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Entropy

Surprisal Analysis-Based Compaction of Entangled Molecular States of Maximal Entropy

James Russell Hamilton, F. Remacle, Raphael David Levine

Published Feb 9, 2026• 1 citation
Publisher

Abstract

An attosecond optical pulse can entangle coherently related states of different characters, such as electronic and vibrational, in a molecular system. Using a quantum information theoretic approach, we explicitly define and discuss the surprisal of such a system in the maximal entropy formalism and identify the constraints and their conjugate Lagrange multipliers. Surprisal analysis shows how these constraints become fewer and simpler in the sudden approximation of the dynamics, a limit often valid for an ultrafast excitation. The optically accessible lower electronic states of N2 are used as a numerical example to show the compaction of the dynamics from On2 down to On constraints, where n is the number of vibronic states. The von Neumann entropy is used to confirm the fidelity of the compaction.

DOI
10.3390/e28020192

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An attosecond optical pulse can entangle coherently related states of different characters, such as electronic and vibrational, in a molecular system. Using a quantum information theoretic approach, we explicitly define and discuss the surprisal of such a system in the maximal entropy formalism and identify the constraints and their conjugate Lagrange multipliers. Surprisal analysis shows how these constraints become fewer and simpler in the sudden approximation of the dynamics, a limit often valid for an ultrafast excitation. The optically accessible lower electronic states of N2 are used as a numerical example to show the compaction of the dynamics from On2 down to On constraints, where n is the number of vibronic states. The von Neumann entropy is used to confirm the fidelity of the compaction.
Progress toward quantum technologies continues to provide essential new insights into the microscopic dynamics of systems in phase space. This highlights coherence effects whether these are due to ultrafast lasers whose energy width spans several states all the way to the output of quantum computing. Surprisal analysis has provided seminal insights into the probability distributions of quantum systems from elementary particle and also nuclear physics through molecular reaction dynamics to system biology. It is therefore necessary to extend surprisal analysis to the full quantum regime where it characterizes not only the probabilities of states but also their coherence. In principle, this can be done by the maximal entropy formalism, but in the full quantum regime, its application is far from trivial [S. Dagan and Y. Dothan, Phys. Rev. D 26, 248 (1982)] because an exponential function of non-commuting operators is not easily accommodated. Starting from an exact dynamical approach, we develop a description of the dynamics where the quantum mechanical surprisal, a linear combination of operators, plays a central role. We provide an explicit route to the Lagrange multipliers of the system and identify those operators that act as the dominant constraints.
We characterize both entanglement and quantum coherence in a molecular system by connecting the linear entropy of electronic-nuclear entanglement with Wigner-Yanase skew information measuring vibronic coherence and local quantum uncertainty on electronic energy. Linear entropy of entanglement and quantifiers of quantum coherence are derived for a molecular system described in a bipartite Hilbert space H=Hel⨂Hvib\mathcal{H}={\mathcal{H}}_{el}⨂{\mathcal{H}}_{vib}H=Hel​⨂Hvib​ of finite dimension ${N}_{el}\ifmmode\times\else\texttimes\fi{}{N}_{v}$, and relations between them are established. For the specific case of the electronic-vibrational entanglement, we find the linear entropy of entanglement as having a more complex informational content than the von Neumann entropy. By keeping the information carried by the vibronic coherences in a molecule, linear entropy seizes vibrational motion in the electronic potentials as entanglement dynamics. We analyze entanglement oscillations in an isolated molecule, and show examples for the control of entanglement dynamics in a molecule through the creation of coherent vibrational wave packets in several electronic potentials by using chirped laser pulses.
We introduce a practical method for compacting the time evolution of the quantum state of a closed physical system. The density matrix is specified as a function of a few time-independent observables where their coefficients are time-dependent. The key mathematical step is the vectorization of the surprisal, the logarithm of the density matrix, at each time point of interest. The time span used depends on the required spectral resolution. The entire course of the system evolution is represented as a matrix where each column is the vectorized surprisal at the given time point. Using singular value decomposition, SVD, of this matrix we generate realistic approximations for the time-independent observables and their respective time dependent coefficients. This allows a simplification of the algebraic procedure for determining the dominant constraints (the time-independent observables) in the sense of the maximal entropy approach. A nonstationary coherent initial state of a Morse oscillator is used to introduce the approach. We derive analytical exact expression for the surprisal as a function of time and this offers a benchmark for comparison with the accurate but approximate SVD results. We discuss two examples of a Morse potential of different anharmonicities, the H2 and I2 molecules. We further demonstrate the approach for a two coupled electronic states problem, the well studied non radiative decay of pyrazine from its bright state. Five constraints are found to be enough to capture the ultrafast electronic population exchange and to recover the dynamics of the wave packet in both electronic states.
We investigate the entanglement between electronic and vibrational degrees of freedom produced by a vibronic coupling in a molecular system described in the Born-Oppenheimer approximation. Entanglement in a pure state of the Hilbert space H=Hel⨂Hvib\mathcal{H}={\mathcal{H}}_{\mathrm{el}}⨂{\mathcal{H}}_{\mathrm{vib}}H=Hel​⨂Hvib​ is quantified using the von Neumann entropy of the reduced density matrix and the reduced linear entropy. Expressions for these entanglement measures are derived for the $2\ifmmode\times\else\texttimes\fi{}{N}_{v}$ and $3\ifmmode\times\else\texttimes\fi{}{N}_{v}$ cases of the bipartite entanglement, where 2 and 3 are the dimensions of the electronic Hilbert space H\mathcal{H}Hel{}_{\mathrm{el}}el​, and Nv{N}_{v}Nv​ is the dimension of H\mathcal{H}Hvib{}_{\mathrm{vib}}vib​. We study the entanglement dynamics for two electronic states coupled by a laser pulse (a $2\ifmmode\times\else\texttimes\fi{}{N}_{v}$ case), taking as an example a coupling between the ${a}^{3}{{\ensuremath{\Sigma}}_{u}}^{+}(6s,6s)$ and 1g(6s,6p3/2){1}_{g}(6s,6{p}_{3/2})1g​(6s,6p3/2​) states of the Cs2{}_{2}2​ molecule. The reduced linear entropy expression obtained for the $3\ifmmode\times\else\texttimes\fi{}{N}_{v}$ case is used to follow the entanglement evolution in a scheme proposed for the control of the vibronic dynamics in a Cs2{}_{2}2​ cold molecule, implying the ${a}^{3}{{\ensuremath{\Sigma}}_{u}}^{+}(6s,6s)$, ${0}_{g}{}^{\ensuremath{-}}(6s,6{p}_{3/2})$, and ${0}_{g}{}^{\ensuremath{-}}(6s,5d)$ electronic states, which are coupled by a nonadiabatic radial coupling and a sequence of chirped laser pulses.
We consider the quantum entanglement of the electronic and vibrational degrees of freedom in molecules with tendencies towards double welled potentials. In these bipartite systems, the von Neumann entropy of the reduced density matrix is used to quantify the electron-vibration entanglement for the lowest two vibronic wavefunctions obtained from a model Hamiltonian based on coupled harmonic diabatic potential-energy surfaces. Significant entanglement is found only in the region in which the ground vibronic state contains a density profile that is bimodal (i.e., contains two separate local maxima). However, in this region two distinct types of density and entanglement profiles are found: one type arises purely from the degeneracy of energy levels in the two potential wells and is destroyed by slight asymmetry, while the other arises through strong interactions between the diabatic levels of each well and is relatively insensitive to asymmetry. These two distinct types are termed fragile degeneracy-induced entanglement and persistent entanglement, respectively. Six classic molecular systems describable by two diabatic states are considered: ammonia, benzene, BNB, pyridine excited triplet states, the Creutz-Taube ion, and the radical cation of the "special pair" of chlorophylls involved in photosynthesis. These chemically diverse systems are all treated using the same general formalism and the nature of the entanglement that they embody is elucidated.
We review the framework of operator Hilbert space and introduce the one- and two-particle super reduced density matrices (1-SRDMs and 2-SRDMs), as well as the super mutual information (SMI). The eigenvectors of the 1-SRDMs define what we term natural single particle operator bases, and provide a way to compress vibrational and vibronic Hamiltonians with controlled error. The SMI is defined from the operator entanglement entropy of the 1-SRDMs and 2-SRDMs, and captures the correlation between operators acting on different one-mode subspaces, which may be used to reveal and quantify both direct and indirect couplings that might otherwise be difficult to extract. Efficient numerical approaches for the calculation of SRDMs and the SMI are developed and applied to a set of prototypical vibrational and vibronic Hamiltonians, as well as approximations to the corresponding time-evolution operators. Through this, we demonstrate that: (i) commonly used vibronic Hamiltonians are amenable to extremely high levels of compression without compromising accuracy, and; (ii) SMI analysis can be used to systematically and quantitatively reveal both direct and indirect couplings that might otherwise be difficult to extract, including indirect couplings of vibrational modes via intermediary electronic-vibrational interactions.