Carcassi, Landini, and Aidala (Physica Scripta 101, 065105 (2026); arXiv:2411.00972) showed that classical mechanics is recovered as the high-entropy limit of quantum mechanics: an explicitly constructed completely positive trace-preserving 'stretching map' TQT_Q — Lindblad evolution with jump operator a†a^{\dagger} and vanishing Hamiltonian, which rescales every anti-normally ordered monomial an(a†)ma^{n}(a^{\dagger})^{m} by (λ)n+m(\sqrt{\lambda})^{n+m} — increases the von Neumann entropy of every state, realizes the Weyl-Heisenberg group contraction that rescales the canonical commutator to iℏ/λi\hbar/\lambda, and drives the Wigner function toward a positive classical distribution. The construction leaves four interlinked questions open, posed here as one problem: (1) whether every quantum map performing the same group contraction factorizes as TQ∘UT_Q\circ U with UU unitary (or what the classification of such maps is); (2) whether the entropy increment of TQT_Q is uniform over all states at finite λ\lambda; (3) whether phase-space stretching is necessary, not merely sufficient, for unbounded entropy increase; and (4) whether quantum mechanics is the unique theory repairing classical mechanics at low entropy. No resolution of these questions is known in the literature to date.

Background

The relationship between quantum and classical mechanics is usually expressed through the limit ℏ→0\hbar\to 0. Carcassi, Landini, and Aidala (Physica Scripta 101, 065105 (2026)) proposed a physically sharper reading: classical mechanics is recovered as the high-entropy limit of quantum mechanics. The von Neumann entropy S(ρ)=−tr(ρln⁡ρ)S(\rho) = -\mathrm{tr}(\rho\ln\rho) of a quantum state ρ\rho equals zero exactly on pure states and takes arbitrarily large values on mixed states; the authors show that mixed states of sufficiently high entropy are well approximated by classical probability distributions over phase space, so that taking ℏ→0\hbar\to 0 mathematically corresponds to sending the entropy of pure states to −∞-\infty, just as the non-relativistic limit c→∞c\to\infty corresponds physically to v≪cv\ll c.

Two phase-space representations are central. The Wigner function W(α)W(\alpha) of a state is a quasi-probability distribution over phase space (labeled here by the complex coordinate α\alpha combining position and momentum): it is real and normalized but can take negative values, which encode quantum interference. The Husimi function Q(α)=⟨α∣ρ∣α⟩Q(\alpha) = \langle\alpha|\rho|\alpha\rangle is the expectation of ρ\rho on coherent states ∣α⟩|\alpha\rangle (minimal-uncertainty wave packets saturating σxσp=ℏ/2\sigma_x\sigma_p = \hbar/2); it is the Wigner function convolved with a Gaussian (a Weierstrass transform) and is therefore non-negative. Anti-normal ordering — placing annihilation operators to the left of creation operators, as in an(a†)ma^{n}(a^{\dagger})^{m} — is the operator ordering whose expectation values are the moments of the Husimi function.

Mathematically, both the relativistic and the classical limit are group contractions: rescalings under which one Lie group degenerates to another, as the Poincaré group contracts to the Galilei group when c→∞c\to\infty. In the quantum case the Weyl-Heisenberg group of phase-space translations contracts to the abelian classical translation group as the canonical commutator [X,P]=iℏ[X,P] = i\hbar is rescaled toward zero; equivalently, the Moyal bracket (the quantum generalization of the Poisson bracket that governs the evolution of the Wigner function) degenerates to the classical Poisson bracket. A symplectomorphism is a canonical transformation of classical phase space, i.e., a transformation preserving the symplectic form; in quantum mechanics, linear symplectomorphisms are implemented by unitary operators.

The paper's dynamical construction is a pure quantum stretching map TQT_Q. Classically, uniformly stretching phase space by a factor λ\lambda (area scaling by λ\lambda per degree of freedom) raises the Gibbs entropy of every distribution by exactly ln⁡λ\ln\lambda and dilutes the ℏ\hbar-scale quantum structure. The quantum analogue must be a completely positive trace-preserving (CPTP) map — a quantum channel, the general evolution of an open quantum system — since it must map mixed states to mixed states and increase entropy for all states. The authors realize it as the solution of a Lindblad master equation (the general Markovian evolution of a density operator) with vanishing Hamiltonian and a single jump operator L=a†L = a^{\dagger} (the creation operator of the harmonic oscillator that labels phase space by α\alpha), run for a time set by γ=ln⁡λ/Δt\gamma = \ln\lambda/\Delta t. In the Heisenberg picture each anti-normally ordered monomial is an eigenvector of the evolution, an(a†)m↦(λ)n+man(a†)ma^{n}(a^{\dagger})^{m}\mapsto(\sqrt{\lambda})^{n+m}a^{n}(a^{\dagger})^{m}, so the Husimi function is stretched exactly as in the classical case. The map increases the von Neumann entropy of every state because a†a≤aa†a^{\dagger}a \le aa^{\dagger} (Lindblad's operator criterion for entropy-increasing quantum dynamics). On the rescaled operators X^=X/λ\hat X = X/\sqrt{\lambda} and P^=P/λ\hat P = P/\sqrt{\lambda} the commutator becomes [X^,P^]=iℏ/λ[\hat X,\hat P] = i\hbar/\lambda: the group contraction is realized exactly, and the stretched Wigner function tends, for λ≫1\lambda\gg 1, to the stretched Husimi function QλQ_{\lambda}, a positive classical distribution, so quantum negativity is suppressed. This channel coincides with the quantum-limited phase-insensitive linear amplifier of quantum optics (gain λ\lambda with a vacuum environment). Composing with unitaries, RQ=TQ∘UR_Q = T_Q\circ U, preserves unitary evolution, so classical Hamiltonian dynamics is recovered alongside classical states: high entropy is the one and only requirement for classicality, independent of the mechanism producing the entropy increase.

Problem Statement

The problem is to characterize completely the entropy-increasing quantum dynamics that realize the classical (high-entropy) limit constructed by Carcassi, Landini, and Aidala, through the four questions below, which that construction leaves open. Each of the four questions below is an independently resolvable or refutable part of this single problem and states its own acceptance standard, so resolving any one part counts as a valid resolution of that part; the problem as a whole is characterized only once all four parts are settled. The construction to be characterized is the pure quantum stretching map TQT_Q — the completely positive trace-preserving channel generated by the Lindblad evolution with jump operator a†a^{\dagger} and vanishing Hamiltonian — which acts on anti-normally ordered monomials as an(a†)m↦(λ)n+man(a†)ma^{n}(a^{\dagger})^{m}\mapsto(\sqrt{\lambda})^{n+m}a^{n}(a^{\dagger})^{m} and, on the rescaled canonical operators, realizes the Weyl-Heisenberg group contraction [X^,P^]=iℏ/λ[\hat X,\hat P] = i\hbar/\lambda.

(1) Classification and factorization. Is every quantum map that performs the same group contraction — i.e., that induces the same rescaling of the effective commutator between the canonical operators, realizing the contraction of the Weyl-Heisenberg group to the abelian classical phase-space group — expressible as TQ∘UT_Q\circ U with UU a unitary (implementing a symplectomorphism on classical phase space)? If not, classify all such maps or characterize the obstruction. In the limit λ→∞\lambda\to\infty factorization always holds, because the classical limit can always be factorized as a pure stretching map following a symplectomorphism; the open question concerns the exact structure at finite λ\lambda and the uniqueness of the factorization.

(2) The entropy increment. TQT_Q is known to increase the von Neumann entropy of every state. Determine, as a function of the state ρ\rho, the entropy gain S(TQ(ρ))−S(ρ)S(T_Q(\rho)) - S(\rho) at finite stretch factor λ\lambda, and whether this gain is independent of ρ\rho.

(3) Necessity of phase-space stretching. The stretching map is sufficient to drive arbitrary states to high entropy and to suppress Wigner-function negativity. Is stretching of phase space also necessary, in the sense that any process capable of driving the entropy of arbitrary states arbitrarily high must asymptotically stretch phase space? Or do alternative entropy-increasing mechanisms exist (for example, mechanisms that translate distributions into unexplored phase-space regions, or otherwise distort phase space) that equally realize the classical limit? The established partial result covers only iterated maps: a map that fails to stretch some phase-space region cannot drive entropy to infinity when reapplied, because a finite phase-space region supports only finite-entropy states.

(4) Uniqueness of quantum mechanics. Can it be proven that quantum mechanics is the only theory that repairs classical mechanics at low entropy — that is, the only consistent extension of classical mechanics in which states of sufficiently high entropy are approximated by classical distributions, recovering classical mechanics as the high-entropy limit? A resolution of this part must fix an admissible comparison class of candidate theories explicitly and then either prove uniqueness within that class or exhibit one concrete alternative theory with the same recovery property. The class is admissible only if it is fixed by conditions independent of quantum mechanics' own postulates and of the uniqueness property being tested, so that a class reduced to quantum mechanics alone, or otherwise tailored to the conclusion, cannot make the claim vacuous; natural frameworks for constructing an admissible class are generalized probabilistic theories (operational frameworks that abstract the probabilistic structure of quantum mechanics) and deformation quantization (the study of deformations of the algebra of classical observables, of which the Moyal bracket is the canonical example).

References

  1. Gabriele Carcassi, Manuele Landini, Christine A. Aidala, 'Classical mechanics as the high-entropy limit of quantum mechanics', Physica Scripta 101 (6), 065105 (2026). DOI: 10.1088/1402-4896/ae3a20, https://doi.org/10.1088/1402-4896/ae3a20 (preprint arXiv:2411.00972, https://arxiv.org/abs/2411.00972)
  2. Gabriele Carcassi, Tobias Thrien, Christine A. Aidala, 'Reverse Quantum Mechanics', arXiv:2608.27543 (2026), https://arxiv.org/abs/2608.27543
  3. Gabriel Rabelo-Soares, Gojko Vujanovic, Giorgio Torrieri, 'Hydrodynamical Initial State in Small Systems From the Phase-Space Entropy', arXiv:2511.03851 (2025), https://arxiv.org/abs/2511.03851
  4. Petr Hajicek, 'Realism-Completeness-Universality interpretation of quantum mechanics', arXiv:1509.05547 (2015), https://arxiv.org/abs/1509.05547