Classical–Quantum Duality Map

• The Classical–Quantum Duality Map is a structure-preserving correspondence that links classical variables and quantum duals via constructions like Planck-scale inversion and oscillator embeddings.
• It employs varied frameworks—ranging from stochastic-process maps and convex duality to operational protocols—to translate spectral, compositional, and geometric properties between theories.
• This approach bridges dynamics and symmetry, smoothing the classical–quantum transition across regimes and highlighting dual formulations with a characteristic two-branch structure.

Searching arXiv for the supplied papers to ground the article in the current literature. A classical–quantum duality map is a structure-preserving correspondence between classical and quantum descriptions, but the literature uses the term for several distinct constructions rather than for a single canonical formalism. In one line of work, the map is an inversion through the Planck scale relating classical gravitational variables and their quantum duals; in another, it is an exact rewriting of quantum state evolution as classical oscillator dynamics; elsewhere it appears as an operational state-transfer protocol, a correspondence between stochastic matrices and qubit channels, a convex duality theorem for hybrid classical–quantum states, or a geometric relation between classical action periods and quantum spectra (Sanchez, 2018, Skinner, 2013, Ashkenazi et al., 2021, Karimipour et al., 2011, Barthe et al., 8 Oct 2025, Kreshchuk et al., 2016). Taken together, these papers suggest that the expression names a family of maps that preserve selected structures—such as spectra, composition laws, matrix elements, or generalized actions—under sharply different assumptions.

1. Terminological scope and recurring structures

The broad literature attaches the phrase to several non-equivalent but technically precise constructions. Some maps relate variables across regimes, some embed one dynamics into another, and some express convex or operational duality rather than theory equivalence.

Formulation	Mapped objects	Representative paper
Planck-scale inversion map	Classical/gravity variables and quantum dual variables	(Sanchez, 2018)
Exact dynamical embedding	Quantum states and classical oscillators	(Skinner, 2013)
Finite-state correspondence	Classical reversible dynamics and finite-energy quantum dynamics	(Margolus, 2011)
Operational duality protocol	Dual lattice-model states on different Hilbert spaces	(Ashkenazi et al., 2021)
Hybrid convex duality	Classical–quantum couplings and split witnesses	(Barthe et al., 8 Oct 2025)
Stochastic-process map	$4\times 4$ stochastic matrices and qubit channels	(Karimipour et al., 2011)

The common structural motifs are explicit. Many constructions preserve composition, as in the stochastic-process map and the oscillator embedding; others preserve matrix elements of observables, as in the duality-as-transformation protocol; still others preserve spectra, as in the stochastic and action-based formulations. This suggests that “duality map” often denotes a controlled translation of invariant content rather than a reduction of quantum theory to classical mechanics.

2. Planck-scale inversion and the $QG$ variables

The most explicit use of the expression appears in the proposal that the classical–quantum duality underlying ordinary quantum theory extends to the Planck domain (Sanchez, 2018). The basic statement is that corresponding gravity and quantum variables are related by inversion through the appropriate Planck quantity: $O_Q=o_P^2\,O_G^{-1}, \qquad O_G=o_P^2\,O_Q^{-1}.$ The paper applies this componentwise to

$$O_G=(L_G,\,M,\,\mathcal K_G,\,T_e),\qquad O_Q=(L_Q,\,M_Q,\,\mathcal K_Q,\,T_{eQ}),$$

so that the construction is a length–mass–time–temperature–acceleration duality through the Planck scale (Sanchez, 2018).

For length and mass, the explicit relations are

$$L_G=\frac{G}{c^2}M,\qquad L_Q=\frac{\hbar}{Mc},\qquad L_Q=\frac{l_P^2}{L_G},\qquad M_Q=\frac{m_P^2}{M}.$$

The Planck scale is defined by

$$\frac{l_P}{m_P}=\frac{G}{c^2},\qquad l_Pm_P=\frac{\hbar}{c},$$

and is the self-dual point: $O_G=O_Q=o_P.$ In the elementary-particle regime,

$$0\le M<m_P,\qquad 0\le L_G<l_P,\qquad l_P<L_Q\le\infty,$$

whereas in the classical gravity regime,

$$m_P<M<\infty,\qquad l_P<L_G<\infty,\qquad 0\le L_Q<l_P.$$

At $M=m_P$, one has $QG$0 (Sanchez, 2018).

The distinctive construction is the introduction of variables that combine both sectors: $QG$1 For normalized length and mass this gives

$QG$2

and the same normalized expression is asserted for mass, acceleration, and temperature: $QG$3 This is invariant under the exchange $QG$4, so

$QG$5

The paper interprets this as a complete variable spanning the classical, semiclassical, quantum, and Planck regimes in a single expression (Sanchez, 2018).

A central feature is the two-branch structure. Solving

$QG$6

yields inverse-related branches,

$QG$7

which the paper reads as “two different and dual ways of reaching the Planck scale” (Sanchez, 2018).

3. Analytic extension, Kruskal structure, and “quantum dressed” horizons

The same proposal introduces a time-type variable from a difference rather than a sum: $QG$8 so that

$QG$9

Thus $O_Q=o_P^2\,O_G^{-1}, \qquad O_G=o_P^2\,O_Q^{-1}.$0 is even under inversion while $O_Q=o_P^2\,O_G^{-1}, \qquad O_G=o_P^2\,O_Q^{-1}.$1 is odd (Sanchez, 2018).

The analytic extension is built through star coordinates and null variables: $O_Q=o_P^2\,O_G^{-1}, \qquad O_G=o_P^2\,O_Q^{-1}.$2 which in $O_Q=o_P^2\,O_G^{-1}, \qquad O_G=o_P^2\,O_Q^{-1}.$3 variables become

$O_Q=o_P^2\,O_G^{-1}, \qquad O_G=o_P^2\,O_Q^{-1}.$4

The explicit “exterior” formulas are

$O_Q=o_P^2\,O_G^{-1}, \qquad O_G=o_P^2\,O_Q^{-1}.$5

with

$O_Q=o_P^2\,O_G^{-1}, \qquad O_G=o_P^2\,O_Q^{-1}.$6

Four patches $O_Q=o_P^2\,O_G^{-1}, \qquad O_G=o_P^2\,O_Q^{-1}.$7 are required to cover the full $O_Q=o_P^2\,O_G^{-1}, \qquad O_G=o_P^2\,O_Q^{-1}.$8 manifold, and the paper explicitly compares $O_Q=o_P^2\,O_G^{-1}, \qquad O_G=o_P^2\,O_Q^{-1}.$9 to complete Kruskal coordinates and $$O_G=(L_G,\,M,\,\mathcal K_G,\,T_e),\qquad O_Q=(L_Q,\,M_Q,\,\mathcal K_Q,\,T_{eQ}),$$0 or $$O_G=(L_G,\,M,\,\mathcal K_G,\,T_e),\qquad O_Q=(L_Q,\,M_Q,\,\mathcal K_Q,\,T_{eQ}),$$1 to incomplete Schwarzschild or Rindler coordinates (Sanchez, 2018).

Applied to Schwarzschild–Kruskal spacetime, the map reinterprets the exterior regions as classical or semiclassical gravity domains and the interior as “totally quantum.” The null horizons are

$$O_G=(L_G,\,M,\,\mathcal K_G,\,T_e),\qquad O_Q=(L_Q,\,M_Q,\,\mathcal K_Q,\,T_{eQ}),$$2

while the singularity $$O_G=(L_G,\,M,\,\mathcal K_G,\,T_e),\qquad O_Q=(L_Q,\,M_Q,\,\mathcal K_Q,\,T_{eQ}),$$3 is mapped to

$$O_G=(L_G,\,M,\,\mathcal K_G,\,T_e),\qquad O_Q=(L_Q,\,M_Q,\,\mathcal K_Q,\,T_{eQ}),$$4

in the interior. More generally, the four hyperbolae

$$O_G=(L_G,\,M,\,\mathcal K_G,\,T_e),\qquad O_Q=(L_Q,\,M_Q,\,\mathcal K_Q,\,T_{eQ}),$$5

are interpreted as the boundaries of the quantum-gravity region, with $$O_G=(L_G,\,M,\,\mathcal K_G,\,T_e),\qquad O_Q=(L_Q,\,M_Q,\,\mathcal K_Q,\,T_{eQ}),$$6 understood as $$O_G=(L_G,\,M,\,\mathcal K_G,\,T_e),\qquad O_Q=(L_Q,\,M_Q,\,\mathcal K_Q,\,T_{eQ}),$$7 in Planck units (Sanchez, 2018).

From this construction comes the paper’s strongest interpretive claim: the horizon becomes “quantum dressed” or “l’horizon habillé.” Instead of a single null surface $$O_G=(L_G,\,M,\,\mathcal K_G,\,T_e),\qquad O_Q=(L_Q,\,M_Q,\,\mathcal K_Q,\,T_{eQ}),$$8, the Planck-scale boundaries appear as

$$O_G=(L_G,\,M,\,\mathcal K_G,\,T_e),\qquad O_Q=(L_Q,\,M_Q,\,\mathcal K_Q,\,T_{eQ}),$$9

The stated physical meaning is that near the horizon the distinction between classical exterior and quantum interior is smoothed by a Planck-width structure (Sanchez, 2018).

The symmetry content is also enlarged. In normalized form,

$$L_G=\frac{G}{c^2}M,\qquad L_Q=\frac{\hbar}{Mc},\qquad L_Q=\frac{l_P^2}{L_G},\qquad M_Q=\frac{m_P^2}{M}.$$0

while in the Kruskal setting

$$L_G=\frac{G}{c^2}M,\qquad L_Q=\frac{\hbar}{Mc},\qquad L_Q=\frac{l_P^2}{L_G},\qquad M_Q=\frac{m_P^2}{M}.$$1

The paper further states that spacetime reflections, antipodal symmetry, and PT/CPT symmetry are contained in the broader $$L_G=\frac{G}{c^2}M,\qquad L_Q=\frac{\hbar}{Mc},\qquad L_Q=\frac{l_P^2}{L_G},\qquad M_Q=\frac{m_P^2}{M}.$$2 symmetry (Sanchez, 2018). Its own limitations are explicit: the framework is heuristic and kinematical, not a derivation of dynamics from a specific quantum-gravity theory (Sanchez, 2018).

4. Exact dynamical correspondences

A distinct class of classical–quantum duality maps rewrites one dynamics exactly in the variables of another. Skinner constructs an exact mapping between arbitrary finite-dimensional quantum evolution and networks of classical coupled oscillators (Skinner, 2013). In the density-matrix formulation,

$$L_G=\frac{G}{c^2}M,\qquad L_Q=\frac{\hbar}{Mc},\qquad L_Q=\frac{l_P^2}{L_G},\qquad M_Q=\frac{m_P^2}{M}.$$3

with $$L_G=\frac{G}{c^2}M,\qquad L_Q=\frac{\hbar}{Mc},\qquad L_Q=\frac{l_P^2}{L_G},\qquad M_Q=\frac{m_P^2}{M}.$$4 real antisymmetric for a Hermitian operator basis, so $$L_G=\frac{G}{c^2}M,\qquad L_Q=\frac{\hbar}{Mc},\qquad L_Q=\frac{l_P^2}{L_G},\qquad M_Q=\frac{m_P^2}{M}.$$5 is real symmetric and negative semidefinite. The quantum dynamics therefore become the dynamics of coupled harmonic oscillators. For density matrices the mapping requires $$L_G=\frac{G}{c^2}M,\qquad L_Q=\frac{\hbar}{Mc},\qquad L_Q=\frac{l_P^2}{L_G},\qquad M_Q=\frac{m_P^2}{M}.$$6 or $$L_G=\frac{G}{c^2}M,\qquad L_Q=\frac{\hbar}{Mc},\qquad L_Q=\frac{l_P^2}{L_G},\qquad M_Q=\frac{m_P^2}{M}.$$7 real coordinates; for Schrödinger states it uses only $$L_G=\frac{G}{c^2}M,\qquad L_Q=\frac{\hbar}{Mc},\qquad L_Q=\frac{l_P^2}{L_G},\qquad M_Q=\frac{m_P^2}{M}.$$8 real oscillator coordinates, corresponding to the real and imaginary parts of the amplitudes: $$L_G=\frac{G}{c^2}M,\qquad L_Q=\frac{\hbar}{Mc},\qquad L_Q=\frac{l_P^2}{L_G},\qquad M_Q=\frac{m_P^2}{M}.$$9 The correspondence is exact at the level of state evolution, including expectation values and measurement probabilities, but the paper explicitly does not claim that measurement collapse or entanglement ontology thereby become classical (Skinner, 2013).

Margolus gives an exact map in the opposite direction for finite-state, invertible, discrete-time classical dynamics (Margolus, 2011). Each classical configuration $$\frac{l_P}{m_P}=\frac{G}{c^2},\qquad l_Pm_P=\frac{\hbar}{c},$$0 is identified with an orthogonal basis vector $$\frac{l_P}{m_P}=\frac{G}{c^2},\qquad l_Pm_P=\frac{\hbar}{c},$$1, the classical update becomes a unitary shift

$$\frac{l_P}{m_P}=\frac{G}{c^2},\qquad l_Pm_P=\frac{\hbar}{c},$$2

and a Hamiltonian is defined in the Fourier basis by

$$\frac{l_P}{m_P}=\frac{G}{c^2},\qquad l_Pm_P=\frac{\hbar}{c},$$3

Then the continuous Schrödinger evolution reproduces the classical update exactly at the sampling times, while the intermediate motion is a bandlimited interpolation: $$\frac{l_P}{m_P}=\frac{G}{c^2},\qquad l_Pm_P=\frac{\hbar}{c},$$4 A major identification in this map is that average quantum energy becomes classical update rate: $$\frac{l_P}{m_P}=\frac{G}{c^2},\qquad l_Pm_P=\frac{\hbar}{c},$$5 The paper presents this as an exact isomorphism for finite-state reversible dynamics, not as standard quantization of a classical phase space (Margolus, 2011).

For free bosonic fields in time-dependent classical backgrounds, the “classical-quantum correspondence” maps $$\frac{l_P}{m_P}=\frac{G}{c^2},\qquad l_Pm_P=\frac{\hbar}{c},$$6 quantum degrees of freedom to a classical matrix oscillator system with $$\frac{l_P}{m_P}=\frac{G}{c^2},\qquad l_Pm_P=\frac{\hbar}{c},$$7 simple harmonic oscillators (Vachaspati et al., 2018). Starting from

$$\frac{l_P}{m_P}=\frac{G}{c^2},\qquad l_Pm_P=\frac{\hbar}{c},$$8

the Heisenberg evolution is parametrized by Bogoliubov matrices and rewritten in terms of classical matrix variables $$\frac{l_P}{m_P}=\frac{G}{c^2},\qquad l_Pm_P=\frac{\hbar}{c},$$9 and $O_G=O_Q=o_P.$0 obeying

$O_G=O_Q=o_P.$1

The classical Hamiltonian is

$O_G=O_Q=o_P.$2

and the paper states that the quantum vacuum energy equals the classical energy,

$O_G=O_Q=o_P.$3

The map is exact for Gaussian free-field dynamics and is proposed as a practical way to treat quantum radiation and backreaction within a larger classical system (Vachaspati et al., 2018).

A related reduced-dynamics correspondence appears in the quantum Bernoulli map, where the quantum baker map is projected onto diagonal density matrices in the position basis. The resulting quantum Bernoulli map admits a quantum analogue of the classical generalized spectral representation in terms of decaying modes, with quantum Bernoulli polynomials that become classical Bernoulli polynomials in the limit $O_G=O_Q=o_P.$4 (Ordonez et al., 2011).

5. Operational, stochastic, hybrid, and optimization-based formulations

Some recent formulations shift the emphasis from variable identification to executable procedures or convex duality. In the work on duality as a feasible physical transformation, duality is defined by preservation of matrix elements of observables,

$O_G=O_Q=o_P.$5

and implemented for Abelian lattice models by preparing an auxiliary register, applying a finite-depth local unitary $O_G=O_Q=o_P.$6, and measuring/postselecting the original register (Ashkenazi et al., 2021). For the $O_G=O_Q=o_P.$7-dimensional $O_G=O_Q=o_P.$8 clock model this yields the familiar self-duality

$O_G=O_Q=o_P.$9

but now as a physical state-transfer protocol using local two-body gates and single-site measurements (Ashkenazi et al., 2021).

Karimipour and Memarzadeh construct another exact bridge by mapping four-dimensional classical stochastic matrices to qubit channels (Karimipour et al., 2011). A probability vector is parametrized by a “Bloch tetrahedron”

$$0\le M<m_P,\qquad 0\le L_G<l_P,\qquad l_P<L_Q\le\infty,$$0

and a stochastic matrix $$0\le M<m_P,\qquad 0\le L_G<l_P,\qquad l_P<L_Q\le\infty,$$1 induces an affine map $$0\le M<m_P,\qquad 0\le L_G<l_P,\qquad l_P<L_Q\le\infty,$$2. The corresponding qubit map is

$$0\le M<m_P,\qquad 0\le L_G<l_P,\qquad l_P<L_Q\le\infty,$$3

with

$$0\le M<m_P,\qquad 0\le L_G<l_P,\qquad l_P<L_Q\le\infty,$$4

The map preserves composition,

$$0\le M<m_P,\qquad 0\le L_G<l_P,\qquad l_P<L_Q\le\infty,$$5

and preserves spectrum. For the normal form $$0\le M<m_P,\qquad 0\le L_G<l_P,\qquad l_P<L_Q\le\infty,$$6, complete positivity of the qubit map is equivalent to positivity of the classical doubly stochastic matrix, and symmetric classical generators correspond to Lindblad generators (Karimipour et al., 2011).

The hybrid-dynamics literature uses “map” in a narrower sense. The paper on the two classes of hybrid classical-quantum dynamics does not construct a duality between a purely classical theory and a purely quantum theory; instead it gives the most general completely positive, norm-preserving Markovian evolution on a joint classical–quantum state space (Oppenheim et al., 2022). The state is

$$0\le M<m_P,\qquad 0\le L_G<l_P,\qquad l_P<L_Q\le\infty,$$7

and the main classification theorem states that memoryless hybrid dynamics falls into exactly two classes: finite-sized jumps in classical phase space or continuous dynamics. In the continuous case, nontrivial classical–quantum back-reaction requires both quantum decoherence and classical diffusion: $$0\le M<m_P,\qquad 0\le L_G<l_P,\qquad l_P<L_Q\le\infty,$$8 The paper explicitly presents this as a bridge for coupling, not as a theory-to-theory duality map (Oppenheim et al., 2022).

A convex-analytic version of the term appears in the duality theorem for classical–quantum states (Barthe et al., 8 Oct 2025). For countable classical sets $$0\le M<m_P,\qquad 0\le L_G<l_P,\qquad l_P<L_Q\le\infty,$$9 and finite-dimensional Hilbert spaces $$m_P<M<\infty,\qquad l_P<L_G<\infty,\qquad 0\le L_Q<l_P.$$0, a classical–quantum state is a function

$$m_P<M<\infty,\qquad l_P<L_G<\infty,\qquad 0\le L_Q<l_P.$$1

with positive values and total trace at most $$m_P<M<\infty,\qquad l_P<L_G<\infty,\qquad 0\le L_Q<l_P.$$2. The theorem states that for bounded assertions $$m_P<M<\infty,\qquad l_P<L_G<\infty,\qquad 0\le L_Q<l_P.$$3,

$$m_P<M<\infty,\qquad l_P<L_G<\infty,\qquad 0\le L_Q<l_P.$$4

where

$$m_P<M<\infty,\qquad l_P<L_G<\infty,\qquad 0\le L_Q<l_P.$$5

Here the duality is between relational couplings and split witnesses, and it underlies the soundness and completeness of the logic $$m_P<M<\infty,\qquad l_P<L_G<\infty,\qquad 0\le L_Q<l_P.$$6 (Barthe et al., 8 Oct 2025).

A closely related optimization-oriented development is the quantum-classical moment duality based on degree-$$m_P<M<\infty,\qquad l_P<L_G<\infty,\qquad 0\le L_Q<l_P.$$7 Sum-of-Squares (Daskin, 19 Jun 2026). For any $$m_P<M<\infty,\qquad l_P<L_G<\infty,\qquad 0\le L_Q<l_P.$$8-qubit density matrix $$m_P<M<\infty,\qquad l_P<L_G<\infty,\qquad 0\le L_Q<l_P.$$9, the matrix

$M=m_P$0

is automatically feasible for the classical Goemans–Williamson relaxation. Applying random hyperplane rounding yields the certificate

$M=m_P$1

The same moment matrix is then used for correlation-based circuit cutting with rigorous error bounds (Daskin, 19 Jun 2026).

6. Geometric, algebraic, and field-theoretic realizations

Other papers use the phrase for more specialized geometric or algebraic correspondences. Allegretti and Kim construct a quantum duality map for cluster varieties associated with punctured surfaces (Allegretti et al., 2015). The map

$M=m_P$2

sends tropical integral laminations to elements of the quantum cluster algebra. Its classical limit is the Fock–Goncharov map,

$M=m_P$3

and its highest term is

$M=m_P$4

Here the duality is between tropical lamination data and canonical quantum functions on $M=m_P$5-cluster varieties (Allegretti et al., 2015).

In noncommutative gauge theory, the $M=m_P$6-exact Seiberg–Witten map is shown to survive quantization perturbatively in the coupling and exactly in $M=m_P$7 (Martin et al., 2016). The core statement is equality of on-shell DeWitt effective actions,

$M=m_P$8

The proof uses the triviality of the Jacobians $M=m_P$9 and $QG$00 in dimensional regularization. In this setting, “quantum duality” means equality of the physical perturbative quantum theory on shell, not identity of arbitrary off-shell Green functions (Martin et al., 2016).

The action-based formulation developed for sextic and Lamé systems traces spectral reflection symmetry back to classical period relations on the Riemann surface of the classical momentum (Kreshchuk et al., 2016). The classical action

$QG$01

lifts to the quantum action

$QG$02

with

$QG$03

For the sextic quasi-exactly solvable sector, the quantum relation

$QG$04

implies

$QG$05

For Lamé,

$QG$06

The paper interprets this as a map from classical cycle duality of $QG$07 to reflection symmetry of the quantum spectrum (Kreshchuk et al., 2016).

The Newman–Penrose map gives another field-theoretic example (Elor et al., 2020). Starting from a real four-dimensional Kerr–Schild spacetime whose null vector generates an expanding shear-free null geodesic congruence, one extracts a harmonic scalar $QG$08 and applies

$QG$09

The resulting $QG$10 is a self-dual vacuum Maxwell field on flat space,

$QG$11

For Schwarzschild and Kerr, the real part agrees up to gauge transformation with the usual single-copy gauge field, while the imaginary part gives the electromagnetic dual counterpart (Elor et al., 2020).

A more explicitly classical-network realization appears in the construction of “quantum-like product states” from oscillator graphs (Scholes et al., 2024). The exact theorem is a one-to-one map between the product basis of $QG$12-qubit states and the emergent eigenstates of classical oscillator networks built from Cartesian products of graphs. The key spectral identity is

$QG$13

so if $QG$14 and $QG$15, then $QG$16 is an eigenvector of the product graph with eigenvalue $QG$17. The paper extends this to “quantum-like gates,” but efficient entangling constructions preserve the product structure only approximately (Scholes et al., 2024).

7. Limits, interpretive disputes, and open directions

The literature is unusually explicit about limits. The Planck-scale $QG$18 variables are presented as heuristic and kinematical, not as a derivation of dynamics from loop gravity, string theory, or another specific microscopic framework (Sanchez, 2018). Skinner’s oscillator map is exact for state evolution, but the paper stresses that it does not explain measurement collapse, Born-rule ontology, or the meaning of entanglement as a nonclassical resource (Skinner, 2013). The field-theory correspondence for $QG$19 variables is exact only for free real bosonic fields in classical backgrounds; interactions and genuinely quantum backgrounds lie outside its exact domain (Vachaspati et al., 2018).

Some proposals are bridges rather than dualities in the stronger sense. The hybrid master-equation framework explicitly does not give a map

$QG$20

but rather the most general consistent hybrid evolution law (Oppenheim et al., 2022). The classical–quantum state duality theorem in program semantics is a convex duality between couplings and witnesses, not a claim that classical and quantum theories are equivalent (Barthe et al., 8 Oct 2025). The moment-duality construction based on degree-$QG$21 SoS uses only $QG$22-moments and therefore discards most state information even though it yields a universal optimization certificate (Daskin, 19 Jun 2026).

Interpretive proposals are similarly circumscribed. The metaplectic “classical dual states” program argues that ontological states are already present inside quantum theory as dual classical states, obtained by projection onto circle, cylinder, or coherent-state representations, with $QG$23 as the symmetry of the classical–quantum duality (Cirilo-Lombardo et al., 19 Jan 2025). Yet the construction is representational and state-projection based, not a Bohmian hidden-variable theory with trajectories or a full dynamical equivalence between quantum and classical mechanics (Cirilo-Lombardo et al., 19 Jan 2025). The graph-based “quantum-like” constructions reproduce tensor-product kinematics and some gate analogues, but not a full account of measurement, Born probabilities, or nonlocality, and exact efficient entangling implementations are not established (Scholes et al., 2024).

Taken together, these limitations indicate that “classical-quantum duality map” names a recurrent research strategy rather than a settled doctrine. In one branch it denotes exact mathematical embeddings; in another, operational state-transfer protocols; in another, geometric period relations; in another, convex duality for hybrid objects. A plausible implication is that the most durable content of the term lies not in the slogan of a universal classical–quantum equivalence, but in explicit maps that preserve a sharply delimited set of structures under clearly stated dynamical, geometric, or algebraic assumptions (Sanchez, 2018, Ashkenazi et al., 2021, Kreshchuk et al., 2016, Barthe et al., 8 Oct 2025).