In-Plane Quantization (IPQ) State

• The IPQ state is a framework defined by applying group-theoretic quantization, coherent state techniques, and POVM-based methods to capture noncommutative features on the 2D plane.
• It utilizes a modified symplectic structure and shifted canonical commutation relations to quantify the intrinsic quantum of area, effectively linking classical and quantum regimes.
• IPQ state constructions are instrumental in modeling transport phenomena in condensed matter systems, such as the quantized conductance observed in magnetic topological insulators.

The In Plane Quantization (IPQ) state encompasses a set of rigorous mathematical and physical constructions that arise in quantum mechanics, quantum information, and condensed matter physics contexts. It specifically addresses quantization procedures, group-theoretic symmetry extensions, and the role of noncommutative geometry, often on the Euclidean plane ℝ² or variants such as the punctured plane. The term "IPQ state" denotes states constructed within these frameworks—via canonical group quantization, positive operator-valued measures (POVM), coherent-state methods, or in the context of universal transport regimes—encoding quantized features tied to the geometry and symplectic structure of the plane.

1. Symplectic Structure and Canonical Group Quantization

In canonical quantization, the classical phase space $T^*\mathbb{R}^2 \cong \{(q^i, p_i)\}_{i=1,2}$ is endowed with a modified symplectic two-form to model noncommuting coordinates: $$\omega = dq^i \wedge dp_i + \frac{1}{2}\,\theta^{ij}\,dp_i \wedge dp_j, \quad \theta^{ij} = -\theta^{ji}.$$ In two dimensions, after fixing $\theta^{12} = -\theta^{21} = \theta$, this becomes

$$\omega = dq^i \wedge dp_i + \frac{\theta}{2}\,\varepsilon^{ij}\,dp_i \wedge dp_j,$$

with $\varepsilon^{12} = 1$. This dual (“magnetic field”-like) term signals noncommutativity in the configuration coordinates and drives the extension of the translation group from abelian $\mathbb{R}^2 \times \mathbb{R}^2$ to a nonabelian, centrally extended group $G_\theta$.

Quantization via Isham’s canonical-group construction then yields a deformed Heisenberg group $G_\theta$ whose Lie algebra generators and their commutation relations directly incorporate the noncommutativity parameter $\theta$. Explicitly, on shifted variables

$$q'^i = q^i - \frac{1}{2}\theta^{ij}p_j, \quad p'_i = p_i,$$

the Poisson brackets are

$$\{q'^i, q'^j\} = \theta^{ij}, \quad \{q'^i, p'_j\} = \delta^i_j, \quad \{p'_i, p'_j\} = 0.$$

Quantization yields the canonical commutation relations: $$[\widehat Q_i, \widehat Q_j] = i\theta\,\varepsilon_{ij},\quad [\widehat Q_i, \widehat P_j] = i\hbar\,\delta_{ij},\quad [\widehat P_i, \widehat P_j] = 0.$$ This formalism underpins group-theoretic IPQ states as vectors in the Schrödinger representation of $G_\theta$ (Umar et al., 2018, Chowdhury et al., 2012).

2. Coherent State and POVM Quantization Schemes

A parallel, operationally motivated approach constructs IPQ states via families of labeled density operators or coherent states, often exploiting overcompleteness and a resolution-of-identity property.

In the POVM quantization framework, the plane is identified with $\mathbb{C}$, and the thermal/Gibbs density operator at temperature $T$ is “displaced” to define

$$\rho_T(z) = D(z)\,\rho_T\,D(z)^\dagger,\qquad D(z) = e^{z\,a^\dagger - \overline{z}\,a},$$

where $a, a^\dagger$ act on a harmonic-oscillator Hilbert space. At zero temperature, these become pure-state projectors onto canonical coherent states $|z\rangle$. The collection $\{ \rho_T(z) \}_{z\in\mathbb{C}}$ resolves the identity: $\int_{\mathbb{C}} \rho_T(z)\,\frac{d^2z}{\pi} = I.$ Quantization of functions $f(z)$ proceeds via

$$A_f = \int_{\mathbb{C}} f(z)\,\rho_T(z)\,\frac{d^2z}{\pi}.$$

This approach preserves the canonical commutation relation $[A_q, A_p] = iI$, even in the “fuzzy,” temperature-smoothened regime (for $T > 0$) (Gazeau et al., 2014, Beneduci et al., 2021).

In the circular/plane orientation model, an IPQ state can also be realized as a density matrix $\rho_{r, \phi_0}$ on $\mathbb{R}^2$, rotated by $SO(2)$ actions and integrated against probability densities over the orientation variable (Beneduci et al., 2021). This establishes a direct link with the Toeplitz quantization picture and shows that IPQ states are generally associated with mixtures, pure states, and POVMs derived from in-plane geometric decompositions.

3. Group-Theoretic and Coherent-State Properties

A substantial formal foundation for IPQ states derives from group-theoretic quantization of the plane and its variants. Various central extensions of symmetry groups—such as the two-fold central extension of the Galilei group or the translation group $\mathbb{R}^4$—generate unitary irreducible representations on Hilbert space $L^2(\mathbb{R}^2)$. Coherent states $|n_{q,p}\rangle$ at phase-space points $(q,p)$ are built from a fixed fiducial state via group action: $$|q,p\rangle = U_{m,\lambda}\bigl(0,0,\text{Id},0, p/m, q\bigr) |\chi\rangle.$$ Resolution of the identity holds: $$\int_{\mathbb{R}^4} |n_{q,p}\rangle \langle n_{q,p}|\,dq\,dp = \mathrm{Id}_{L^2(\mathbb{R}^2)}.$$ IPQ states are typically identified as either the “vacuum” $|n_{0,0}\rangle$ or as unique cyclic vectors for the symmetry group action. Expectation values and uncertainties in these states saturate the noncommutative uncertainty relations: $$\Delta Q_1^2 = \sigma^2 + \frac{\theta^2}{4\sigma^2},\quad \Delta Q_2^2 = \sigma^2 + \frac{\theta^2}{4\sigma^2},\quad \Delta P_i^2 = \frac{1}{4\sigma^2}$$ with minimal area $\Delta Q_1\,\Delta Q_2 = |\theta|/2$ (Chowdhury et al., 2012).

When quantizing the punctured plane $\mathbb{R}^2 \setminus \{0\}$, the canonical group includes rotations and dilations, yielding IPQ basis states labeled by angular and dilation quantum numbers, with explicit wavefunctions

$$\psi_{m,k}(\rho,\phi) = (2\pi)^{-1/2}\,\rho^{i k}\,e^{i m\phi},$$

diagonalizing angular momentum and dilation generators (Somvanshi et al., 28 Oct 2025).

4. Physical Interpretation and Parameter Regimes

IPQ states encode noncommutative geometry via the parameter $\theta$, which strictly quantifies the intrinsic “quantum of area” in configuration space. The limit $\theta \rightarrow 0$ recovers ordinary quantum mechanics, while $\theta \gg \hbar$ corresponds to strong noncommutativity, analogous to lowest-Landau-level physics in strong magnetic fields.

In IPQ states, the uncertainty principle acquires a nontrivial geometry-driven lower bound, i.e.: $$\Delta Q_1\,\Delta Q_2 \geq \frac{|\theta|}{2},\quad \Delta Q_i\,\Delta P_i \geq \frac{\hbar}{2}.$$ The parameter $\theta$ thus relays an interpolation between the classical and “quantum-geometric” regimes.

In the context of plane quantization for quantum simulation (e.g., first-quantized electronic structure), “IPQ state” may refer to the preparation of antisymmetrized many-body wavefunctions (Slater determinants) in the plane-wave basis. Efficient mappings of molecular orbitals constructed from Gaussians onto plane waves enable scalable quantum computation of molecular states with a cost only logarithmic in basis size (Huggins et al., 28 Jun 2024).

5. IPQ States in Condensed Matter and Topological Phases

The term “IPQ state” also denotes a unique transport regime discovered in trilayer magnetic topological insulators (MTIs), where under strong in-plane magnetic fields the longitudinal conductivity is quantized at $\sigma_{xx} = e^2/h$ and Hall conductivity vanishes $\sigma_{xy} \approx 0$. This regime occurs precisely at the quantum critical point between quantum anomalous Hall (QAH) phases, with the IPQ state realized when the mass gap of Dirac surface bands closes as the magnetization tilts in the plane. Charge inhomogeneities and disorder-induced electron-hole puddles dominate transport, enforcing the universal quantization via a self-consistent approximation for Dirac carriers;

$$\sigma_{\min} \simeq C_0\left( \frac{n_*}{n_{\mathrm{imp}}} \right)\left( \frac{e^2}{h} \right), \qquad n_*/n_{\mathrm{imp}}\approx \frac{1}{150}.$$

The IPQ state (with $\sigma_{xx}=e^2/h$, $\sigma_{xy}\approx0$) marks the “midpoint” of the semicircular renormalization group flow interpolating between Chern-number $\pm1$ states (Yang et al., 6 Nov 2025).

6. Methodological Variants and Extensions

Multiple methodological variants of in-plane quantization exist:

• POVM-based IPQ: Quantization is achieved via families of density matrices (thermal or pure-state coherent projectors), integral quantization maps, and their induced probability and measurement structures. Compatibility, purity, and joint measurability are determined via Naimark dilation and POVM theory (Gazeau et al., 2014, Beneduci et al., 2021).
• Group-Theoretic IPQ: State construction rests on identifying the canonical symmetry group and its central extensions, then building (coherent) states as orbits in the Hilbert space. The quantization map transforms phase-space observables into self-adjoint operators exhibiting noncommutative commutation relations (Chowdhury et al., 2012, Somvanshi et al., 28 Oct 2025, Umar et al., 2018).
• IPQ in Quantum Algorithms: First-quantized molecular IPQ states are prepared efficiently using MPS (matrix product state) factorization, overlapping Gaussian orbitals and plane-wave grids, and low-depth change-of-basis isometries, yielding polylogarithmic resource scaling versus the plane-wave basis dimension $N$ (Huggins et al., 28 Jun 2024).
• IPQ in Topological Transport: The experimental realization of IPQ states as quantized critical transport regimes in MTIs links microscopic Hamiltonians, disorder statistics, and topological invariants, bridging quantum Hall universality with controllable material systems (Yang et al., 6 Nov 2025).

7. Table: Representative IPQ State Constructions

Method/Context	State Representation/Layout	Characteristic Property
Group-theoretic (Heisenberg, Galilei)	$$\psi_0(q) = \exp(-\tfrac12 q^T\Sigma^{-1}q)/\sqrt{\pi\det{\Sigma}^{1/2}}$$	Minimal area, noncommutative uncertainty
POVM/Coherent state	$\rho_T(z) = D(z) \rho_T D(z)^\dagger$	Resolution of the identity, commutation relations
Punctured-plane quantization	$$\psi_{m,k}(\rho, \phi) = (2\pi)^{-1/2}\rho^{ik}e^{im\phi}$$	Diagonalizes angular momentum, dilation
Condensed matter transport	Not a wavefunction: quantized conductivities at QCP	$\sigma_{xx}=e^2/h,\, \sigma_{xy}\approx 0$
Quantum simulation	Antisymmetrized MPS, plane-wave basis	Logarithmic resource scaling

Conclusion

In Plane Quantization (IPQ) states unify several threads in mathematical physics, quantum information, and condensed matter: they arise naturally from group-theoretic extensions capturing noncommutative geometry of the plane, from overcomplete sets of density operators or coherent states, from geometric quantization on domains with topological features (e.g., the punctured plane), and from critical phenomena in topological materials characterized by quantized transport. In all cases, the nomenclature signals a quantization scheme, state or measurement on the two-dimensional plane that crucially reflects the underlying symmetry, noncommutativity, or universal behavior of the system.