Many quantum chemical similarity measures have been derived and substantiated by applying concepts and quantities from information theory to the electron density. To justify the use of information theory, the electron density is usually equated to a probability distribution, despite the fact that such an assumption can give rise to inconsistencies such as negative Kullback-Leibler divergences. In this work we show, using quantum information theory, that both interpreting electron densities as probability distributions as well as any pragmatic normalization thereof gives rise to far-reaching information theoretical inconsistencies. By applying the theory of open quantum subsystems to quantum chemical partitions, we show using thought experiments that many statements concerning the information content of atoms in molecules need to be reconsidered. This study represents a step towards generalizing and reformulating the information theoretical approach to conceptual quantum chemistry beyond its classical perspective.
Many quantum chemical similarity measures have been derived and substantiated by applying concepts and quantities from information theory to the electron density. To justify the use of information theory, the electron density is usually equated to a probability distribution, despite the fact that such an assumption can give rise to inconsistencies such as negative Kullback-Leibler divergences. In this work we show, using quantum information theory, that both interpreting electron densities as probability distributions as well as any pragmatic normalization thereof gives rise to far-reaching information theoretical inconsistencies. By applying the theory of open quantum subsystems to quantum chemical partitions, we show using thought experiments that many statements concerning the information content of atoms in molecules need to be reconsidered. This study represents a step towards generalizing and reformulating the information theoretical approach to conceptual quantum chemistry beyond its classical perspective.
Scientists often refer to matter, energy, and information as the three basic categories, though they are related by various scientific laws. Compared to the concepts of matter and energy, the scientific definition of information is relatively new: it was only in the 1940s that Claude Shannon introduced the definition of information as a reduction of uncertainty, laying the foundation for modern information theory. In molecular electronic structure theory, we routinely encounter nonnegative quantities that sum to unity, forming valid probability distributions. These include properly normalized electron densities, eigenvalues of reduced density matrices, squared wave function coefficients in orthonormal bases, and related quantities. The incorporation of information-theoretic concepts into quantum chemistry has yielded profound insights into electronic system behavior. The two dominant frameworks in electronic structure theory are quantum many-body theory and density functional theory. This thesis systematically integrates these approaches with: classical information theory, yielding the information-theoretic approach, and quantum information theory, generating distinct tools like orbital entanglement measures and quantum fidelity metrics. In this thesis, we subsequently demonstrate applications of these information-theoretic methods in quantum chemistry for analyzing electronic structure and chemical phenomena.
Background: The resultant measures of the information content in molecular electronic states are revisited. In these overall measures the current-related terms complement the probability functionals of classical Information Theory. The nonclassical Shannon entropy reflects the average magnitude of the state phase distribution, while the related current term in the complementary Fisher measure accounts for the average phase-gradient. Method: These generalized Fisher and Shannon descriptors of the information content in complex electronic states are applied to determine the phase-equilibria in molecules. The vertical and horizontal equilibria in molecules, which mark the extrema of the nonclassical and resultant entropy/information descriptors, respectively, are explored. The requirement, that the resultant measures have common solutions of the associated extreme entropy/information principles, calls for the negative sign of the nonclassical gradient measure. Conclusion: These extrema imply the phase-transformation of molecular electronic states. A separation of the density (modulus) and current (phase) factors in many-electron states is effected using the Harriman-Zumbach-Maschke construction of complex antisymmetric states yielding the specified electron density. A combined description of molecular systems, which accounts for the density and current degrees-of-freedom of electronic states is summarized and continuity equations for the state phase and resultant entropy are discussed. Keywords: Continuity equations, current-information, information continuity, molecular equilibria, phase-entropy/information, entropy.
The entropic perspective on the molecular electronic structure is investigated. Information-theoretic description of electron probabilities is extended to cover the complex amplitudes (wave functions) of quantum mechanics. This analysis emphasizes the entropic concepts due to the phase part of electronic states, which generates the probability current density, thus allowing one to distinguish the information content of states generating the same electron density and differing in their current densities. The classical information measures of Fisher and Shannon, due to the probability/density distributions themselves, are supplemented by the nonclassical terms generated by the wave-function phase or the associated probability current. A complementary character of the Fisher and Shannon information measures is explored and the relationship between these classical information densities is derived. It is postulated to characterize also their nonclassical (phase/current-dependent) contributions. The continuity equations of the generalized information densities are examined and the associated nonclassical information sources are identified. The variational rules involving the quantum-generalized Shannon entropy, which generate the stationary and time-dependent Schrödinger equations from the relevant maximum entropy principles, are discussed and their implications for the system “thermodynamic” equilibrium states are examined. It is demonstrated that the lowest, stationary “thermodynamic” state differs from the true ground state of the system, by exhibiting the space-dependent phase, linked to the modulus part of the wave function, and hence also a nonvanishing probability current.
The classical Information Theory (IT) deals with entropic descriptors of the probability distributions and probability-propagation (communication) systems, e.g., the electronic channels in molecules reflecting the information scattering via the system chemical bonds. The quantum IT additionally accounts for the non-classical (current/phase)-related contributions in the resultant information content of electronic states. The classical and non-classical terms in the quantum Shannon entropy and Fisher information are reexamined. The associated probability-propagation and current-scattering networks are introduced and their Fisher- and Shannon-type descriptors are identified. The non-additive and additive information descriptors of the probability channels in both the Atomic Orbital and local resolution levels are related to the network conditional-entropy and mutual-information, which represent the IT covalency and ionicity components in the classical communication theory of the chemical bond. A similar partition identifies the associated bond indices in the molecular current/phase channels. The resultant bond descriptors combining the classical and non-classical terms, due to the probability and current distributions, respectively, are proposed as generalized communication-noise (covalency) and information-flow (iconicity) concepts in the quantum IT.
It is the ultimate goal of this work to foster synergy between quantum chemistry and the flourishing field of quantum information theory. For this, we first translate quantum information concepts, such as entanglement and correlation, into the context of quantum chemical systems. In particular, we establish two conceptually distinct perspectives on 'electron correlation', leading to a notion of orbital and particle correlation. We then demonstrate that particle correlation equals total orbital correlation minimized over all orbital bases. Accordingly, particle correlation resembles the minimal, thus intrinsic, complexity of many-electron wave functions, while orbital correlation quantifies their complexity relative to a basis. We illustrate these concepts of intrinsic and extrinsic correlation complexity in molecular systems, which also manifests the crucial link between the two correlation pictures. Our results provide theoretical justification for the long-favored natural orbitals for simplifying electronic structures, and open new pathways for developing more efficient approaches towards the electron correlation problem.
A recent development in quantum chemistry has established the quantum mutual information between orbitals as a major descriptor of electronic structure. This has already facilitated remarkable improvements in numerical methods and may lead to a more comprehensive foundation for chemical bonding theory. Building on this promising development, our work provides a refined discussion of quantum information theoretical concepts by introducing the physical correlation and its separation into classical and quantum parts as distinctive quantifiers of electronic structure. In particular, we succeed in quantifying the entanglement. Intriguingly, our results for different molecules reveal that the total correlation between orbitals is mainly classical, raising questions about the general significance of entanglement in chemical bonding. Our work also shows that implementing the fundamental particle number superselection rule, so far not accounted for in quantum chemistry, removes a major part of correlation and entanglement seen previously. In that respect, realizing quantum information processing tasks with molecular systems might be more challenging than anticipated.