator: From squeezed states to coherent-state superpositions. Phys. Rev. Lett.
**1988**, 60, 1836. [Google Scholar] - Hach Edwin, E., III; Gerry Christopher, C. Generation of mixtures of Schrödinger-cat states from a competitive two-photon process. Phys. Rev. A
**1994**, 49, 490. [Google Scholar] [CrossRef] - Gilles, L.; Garraway, B.M.; Knight, P.L. Generation of nonclassical light by dissipative two-photon processes. Phys. Rev. A
**1994**, 49, 2785. [Google Scholar] [CrossRef] [PubMed] - Teh, R.Y.; Sun, F.-X.; Polkinghorne, R.; He, Q.; Gong, Q.; Drummond, P.D.; Reid, M.D. Dynamics of transient cat states in degenerate parametric oscillation with and without nonlinear kerr interactions. Phys. Rev. A
**2020**, 101, 043807. [Google Scholar] [CrossRef] - Ku, H.-Y.; Lambert, N.; Chan, F.-J.; Emary, C.; Chen, Y.-N.; Nori, F. Experimental test of non-macrorealistic cat states in the cloud. npj Quantum Inf.
**2020**, 6, 98. [Google Scholar] [CrossRef] - Omran, A.; Levine, H.; Keesling, A.; Semeghini, G.; Wang, T.T.; Ebadi, S.; Bernien, H.; Zibrov, A.S.; Pichler, H.; Choi, S.; et al. Generation and manipulation of Schrödinger cat states in Rydberg atom arrays. Science
**2019**, 365, 570. [Google Scholar] [CrossRef] [PubMed] - Wright, E.; Walls, D.; Garrison, J. Collapses and Revivals of Bose-Einstein Condensates Formed in Small Atomic Samples. Phys. Rev. Lett.
**1996**, 77, 2158. [Google Scholar] [CrossRef] - Greiner, M.; Mandel, O.; Bloch, T.H.I. Collapse and revival of the matter wave field of a Bose-Einstein condensate. Nature
**2002**, 419, 51. [Google Scholar] [CrossRef] - Bell, J.S. On the Einstein-Podolsky-Rosen paradox. Physics
**1964**, 1, 195. [Google Scholar] [CrossRef] - Bell, J.S. On the Problem of Hidden Variables in Quantum Mechanics. Rev. Mod. Phys.
**1966**, 38, 447–452. [Google Scholar] [CrossRef] - Kochen, S.; Specker, E. The problem of hidden variables in quantum mechanics. J. Math. Andm.
**1967**, 17, 59. [Google Scholar] [CrossRef] - Maroney, O.J.E.; Timpson, C.G. Quantum- vs. Macro- Realism: What does the Leggett–Garg Inequality actually test? arXiv
**2017**, arXiv:1412.6139. [Google Scholar] - Bohm, D. A suggested interpretation of quantum theory in terms of “hidden” variables. Phys. Rev.
**1952**, 85, 166. [Google Scholar] [CrossRef] - Budiyono, A.; Rohrlich, D. Quantum mechanics as classical statistical mechanics with an ontic extension and an epistemic restriction. Nat. Commun.
**2017**, 8, 1306. [Google Scholar] [CrossRef] [PubMed] - Husimi, K. Some Formal Properties of the Density Matrix. Proc. Phys. Math. Soc. Jpn.
**1940**, 22, 264. [Google Scholar]

**Figure 1.**Realization of a nonlinear beam splitter: solutions are shown for the Hamiltonian ${H}_{kl}$ after a time t with initial state ${|N\rangle}_{k}{|0\rangle}_{l}$. Here $N=2$, $\kappa =1$, and $g=30$ (

**left**), and $N=5$, $\kappa =20$, and $g=333.33$ (

**right**). ${P}_{N}$ (black solid line) is the probability for all N bosons to be in mode k; ${P}_{0}$ (blue dashed line) is the probability for all N bosons to be in mode l. The parameters identify regimes optimal, or nearly optimal, for the nonlinear beam splitter interaction, where ${P}_{N}+{P}_{0}\sim 1$ and ${P}_{N}\sim {cos}^{2}{\omega}_{N}t$.

**Figure 2.**Creation of the superposition $|{\psi}_{sup}\rangle $ from $|3\rangle $, for $N=2$. The values of $\kappa $ and g are chosen as in Figure 1. Each sequence shows the initial state $|3\rangle $ (

**left**), the intermediate state ${U}_{1i}|3\rangle $ (

**center**), and the final state ${U}_{2i}{U}_{1i}|3\rangle $ (

**right**), where ${U}_{1i}={e}^{-i{H}_{32}{t}_{1i}/\hslash}$ and ${U}_{2i}={e}^{-i{H}_{21}{t}_{2i}/\hslash}$ for suitable choices of times ${t}_{1i}$ and ${t}_{2i}$. The probability that the system is in state $|k,l,m\rangle \equiv {|k\rangle}_{1}{|l\rangle}_{2}{|m\rangle}_{3}$ is depicted at the given time in the sequence. The probability that the system is in a state different to $|1\rangle $, $|2\rangle $, or $|3\rangle $ is less than $3\times {10}^{-3}$.

**Figure 3.**Creation of the superposition $|3\rangle $ from the post-selected state $|{\psi}_{f}\rangle $ using the operations ${U}_{f}$, for $N=5$. The values of $\kappa $ and g are chosen as in Figure 1. Each sequence shows the initial state $|{\psi}_{f}\rangle $ (

**left**), the intermediate state ${U}_{2f}^{-1}|{\psi}_{f}\rangle $ (

**center**), and the final state ${U}_{1f}^{-1}{U}_{2f}^{-1}|{\psi}_{f}\rangle $ (

**right**), where ${U}_{1f}\equiv {U}_{32}$ and ${U}_{2f}\equiv {U}_{21}$, as defined in the text. The probability that the system is in state $|k,l,m\rangle \equiv {|k\rangle}_{1}{|l\rangle}_{2}{|m\rangle}_{3}$ is depicted at the given time in the sequence. The probability that the system is in a state different to $|1\rangle $, $|2\rangle $, or $|3\rangle $ is less than $6\times {10}^{-3}$.

**Figure 4.**The dynamics corresponding to Alice’s transformations ${U}_{f}$ if (

**top row**) Bob detects N photons in box 1 at time ${t}_{2}$, or (

**lower row**) if Bob detects the photons are not in box 1 at time ${t}_{2}$. The histograms give the probabilities for detecting N photons in box K. The probability that the system is in state $|k,l,m\rangle \equiv {|k\rangle}_{1}{|l\rangle}_{2}{|m\rangle}_{3}$ is depicted at the given time in the sequence. Here, we show the initial state after Bob’s measurement at time ${t}_{2}$ (

**left**), the state generated at time ${t}_{2}^{\prime}$ after Alice’s transformation ${U}_{2f}^{-1}$ (

**center**), and the state generated at time ${t}_{3}$ after Alice’s further transformation ${U}_{1f}^{-1}$ (

**right**). The final state after Alice’s total transformation ${U}_{f}$ is (35) (top) or (37) (lower) to an excellent approximation. The probability that the system is in any other state apart from $|00N\rangle $, $|0N0\rangle $ or $|N00\rangle $ is less than $8\times {10}^{-3}$. The solutions are for $N=2$.

**Figure 5.**Dynamics of Alice’s transformation ${U}_{f}$ performed on the system in state $|{\psi}_{f}\rangle =-|1\rangle +|2\rangle +|3\rangle +|4\rangle $ at time ${t}_{2}$. The operation ${U}_{f1}$ is carried out by evolving under ${H}_{NL}$ for a time $t=3\pi /2\Omega $. Next, the operation ${U}_{f2}$ is carried out by evolving under ${H}_{NL}$ for a further time $t=\pi /\Omega $. At time ${t}_{3}$ after the combined operation ${U}_{f}={U}_{f2}{U}_{f1}$, the system is in state $|3\rangle $.

**Figure 6.**Dynamics of Alice’s transformation ${U}_{f}$ performed at time ${t}_{2}$ after Bob makes his measurement. Here, we take the case where Bob obtains the following results: (

**top**) that the ball is in box 1 (implying the system is in state $|1\rangle $ at time ${t}_{2}$) or (

**lower**) that the ball is in box 4 (implying the system is in state $|4\rangle $ at time ${t}_{2}$). The figures show contour plots of $Q\left(\alpha \right)$ for ${\alpha}_{0}=3$ and $k=3$. Here, ${U}_{f}={U}_{f2}{U}_{f1}$. The plots show the state after the first transformation ${U}_{f1}$ at time ${t}_{2}+3\pi /2$. In both cases, there is a finite probability of Alice finding the ball in box 3.

**Figure 7.**Dynamics of Alice’s transformation ${U}_{f}$ performed at time ${t}_{2}$ after Bob makes his measurement. Here, we take the case where Bob obtains the following results: (

**top**) that the ball is not in box 1 or 4 (implying the system in state $-|2\rangle +|3\rangle $ at time ${t}_{2}$) or (

**lower**) that the ball is not in box 2 or 4 (so that the system in state $|1\rangle +|3\rangle $ at time ${t}_{2}$). The figures show contour plots of $Q\left(\alpha \right)$ for ${\alpha}_{0}=3$ and $k=3$. In both cases, there is zero probability of Alice finding the ball in box 3.

**Figure 8.**Schematic depicting the assumptions of the weak macroscopic realism (wMR) model for the three-box paradox. The ball is placed in box 3 at time ${t}_{0}$. After some unitary transformations, the system at time ${t}_{1}$ is in a superposition $|{\psi}_{sup}\rangle $ of being in one of the three boxes. Bob’s measurement of the system at time ${t}_{1}$ consists of opening either box 1 or box 2. Alice performs the unitary transformations ${U}_{f}$ on the system at time ${t}_{2}$, after Bob’s measurement, and opens box 3 at time ${t}_{3}$. According to wMR, at each time ${t}_{k}$, the outcome of finding the ball in a given box I or not is fixed, prior to the observer opening the box I. The predetermined outcome can be denoted by the variable ${\lambda}_{I,k}$. Also, wMR implies the ball will be found in only one box, if all boxes were to be opened. The predetermined outcome for which box the ball would be found in at time ${t}_{k}$ can be denoted by the variable ${\tilde{\lambda}}_{k}$. Bob’s measurement is assumed accurate, and his outcome will be determined by either ${\lambda}_{1,1}$ or ${\lambda}_{2,1}$. The values of ${\lambda}_{I,k}$ and ${\tilde{\lambda}}_{k}$ (but not the details about the states) are unchanged on measurement.

**Figure 9.**Schematic depicting the assumptions of the weak macroscopic realism (wMR) model. The model is as for Figure 8. Here, Alice’s transformation occurs in two stages. The first unitary acts only on boxes 1 and 2. According to wMR, premise (2), the value for ${\lambda}_{3,2}$ cannot change due to this operation (although nothing is inferred about the state of box 3 changing). Similarly, the second stage of Alice’s operation involving only boxes 2 and 3 cannot change the value of ${\lambda}_{1,{2}^{\prime}}$.

**Figure 10.**Measurable macroscopic differences occur for the probabilities of outcomes at time ${t}_{3}$, depending on whether Bob makes a measurement or not, but the differences emerge only after the dynamics of Alice’s operations (shuffling). This is consistent with weak macroscopic realism (wMR) since in the wMR model, the values ${\lambda}_{I,1}$ are assumed not to change with Bob’s measurement (Figure 8). The probabilities $P\left({n}_{k}\right)$ of obtaining N on measuring the photon number $\widehat{n}$ of mode k at time t are plotted. The top sequence shows the sequence of probabilities if Bob makes no measurement. The lower sequence shows the probabilities if Bob makes a measurement between times ${t}_{1}$ and ${t}_{2}$. The plots on the right show the probabilities at time ${t}_{3}$, after Alice’s operation ${U}_{f}$, which takes place between times ${t}_{2}$ and ${t}_{3}$.

**Figure 11.**Validating the predictions of weak macroscopic realism (wMR): the sequence as Alice makes the operations ${U}_{f}$ on the states (

**top**) $|{\psi}_{sup}\rangle =|{\psi}_{1}\rangle $ and (

**lower**) ${\rho}_{3,mix}$, which is the state after Bob’s measurement detects the system in state $|3\rangle $ at time ${t}_{2}$. Alice first transforms (shuffling between boxes 1 and 2) according to ${U}_{f2}^{-1}={U}_{21}$, leaving box 3 untouched. Then, she transforms (shuffling between boxes 3 and 2) by ${U}_{1f}^{-1}={U}_{32}$ (see Figure 9). Far left are the plots of $P\left({n}_{k}\right)$ for the initial states (top) $|{\psi}_{sup}\rangle $ and (lower) ${\rho}_{mix}$. Second from left are the states after Alice’s transformation ${U}_{21}$. The predictions for $|{\psi}_{sup}\rangle $ and the mixture ${\rho}_{3,mix}$ are identical. Far right are the states after Alice’s transformation (shuffling between boxes 2 and 3) ${U}_{32}$, where the predictions diverge. The results are in agreement with the predictions of wMR (2), which posits that the system has a definite property ${\lambda}_{3,2}$ for box 3 (the ball is in the box or not) at time ${t}_{2}$, and this property cannot be changed by any shuffling ${U}_{21}$ on boxes 2 and 1. This is because throughout the shuffling ${U}_{21}$, the predictions are identical to those of the mixture ${\rho}_{3,mix}$, consistent with the definite value ${\lambda}_{3,2}$.

**Figure 12.**Q functions for the superposition $|{\psi}_{sup}\rangle $(Equation (44)) (

**left**), and the mixture ${\rho}_{mix,14}$(Equation (86)) (

**right**) become increasingly indistinguishable as the system size ${\alpha}_{0}$ increases. The top and lower pairs are for ${\alpha}_{0}=2$ and ${\alpha}_{0}=6$, respectively.

**Figure 13.**The dynamics of the macroscopic three-box paradox. The dynamics induced by Alice’s transformations ${U}_{f}$ are macroscopically sensitive to whether or not Bob has made a prior measurement of the system at time ${t}_{1}$, despite Bob’s measurement being seemingly non-invasive. The figure shows contour plots of $Q\left(\alpha \right)$ for the system prepared at time ${t}_{1}$ in thesuperposition state $|{\psi}_{sup}\rangle $ (Equation (44)) (

**far left**) as it evolves under the action of Bob and Alice’s operations. The top sequence shows the dynamics ifthere is no measurement made by Bob. Alice makes her transformation ${U}_{f}$ on the system starting at time ${t}_{2}$ (

**second from left**). The dynamics of the transformation ${U}_{f}$ are completed at time ${t}_{3}$. The lower sequence shows the evolution if Bob makes a measurement of the system at time ${t}_{1}$ (the outcome revealing whether the system is in the state $|1\rangle $ or $|4\rangle $, or not). At time ${t}_{2}$, after Bob’s measurement, the system is in the mixed state ${\rho}_{mix,14}$. While the Q function for ${\rho}_{mix,4}$ (

**lower second from left**) is indistinguishable from that of $|{\psi}_{sup}\rangle $ (

**top second from left**) as $\alpha \to \infty $, indicating a non-invasive measurement, there is a macroscopic difference between the final probabilities (

**top far right**and

**lower far right**) after the dynamics of Alice’s transformations, at time ${t}_{3}$. Here, ${\alpha}_{0}=6$ and $k=3$.

**Figure 14.**Schematic depicting how the weak macroscopic realism (wMR) model can be consistent with the strong Maroney–Leggett–Garg (MLG) test of macro-realism. The notation is as for Figure 8. In the proposed MLG test, the variable ${\lambda}_{k}$ takes the value $-1$ if the ball is in box 1 or 2, and $+1$ if the ball is in box 3. The MLG test gives justification of non-invasive measurability (NIM) because it can be verified that $\langle {\lambda}_{0}{\lambda}_{3}\rangle $ is unchanged depending on whether Bob makes a measurement or not (Condition (89)). Bob cannot fully evaluate ${\lambda}_{1}$ by his measurement in which he opens either box 1 or 2. His measurement does not change the value of ${\lambda}_{1}$, but it may, however, change the state of the system. This implies that the unitary operations ${U}_{f}$ employed by Alice can result in a different state of the system at time ${t}_{3}$, and hence different correlations $\langle {\lambda}_{1}{\lambda}_{3}\rangle $, depending on whether Bob makes a measurement.

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**MDPI and ACS Style**

Hatharasinghe, C.; Thenabadu, M.; Drummond, P.D.; Reid, M.D.
A Macroscopic Quantum Three-Box Paradox: Finding Consistency with Weak Macroscopic Realism. *Entropy* **2023**, *25*, 1620.
https://doi.org/10.3390/e25121620

**AMA Style**

Hatharasinghe C, Thenabadu M, Drummond PD, Reid MD.
A Macroscopic Quantum Three-Box Paradox: Finding Consistency with Weak Macroscopic Realism. *Entropy*. 2023; 25(12):1620.
https://doi.org/10.3390/e25121620

**Chicago/Turabian Style**

Hatharasinghe, Channa, Manushan Thenabadu, Peter D. Drummond, and Margaret D. Reid.
2023. "A Macroscopic Quantum Three-Box Paradox: Finding Consistency with Weak Macroscopic Realism" *Entropy* 25, no. 12: 1620.
https://doi.org/10.3390/e25121620