Quantum Disk Holography

• Holography on the quantum disk is an approach that leverages quantum and noncommutative geometries to relate bulk dynamics with boundary conformal data.
• It encompasses applications such as quantum spin holography, entangled-photon imaging, and quantum gravity, connecting rigorous mathematical models to experimental practices.
• The topic employs q-deformed algebras, matrix models, and scattering formalisms to quantify holographic encoding and retrieve quantum information from disk geometries.

Holography on the quantum disk refers to the collection of theoretical, experimental, and computational frameworks in which the principles of holographic duality are realized or analyzed within geometric, physical, or operator-theoretic models involving the "disk," most frequently in two dimensions, but also encompassing its quantum, noncommutative, and topological connections. The term encompasses both (i) the application of holographic encoding—where boundary data reconstructs bulk information—in models governed by quantum mechanical laws, and (ii) explicit realizations in physical or mathematical systems where the disk geometry and its quantum analogs play a central role.

1. Quantum Disk Geometries and Noncommutative Structures

The quantum disk is a noncommutative deformation of the classical hyperbolic disk. Instead of commuting coordinates, generators $z$ and $z^*$ satisfy $z^* z = q^2 z z^* + 1 - q^2$, and more generally, all functions form a noncommutative algebra. The isometry group is replaced by the quantum group $SU_q(1,1)$ (Almheiri et al., 10 Jan 2024). The associated universal enveloping algebra $U_q(\mathfrak{su}_{1,1})$ has generators $K$, $E$, $F$ obeying:

$$K E = q^2 E K, \quad K F = q^{-2} F K, \quad [E, F] = \frac{K - K^{-1}}{q - q^{-1}}$$

with a coproduct encoding the nontrivial quantum group action:

$$\Delta(K) = K \otimes K,\quad \Delta(E) = E \otimes 1 + K \otimes E, \quad \Delta(F) = F \otimes K^{-1} + 1 \otimes F$$

This noncommutativity and Hopf algebra structure source the discrete, "braided" nature of quantum geometry in the bulk.

2. Holographic Duality and Bulk–Boundary Correspondence

Holography on the quantum disk studies how quantum or noncommutative bulk dynamics translate into boundary theory. For the noncommutative quantum disk, bulk fields $\phi(z, z^*)$ admit boundary expansions of the form $\phi(z, z^*) \sim [f(z) + f^*(z^*)]\, y^\Delta$, with $y = 1 - z z^*$. The $q$-deformed Casimir operator,

$$C_q = q^{-1}(1-z z^*)^2 \frac{\partial}{\partial z}\frac{\partial}{\partial z^*}$$

controls the spectrum and determines the scaling dimensions of boundary operators. The boundary limit (where $y \to 0$) effectively projects out the noncommutativity, resulting in a boundary theory with commutative coordinates and conformal invariance. Explicit two-point functions on the boundary involve $q$-Pochhammer symbols and reduce to standard conformal correlators as $q \to 1$:

$$\langle O(\varphi) O(\theta) \rangle \propto \left(1 / \sin[(\varphi-\theta)/2] \right)^{2\Delta}$$

(Almheiri et al., 10 Jan 2024). Thus, the noncommutative bulk geometry leaves a deformation imprint on local boundary dynamics without spoiling the essential features of holographic duality.

3. Quantum Spin Holography and Information Encoding

Quantum spin holography uses surface-state electrons and spin-dependent scattering to encode and retrieve information on quantum disks. Spin-polarized electrons (majority $\uparrow$, minority $\downarrow$) scatter off engineered arrays of atoms or molecules, producing distinct local density of states (LDOS) patterns per spin channel. The multiple scattering formalism describes the process:

$$a_\text{out} = a_\text{in} \frac{e^{2i\eta}-1}{2i}, \quad \eta = \delta + i \frac{\ln \alpha}{2}$$

with propagation amplitude between scatterers modeled as $$\left(\frac{2}{\pi k \Delta r}\right)^{1/2} e^{i k \Delta r}$$. By arrangement optimization (e.g. genetic algorithms), two independent pages of bit-encoded information can be stored for the same physical region—one for each spin species—effectively doubling storage density. Experimental implementations on Co/Cu(111) quantum disks with Cu capping exploit intrinsic spin polarization and magnetic scatterers; the readout requires spin-polarized STM (Brovko et al., 2012).

4. Quantum Holography via Entangled States and Hybrid Degrees of Freedom

Quantum holography can leverage entangled photons in various degrees of freedom including polarization, spatial modes (OAM), and via hybrid entanglement (metasurface-assisted). Unlike classical holography based on first-order coherence, quantum schemes encode phase information in second-order (coincedence) correlations, as in:

• Polarization-hyperentangled photons: Phase images are remotely reconstructed even under high classical noise, with enhanced spatial resolution (Defienne et al., 2019).
• OAM-entangled photons: High-dimensional channel capacity allows multiplexed image encoding and quantum encryption; superposition keys on the OAM Poincaré sphere offer security greater than binary schemes (Kong et al., 2023).
• Metasurface-generated polarization-hologram hybrid entanglement: A metasurface imprints two holograms correlated with photon circular polarization, enabling remote erasure or selective retrieval of hologram content depending on idler polarization measurement (quantum holographic eraser) (Liang et al., 20 Aug 2024).

Table: Quantum Holography Modalities

Physical Degree	Storage Capacity	Robustness/Selectivity
Spin (QSH)	doubled (by spin)	controlled by spin alignment
Polarization	2D (HH/VV)	robust to phase disorder
OAM	High-dimensional (N>2)	security via superpositions
Hybrid (Metasurface)	amplitude+phase+pol state	selective erasure by detection

5. Boundary Encoding in Quantum Gravity and Topological Models

In loop quantum gravity (LQG), the quantum holographic principle states that each puncture (elementary area unit) at the boundary encodes a qubit per Planck cell, not merely a classical bit. Fermion fields in the bulk are regularized by doubling degrees of freedom (environment and system), with Bogoliubov transformations entangling spin states at boundary punctures. On fuzzy spheres ($N=2$ representation), this discretization reduces infinite-dimensional operator content to qubit-sized "cells," and an area–information relation

$$2^N \leq \frac{A}{4L_p^2} \Rightarrow N = -\frac{1}{\ln2} \ln\left(\frac{A}{4L_p^2}\right)$$

emerges, yielding an exponential quantum information capacity per boundary area (Zizzi, 2021).

Topological holography extends the concept in (1+1)D disk-like systems, embedding quantum phases or critical points on the boundary of (2+1)D modular fusion categories. The sandwich construction separates symmetry (Lagrangian algebra data) from local dynamics, constrains anomalies, and realizes partition functions as inner products between bulk and boundary states:

$Z = \langle \mathcal{A} | \Psi \rangle$

where $|\mathcal{A}\rangle$ describes topological boundary and $|\Psi\rangle$ the physical boundary (gapped/gapless/Cardy state) (Huang et al., 2023).

6. Quantum Disk in 2D Gravity, Minimal String, and Matrix Model Correspondence

In 2D quantum gravity, particularly Liouville gravity on the disk, holographic duality manifests through gravitationally dressed correlators. For the fixed-length disk boundary, bulk amplitudes—expressed as integrals over a quantum-deformed group measure—mirror observables in 2D BF theory and are closely related to the modular double of $U_q(\mathfrak{sl}(2,\mathbb{R}))$ (Mertens et al., 2020):

• Bulk one-point:

$$\langle \mathcal{T}_\alpha \rangle_\ell = \frac{2}{b} \int_0^\infty ds\, e^{-\ell \mu_B(s)} \cos(4\pi P s)$$

• Boundary two-point:

$$\mathcal{A}_\beta(\ell_1, \ell_2) = N_\beta \int ds_1 ds_2\, \rho(s_1)\rho(s_2) e^{-\mu_B(s_1)\ell_1} e^{-\mu_B(s_2)\ell_2} \ldots$$

Disk amplitudes computed in continuum field theory match matrix model results for minimal string theory after appropriate degeneration and limiting procedures, providing a non-perturbative check of holography on the disk. Multi-boundary topologies are assembled by gluing bulk functions with quantum-deformed Weil-Petersson volumes, and the theory flows to JT gravity in a parametric limit.

7. Quantum Gravity, Fermion Backreaction, and Holographic Phases on the Disk

For triangulated two-dimensional disks with $N$ Kähler–Dirac fermion flavors (Asaduzzaman et al., 27 Jan 2024), imposition of negative average bulk curvature, Dirichlet boundary conditions, and a sum over triangulations leads to holographic scaling in the large-$N$ limit. Power-law decay for boundary correlators of the form

$\langle \phi_i \phi_j \rangle \sim 1/r^{2\Delta}$

is observed, with $\Delta(\Delta-d) = (mL)^2$ for dimension $d=1$ (corresponding to the standard AdS/CFT scaling relation). For finite $N$ and massive fermions, quantum fluctuations disorder the geometry, breaking holographic scaling; massless fermions preserve a conformal (hence holographic) spectrum on the boundary, consistent with quantum Liouville theory.

• Quantum spin holography with surface state electrons (Brovko et al., 2012)
• Experimental observation of quantum holographic imaging (Song et al., 2013)
• Quantum renormalization group and holography (Lee, 2013)
• Hologram of a single photon (Chrapkiewicz et al., 2015)
• The disk-based origami theorem and a glimpse of holography for traversing flows (Katz, 2018)
• Polarization entanglement-enabled quantum holography (Defienne et al., 2019)
• Liouville quantum gravity -- holography, JT and matrices (Mertens et al., 2020)
• Quantum holography from fermion fields (Zizzi, 2021)
• High-dimensional entanglement-enabled holography for quantum encryption (Kong et al., 2023)
• Digital holographic imaging via direct quantum wavefunction reconstruction (Hu et al., 2023)
• Topological holography, quantum criticality, and boundary states (Huang et al., 2023)
• Holography on the quantum disk (Almheiri et al., 10 Jan 2024)
• Fermions, quantum gravity and holography in two dimensions (Asaduzzaman et al., 27 Jan 2024)
• Metasurface-enabled quantum holograms with hybrid entanglement (Liang et al., 20 Aug 2024)