#
Experimental Implications of Negative Quantum Conditional Entropy—H_{2} Mobility in Nanoporous Materials

## Abstract

**:**

**2011**, 474, 61]. Extending this existing work, here we provide evidence of the significance of negative conditional entropy in a concrete experimental context: Incoherent Neutron Scattering (INS) from protons of H${}_{2}$ in nano-scale environments; e.g., in INS from H${}_{2}$ in C-nanotubes, the data of the H${}_{2}$ translational motion along the nanotube axis seems to show that the neutron apparently scatters from a fictitious particle with mass of 0.64 atomic mass units (a.m.u.)—instead of the value of 2 a.m.u. as conventionally expected. An independent second experiment confirms this finding. However, taking into account the possible negativity of conditional entropy, we explain that this effect has a natural interpretation in terms of quantum thermodynamics. Moreover, it is intrinsically related to the number of qubits capturing the interaction of the two quantum systems H${}_{2}$ and C-nanotube. The considered effect may have technological applications (e.g., in H-storage materials and fuel cells).

## 1. Introduction: “Information is Physical”

## 2. The Elitzur-Vaidman Effect, Google’s Quantum Processor, and the Physical Meaning of the Quantum State Vector

## 3. INS Measurement—Outline of Some Results

#### 3.1. Conventional Theory: Momentum and Energy Conservation in Two-Body Collisions

#### 3.2. Experimental Determination of Scatterer’s Mass

#### 3.3. INS from Bulk Ice-Ih—Conventional Theory

_{2}O molecules of ice.

## 4. The New Scattering Effect

#### 4.1. Example: INS from Single H${}_{2}$ Molecules in C-Nanotubes

#### 4.2. Second Example—INS from Single H${}_{2}$ Molecules in the Metal-Organic Framework Material “HKUST-1”

#### 4.3. Comparison of the Two Experiments

## 5. Quantum Heating and Quantum Cooling Due to Quantum Environment—Kurizki’s Model

## 6. Thermodynamic Meaning of Negative Conditional Entropy, Generalized Landauer’s Principle, and Interpretation of the INS Effect

#### 6.1. Interpretation of the E-Excess INS Effect

## 7. Quantum Correlations in Scattering Dynamics

## 8. Excess E-Transfer in Neutron-H Collision—Number of “Consumed” Qubits

## 9. Additional Remarks and Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

a.m.u. | Atomic Mass Unit |

FSE | Final-State Effects |

IFM | Interaction Free Measurement |

IA | Impulse Approximation |

IINS | Inelastic Incoherent Neutron Scattering |

INS | Equivalent to IINS |

IT | Information theory |

meV | Milli-Electron Volt |

MZI | Mach–Zehnder Interferometer |

NP | nondeterministic polynomial |

QE | Quantum Entanglement |

QTD | Quantum Thermodynamics |

QIT | Quantum Information Theory |

QoC | Quantumness of Correlations |

QND | Quantum Non-Demolition |

TOF | Time-of-Flight |

TSVF | Two-State Vector Formalism |

WV | Weak Value |

## Appendix A. Experimental Context—Incoherent Neutron Scattering

#### Appendix A.1. Incoherent Inelastic Neutron Scattering from Protons

**Figure A1.**Scheme of a time-of-flight neutron scattering setup. (Taken from [26], with permission of Quanta).

#### Appendix A.2. Details of Scattering Spectrometer, Calibration and “What Is Measured”

## References

- Horodecki, R.; Horodecki, P.; Horodecki, M.; Horodecki, K. Quantum entanglement. Rev. Mod. Phys.
**2009**, 81, 865–942. [Google Scholar] [CrossRef][Green Version] - Vedral, V. Quantum entanglement. Nat. Phys.
**2014**, 10, 256–258. [Google Scholar] [CrossRef] - Henderson, L.; Vedral, V. Classical, quantum and total correlations. J. Phys. A Math. Gen.
**2001**, 34, 6899–6905. [Google Scholar] [CrossRef] - Ollivier, H.; Zurek, W.H. Quantum discord: A measure of the quantumness of correlations. Phys. Rev. Lett.
**2002**, 88, 017901. [Google Scholar] [CrossRef] [PubMed] - Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Landauer, R. Information is Physical. Phys. Today
**1991**, 44, 23. [Google Scholar] [CrossRef] - Feynman, R.P.; Leighton, R.B.; Sands, M. The Feynman Lectures on Physics, Volume III, Quantum Mechanics; Addison-Wesley: Reading, MA, USA, 1965. [Google Scholar]
- Squires, G.L. Introduction to the Theory, of Thermal Neutron Scattering, 2nd ed.; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- van Hove, L. Correlations in space and time and Born approximation scattering in systems of interacting particles. Phys. Rev.
**1954**, 95, 249–262. [Google Scholar] [CrossRef][Green Version] - Watson, G.I. Neutron Compton scattering. J. Phys. Condens. Matter
**1996**, 8, 5955–5975. [Google Scholar] [CrossRef][Green Version] - Mitchell, P.C.H.; Parker, S.F.; Ramirez-Cuesta, A.J.; Tomkinson, J. Vibrational Spectroscopy with Neutrons; World Scientific: Singapore, 2005. [Google Scholar]
- Cerf, N.J.; Adami, C. Negative entropy and information in quantum mechanics. Phys. Rev. Lett.
**1997**, 79, 5194–5197. [Google Scholar] [CrossRef][Green Version] - del Rio, L.; Åberg, J.; Renner, R.; Dahlsten, O.; Vedral, V. The thermodynamic meaning of negative entropy. Nature
**2011**, 474, 61–63, (See also Addendum at doi:10.1038/nature10395). [Google Scholar] - Modi, K.; Brodutch, A.; Cable, H.; Paterek, T.; Vedral, V. The classical-quantum boundary for correlations: Discord and related measures. Rev. Mod. Phys.
**2012**, 84, 1655–1707. [Google Scholar] [CrossRef][Green Version] - Available online: https://neutrons.ornl.gov/ARCS (accessed on 19 October 2020).
- Available online: https://www.isis.stfc.ac.uk/Pages/mari.aspx (accessed on 19 October 2020).
- Chatzidimitriou-Dreismann, C.A. Quantumness of correlations and Maxwell’s demon in molecular excitations created by neutron scattering. Int. J Quantum Chem.
**2015**, 115, 909–929. [Google Scholar] [CrossRef] - Chatzidimitriou-Dreismann, C.A. Quantum Confinement Effects of Hydrogen in Nanocavities–Experimental INS Results and New Insights. Recent Prog. Mater.
**2020**, 2, 53. [Google Scholar] [CrossRef] - Leifer, M.S. Is the Quantum State Real? An Extended Review of ψ-ontology Theorems. Quanta
**2014**, 3, 67–155. [Google Scholar] [CrossRef] - Elitzur, A.C.; Vaidman, L. Quantum mechanical interaction-free measurements. Found. Phys.
**1993**, 23, 987–997. [Google Scholar] [CrossRef][Green Version] - Arute, F.; Arya, K.; Babbush, R.; Bacon, D.; Bardin, J.C.; Barends, R.; Biswas, R.; Boixo, S.; Brandao, F.G.S.L.; Buell, D.A.; et al. Quantum supremacy using a programmable superconducting processor. Nature
**2019**, 574, 505–510. [Google Scholar] [CrossRef][Green Version] - Wang, H.; Qin, J.; Ding, X.; Chen, M.C.; Chen, S.; You, X.; He, Y.M.; Jiang, X.; You, L.; Wang, Z.; et al. Boson sampling with 20 input photons and a 60-Mode interferometer in a 10
^{14}-dimensional Hilbert space. Phys. Rev. Lett.**2019**, 123, 250503. [Google Scholar] [CrossRef][Green Version] - Arora, S.; Barak, B. Computational Complexity—A Modern Approach; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Fortnow, L. The status of the P versus NP problem. Commun. ACM
**2009**, 52, 78–86. [Google Scholar] [CrossRef][Green Version] - Diallo, S.O.; Azuah, R.T.T.; Abernathy, D.L.; Rota, R.; Boronat, J.; Glyde, H.R. Bose–Einstein condensation in liquid
^{4}He near the liquid-solid transition line. Phys. Rev. B**2012**, 85, 140505. [Google Scholar] [CrossRef][Green Version] - Chatzidimitriou-Dreismann, C.A. Weak measurement and Two-State-Vector formalism: Deficit of momentum transfer in scattering processes. Quanta
**2016**, 5, 61–84. [Google Scholar] [CrossRef][Green Version] - Kearley, G.J.; Fillaux, F.; Baron, M.H.; Bennington, S.; Tomkinson, J. A new look at proton transfer dynamics along the hydrogen bonds in amides and peptides. Science
**1994**, 264, 1285–1289. [Google Scholar] [CrossRef] - Olsen, R.J.; Beckner, M.; Stone, M.B.; Pfeifer, P.; Wexler, C.; Taub, H. Quantum excitation spectrum of hydrogen adsorbed in nanoporous carbons observed by inelastic neutron scattering. Carbon
**2013**, 58, 46–58. [Google Scholar] [CrossRef] - Callear, S.K.; Ramirez-Cuesta, A.J.; David, W.I.F.; Millange, F.; Walton, R.I. High-resolution inelastic neutron scattering and neutron powder diffraction study of the adsorption of dihydrogen by the Cu(II) metal–organic framework material HKUST-1. Chem. Phys.
**2013**, 427, 9–17. [Google Scholar] [CrossRef] - Erez, N.; Gordon, G.; Nest, M.; Kurizki, G. Thermodynamic constrol of frequent quantum measurements. Nature
**2008**, 452, 724–727. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gordon, G.; Bensky, G.; Gelbwaser-Klimovsky, D.; Bhaktavatsala Rao, D.D.; Erez, N.; Kurizki, G. Cooling down quantum bits on ultrashort time scales. New J. Phys.
**2009**, 11, 123025. [Google Scholar] [CrossRef] - Scully, M.O.; Zubairy, M.S. Quantum Optics; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- von Neumann, J. Mathematical Foundations of Quantum Mechanics; Princeton University Press: Princeton, NJ, USA, 1955. [Google Scholar]
- Shannon, C.E.; Weaver, W. The Mathematical Theory of Communication; University of Illinois: Urbana, IL, USA, 1949. [Google Scholar]
- Landauer, R. Dissipation and heat generation in the computing process. IBM J. Res. Develop.
**1961**, 5, 148–156. [Google Scholar] [CrossRef] - Bennett, C.H. The thermodynamics of computation—A review. Int. J. Theor. Phys.
**1982**, 21, 905–940. [Google Scholar] [CrossRef] - Bennett, C.H. Notes on Landauer’s principle, reversible computation and Maxwell’s demon. Stud. Hist. Philos. Mod. Phys.
**2003**, 34, 501–510. [Google Scholar] [CrossRef][Green Version] - Chatzidimitriou-Dreismann, C.A.; Gray, E.M.; Blach, T.P. Distinguishing new science from calibration effects in the electron-volt neutron spectrometer Vesuvio at ISIS. Nucl. Instr. Meth. A
**2012**, 676, 120–125. [Google Scholar] [CrossRef] - Chatzidimitriou-Dreismann, C.A.; Gray, E.M.; Blach, T.P. Indications of energetic consequences of decoherence at short times for scattering from open quantum systems. AIP Adv.
**2011**, 1, 022118. [Google Scholar] [CrossRef][Green Version] - Chatzidimitriou-Dreismann, C.A. Weak values and two-state-vector formalism in elementary scattering and reflectivity—A new effect. Universe
**2019**, 5, 58. [Google Scholar] [CrossRef][Green Version] - Aharonov, Y.; Rohrlich, D. Quantum Paradoxes: Quantum Theory for the Perplexed; WILEY-VCH: Weinheim, Germany, 2005. [Google Scholar]
- Dressel, J.; Malik, M.; Miatto, F.M.; Jordan, A.N.; Boyd, R.W. Colloquium: Understanding quantum weak values: Basics and applications. Rev. Mod. Phys.
**2014**, 86, 307–316. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Schematic representation (blue points) of measured dynamic structure factor $S(K,E)$ of liquid helium [25]. The red line is the calculated recoil parabola, for the mass of ${}^{4}$He, shown as a guide to the eye. The white-blue ribbon around the recoil parabola represent data points measured with the time-of-flight spectrometer ARCS [15] (Figure taken from reference [26] with permission of Quanta).

**Figure 2.**Experimental $S(K,E)$ intensity map of ice Ih at 20 K and 1 bar, recorded on MARI [16]. ${E}_{0}=750$ meV. The shown black line (parabola) is the conventional recoil trajectory of a free H. The strong intensity ribbon at $E\approx 0$ is mainly due to the aluminum cell containing the water (ice). The intensity peak centered at $E\approx 420$ meV and $K\approx 14$ Å${}^{-1}$ represents the vibrational stretching OH-vibrations. See the discussion in the text. (Reproduced from reference [17] with permission.)

**Figure 3.**(

**a**) Cartoon of various molecular motions of H${}_{2}$ in a nanotube, as conventionally expected. (

**b**) Experimental Incoherent Neutron Scattering (INS) results from H${}_{2}$ in carbon nanotubes. Incident neutron energy: ${E}_{0}=90$ meV; Figure 1 of [28]. The translation motion of the recoiling H${}_{2}$ molecules causes the observed continuum of intensity, usually called “roto-recoil” (yellow-orange ribbon). The $K-E$ position of the first rotational excitation of H${}_{2}$ (dark red-brown ellipsoid). agrees with conventional theoretical expectations. In clear contrast, a detailed fit (green parabola) to the roto-recoil data yields the effective mass of translating H${}_{2}$ to be ${M}_{eff}\approx 0.64$ a.m.u. The red parabola on the right shows the conventional-theoretical parabola associated with the H${}_{2}$-mass $M=2$ a.m.u. See the text for more explanations. (Reproduced from [28], with permission of Elsevier Ltd.).

**Figure 4.**(

**a**): The $S(Q,E)$ map of hydrogen for 1(p-H2):Cu in HKUST-1 for an incident energy of 75 meV measured using the MARI spectrometer. The green broad intensity ribbon is due to the translational quantum dynamics of H${}_{2}$ and corresponds to the associated broad intensity ribbon shown in the preceding Figure 3. (

**b**): Series of cuts along momentum transfer at a series of energy transfers corresponding to the intensity peaks shown in the right panel, i.e., the integrated (over momentum transfer) intensity for along energy transfer for the $S(Q,E)$ map shown in panel (

**a**). (Reproduced from [29], with permission of Elsevier Ltd.).

**Figure 5.**Comparison of the two experimental results: Experimental $S(K,E)$ maps of the two original papers, appropriately stretched/scaled to facilitate comparison. (

**a**): data from [29]; (

**b**) data from [28]. The shown two vertical dashed lines (white) are guides to the eye. The two broad continuous ribbons due to the translational modes are virtually identically positioned in the momentum-energy plane $K-E$.

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Chatzidimitriou-Dreismann, C.A.
Experimental Implications of Negative Quantum Conditional Entropy—H_{2} Mobility in Nanoporous Materials. *Appl. Sci.* **2020**, *10*, 8266.
https://doi.org/10.3390/app10228266

**AMA Style**

Chatzidimitriou-Dreismann CA.
Experimental Implications of Negative Quantum Conditional Entropy—H_{2} Mobility in Nanoporous Materials. *Applied Sciences*. 2020; 10(22):8266.
https://doi.org/10.3390/app10228266

**Chicago/Turabian Style**

Chatzidimitriou-Dreismann, C. Aris.
2020. "Experimental Implications of Negative Quantum Conditional Entropy—H_{2} Mobility in Nanoporous Materials" *Applied Sciences* 10, no. 22: 8266.
https://doi.org/10.3390/app10228266