Classical Quantum System

• Classical quantum system is a family of constructions bridging classical dynamics with quantum mechanics via constrained trajectories, coarse graining, and effective mapping.
• It employs methodologies like projecting Schrödinger evolution onto classical manifolds, reproducing classical laws such as the Ehrenfest theorem under narrow-packet conditions.
• The framework extends to equilibrium and hybrid dynamics, allowing classical representations of quantum observables through effective potentials, thermodynamic equivalence, and stochastic coupling.

Searching arXiv for the papers on arXiv and closely related work to ground the article in the cited literature. A classical quantum system is not a single standardized object but a family of constructions in which classical mechanics is embedded in, extracted from, or consistently coupled to quantum theory. In the recent arXiv literature, the phrase refers to projected classical flows on manifolds of “classical” quantum states, coherent-state bridges between classical and quantum actions, equilibrium classical models reproducing quantum thermodynamics and pair structure, Markovian quantum–classical hybrid dynamics, and macroscopic sectors of quantum theory whose observables are effectively classical (Sperling et al., 2018, Klauder, 2020, Dufty et al., 2012, Barchielli, 2024, Bóna, 2019). It also appears in operational settings, such as classical control of untrusted quantum devices and communication tasks contrasting single classical and quantum systems (Reichardt et al., 2012, Frenkel et al., 2013, Patra et al., 2022).

1. Conceptual scope and principal meanings

Across this literature, the term denotes several non-equivalent constructions. One line asks how a set of states deemed “classical” can evolve inside a quantum Hilbert space without ever leaving a prescribed manifold, so that classical dynamics appears as constrained Schrödinger evolution. A second line asks how a quantum equilibrium state can be replaced by an effective classical system with an effective temperature, local chemical potential, and pair potential chosen to reproduce thermodynamic and structural observables. A third line studies genuine hybrid systems in which a classical process and a quantum subsystem have their own dynamics and interact through coupled stochastic equations. A fourth line derives classical mechanics from quantum mechanics by coarse graining, by symplectic reduction on projective Hilbert space, or by passing to macroscopic observables in infinite systems (Sperling et al., 2018, Dufty et al., 2012, Barchielli, 2024, Radonjić et al., 2011, Bóna, 2019).

These meanings differ in ontology and purpose. In constrained-trajectory approaches, the underlying description remains fully quantum and “classicality” is a restriction of admissible states. In equilibrium mappings, the classical model is a computational surrogate for static quantum observables. In hybrid dynamics, classical and quantum components are treated as distinct subsystems coupled by Markovian dynamics. In macroscopic-emergence approaches, classical variables arise as collective or central observables of a quantum many-body system. This diversity suggests that “classical quantum system” is best understood as a structural category rather than a single formalism.

2. Classical dynamics embedded within quantum mechanics

A systematic construction of classical evolution inside quantum theory starts from a family of pure states $|\psi(\boldsymbol q)\rangle$ regarded as classical, with mixed classical states written as

$$\hat\rho = \int dP(\boldsymbol q)\,|\psi(\boldsymbol q)\rangle\langle\psi(\boldsymbol q)|.$$

Restricting the quantum action

$$\mathcal L = \frac{i\hbar}{2}\big(\langle\Psi|\dot\Psi\rangle-\langle\dot\Psi|\Psi\rangle\big)-\langle\Psi|\hat H|\Psi\rangle$$

to this manifold yields equations of motion for $\boldsymbol q(t)$,

$$i\hbar\,\langle\nabla_{\boldsymbol q}\psi|\nabla_{\boldsymbol q}\psi\rangle\,\dot{\boldsymbol q} = \nabla_{\boldsymbol q^\ast}\langle\psi|\hat H|\psi\rangle,$$

equivalently,

$$\langle\nabla_{\boldsymbol q}\psi| \Big(i\hbar\tfrac{d}{dt}|\psi\rangle-\hat H|\psi\rangle\Big)=\boldsymbol 0.$$

The resulting trajectory stays on the classical manifold for all times, and the construction requires neither a semiclassical limit $\hbar\to0$ nor large-$N$ assumptions. Newton’s second law appears as a special case: for displaced wave packets, the projected equations reproduce the Ehrenfest theorem exactly and reduce to $m\ddot X=-V'(X)$ under the usual narrow-packet conditions (Sperling et al., 2018).

The same paper shows that this projected dynamics unifies several familiar approximations. Coherent-state dynamics in nonlinear optics yields a classical phase-space amplitude equation structurally parallel to the Heisenberg equation for $\hat a$. Product-state constraints give self-consistent semiclassical hybrid equations. For multipartite systems, projection to the manifold of $$\hat\rho = \int dP(\boldsymbol q)\,|\psi(\boldsymbol q)\rangle\langle\psi(\boldsymbol q)|.$$0-separable states produces the separability Schrödinger equation, in which each subsystem evolves under a Hamiltonian depending parametrically on the states of the others while the global state remains separable (Sperling et al., 2018).

A closely related but more geometric formulation uses coherent states to connect quantum and classical action functionals. Starting from

$$\hat\rho = \int dP(\boldsymbol q)\,|\psi(\boldsymbol q)\rangle\langle\psi(\boldsymbol q)|.$$1

restriction to canonical coherent states $$\hat\rho = \int dP(\boldsymbol q)\,|\psi(\boldsymbol q)\rangle\langle\psi(\boldsymbol q)|.$$2 gives

$$\hat\rho = \int dP(\boldsymbol q)\,|\psi(\boldsymbol q)\rangle\langle\psi(\boldsymbol q)|.$$3

The same pattern extends to spin coherent states and affine coherent states for constrained configuration spaces such as $$\hat\rho = \int dP(\boldsymbol q)\,|\psi(\boldsymbol q)\rangle\langle\psi(\boldsymbol q)|.$$4. The semi-classical action is therefore a literal restriction of the quantum action, while coherent-state resolutions of the identity allow the restricted action to be “fleshed out” back to the full quantum action. This construction is presented as a model-independent alternative to path-integral routes and as a smooth passage between classical and quantum descriptions (Klauder, 2020).

Other papers push the embedding further in the opposite direction: quantum dynamics can be represented by an underlying classical Hamiltonian or transport system. A canonical transformation from real phase-space variables $$\hat\rho = \int dP(\boldsymbol q)\,|\psi(\boldsymbol q)\rangle\langle\psi(\boldsymbol q)|.$$5 to complex canonical variables $$\hat\rho = \int dP(\boldsymbol q)\,|\psi(\boldsymbol q)\rangle\langle\psi(\boldsymbol q)|.$$6 turns a bilinear Hamiltonian $$\hat\rho = \int dP(\boldsymbol q)\,|\psi(\boldsymbol q)\rangle\langle\psi(\boldsymbol q)|.$$7 into the Schrödinger equation $$\hat\rho = \int dP(\boldsymbol q)\,|\psi(\boldsymbol q)\rangle\langle\psi(\boldsymbol q)|.$$8, while suitable stochastic extensions reproduce the von Neumann and Lindblad equations (Oliveira, 2023). Likewise, deterministic transport equations with probabilistic initial conditions can be written in terms of a real classical wave function $$\hat\rho = \int dP(\boldsymbol q)\,|\psi(\boldsymbol q)\rangle\langle\psi(\boldsymbol q)|.$$9 with $$\mathcal L = \frac{i\hbar}{2}\big(\langle\Psi|\dot\Psi\rangle-\langle\dot\Psi|\Psi\rangle\big)-\langle\Psi|\hat H|\Psi\rangle$$0, obeying

$$\mathcal L = \frac{i\hbar}{2}\big(\langle\Psi|\dot\Psi\rangle-\langle\dot\Psi|\Psi\rangle\big)-\langle\Psi|\hat H|\Psi\rangle$$1

so that superposition, interference, non-commuting operators, and unitary time evolution arise within a probabilistic classical transport framework (Wetterich, 15 May 2026).

3. Equilibrium classical representations

At equilibrium, the classical representation program constructs a classical grand-canonical system with effective parameters $$\mathcal L = \frac{i\hbar}{2}\big(\langle\Psi|\dot\Psi\rangle-\langle\dot\Psi|\Psi\rangle\big)-\langle\Psi|\hat H|\Psi\rangle$$2, $$\mathcal L = \frac{i\hbar}{2}\big(\langle\Psi|\dot\Psi\rangle-\langle\dot\Psi|\Psi\rangle\big)-\langle\Psi|\hat H|\Psi\rangle$$3, and $$\mathcal L = \frac{i\hbar}{2}\big(\langle\Psi|\dot\Psi\rangle-\langle\dot\Psi|\Psi\rangle\big)-\langle\Psi|\hat H|\Psi\rangle$$4 by requiring

$$\mathcal L = \frac{i\hbar}{2}\big(\langle\Psi|\dot\Psi\rangle-\langle\dot\Psi|\Psi\rangle\big)-\langle\Psi|\hat H|\Psi\rangle$$5

The goal is not to reproduce all quantum observables but to match pressure, density profile, and pair correlations exactly at the level of definitions. This makes strong-coupling classical methods—molecular dynamics, classical Monte Carlo, integral-equation theory, and classical density functional theory—available for quantum systems once the effective classical parameters are known (Dufty et al., 2012).

The inversion is carried out with classical liquid-state relations. In HNC form, the effective pair potential is obtained from the quantum hole function $$\mathcal L = \frac{i\hbar}{2}\big(\langle\Psi|\dot\Psi\rangle-\langle\dot\Psi|\Psi\rangle\big)-\langle\Psi|\hat H|\Psi\rangle$$6 and the associated direct correlation function $$\mathcal L = \frac{i\hbar}{2}\big(\langle\Psi|\dot\Psi\rangle-\langle\dot\Psi|\Psi\rangle\big)-\langle\Psi|\hat H|\Psi\rangle$$7 through

$$\mathcal L = \frac{i\hbar}{2}\big(\langle\Psi|\dot\Psi\rangle-\langle\dot\Psi|\Psi\rangle\big)-\langle\Psi|\hat H|\Psi\rangle$$8

with $$\mathcal L = \frac{i\hbar}{2}\big(\langle\Psi|\dot\Psi\rangle-\langle\dot\Psi|\Psi\rangle\big)-\langle\Psi|\hat H|\Psi\rangle$$9 determined by the Ornstein–Zernike equation. The effective local chemical potential is expressed analogously as a functional of $\boldsymbol q(t)$0 and $\boldsymbol q(t)$1, and $\boldsymbol q(t)$2 is then fixed by pressure equivalence. Applied to the ideal Fermi gas, this mapping generates a nontrivial Pauli potential, $\boldsymbol q(t)$3 in the nondegenerate regime, and a finite $\boldsymbol q(t)$4 as $\boldsymbol q(t)$5, thereby encoding degeneracy pressure in an effective classical temperature (Dufty et al., 2011, Dufty et al., 2012).

For jellium, the effective potential is decomposed as

$\boldsymbol q(t)$6

where the first term is the ideal-gas Pauli part and $\boldsymbol q(t)$7 is built from weak-coupling quantum input. In the application paper, $\boldsymbol q(t)$8 is obtained from RPA structure factors; it has the exact Coulomb tail with effective coupling $\boldsymbol q(t)$9, is finite at the origin, and reduces to the Kelbg potential in the high-temperature, low-density limit. Solving the full HNC equations with this potential yields $$i\hbar\,\langle\nabla_{\boldsymbol q}\psi|\nabla_{\boldsymbol q}\psi\rangle\,\dot{\boldsymbol q} = \nabla_{\boldsymbol q^\ast}\langle\psi|\hat H|\psi\rangle,$$0, $$i\hbar\,\langle\nabla_{\boldsymbol q}\psi|\nabla_{\boldsymbol q}\psi\rangle\,\dot{\boldsymbol q} = \nabla_{\boldsymbol q^\ast}\langle\psi|\hat H|\psi\rangle,$$1, and pressure in good agreement with restricted PIMC and diffusion Monte Carlo over a broad range of $$i\hbar\,\langle\nabla_{\boldsymbol q}\psi|\nabla_{\boldsymbol q}\psi\rangle\,\dot{\boldsymbol q} = \nabla_{\boldsymbol q^\ast}\langle\psi|\hat H|\psi\rangle,$$2 and $$i\hbar\,\langle\nabla_{\boldsymbol q}\psi|\nabla_{\boldsymbol q}\psi\rangle\,\dot{\boldsymbol q} = \nabla_{\boldsymbol q^\ast}\langle\psi|\hat H|\psi\rangle,$$3 (Dutta et al., 2012).

The same framework also addresses inhomogeneous systems. For harmonically confined charges, the effective external potential derived from the ideal trapped Fermi gas and the effective pair potential from uniform jellium generate shell structure in the equilibrium density. The analysis isolates two distinct quantum mechanisms: diffraction-induced softening of the interaction and exchange-induced reshaping of the confinement potential. In the mean-field limit, both can generate radial shells even when the classical mean-field problem does not (Dutta et al., 2012).

4. Hybrid quantum–classical dynamics

A rigorous Markovian framework for quantum–classical hybrids treats the classical component as a stochastic process $$i\hbar\,\langle\nabla_{\boldsymbol q}\psi|\nabla_{\boldsymbol q}\psi\rangle\,\dot{\boldsymbol q} = \nabla_{\boldsymbol q^\ast}\langle\psi|\hat H|\psi\rangle,$$4 and the quantum component as a state on a separable Hilbert space $$i\hbar\,\langle\nabla_{\boldsymbol q}\psi|\nabla_{\boldsymbol q}\psi\rangle\,\dot{\boldsymbol q} = \nabla_{\boldsymbol q^\ast}\langle\psi|\hat H|\psi\rangle,$$5. The coupled dynamics is built from two stochastic differential equations: one for $$i\hbar\,\langle\nabla_{\boldsymbol q}\psi|\nabla_{\boldsymbol q}\psi\rangle\,\dot{\boldsymbol q} = \nabla_{\boldsymbol q^\ast}\langle\psi|\hat H|\psi\rangle,$$6, driven by Wiener and Poisson processes, and one for an unnormalized quantum state or density operator with coefficients $$i\hbar\,\langle\nabla_{\boldsymbol q}\psi|\nabla_{\boldsymbol q}\psi\rangle\,\dot{\boldsymbol q} = \nabla_{\boldsymbol q^\ast}\langle\psi|\hat H|\psi\rangle,$$7, $$i\hbar\,\langle\nabla_{\boldsymbol q}\psi|\nabla_{\boldsymbol q}\psi\rangle\,\dot{\boldsymbol q} = \nabla_{\boldsymbol q^\ast}\langle\psi|\hat H|\psi\rangle,$$8, and $$i\hbar\,\langle\nabla_{\boldsymbol q}\psi|\nabla_{\boldsymbol q}\psi\rangle\,\dot{\boldsymbol q} = \nabla_{\boldsymbol q^\ast}\langle\psi|\hat H|\psi\rangle,$$9. After a change of probability measure, the normalized conditional quantum state obeys a nonlinear stochastic master equation, while the classical process acquires state-dependent drift and jump intensities. The same dynamics defines a completely positive hybrid dynamical semigroup on

$$\langle\nabla_{\boldsymbol q}\psi| \Big(i\hbar\tfrac{d}{dt}|\psi\rangle-\hat H|\psi\rangle\Big)=\boldsymbol 0.$$0

which reduces to a Lindblad semigroup in the purely quantum limit and to Liouville or Kolmogorov–Fokker–Planck equations in the purely classical limit. An important structural conclusion is that if information flows from quantum to classical, the dynamics is necessarily dissipative (Barchielli, 2024).

The same work emphasizes that hybrid trajectories are not merely formal. They provide a measurement-theoretic interpretation in which $$\langle\nabla_{\boldsymbol q}\psi| \Big(i\hbar\tfrac{d}{dt}|\psi\rangle-\hat H|\psi\rangle\Big)=\boldsymbol 0.$$1 can represent a classical record or control signal, and they support nontrivial phenomena such as hidden entanglement: ensembles of conditional quantum states can retain or generate entanglement even when the a priori mixed state obeys a dissipative master equation with little or no visible entanglement (Barchielli, 2024).

Not all hybrid schemes are equally successful. A mean-field treatment of the hydrogen atom with a quantum electron and a classical proton yields an electron dynamics only marginally different from the full quantum two-body description, but the proton behaves like a free particle rather than orbiting the center of mass. The failure is linked to the absence of momentum conservation in the mean-field hybrid equations, and the analysis argues that this pathology is likely generic for Born–Oppenheimer-type hybrid approaches (Zhan et al., 2013).

Hybridity also appears as an engineered resource. A quantum reservoir processor built from a network of quantum dots receives quantum input states through cascaded modes and classical data through a coherent drive $$\langle\nabla_{\boldsymbol q}\psi| \Big(i\hbar\tfrac{d}{dt}|\psi\rangle-\hat H|\psi\rangle\Big)=\boldsymbol 0.$$2. The same reservoir supports multitasking, including quantum state tomography, nonlinear equalization of classical channels, closed-loop prediction when external classical access is interrupted, and training of depolarizing quantum channels via quantum readout. In this setting, classical and quantum information are not separate pre- and post-processing layers but simultaneous drivers of a single open quantum dynamical system (Tran et al., 2022).

5. Information, control, and communication

A different meaning of classical–quantum coupling arises when a purely classical agent controls or certifies a quantum device. In the CHSH-rigidity framework, a classical verifier exchanges only classical bits with two non-communicating quantum devices. If the observed CHSH winning probability is close to the Tsirelson optimum $$\langle\nabla_{\boldsymbol q}\psi| \Big(i\hbar\tfrac{d}{dt}|\psi\rangle-\hat H|\psi\rangle\Big)=\boldsymbol 0.$$3, the underlying state and observables must be close, up to local isometries and ancillary junk, to EPR pairs measured with the ideal CHSH reflections. This “classical leash” supports device-independent QKD, nonlocal circuit implementation, and complexity-theoretic consequences such as $$\langle\nabla_{\boldsymbol q}\psi| \Big(i\hbar\tfrac{d}{dt}|\psi\rangle-\hat H|\psi\rangle\Big)=\boldsymbol 0.$$4 (Reichardt et al., 2012).

The distinction between classical and quantum correlations inside a quantum state is another operational strand. For a bipartite state $$\langle\nabla_{\boldsymbol q}\psi| \Big(i\hbar\tfrac{d}{dt}|\psi\rangle-\hat H|\psi\rangle\Big)=\boldsymbol 0.$$5, total correlation is measured by

$$\langle\nabla_{\boldsymbol q}\psi| \Big(i\hbar\tfrac{d}{dt}|\psi\rangle-\hat H|\psi\rangle\Big)=\boldsymbol 0.$$6

classical correlation by

$$\langle\nabla_{\boldsymbol q}\psi| \Big(i\hbar\tfrac{d}{dt}|\psi\rangle-\hat H|\psi\rangle\Big)=\boldsymbol 0.$$7

and quantum discord by $$\langle\nabla_{\boldsymbol q}\psi| \Big(i\hbar\tfrac{d}{dt}|\psi\rangle-\hat H|\psi\rangle\Big)=\boldsymbol 0.$$8. A variational hybrid quantum–classical algorithm implements this optimization by vectorizing $$\langle\nabla_{\boldsymbol q}\psi| \Big(i\hbar\tfrac{d}{dt}|\psi\rangle-\hat H|\psi\rangle\Big)=\boldsymbol 0.$$9 into a pure state on a doubled Hilbert space and variationally searching over measurement operators on one subsystem; the reported outputs agree with exact calculations for the tested two-qubit states (Mahdian et al., 2021).

Communication theory supplies a sharper contrast. In one single-use storage model, the achievable channel sets for a classical $\hbar\to0$0-state system and an $\hbar\to0$1-level quantum system coincide: $\hbar\to0$2 Hence, for every input distribution and reward function, the maximum expected reward obtainable with an $\hbar\to0$3-level quantum system equals that obtainable with a classical $\hbar\to0$4-state system, and the maximal mutual information is the same in both cases (Frenkel et al., 2013). In a different operational setting, however, restaurant games $\hbar\to0$5 show that a noiseless 1-bit classical channel can be insufficient without shared randomness, the same 1-bit channel assisted by classical shared randomness can achieve the goal, and a single qubit without shared randomness can also achieve it. The advantageous quantum strategy must invoke quantum interference at both encoding and decoding (Patra et al., 2022). The literature therefore does not attribute a uniform communication advantage or disadvantage to “a single quantum system”; the outcome depends on the game definition and on which auxiliary resources are treated as free.

6. Coarse graining, macroscopic variables, and emergent classicality

One route from quantum to classical mechanics constrains the quantum dynamics to the manifold of coherent states. For a system of nonlinear oscillators with canonical operators $\hbar\to0$6, the coarse-grained observables are the expectation values $\hbar\to0$7 and $\hbar\to0$8. States with the same $\hbar\to0$9 are treated as equivalent, and the constrained dynamics is chosen so that the evolution preserves constant and minimal quantum fluctuations of the fundamental observables. On the coherent-state manifold, the effective Hamiltonian has the form of a classical Hamiltonian plus correction terms that vanish for large mass or small dispersion, yielding the corresponding classical nonlinear oscillator system on sufficiently large scales (Radonjić et al., 2011).

A broader geometric and algebraic program derives classical systems from quantum mechanics in two ways. For finite systems, projective Hilbert space $N$0 is a symplectic manifold, and suitable group orbits reduce to coadjoint orbits in $N$1, which serve as classical phase spaces with Hamiltonian dynamics induced from the quantum theory. For infinite systems, macroscopic variables arise from the center of the observable algebra, so that the relevant W$N$2-subalgebra is commutative and its spectrum is a genuine classical phase space for macroscopic observables. Measurement models then couple a microscopic quantum system to such a macroscopic quantum subsystem, whose central observables behave classically (Bóna, 2019).

Two further lines reinforce the same theme. In one, deterministic transport equations with probabilistic initial data are rewritten as quantum systems: the real classical wave function $N$3 yields a Hermitian Hamiltonian, non-commuting statistical observables, conserved quantities such as quantum energy $N$4 and quantum angular momentum, and a complex functional integral formulation. In another, a classical Hamiltonian system is mapped by a complex canonical transformation to a Hilbert-space dynamics in which the real and imaginary parts of the wave function are classical canonical variables, and suitable stochastic extensions generate the Schrödinger, von Neumann, and Lindblad equations (Wetterich, 15 May 2026, Oliveira, 2023).

Taken together, these developments show that “classical quantum system” names a spectrum of constructions rather than a doctrine. In some, classicality is a constrained manifold inside Hilbert space; in some, it is an effective equilibrium surrogate; in some, it is a stochastic subsystem coupled to a quantum one; and in some, it is the macroscopic or central sector of a fully quantum many-body theory. The common thread is that classical behavior is not treated as external to quantum mechanics: it is represented as a distinguished structure within quantum formalism, or as a rigorously controlled approximation to it.