Quantum–Classical Entropy Analysis for Nonlinearly-Coupled Continuous-Variable Bipartite Systems

A recent series of papers has shown that a symmetrical quasi-classical (SQC) windowing procedure applied to the Meyer-Miller (MM) classical vibronic Hamiltonian provides a very good treatment of electronically nonadiabatic processes in a variety of benchmark model systems, including systems that exhibit strong quantum coherence effects and some which other approximate approaches have difficulty in describing correctly. In this paper, a different classical electronic Hamiltonian for the treatment of electronically nonadiabatic processes is proposed (and "quantized" via the SQC windowing approach), which maps the dynamics of F coupled electronic states to a set of F spin-(1)/2 degrees of freedom (DOF), similar to the Fermionic spin model described by Miller and White (J. Chem. Phys. 1986, 84, 5059). It is noted that this spin-mapping (SM) Hamiltonian is an exact Hamiltonian if treated as a quantum mechanical (QM) operator-and thus QM'ly equivalent to the MM Hamiltonian-but that an analytically distinct classical analogue is obtained by replacing the QM spin-operators with their classical counterparts. Due to their analytic differences, a practical comparison is then made between the MM and SM Hamiltonians (when quantized with the SQC technique) by applying the latter to many of the same benchmark test problems successfully treated in our recent work with the SQC/MM model. We find that for every benchmark problem the MM model provides (slightly) better agreement with the correct quantum nonadiabatic transition probabilities than does the new SM model. This is despite the fact that one might expect, a priori, a more natural description of electronic state populations (occupied versus unoccupied) to be provided by DOF with only two states, i.e., spin-(1)/2 DOF, rather than by harmonic oscillator DOF which have an infinite manifold of states (though only two of these are ever occupied).

ABSTRACTClassical dynamics is generally considered as the prototype of a deterministic theory; the equations of motion determine the coordinates q and the momenta p at any time provided they are given at an initial instant. It is pointed out that this is an unrealistic assumption, for it is inevitable that there should be small uncertainties Δq, Δp; a point of the mathematical continuum has no physical significance. The uncertainties can be taken into account without violating the deterministic equations by introducing a probability density P in phase space p, q. The function P satisfies the partial differential equation which expresses Liouville's theorem. Thus it can be shown that an initial uncertainty spreads out in phase space, so that finally all states of the system consistent with the mean initial values of the constants of the motion are equally probable. This property holds for one degree of freedom just as well as for many, and is not a consequence of our ignorance concerning large numbers of particles. The essential difference between classical mechanics and quantum mechanics consists not in the physical necessity of a statistical interpretation, but in the further introduction of a probability amplitude, the square of which is the probability density; thisimplies the restriction of the initial uncertainties as expressed by the uncertainty laws, and the phenomenon of interference of probabilities which makes necessary a revision of our concepts of physical reality. It is remarkable that the product of the uncertainties of a properly chosen pair of variables is constant also in classical mechanics, although of course its value is not a universal constant as in the quantum case.

Correlation and entanglement of a three-level atom inside a dissipative cavity

Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 and Phys. Rev. A 96, 022117] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function -- the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics -- extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. Firstly we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space -- putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of "parity" and "displacement" in a natural broadening of previously developed phase space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.

Quantum information theory has provided the formal framework for describing how information is stored, transmitted, and transformed in physical quantum systems [Renes, 2022; Tomamichel, 2015; Wilde, 2013]. Its entropic formulations underpin our understanding of quantum computation, communication, and cryptography. Yet this theory traditionally treats all quantum operations as freely available, ignoring computational restrictions. In practice, however, any manipulation of quantum information must be performed by devices of bounded complexity and runtime. Capturing such realistic constraints requires extending quantum information theory to include computational efficiency as a fundamental component. This work takes a first step toward building a computational version of quantum information theory, one that treats efficiency as part of the theory itself. The goal is to understand how the behavior of quantum information changes when the parties involved can only perform computationally efficient operations. This approach bridges the abstract, ideal setting of quantum information theory with the practical limitations of real quantum devices, offering a means to study information processing under realistic resource constraints. At the center of this work are two new quantities: the quantum computational min-entropy and the quantum computational max-entropy. These entropies extend standard quantum entropies by explicitly limiting the computational power of the observer or adversary. The quantum computational min-entropy captures how unpredictable a quantum system A remains to an observer holding system B, when that observer is restricted to quantum circuits of bounded size. Formally, for a bipartite state ρ_{AB}, we define {H^c}^s_{min}(A|B)_{ρ} ≔ -log d_A max_{ℰ^s_{B→A'}} F((𝕀_A ⊗ ℰ^s)(ρ_{AB}),|Φ_{AA'}⟩⟨Φ_{AA'}|) , where the maximization is over quantum channels that can be implemented by circuits of size at most s, and F denotes fidelity with a maximally entangled state. In the classical setting, the min-entropy can be expressed through the maximal probability of correctly guessing a random variable given some side-information. In the fully quantum setting, this idea extends to uncertainty about quantum information [König et al., 2009], quantifying how well one system can be inferred from another using local quantum operations. Our definition generalizes this operational viewpoint by restricting the computational power of the observer to efficient quantum circuits. This definition extends the operational meaning of the information-theoretic quantum min-entropy [König et al., 2009] by incorporating computational constraints, and it provides the fully quantum counterpart of the classical unpredictability entropy [Hsiao et al., 2007]. We establish fundamental properties for the computational min-entropy, including monotonicity in the circuit size and smoothing parameters, efficient data-processing inequalities, and fully quantum leakage and purification chain rules, which were left as open questions in earlier definitions of quantum computational entropies [Yi-Hsiu Chen et al., 2017; Munson et al., 2025]. For classical–quantum states, it coincides with the previously defined quantum computational unpredictability entropy [Noam Avidan and Rotem Arnon, 2025], showing that the new definition correctly generalizes known results. We also introduce the quantum computational max-entropy through a duality relation [Tomamichel et al., 2010] with the min-entropy using a fixed purification. Finally, we prove unconditional separations between the computational and information-theoretic entropies, demonstrating that computational restrictions can fundamentally alter entropic behavior even for simple states. These results establish the fundamental mathematical framework for studying quantum information within realistic computational constraints. By integrating efficiency directly into entropic quantities, they open the door to a fully developed computational quantum information theory that parallels its information-theoretic counterpart. Such a framework provides the foundation for analyzing cryptographic security against computationally bounded quantum adversaries [Noam Avidan and Rotem Arnon, 2025] and the limits of efficient quantum state manipulation. More broadly, it suggests that many core notions in quantum information theory may have refined computational analogues yet to be explored.

In periodically-driven quantum systems, resonances can induce exotic nonequilibrium behavior and new phases of matter without static analog. We report on the emergence of fractional and integer resonances in a broad class of many-body Hamiltonians with a modulated hopping with a frequency that is either a fraction or an integer of the on-site interaction. We contend that there is a fundamental difference between these resonances when interactions bring the system to a Floquet prethermal state. Second-order processes dominate the dynamics in the fractional resonance case, leading to less entanglement and more localized quantum states than in the integer resonance case dominated by first-order processes. We demonstrate the dominating emergence of fractional resonances using the Magnus expansion of the effective Hamiltonian and quantify their effects on the many-body dynamics via quantum states' von Neumann entropy and Loschmidt echo. Our findings reveal novel features of the nonequilibrium quantum many-body system, such as the coexistence of Floquet prethermalization and localization, that may allow to development of quantum memories for quantum technologies and quantum information processing.

We consider the effect of a thermal bath on quantum correlations induced by the gravitational interaction in the weak field limit between two massive cat states, called gravitational cat (gravcat) states. The main goal of this paper is to provide a good understanding of the effects of temperature and several parameters in the entanglement (measured by the concurrence) and quantum coherence (measured by the l1-norm that is defined from the minimal distance between the quantum state and the set of incoherent states) which are derived from the thermal quantum density operator. Our results show that the thermal concurrence and l1-norm can be significantly optimized by increasing the masses or decreasing the distance between them. We investigate and discuss the behavior of these quantities under temperature variations in different regimes, including some that are expected to be experimentally feasible in the future. In particular, we observe that thermal fluctuations raise non-entangled quantum correlations when entanglement suddenly drops.

The Ashtekar-Streubel fluxes give a proposed definition of the angular momentum emitted by an isolated gravitationally radiating system. This was based on identifying a "phase space of radiative modes," independent of any internal degrees of freedom, and using the Hamiltonian functions conjugate to the action of the Bondi-Metzner-Sachs (BMS) group as the energy-momentum, supermomentum and angular momentum. However, there are some difficulties in formulating this phase space so as to get the proper degrees of freedom. I consider how to address this point, and also to identify circumstances in which the radiative modes are sufficiently decoupled that they can be assigned their own angular momentum. Two different phase spaces are considered. One, which seems to reflect what previous workers have done (it leads to the usual formulas for the Ashtekar-Streubel fluxes), is mathematically simpler, but this has unwanted degrees of freedom and is difficult to interpret physically. The second is a quotient of the first; it is better justified (at least in terms of degrees of freedom), and plausibly decouples the radiative angular momentum from internal modes (at least for systems which ultimately become stationary). The symplectic form for this quotient, and the corresponding angular momentum flux, involve highly non-local correlations. Both phase spaces are shown to have Poisson brackets implementing the BMS algebra. For both phase spaces, for axisymmetric space-times vacuum near null infinity, there can be no gravitational radiation of angular momentum about the axis of symmetry, although matter can carry off angular momentum in such cases.

We present Bell-type tests of nonclassicality and non-Gaussianity for single-mode fields employing a generalized quasiprobability function. Our nonclassicality tests are based on the observation that two orthogonal quadratures in phase space (position and momentum) behave as independent realistic variables for a coherent state. Taking four (three) points at the vertices of a rectangle (right triangle) in phase space, our tests detect every pure nonclassical Gaussian state and a range of mixed Gaussian states. These tests also set an upper bound for all Gaussian states and their mixtures, which thereby provide criteria for genuine quantum non-Gaussianity. We optimize the non-Gaussianity tests by employing a squeezing transformation in phase space that converts a rectangle (right triangle) to a parallelogram (triangle), which enlarges the set of non-Gaussian states detectable in our formulation. We address fundamental and practical limits of our generalized phase-space tests by looking into their relation with decoherence under a lossy Gaussian channel and their robustness against finite data and nonoptimal choice of phase-space points. Furthermore, we demonstrate that our parallelogram test can identify useful resources for nonlocality testing in phase space.

Variational quantum algorithms (VQAs) that estimate values of widely used physical quantities such as the rank, the quantum entropies, the Bures fidelity, and the quantum Fisher information of mixed quantum states are developed. In addition, variations of these VQAs are also adapted to perform other useful functions such as quantum state learning and approximate fractional inverses. The common theme shared by the proposed algorithms is that their cost functions are all based on minimizing the quantum purity of a quantum state. Strategies to mitigate or avoid the problem of exponentially vanishing cost function gradients are also discussed.

We propose coherent (‘Schrödinger cat-like’) states adapted to the parity symmetry providing a remarkable variational description of the ground and first excited states of vibron models for finite (N)-size molecules. Vibron models undergo a quantum shape phase transition (from linear to bent) at a critical value ξc of a control parameter. These trial cat states reveal a sudden increase in vibration–rotation entanglement linear (L) and von Neumann (S) entropies from zero to (to be compared with L(N)max(ξ) = 1 − 1/(N + 1)) and , respectively, above the critical point, ξ > ξc, in agreement with exact numerical calculations. We also compute inverse participation ratios, for which these cat states capture a sudden delocalization of the ground-state wave packet across the critical point. Analytic expressions for entanglement entropies and inverse participation ratios of variational states, as functions of N and ξ, are given in terms of hypergeometric functions.

Assume that $\mathcal {S}(H)$ is the set of all quantum states of a quantum system associating a separable complex Hilbert space H. Denote by S(ρ) and Sp(ρ) (p > 0 with p≠ 1) respectively the von Neumann entropy and Tsallis p-entropy of ρ. In this paper, a sufficient condition for two quantum states being equal is given by von Neumann entropy and Tsallis p-entropy, and we show that, for $\rho ,\sigma \in \mathcal {S}(H)$ , $S(\alpha \rho +(1-\alpha )P)=S(\alpha \sigma +(1-\alpha )P)<\infty $ for all pure states $P\in \mathcal {P}_{1}(H)$ and $S_{p}(\alpha \rho +(1-\alpha )P)=S_{p}(\alpha \sigma +(1-\alpha )P)<\infty $ holds for all pure states $P\in \mathcal {P}_{1}(H)$ respectively implies ρ = σ, where α ∈ (0,1) is any fixed number. In addition, the maps on $\mathcal {S}(H)$ preserving the von Neumann entropy and Tsallis p-entropy of a convex combination are also respectively characterized.

Quantum state reconstruction, or state reconstruction for short, aims at identifying an unknown quantum state ( states in quantum mechanics) on the basis of experimentally accessible data. The Quantum Optics community usually refers to this inverse problem as quantum (state) tomography while the expression quantum state estimation is often used in the field of Quantum computation. Reconstruction procedures depend on the physical context defined by the system carrying the unknown state, the experimentally accessible observables, the size of the ensemble of systems prepared in the unknown state, and the precision of the measured data. A two-level system (such as a spin-1/2, a qubit, or the two polarizations of a photon) prepared in a state with density matrix ρ is sufficient to illustrate the idea of state reconstruction. With two non-negative eigenvalues summing to one, the density matrix is a positive operator, and it depends on three real parameters. In the Bloch representation, the parameters combine to a real vector n with length |n| 1,

How many degrees of freedom are evident in a physical process represented by f(s, t)? In some form questions about “degrees of freedom” (d.o.f.) are common in mathematics, physics, statistics, and geophysics. This would mean, for instance, in how many independent directions a weight suspended from the ceiling could move. Dofs are important for three reasons that will become apparent in the remaining chapters. First, dofs are critically important in understanding why natural analogues can (or cannot) be applied as a forecast method in a particular problem (Chapter 7). Secondly, understanding dofs leads to ideas about truncating data sets efficiently, which is very important for just about any empirical prediction method (Chapters 7 and 8). Lastly, the number of dofs retained is one aspect that has a bearing on how nonlinear prediction methods can be (Chapter 10). In view of Chapter 5 one might think that the total number of orthogonal directions required to reproduce a data set is the dof. However, this is impractical as the dimension would increase (to infinity) with ever denser and slightly imperfect observations. Rather we need a measure that takes into account the amount of variance represented by each orthogonal direction, because some directions are more important than others. This allows truncation in EOF space without lowering the “effective” dof very much. We here think schematically of the total atmospheric or oceanic variance about the mean state as being made up by N equal additive variance processes. N can be thought of as the dimension of a phase space in which the atmospheric state at one moment in time is a point. This point moves around over time in the N-dimensional phase space. The climatology is the origin of the phase space. The trajectory of a sequence of atmospheric states is thus a complicated Lissajous figure in N dimensions, where, importantly, the range of the excursions in each of the N dimensions is the same in the long run. The phase space is a hypersphere with an equal probability radius in all N directions.

Classical statistical mechanics, as well as quantum statistical mechanics, are both acknowledgedly statistical theories even if one subscribes to the conventional point of view about the intrinsically disparate natures of classical and quantum mechanics, respectively. Yet, the conventional mathematical frameworks for these two theories are remarkably dissimilar even at the most fundamental level of description of states of ensembles: in the classical case states are described by distribution functions ƒ(q, p) on phase space, whereas in the quantum case they are given by density operators ρ.