During the last few decades, considerable advances in quantum information theory have shown deep existing connections between quantum correlation effects (like entanglement and quantum discord) and thermodynamics. Here the concept of conditional entropy plays a considerable role. In contrast to the classical case, quantum conditional entropy can take negative values. This counter-intuitive feature, already well understood in the context of information theory, was recently shown theoretically to also have a physical meaning in quantum thermodynamics [del Rio et al. Nature 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 H2 in nano-scale environments; e.g., in INS from H2 in C-nanotubes, the data of the H2 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 H2 and C-nanotube. The considered effect may have technological applications (e.g., in H-storage materials and fuel cells).
During the last few decades, considerable advances in quantum information theory have shown deep existing connections between quantum correlation effects (like entanglement and quantum discord) and thermodynamics. Here the concept of conditional entropy plays a considerable role. In contrast to the classical case, quantum conditional entropy can take negative values. This counter-intuitive feature, already well understood in the context of information theory, was recently shown theoretically to also have a physical meaning in quantum thermodynamics [del Rio et al. Nature 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 H2 in nano-scale environments; e.g., in INS from H2 in C-nanotubes, the data of the H2 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 H2 and C-nanotube. The considered effect may have technological applications (e.g., in H-storage materials and fuel cells).
Current developments of non-relativistic quantum mechanics appear to predict, and reveal, counter-intuitive dynamical effects of hydrogen in nanostructured materials that are of considerable importance for basic research as well as for technological applications. In this review, the experimental focus is on H2O and H molecules in carbon nanotubes and other nanocavities that have been experimentally investigated using the well-established technique of incoherent inelastic neutron scattering (INS). For instance, the momentum and energy transfers, as obtained from the commonly used standard data analysis techniques, from a (I) H2 molecule in a C-nanotube resulting in a roto-translational motion along the nanotube axis seems to (1) either violate the standard conservation laws or (2) to attribute to the H molecule undergoing translation the effective mass a.m.u. (atomic mass units) instead of the expected 2 a.m.u. A similar striking anomalous effect has been found in the neutron-H scattering from the (II) H2O molecules in nano-channels of some solid materials, in which O-H stretching vibrations along the channel axis are created. The results of this scattering process seem to once again either violate the standard conservation laws or to attribute to the effective mass of the struck H2 molecule as a.m.u. instead of the expected value of 1 a.m.u. We show that these counterintuitive observations from the INS studies have no conventional interpretation within the standard non-relativistic scattering theory. However, they can be qualitatively interpreted "from first principles" within the framework of modern theories of (III) time-symmetric quantum dynamics, as provided by the weak values (WV) and two-state- vector formalism (TSVF), and/or (IV) quantum correlations, especially quantum discord (QD) and quantum thermodynamics (QTD). The theoretical analysis provides an intuitive understanding of the experimental results, gives strong evidence that the nano-structured cavities do represent quantum systems which participate significantly in the dynamics of the neutron-H scattering, and, surprisingly, shows that new physical information can be derived from the experimental data. This latter point may also have far-reaching consequences for technology and material sciences (e.g., fuel cells, H storage materials, etc.). Moreover, novel insights into the short-lived quantum dynamics and/or quantum information theory can be gained.
Quantum correlations phenomena, such as entanglement, quantum discord and quantum coherence, are ubiquitous effects caused by interactions between physical systems—such as electrons and ions in a piece of metal, or H atoms/molecules adsorbed in nanoporous materials. Here, we address time-asymmetric quantumness of correlations (QoC), with particular emphasis on their energetic consequences for dynamics and non-equilibrium thermodynamics in condensed matter and/or many-body systems. Some known theoretical models—for example, the quantum Zeno effect and GKSL-type Markovian equations-of-motion, all of them being time-asymmetric—are shortly considered, with emphasis on the general character of one of their common and most intriguing result. Namely, that in clear contradistinction to conventional expectations, degradation (or destruction, decoherence, consumption, smearing out, coarse-graining) of quantum correlations can be a source of work (instead of heat production). The experimental relevance of the theoretical considerations is shown with the aid of a recent scattering experiment exploring the impulsively driven (by neutron collisions) translational dynamics of H$ _2 $ molecules in carbon nanotubes and other nanostructured materials—a topic of immediate relevance for material sciences and related technologies.
The concepts of Weak Values (WV) and Two-State Vector Formalism (TSVF) appear to motivate new experiments and to offer novel insights into dynamical processes in various materials of several scientific and technological fields. To support this view, here we consider the dynamics of hydrogen atoms and/or molecules in nanostructured materials like e.g., carbon nanotubes. The experimental method applied is incoherent scattering of thermal (i.e., non-relativistic) neutrons (INS). In short, the main finding consists in the following effect: the measured energy and momentum transfers are shown to contradict even qualitatively the associated expectations of conventional scattering theory. This effect was recently observed in INS experiments, e.g., in H2 adsorbed in carbon nanotubes, where a large momentum transfer deficit was found. Due to the broad abundance of hydrogen, these findings may be also of technological importance, since they indicate a considerably enhanced H mobility in specific structured material environments. A new INS experiment is proposed concerning the H mobility of an ultra-fast proton conductor (H3OSbTeO6) being of technological relevance. Further neutron scattering investigations on other systems (metallic hydrides and H2 encapsulated inside C60) are proposed. As concerns theoretical implications, the analysis of the experimental results strongly supports the view that the wavefunction (or state vector) represents an ontological physical entity of a single quantum system.
Chapter 7 discusses quantum conditional entropy, extending the concept of conditional majorization and introducing the notion of negative quantum conditional entropy. The chapter starts with the basic definition of conditional entropy, exploring its key properties like monotonicity and additivity. It further delves into the concepts of conditional min- and max-entropies, emphasizing their roles in quantifying uncertainty in quantum states and their operational significance in quantum information theory. The text presents conditional entropy as a measure sensitive to the effects of entanglement, showing that negative conditional entropy is a distinctive feature of quantum systems, contrasting with the classical domain where entropy values are nonnegative. This negativity is particularly pronounced in the context of maximally entangled states and is connected to the fundamental differences between classical and quantum information processing. Moreover, the chapter includes theorems and exercises to solidify understanding, like the invariance of conditional entropy under local isometric channels and its reduction to entropy for product states. It concludes by underscoring the inevitability of negative conditional entropy in quantum systems, a topic of both theoretical and practical importance in the quantum domain.
Abstract Weak Values (WV) and Two-State-Vector Formalism (TSVF) provide novel insights in various quantum physical and technological fields. In the first part of the paper we consider a new quantum effect of scattering accompanying an elementary collision of two quantum systems A and B, in which the latter interacts with a quantum environment. In clear contrast to a classical environment, the quantum case can exhibit counter-intuitive effects of momentum- and energy-transfer which contradict conventional expectations. Experimental evidence of a new effect—deficit of momentum transfer (equivalently: reduced effective mass) in a neutron-atom collision—is presented and theoretically interpreted. Here, non-relativistic incoherent inelastic neutron scattering (INS) is applied. INS on single H2 molecules confined in multi-walled carbon nanotube channels has been experimentally investigated. Interpreted within conventional theory, the results reveal a counter-intuitive reduced effective mass of the recoiling H2 molecule, i.e. M = 0.64 a.m.u. (atomic mass units). In contrast, this finding has a simple qualitative interpretation within WV and TSVF theory. In the second part of the paper we report on current experimental and theoretical investigations in the field of X-ray diffraction (XRD), which belongs to coherent scattering. Preliminary XRD results from cubic crystalline materials show a surprising variation of the measured lattice parameter (usually called “lattice constant”) with momentum transfer. A first theoretical model of the effect in the light of the new theory is presented. These findings give further evidence for the broad character and significance of the novel WV and TSVF theory.
A colloquial interpretation of entropy is that it is the knowledge gained upon learning the outcome of a random experiment. Conditional entropy is then interpreted as the knowledge gained upon learning the outcome of one random experiment after learning the outcome of another, possibly statistically dependent, random experiment. In the classical world, entropy and conditional entropy take only non-negative values, consistent with the intuition that one has regarding the aforementioned interpretations. However, for certain entangled states, one obtains negative values when evaluating commonly accepted and information-theoretically justified formulas for the quantum conditional entropy, leading to the confounding conclusion that one can know less than nothing in the quantum world. Here, we introduce a physically motivated framework for defining quantum conditional entropy, based on two simple postulates inspired by the second law of thermodynamics (non-decrease of entropy) and extensivity of entropy, and we argue that all plausible definitions of quantum conditional entropy should respect these two postulates. We then prove that all plausible quantum conditional entropies take on negative values for certain entangled states, so that it is inevitable that one can know less than nothing in the quantum world. All of our arguments are based on constructions of physical processes that respect the first postulate, the one inspired by the second law of thermodynamics.