# Entanglement as "access control": a novelty assessment and literature map **The specific thesis that entanglement serves as "access control" for classical discrete information structures has no direct precedent in the published literature.** No peer-reviewed paper or preprint explicitly argues that entanglement is a gating mechanism for classical information rather than a computational resource. However, a rich constellation of work from at least five independent research programs provides substantial adjacent infrastructure—and any paper advancing this thesis must engage carefully with all of them. The Forbes group's 2024–2025 papers on topological classification of OAM entanglement come closest, but frame topology as a robust encoding of *quantum* information generated *by* entanglement, not as evidence that entanglement merely addresses pre-existing classical structures. --- ## The Forbes group's topological program establishes the key empirical foundation Three papers form the core of the most directly relevant body of work: **Ornelas, Nape, de Mello Koch & Forbes, "Non-local skyrmions as topologically resilient quantum entangled states of light," *Nature Photonics* 18, 258–266 (2024).** This landmark paper reports the first non-local quantum entangled state with non-trivial skyrmionic topology. Each individual photon possesses no salient topological structure—**topology emerges exclusively from the entanglement** of the biphoton wavefunction. The skyrmion number *N* (an integer counting wrappings of the Poincaré sphere by the non-local Stokes vector field) serves as a discrete topological invariant that persists under smooth deformations until entanglement itself vanishes. The authors propose skyrmion numbers as a "labelling system for entangled states, akin to an alphabet." **Ornelas, Nape, de Mello Koch & Forbes, "Topological rejection of noise by quantum skyrmions," *Nature Communications* 16, 2934 (2025).** This paper introduces the concept of **"digitization of quantum information"** based on discrete topological observables. The skyrmion number remains constant as concurrence, fidelity, and purity degrade, collapsing only at the maximally mixed state limit. The discrete, integer-valued nature of the invariant means noise must flip the signal between quantized states before registering any effect—directly analogous to digital versus analog signal robustness. No other research group has adopted this "digitization" terminology. **de Mello Koch, Ornelas, Gounden, Lu, Nape & Forbes, "Revealing the topological nature of entangled orbital angular momentum states of light," *Nature Communications* 16, 11095 (2025).** This is the most comprehensive paper and the most relevant to the proposed thesis. For the first time, topology is constructed using OAM alone (without polarization), enabling access to high-dimensional topological structures. The key quantitative results are striking: for *d* = 5, the topological manifold has dimension **24** (matching *d*² − 1) with **2,024 candidate invariants**; for *d* = 7, a **48-dimensional manifold** supports **17,296 candidate invariants**. The framework uses non-Abelian gauge fields of SU(*d*) Yang-Mills theory, decomposing the Lie algebra into Gell-Mann triplets forming SU(2) subalgebras, each supporting an independent topological map—effectively a "sector decomposition" of entanglement topology. A fourth paper, de Mello Koch, Lu, Ornelas, Nape & Forbes, "Quantum skyrmions in general quantum channels," *APL Quantum* 2, 026126 (2025), provides the theoretical framework for topological invariant evolution in general quantum channels, using homotopy arguments to prove topological noise rejection for non-depolarizing channels. Several 2026 preprints extend the program to atmospheric turbulence (arXiv:2603.10618), tripartite entanglement visualization (arXiv:2603.10491), and classical-quantum equivalence of topological behavior (*Nature Communications* 2026, DOI: 10.1038/s41467-026-68751-3). --- ## The *d*² − 1 parameter space and information-theoretic constraints The relationship between state-space dimensionality, accessible information, and topological invariants creates a three-tiered hierarchy that the proposed paper must address. A *d*-dimensional qudit's density matrix requires *d*² − 1 real parameters (the generalized Bloch vector). Comprehensive treatments appear in Bertlmann & Krammer, *J. Phys. A* 41, 235303 (2008) and Loubenets & Kulakov, *J. Phys. A* 54, 195301 (2021). For *d* ≥ 3, the space of valid states is no longer a ball but has nontrivial geometry—a constraint the proposed paper should acknowledge. Hardy's axiomatic framework (*arXiv:quant-ph/0101012*, 2001) shows that quantum state spaces contain *K* = *N*² parameters versus *K* = *N* for classical probability theory, with the Bloch vector functioning as a probability vector in a larger-than-classical space. The **Holevo bound** (1973) constrains extractable classical information to log₂(*d*) bits per measurement, creating an enormous gap with the *d*² − 1 parameter count. For *d* = 7 OAM entanglement, the state lives in a 48-dimensional manifold, but only ~2.8 bits are accessible per measurement. The de Mello Koch et al. (2025) finding that **the topological manifold dimension equals *d*² − 1** is therefore a critical data point: the topological spectrum explores the full parameter space, not just the Holevo-accessible fraction. The proposed paper could argue that topological invariants represent a discrete classical skeleton spanning the *d*² − 1 space, with entanglement providing the "access control" that determines which skeleton elements are physically realized. Key experimental benchmarks include Mair et al., *Nature* 412, 313 (2001) for foundational OAM entanglement; Dada et al., *Nature Physics* 7, 677 (2011) for Bell violations up to *d* = 12; Krenn et al., *PNAS* 111, 6243 (2014) for 100-dimensional entanglement; and Fickler et al., *Science* 338, 640 (2012) for OAM values up to ℓ = ±300. Entanglement certification methods that avoid full tomography include Bavaresco et al., *Nature Physics* 14, 1032 (2018) and Friis et al., *Nature Reviews Physics* 1, 72 (2019). --- ## Classical simulation results suggest entanglement has modest classical cost Several independent results establish that the classical communication cost of reproducing entanglement correlations is surprisingly low—supporting (though not directly arguing) the "access control" interpretation. **Toner & Bacon, *Physical Review Letters* 91, 187904 (2003)** proved that all correlations from local projective measurements on a maximally entangled Bell pair can be exactly simulated with local hidden variables plus **just one bit** of classical communication. Regev & Toner, *SIAM J. Computing* 39, 1562 (2009), extended this to maximally entangled states of arbitrary dimension, requiring only **two bits** for dichotomic measurements. These results quantify the "classical cost" of entanglement: the information that entanglement provides beyond local hidden variables is remarkably small in discrete classical terms. **Vidal, *Physical Review Letters* 91, 147902 (2003)** proved that pure-state quantum computations with restricted entanglement are efficiently classically simulable, with classical resources scaling exponentially only in entanglement. **Jozsa & Linden, *Proc. R. Soc. A* 459, 2011 (2003)** proved multi-partite entanglement with unboundedly growing parties is *necessary* for pure-state exponential speedup, but explicitly stated **"it is nevertheless misleading to view entanglement as a key resource for quantum-computational power."** They noted that mixed-state algorithms (DQC1-type) can achieve speedup essentially without entanglement. The quantum discord literature reinforces this: Datta, Shaji & Caves, *PRL* 100, 050502 (2008) showed the DQC1 algorithm achieves quantum speedup with vanishing entanglement but nonzero discord, and Lanyon et al., *PRL* 101, 200501 (2008) experimentally demonstrated quantum computing without entanglement. Ollivier & Zurek's foundational *PRL* 88, 017901 (2001) established that separable states can have nonzero quantum correlations, fundamentally decoupling entanglement from quantumness. --- ## Classical entanglement in structured light mirrors quantum nonseparability A well-developed research program on **classical nonseparability** provides direct experimental evidence that entanglement-like mathematical structures exist without quantum mechanics. Spreeuw's seminal *Foundations of Physics* 28, 361 (1998) introduced the concept, showing that polarization-spatial mode coupling in classical beams produces nonseparable states mathematically identical to quantum entanglement. Forbes, Aiello & Boyd, *Progress in Optics* 63, 99 (2019) comprehensively reviewed this field, establishing that quantum tools (concurrence, Bell measures) apply directly to classical light, with the crucial caveat that classical entanglement cannot produce *nonlocal* correlations. Aiello et al., *New Journal of Physics* 17, 043024 (2015) and Töppel et al., *NJP* 16, 073019 (2014) showed that classical entanglement brings metrological advantages previously thought to require quantum entanglement. The Forbes group's own 2026 *Nature Communications* paper demonstrates "fundamental equivalence between pure topological quantum states and their classical counterparts" under turbulence, with identical skyrmion number preservation. This classical-quantum topological equivalence is perhaps the strongest existing evidence for the proposed thesis—though the Forbes group frames it as parallel behavior rather than as evidence for classical primacy. Spreeuw, *PRA* 63, 062302 (2001), directly concluded that **"quantum nonlocality is the essential ingredient of a quantum computer, even more so than entanglement"**—a provocative statement distinguishing nonlocality from nonseparability that the proposed paper should engage with. --- ## Spekkens' toy model and epistemic frameworks provide philosophical support **Spekkens, *Physical Review A* 75, 032110 (2007)** constructed a purely classical toy theory with an epistemic restriction ("knowledge balance principle") that reproduces entanglement, no-cloning, teleportation, dense coding, interference, and monogamy—failing only at Bell nonlocality and contextuality. Catani & Browne, *New Journal of Physics* 19, 073035 (2017) proved this toy theory is operationally equivalent to stabilizer quantum mechanics in odd prime dimensions, identifying **contextuality** as the genuine quantum resource. This directly supports the proposed thesis's underlying logic: entanglement-like structures can emerge from classical information under access restrictions. The ψ-epistemic program, formalized by Harrigan & Spekkens, *Foundations of Physics* 40, 125 (2010), treats quantum states as encoding classical information about underlying ontic states. Fuchs, Mermin & Schack's QBism (*AJP* 82, 749, 2014) goes further, treating quantum states as subjective Bayesian probabilities—the Bloch vector becomes entirely a classical encoding of expectations. The PBR theorem (*Nature Physics* 8, 476, 2012) constrains but does not eliminate ψ-epistemic models under preparation independence assumptions. --- ## Topological invariants sit at the classical-quantum boundary The computational complexity of topological invariants reveals a nuanced picture. Freedman, Kitaev, Larsen & Wang, *Bull. AMS* 40, 31 (2003) proved that simulating TQFT is computationally equivalent to quantum computation, suggesting topological invariants are "genuinely quantum." However, Aharonov, Jones & Landau, *Algorithmica* 55, 395 (2009) showed that the Jones polynomial—a classical discrete knot invariant—is BQP-complete to approximate, meaning a classical object becomes the hardest problem quantum computers solve. **Atiyah's foundational paper, *Publ. Math. IHÉS* 68, 175 (1988), contains a remarkably prescient observation**: "fundamental topological aspects of such a quantum field theory should be independent of the parameters and it is therefore reasonable to expect them to be computable (in some sense) by examining the classical limit... such topological information is essentially robust and should be independent of the fine analytical details (and difficulties) of the full quantum theory." This directly supports characterizing topological invariants as classical discrete objects emerging from quantum substrates. Chen et al., *Nature Communications* 10, 1557 (2019) demonstrated a **classical microwave analog of topological entanglement entropy**, reproducing the discrete invariant γ = ln 2 for Z₂ order using entirely classical states. Jiang, Wang & Balents, *Nature Physics* 8, 902 (2012) showed classical DMRG algorithms efficiently compute topological entanglement entropy. Colón Vargas et al. (arXiv:2512.19028, 2024) found that WRT invariants of torus bundles are classically computable in polynomial time, while extracting individual coefficients is #P-complete—the invariant is partly classical, partly quantum. --- ## Novelty assessment and positioning recommendations **The proposed thesis is novel.** No published paper or preprint argues that entanglement serves as "access control" for classical discrete information structures. The closest work—the Forbes group's "digitization of quantum information"—frames topology as a robust encoding of quantum information generated by entanglement, not as evidence that entanglement merely gates pre-existing classical structures. However, the novelty space is tightly bounded. The proposed paper should engage with these five distinct bodies of evidence: - **Forbes group topological program** (2024–2026): The empirical foundation. The *d*² − 1 dimensional topological manifold, the sector decomposition via Gell-Mann triplets, and the "digitization" concept are all directly relevant. The proposed reinterpretation reframes their results. - **Classical simulation costs** (Toner-Bacon, Vidal, Jozsa-Linden): The modest classical communication cost of simulating entanglement supports the access-control interpretation. - **Classical nonseparability** (Spreeuw, Aiello, Forbes): The mathematical equivalence of classical and quantum entanglement structures, with nonlocality as the distinguishing feature. - **Epistemic/toy model frameworks** (Spekkens, Catani-Browne, QBism): The philosophical underpinning that entanglement-like phenomena emerge from classical information under epistemic restrictions, with contextuality as the true quantum resource. - **Topological invariant complexity** (Freedman-Kitaev-Larsen-Wang, Aharonov-Jones-Landau, Atiyah): The nuanced classical-quantum boundary of topological invariants. The strongest version of the proposed paper would likely argue that the Forbes group's topological spectrum constitutes a complete discrete classical skeleton of OAM entangled states, that entanglement determines *which* skeleton elements are physically realized (the "access control" function), and that contextuality—not entanglement—marks the boundary where this classical picture fails. The Toner-Bacon 1-bit simulation result and Spekkens' toy model provide the information-theoretic and foundational support, while the *d*² − 1 coincidence between topological manifold dimension and Bloch-space dimension provides the quantitative anchor.

BibTeX

@article{allodium:10.5281/zenodo.19306509,
  title = {\# Entanglement as "access control": a novelty assessment and literature map  **The specific thesis that entanglement serves as "access control" for classical discrete information structures has no direct precedent in the published literature.** No peer-reviewed paper or preprint explicitly argues that entanglement is a gating mechanism for classical information rather than a computational resource. However, a rich constellation of work from at least five independent research programs provides substantial adjacent infrastructure—and any paper advancing this thesis must engage carefully with all of them. The Forbes group's 2024–2025 papers on topological classification of OAM entanglement come closest, but frame topology as a robust encoding of *quantum* information generated *by* entanglement, not as evidence that entanglement merely addresses pre-existing classical structures.  ---  \#\# The Forbes group's topological program establishes the key empirical foundation  Three papers form the core of the most directly relevant body of work:  **Ornelas, Nape, de Mello Koch \& Forbes, "Non-local skyrmions as topologically resilient quantum entangled states of light," *Nature Photonics* 18, 258–266 (2024).** This landmark paper reports the first non-local quantum entangled state with non-trivial skyrmionic topology. Each individual photon possesses no salient topological structure—**topology emerges exclusively from the entanglement** of the biphoton wavefunction. The skyrmion number *N* (an integer counting wrappings of the Poincaré sphere by the non-local Stokes vector field) serves as a discrete topological invariant that persists under smooth deformations until entanglement itself vanishes. The authors propose skyrmion numbers as a "labelling system for entangled states, akin to an alphabet."  **Ornelas, Nape, de Mello Koch \& Forbes, "Topological rejection of noise by quantum skyrmions," *Nature Communications* 16, 2934 (2025).** This paper introduces the concept of **"digitization of quantum information"** based on discrete topological observables. The skyrmion number remains constant as concurrence, fidelity, and purity degrade, collapsing only at the maximally mixed state limit. The discrete, integer-valued nature of the invariant means noise must flip the signal between quantized states before registering any effect—directly analogous to digital versus analog signal robustness. No other research group has adopted this "digitization" terminology.  **de Mello Koch, Ornelas, Gounden, Lu, Nape \& Forbes, "Revealing the topological nature of entangled orbital angular momentum states of light," *Nature Communications* 16, 11095 (2025).** This is the most comprehensive paper and the most relevant to the proposed thesis. For the first time, topology is constructed using OAM alone (without polarization), enabling access to high-dimensional topological structures. The key quantitative results are striking: for *d* = 5, the topological manifold has dimension **24** (matching *d*² − 1) with **2,024 candidate invariants**; for *d* = 7, a **48-dimensional manifold** supports **17,296 candidate invariants**. The framework uses non-Abelian gauge fields of SU(*d*) Yang-Mills theory, decomposing the Lie algebra into Gell-Mann triplets forming SU(2) subalgebras, each supporting an independent topological map—effectively a "sector decomposition" of entanglement topology.  A fourth paper, de Mello Koch, Lu, Ornelas, Nape \& Forbes, "Quantum skyrmions in general quantum channels," *APL Quantum* 2, 026126 (2025), provides the theoretical framework for topological invariant evolution in general quantum channels, using homotopy arguments to prove topological noise rejection for non-depolarizing channels. Several 2026 preprints extend the program to atmospheric turbulence (arXiv:2603.10618), tripartite entanglement visualization (arXiv:2603.10491), and classical-quantum equivalence of topological behavior (*Nature Communications* 2026, DOI: 10.1038/s41467-026-68751-3).  ---  \#\# The *d*² − 1 parameter space and information-theoretic constraints  The relationship between state-space dimensionality, accessible information, and topological invariants creates a three-tiered hierarchy that the proposed paper must address.  A *d*-dimensional qudit's density matrix requires *d*² − 1 real parameters (the generalized Bloch vector). Comprehensive treatments appear in Bertlmann \& Krammer, *J. Phys. A* 41, 235303 (2008) and Loubenets \& Kulakov, *J. Phys. A* 54, 195301 (2021). For *d* ≥ 3, the space of valid states is no longer a ball but has nontrivial geometry—a constraint the proposed paper should acknowledge. Hardy's axiomatic framework (*arXiv:quant-ph/0101012*, 2001) shows that quantum state spaces contain *K* = *N*² parameters versus *K* = *N* for classical probability theory, with the Bloch vector functioning as a probability vector in a larger-than-classical space.  The **Holevo bound** (1973) constrains extractable classical information to log₂(*d*) bits per measurement, creating an enormous gap with the *d*² − 1 parameter count. For *d* = 7 OAM entanglement, the state lives in a 48-dimensional manifold, but only \textasciitilde{}2.8 bits are accessible per measurement. The de Mello Koch et al. (2025) finding that **the topological manifold dimension equals *d*² − 1** is therefore a critical data point: the topological spectrum explores the full parameter space, not just the Holevo-accessible fraction. The proposed paper could argue that topological invariants represent a discrete classical skeleton spanning the *d*² − 1 space, with entanglement providing the "access control" that determines which skeleton elements are physically realized.  Key experimental benchmarks include Mair et al., *Nature* 412, 313 (2001) for foundational OAM entanglement; Dada et al., *Nature Physics* 7, 677 (2011) for Bell violations up to *d* = 12; Krenn et al., *PNAS* 111, 6243 (2014) for 100-dimensional entanglement; and Fickler et al., *Science* 338, 640 (2012) for OAM values up to ℓ = ±300. Entanglement certification methods that avoid full tomography include Bavaresco et al., *Nature Physics* 14, 1032 (2018) and Friis et al., *Nature Reviews Physics* 1, 72 (2019).  ---  \#\# Classical simulation results suggest entanglement has modest classical cost  Several independent results establish that the classical communication cost of reproducing entanglement correlations is surprisingly low—supporting (though not directly arguing) the "access control" interpretation.  **Toner \& Bacon, *Physical Review Letters* 91, 187904 (2003)** proved that all correlations from local projective measurements on a maximally entangled Bell pair can be exactly simulated with local hidden variables plus **just one bit** of classical communication. Regev \& Toner, *SIAM J. Computing* 39, 1562 (2009), extended this to maximally entangled states of arbitrary dimension, requiring only **two bits** for dichotomic measurements. These results quantify the "classical cost" of entanglement: the information that entanglement provides beyond local hidden variables is remarkably small in discrete classical terms.  **Vidal, *Physical Review Letters* 91, 147902 (2003)** proved that pure-state quantum computations with restricted entanglement are efficiently classically simulable, with classical resources scaling exponentially only in entanglement. **Jozsa \& Linden, *Proc. R. Soc. A* 459, 2011 (2003)** proved multi-partite entanglement with unboundedly growing parties is *necessary* for pure-state exponential speedup, but explicitly stated **"it is nevertheless misleading to view entanglement as a key resource for quantum-computational power."** They noted that mixed-state algorithms (DQC1-type) can achieve speedup essentially without entanglement.  The quantum discord literature reinforces this: Datta, Shaji \& Caves, *PRL* 100, 050502 (2008) showed the DQC1 algorithm achieves quantum speedup with vanishing entanglement but nonzero discord, and Lanyon et al., *PRL* 101, 200501 (2008) experimentally demonstrated quantum computing without entanglement. Ollivier \& Zurek's foundational *PRL* 88, 017901 (2001) established that separable states can have nonzero quantum correlations, fundamentally decoupling entanglement from quantumness.  ---  \#\# Classical entanglement in structured light mirrors quantum nonseparability  A well-developed research program on **classical nonseparability** provides direct experimental evidence that entanglement-like mathematical structures exist without quantum mechanics. Spreeuw's seminal *Foundations of Physics* 28, 361 (1998) introduced the concept, showing that polarization-spatial mode coupling in classical beams produces nonseparable states mathematically identical to quantum entanglement. Forbes, Aiello \& Boyd, *Progress in Optics* 63, 99 (2019) comprehensively reviewed this field, establishing that quantum tools (concurrence, Bell measures) apply directly to classical light, with the crucial caveat that classical entanglement cannot produce *nonlocal* correlations.  Aiello et al., *New Journal of Physics* 17, 043024 (2015) and Töppel et al., *NJP* 16, 073019 (2014) showed that classical entanglement brings metrological advantages previously thought to require quantum entanglement. The Forbes group's own 2026 *Nature Communications* paper demonstrates "fundamental equivalence between pure topological quantum states and their classical counterparts" under turbulence, with identical skyrmion number preservation. This classical-quantum topological equivalence is perhaps the strongest existing evidence for the proposed thesis—though the Forbes group frames it as parallel behavior rather than as evidence for classical primacy.  Spreeuw, *PRA* 63, 062302 (2001), directly concluded that **"quantum nonlocality is the essential ingredient of a quantum computer, even more so than entanglement"**—a provocative statement distinguishing nonlocality from nonseparability that the proposed paper should engage with.  ---  \#\# Spekkens' toy model and epistemic frameworks provide philosophical support  **Spekkens, *Physical Review A* 75, 032110 (2007)** constructed a purely classical toy theory with an epistemic restriction ("knowledge balance principle") that reproduces entanglement, no-cloning, teleportation, dense coding, interference, and monogamy—failing only at Bell nonlocality and contextuality. Catani \& Browne, *New Journal of Physics* 19, 073035 (2017) proved this toy theory is operationally equivalent to stabilizer quantum mechanics in odd prime dimensions, identifying **contextuality** as the genuine quantum resource. This directly supports the proposed thesis's underlying logic: entanglement-like structures can emerge from classical information under access restrictions.  The ψ-epistemic program, formalized by Harrigan \& Spekkens, *Foundations of Physics* 40, 125 (2010), treats quantum states as encoding classical information about underlying ontic states. Fuchs, Mermin \& Schack's QBism (*AJP* 82, 749, 2014) goes further, treating quantum states as subjective Bayesian probabilities—the Bloch vector becomes entirely a classical encoding of expectations. The PBR theorem (*Nature Physics* 8, 476, 2012) constrains but does not eliminate ψ-epistemic models under preparation independence assumptions.  ---  \#\# Topological invariants sit at the classical-quantum boundary  The computational complexity of topological invariants reveals a nuanced picture. Freedman, Kitaev, Larsen \& Wang, *Bull. AMS* 40, 31 (2003) proved that simulating TQFT is computationally equivalent to quantum computation, suggesting topological invariants are "genuinely quantum." However, Aharonov, Jones \& Landau, *Algorithmica* 55, 395 (2009) showed that the Jones polynomial—a classical discrete knot invariant—is BQP-complete to approximate, meaning a classical object becomes the hardest problem quantum computers solve.  **Atiyah's foundational paper, *Publ. Math. IHÉS* 68, 175 (1988), contains a remarkably prescient observation**: "fundamental topological aspects of such a quantum field theory should be independent of the parameters and it is therefore reasonable to expect them to be computable (in some sense) by examining the classical limit... such topological information is essentially robust and should be independent of the fine analytical details (and difficulties) of the full quantum theory." This directly supports characterizing topological invariants as classical discrete objects emerging from quantum substrates.  Chen et al., *Nature Communications* 10, 1557 (2019) demonstrated a **classical microwave analog of topological entanglement entropy**, reproducing the discrete invariant γ = ln 2 for Z₂ order using entirely classical states. Jiang, Wang \& Balents, *Nature Physics* 8, 902 (2012) showed classical DMRG algorithms efficiently compute topological entanglement entropy. Colón Vargas et al. (arXiv:2512.19028, 2024) found that WRT invariants of torus bundles are classically computable in polynomial time, while extracting individual coefficients is \#P-complete—the invariant is partly classical, partly quantum.  ---  \#\# Novelty assessment and positioning recommendations  **The proposed thesis is novel.** No published paper or preprint argues that entanglement serves as "access control" for classical discrete information structures. The closest work—the Forbes group's "digitization of quantum information"—frames topology as a robust encoding of quantum information generated by entanglement, not as evidence that entanglement merely gates pre-existing classical structures.  However, the novelty space is tightly bounded. The proposed paper should engage with these five distinct bodies of evidence:  - **Forbes group topological program** (2024–2026): The empirical foundation. The *d*² − 1 dimensional topological manifold, the sector decomposition via Gell-Mann triplets, and the "digitization" concept are all directly relevant. The proposed reinterpretation reframes their results. - **Classical simulation costs** (Toner-Bacon, Vidal, Jozsa-Linden): The modest classical communication cost of simulating entanglement supports the access-control interpretation. - **Classical nonseparability** (Spreeuw, Aiello, Forbes): The mathematical equivalence of classical and quantum entanglement structures, with nonlocality as the distinguishing feature. - **Epistemic/toy model frameworks** (Spekkens, Catani-Browne, QBism): The philosophical underpinning that entanglement-like phenomena emerge from classical information under epistemic restrictions, with contextuality as the true quantum resource. - **Topological invariant complexity** (Freedman-Kitaev-Larsen-Wang, Aharonov-Jones-Landau, Atiyah): The nuanced classical-quantum boundary of topological invariants.  The strongest version of the proposed paper would likely argue that the Forbes group's topological spectrum constitutes a complete discrete classical skeleton of OAM entangled states, that entanglement determines *which* skeleton elements are physically realized (the "access control" function), and that contextuality—not entanglement—marks the boundary where this classical picture fails. The Toner-Bacon 1-bit simulation result and Spekkens' toy model provide the information-theoretic and foundational support, while the *d*² − 1 coincidence between topological manifold dimension and Bloch-space dimension provides the quantitative anchor.},
  author = {Wright, Craig},
  year = {2026},
  doi = {10.5281/zenodo.19306509},
  url = {https://doi.org/10.5281/zenodo.19306509}
}

RIS

TY  - JOUR
TI  - # Entanglement as "access control": a novelty assessment and literature map  **The specific thesis that entanglement serves as "access control" for classical discrete information structures has no direct precedent in the published literature.** No peer-reviewed paper or preprint explicitly argues that entanglement is a gating mechanism for classical information rather than a computational resource. However, a rich constellation of work from at least five independent research programs provides substantial adjacent infrastructure—and any paper advancing this thesis must engage carefully with all of them. The Forbes group's 2024–2025 papers on topological classification of OAM entanglement come closest, but frame topology as a robust encoding of *quantum* information generated *by* entanglement, not as evidence that entanglement merely addresses pre-existing classical structures.  ---  ## The Forbes group's topological program establishes the key empirical foundation  Three papers form the core of the most directly relevant body of work:  **Ornelas, Nape, de Mello Koch & Forbes, "Non-local skyrmions as topologically resilient quantum entangled states of light," *Nature Photonics* 18, 258–266 (2024).** This landmark paper reports the first non-local quantum entangled state with non-trivial skyrmionic topology. Each individual photon possesses no salient topological structure—**topology emerges exclusively from the entanglement** of the biphoton wavefunction. The skyrmion number *N* (an integer counting wrappings of the Poincaré sphere by the non-local Stokes vector field) serves as a discrete topological invariant that persists under smooth deformations until entanglement itself vanishes. The authors propose skyrmion numbers as a "labelling system for entangled states, akin to an alphabet."  **Ornelas, Nape, de Mello Koch & Forbes, "Topological rejection of noise by quantum skyrmions," *Nature Communications* 16, 2934 (2025).** This paper introduces the concept of **"digitization of quantum information"** based on discrete topological observables. The skyrmion number remains constant as concurrence, fidelity, and purity degrade, collapsing only at the maximally mixed state limit. The discrete, integer-valued nature of the invariant means noise must flip the signal between quantized states before registering any effect—directly analogous to digital versus analog signal robustness. No other research group has adopted this "digitization" terminology.  **de Mello Koch, Ornelas, Gounden, Lu, Nape & Forbes, "Revealing the topological nature of entangled orbital angular momentum states of light," *Nature Communications* 16, 11095 (2025).** This is the most comprehensive paper and the most relevant to the proposed thesis. For the first time, topology is constructed using OAM alone (without polarization), enabling access to high-dimensional topological structures. The key quantitative results are striking: for *d* = 5, the topological manifold has dimension **24** (matching *d*² − 1) with **2,024 candidate invariants**; for *d* = 7, a **48-dimensional manifold** supports **17,296 candidate invariants**. The framework uses non-Abelian gauge fields of SU(*d*) Yang-Mills theory, decomposing the Lie algebra into Gell-Mann triplets forming SU(2) subalgebras, each supporting an independent topological map—effectively a "sector decomposition" of entanglement topology.  A fourth paper, de Mello Koch, Lu, Ornelas, Nape & Forbes, "Quantum skyrmions in general quantum channels," *APL Quantum* 2, 026126 (2025), provides the theoretical framework for topological invariant evolution in general quantum channels, using homotopy arguments to prove topological noise rejection for non-depolarizing channels. Several 2026 preprints extend the program to atmospheric turbulence (arXiv:2603.10618), tripartite entanglement visualization (arXiv:2603.10491), and classical-quantum equivalence of topological behavior (*Nature Communications* 2026, DOI: 10.1038/s41467-026-68751-3).  ---  ## The *d*² − 1 parameter space and information-theoretic constraints  The relationship between state-space dimensionality, accessible information, and topological invariants creates a three-tiered hierarchy that the proposed paper must address.  A *d*-dimensional qudit's density matrix requires *d*² − 1 real parameters (the generalized Bloch vector). Comprehensive treatments appear in Bertlmann & Krammer, *J. Phys. A* 41, 235303 (2008) and Loubenets & Kulakov, *J. Phys. A* 54, 195301 (2021). For *d* ≥ 3, the space of valid states is no longer a ball but has nontrivial geometry—a constraint the proposed paper should acknowledge. Hardy's axiomatic framework (*arXiv:quant-ph/0101012*, 2001) shows that quantum state spaces contain *K* = *N*² parameters versus *K* = *N* for classical probability theory, with the Bloch vector functioning as a probability vector in a larger-than-classical space.  The **Holevo bound** (1973) constrains extractable classical information to log₂(*d*) bits per measurement, creating an enormous gap with the *d*² − 1 parameter count. For *d* = 7 OAM entanglement, the state lives in a 48-dimensional manifold, but only ~2.8 bits are accessible per measurement. The de Mello Koch et al. (2025) finding that **the topological manifold dimension equals *d*² − 1** is therefore a critical data point: the topological spectrum explores the full parameter space, not just the Holevo-accessible fraction. The proposed paper could argue that topological invariants represent a discrete classical skeleton spanning the *d*² − 1 space, with entanglement providing the "access control" that determines which skeleton elements are physically realized.  Key experimental benchmarks include Mair et al., *Nature* 412, 313 (2001) for foundational OAM entanglement; Dada et al., *Nature Physics* 7, 677 (2011) for Bell violations up to *d* = 12; Krenn et al., *PNAS* 111, 6243 (2014) for 100-dimensional entanglement; and Fickler et al., *Science* 338, 640 (2012) for OAM values up to ℓ = ±300. Entanglement certification methods that avoid full tomography include Bavaresco et al., *Nature Physics* 14, 1032 (2018) and Friis et al., *Nature Reviews Physics* 1, 72 (2019).  ---  ## Classical simulation results suggest entanglement has modest classical cost  Several independent results establish that the classical communication cost of reproducing entanglement correlations is surprisingly low—supporting (though not directly arguing) the "access control" interpretation.  **Toner & Bacon, *Physical Review Letters* 91, 187904 (2003)** proved that all correlations from local projective measurements on a maximally entangled Bell pair can be exactly simulated with local hidden variables plus **just one bit** of classical communication. Regev & Toner, *SIAM J. Computing* 39, 1562 (2009), extended this to maximally entangled states of arbitrary dimension, requiring only **two bits** for dichotomic measurements. These results quantify the "classical cost" of entanglement: the information that entanglement provides beyond local hidden variables is remarkably small in discrete classical terms.  **Vidal, *Physical Review Letters* 91, 147902 (2003)** proved that pure-state quantum computations with restricted entanglement are efficiently classically simulable, with classical resources scaling exponentially only in entanglement. **Jozsa & Linden, *Proc. R. Soc. A* 459, 2011 (2003)** proved multi-partite entanglement with unboundedly growing parties is *necessary* for pure-state exponential speedup, but explicitly stated **"it is nevertheless misleading to view entanglement as a key resource for quantum-computational power."** They noted that mixed-state algorithms (DQC1-type) can achieve speedup essentially without entanglement.  The quantum discord literature reinforces this: Datta, Shaji & Caves, *PRL* 100, 050502 (2008) showed the DQC1 algorithm achieves quantum speedup with vanishing entanglement but nonzero discord, and Lanyon et al., *PRL* 101, 200501 (2008) experimentally demonstrated quantum computing without entanglement. Ollivier & Zurek's foundational *PRL* 88, 017901 (2001) established that separable states can have nonzero quantum correlations, fundamentally decoupling entanglement from quantumness.  ---  ## Classical entanglement in structured light mirrors quantum nonseparability  A well-developed research program on **classical nonseparability** provides direct experimental evidence that entanglement-like mathematical structures exist without quantum mechanics. Spreeuw's seminal *Foundations of Physics* 28, 361 (1998) introduced the concept, showing that polarization-spatial mode coupling in classical beams produces nonseparable states mathematically identical to quantum entanglement. Forbes, Aiello & Boyd, *Progress in Optics* 63, 99 (2019) comprehensively reviewed this field, establishing that quantum tools (concurrence, Bell measures) apply directly to classical light, with the crucial caveat that classical entanglement cannot produce *nonlocal* correlations.  Aiello et al., *New Journal of Physics* 17, 043024 (2015) and Töppel et al., *NJP* 16, 073019 (2014) showed that classical entanglement brings metrological advantages previously thought to require quantum entanglement. The Forbes group's own 2026 *Nature Communications* paper demonstrates "fundamental equivalence between pure topological quantum states and their classical counterparts" under turbulence, with identical skyrmion number preservation. This classical-quantum topological equivalence is perhaps the strongest existing evidence for the proposed thesis—though the Forbes group frames it as parallel behavior rather than as evidence for classical primacy.  Spreeuw, *PRA* 63, 062302 (2001), directly concluded that **"quantum nonlocality is the essential ingredient of a quantum computer, even more so than entanglement"**—a provocative statement distinguishing nonlocality from nonseparability that the proposed paper should engage with.  ---  ## Spekkens' toy model and epistemic frameworks provide philosophical support  **Spekkens, *Physical Review A* 75, 032110 (2007)** constructed a purely classical toy theory with an epistemic restriction ("knowledge balance principle") that reproduces entanglement, no-cloning, teleportation, dense coding, interference, and monogamy—failing only at Bell nonlocality and contextuality. Catani & Browne, *New Journal of Physics* 19, 073035 (2017) proved this toy theory is operationally equivalent to stabilizer quantum mechanics in odd prime dimensions, identifying **contextuality** as the genuine quantum resource. This directly supports the proposed thesis's underlying logic: entanglement-like structures can emerge from classical information under access restrictions.  The ψ-epistemic program, formalized by Harrigan & Spekkens, *Foundations of Physics* 40, 125 (2010), treats quantum states as encoding classical information about underlying ontic states. Fuchs, Mermin & Schack's QBism (*AJP* 82, 749, 2014) goes further, treating quantum states as subjective Bayesian probabilities—the Bloch vector becomes entirely a classical encoding of expectations. The PBR theorem (*Nature Physics* 8, 476, 2012) constrains but does not eliminate ψ-epistemic models under preparation independence assumptions.  ---  ## Topological invariants sit at the classical-quantum boundary  The computational complexity of topological invariants reveals a nuanced picture. Freedman, Kitaev, Larsen & Wang, *Bull. AMS* 40, 31 (2003) proved that simulating TQFT is computationally equivalent to quantum computation, suggesting topological invariants are "genuinely quantum." However, Aharonov, Jones & Landau, *Algorithmica* 55, 395 (2009) showed that the Jones polynomial—a classical discrete knot invariant—is BQP-complete to approximate, meaning a classical object becomes the hardest problem quantum computers solve.  **Atiyah's foundational paper, *Publ. Math. IHÉS* 68, 175 (1988), contains a remarkably prescient observation**: "fundamental topological aspects of such a quantum field theory should be independent of the parameters and it is therefore reasonable to expect them to be computable (in some sense) by examining the classical limit... such topological information is essentially robust and should be independent of the fine analytical details (and difficulties) of the full quantum theory." This directly supports characterizing topological invariants as classical discrete objects emerging from quantum substrates.  Chen et al., *Nature Communications* 10, 1557 (2019) demonstrated a **classical microwave analog of topological entanglement entropy**, reproducing the discrete invariant γ = ln 2 for Z₂ order using entirely classical states. Jiang, Wang & Balents, *Nature Physics* 8, 902 (2012) showed classical DMRG algorithms efficiently compute topological entanglement entropy. Colón Vargas et al. (arXiv:2512.19028, 2024) found that WRT invariants of torus bundles are classically computable in polynomial time, while extracting individual coefficients is #P-complete—the invariant is partly classical, partly quantum.  ---  ## Novelty assessment and positioning recommendations  **The proposed thesis is novel.** No published paper or preprint argues that entanglement serves as "access control" for classical discrete information structures. The closest work—the Forbes group's "digitization of quantum information"—frames topology as a robust encoding of quantum information generated by entanglement, not as evidence that entanglement merely gates pre-existing classical structures.  However, the novelty space is tightly bounded. The proposed paper should engage with these five distinct bodies of evidence:  - **Forbes group topological program** (2024–2026): The empirical foundation. The *d*² − 1 dimensional topological manifold, the sector decomposition via Gell-Mann triplets, and the "digitization" concept are all directly relevant. The proposed reinterpretation reframes their results. - **Classical simulation costs** (Toner-Bacon, Vidal, Jozsa-Linden): The modest classical communication cost of simulating entanglement supports the access-control interpretation. - **Classical nonseparability** (Spreeuw, Aiello, Forbes): The mathematical equivalence of classical and quantum entanglement structures, with nonlocality as the distinguishing feature. - **Epistemic/toy model frameworks** (Spekkens, Catani-Browne, QBism): The philosophical underpinning that entanglement-like phenomena emerge from classical information under epistemic restrictions, with contextuality as the true quantum resource. - **Topological invariant complexity** (Freedman-Kitaev-Larsen-Wang, Aharonov-Jones-Landau, Atiyah): The nuanced classical-quantum boundary of topological invariants.  The strongest version of the proposed paper would likely argue that the Forbes group's topological spectrum constitutes a complete discrete classical skeleton of OAM entangled states, that entanglement determines *which* skeleton elements are physically realized (the "access control" function), and that contextuality—not entanglement—marks the boundary where this classical picture fails. The Toner-Bacon 1-bit simulation result and Spekkens' toy model provide the information-theoretic and foundational support, while the *d*² − 1 coincidence between topological manifold dimension and Bloch-space dimension provides the quantitative anchor.
AU  - Wright, Craig
PY  - 2026
DO  - 10.5281/zenodo.19306509
UR  - https://doi.org/10.5281/zenodo.19306509
ER  - 

APA

Craig, W. (2026). # Entanglement as "access control": a novelty assessment and literature map **The specific thesis that entanglement serves as "access control" for classical discrete information structures has no direct precedent in the published literature.** No peer-reviewed paper or preprint explicitly argues that entanglement is a gating mechanism for classical information rather than a computational resource. However, a rich constellation of work from at least five independent research programs provides substantial adjacent infrastructure—and any paper advancing this thesis must engage carefully with all of them. The Forbes group's 2024–2025 papers on topological classification of OAM entanglement come closest, but frame topology as a robust encoding of *quantum* information generated *by* entanglement, not as evidence that entanglement merely addresses pre-existing classical structures. --- ## The Forbes group's topological program establishes the key empirical foundation Three papers form the core of the most directly relevant body of work: **Ornelas, Nape, de Mello Koch & Forbes, "Non-local skyrmions as topologically resilient quantum entangled states of light," *Nature Photonics* 18, 258–266 (2024).** This landmark paper reports the first non-local quantum entangled state with non-trivial skyrmionic topology. Each individual photon possesses no salient topological structure—**topology emerges exclusively from the entanglement** of the biphoton wavefunction. The skyrmion number *N* (an integer counting wrappings of the Poincaré sphere by the non-local Stokes vector field) serves as a discrete topological invariant that persists under smooth deformations until entanglement itself vanishes. The authors propose skyrmion numbers as a "labelling system for entangled states, akin to an alphabet." **Ornelas, Nape, de Mello Koch & Forbes, "Topological rejection of noise by quantum skyrmions," *Nature Communications* 16, 2934 (2025).** This paper introduces the concept of **"digitization of quantum information"** based on discrete topological observables. The skyrmion number remains constant as concurrence, fidelity, and purity degrade, collapsing only at the maximally mixed state limit. The discrete, integer-valued nature of the invariant means noise must flip the signal between quantized states before registering any effect—directly analogous to digital versus analog signal robustness. No other research group has adopted this "digitization" terminology. **de Mello Koch, Ornelas, Gounden, Lu, Nape & Forbes, "Revealing the topological nature of entangled orbital angular momentum states of light," *Nature Communications* 16, 11095 (2025).** This is the most comprehensive paper and the most relevant to the proposed thesis. For the first time, topology is constructed using OAM alone (without polarization), enabling access to high-dimensional topological structures. The key quantitative results are striking: for *d* = 5, the topological manifold has dimension **24** (matching *d*² − 1) with **2,024 candidate invariants**; for *d* = 7, a **48-dimensional manifold** supports **17,296 candidate invariants**. The framework uses non-Abelian gauge fields of SU(*d*) Yang-Mills theory, decomposing the Lie algebra into Gell-Mann triplets forming SU(2) subalgebras, each supporting an independent topological map—effectively a "sector decomposition" of entanglement topology. A fourth paper, de Mello Koch, Lu, Ornelas, Nape & Forbes, "Quantum skyrmions in general quantum channels," *APL Quantum* 2, 026126 (2025), provides the theoretical framework for topological invariant evolution in general quantum channels, using homotopy arguments to prove topological noise rejection for non-depolarizing channels. Several 2026 preprints extend the program to atmospheric turbulence (arXiv:2603.10618), tripartite entanglement visualization (arXiv:2603.10491), and classical-quantum equivalence of topological behavior (*Nature Communications* 2026, DOI: 10.1038/s41467-026-68751-3). --- ## The *d*² − 1 parameter space and information-theoretic constraints The relationship between state-space dimensionality, accessible information, and topological invariants creates a three-tiered hierarchy that the proposed paper must address. A *d*-dimensional qudit's density matrix requires *d*² − 1 real parameters (the generalized Bloch vector). Comprehensive treatments appear in Bertlmann & Krammer, *J. Phys. A* 41, 235303 (2008) and Loubenets & Kulakov, *J. Phys. A* 54, 195301 (2021). For *d* ≥ 3, the space of valid states is no longer a ball but has nontrivial geometry—a constraint the proposed paper should acknowledge. Hardy's axiomatic framework (*arXiv:quant-ph/0101012*, 2001) shows that quantum state spaces contain *K* = *N*² parameters versus *K* = *N* for classical probability theory, with the Bloch vector functioning as a probability vector in a larger-than-classical space. The **Holevo bound** (1973) constrains extractable classical information to log₂(*d*) bits per measurement, creating an enormous gap with the *d*² − 1 parameter count. For *d* = 7 OAM entanglement, the state lives in a 48-dimensional manifold, but only ~2.8 bits are accessible per measurement. The de Mello Koch et al. (2025) finding that **the topological manifold dimension equals *d*² − 1** is therefore a critical data point: the topological spectrum explores the full parameter space, not just the Holevo-accessible fraction. The proposed paper could argue that topological invariants represent a discrete classical skeleton spanning the *d*² − 1 space, with entanglement providing the "access control" that determines which skeleton elements are physically realized. Key experimental benchmarks include Mair et al., *Nature* 412, 313 (2001) for foundational OAM entanglement; Dada et al., *Nature Physics* 7, 677 (2011) for Bell violations up to *d* = 12; Krenn et al., *PNAS* 111, 6243 (2014) for 100-dimensional entanglement; and Fickler et al., *Science* 338, 640 (2012) for OAM values up to ℓ = ±300. Entanglement certification methods that avoid full tomography include Bavaresco et al., *Nature Physics* 14, 1032 (2018) and Friis et al., *Nature Reviews Physics* 1, 72 (2019). --- ## Classical simulation results suggest entanglement has modest classical cost Several independent results establish that the classical communication cost of reproducing entanglement correlations is surprisingly low—supporting (though not directly arguing) the "access control" interpretation. **Toner & Bacon, *Physical Review Letters* 91, 187904 (2003)** proved that all correlations from local projective measurements on a maximally entangled Bell pair can be exactly simulated with local hidden variables plus **just one bit** of classical communication. Regev & Toner, *SIAM J. Computing* 39, 1562 (2009), extended this to maximally entangled states of arbitrary dimension, requiring only **two bits** for dichotomic measurements. These results quantify the "classical cost" of entanglement: the information that entanglement provides beyond local hidden variables is remarkably small in discrete classical terms. **Vidal, *Physical Review Letters* 91, 147902 (2003)** proved that pure-state quantum computations with restricted entanglement are efficiently classically simulable, with classical resources scaling exponentially only in entanglement. **Jozsa & Linden, *Proc. R. Soc. A* 459, 2011 (2003)** proved multi-partite entanglement with unboundedly growing parties is *necessary* for pure-state exponential speedup, but explicitly stated **"it is nevertheless misleading to view entanglement as a key resource for quantum-computational power."** They noted that mixed-state algorithms (DQC1-type) can achieve speedup essentially without entanglement. The quantum discord literature reinforces this: Datta, Shaji & Caves, *PRL* 100, 050502 (2008) showed the DQC1 algorithm achieves quantum speedup with vanishing entanglement but nonzero discord, and Lanyon et al., *PRL* 101, 200501 (2008) experimentally demonstrated quantum computing without entanglement. Ollivier & Zurek's foundational *PRL* 88, 017901 (2001) established that separable states can have nonzero quantum correlations, fundamentally decoupling entanglement from quantumness. --- ## Classical entanglement in structured light mirrors quantum nonseparability A well-developed research program on **classical nonseparability** provides direct experimental evidence that entanglement-like mathematical structures exist without quantum mechanics. Spreeuw's seminal *Foundations of Physics* 28, 361 (1998) introduced the concept, showing that polarization-spatial mode coupling in classical beams produces nonseparable states mathematically identical to quantum entanglement. Forbes, Aiello & Boyd, *Progress in Optics* 63, 99 (2019) comprehensively reviewed this field, establishing that quantum tools (concurrence, Bell measures) apply directly to classical light, with the crucial caveat that classical entanglement cannot produce *nonlocal* correlations. Aiello et al., *New Journal of Physics* 17, 043024 (2015) and Töppel et al., *NJP* 16, 073019 (2014) showed that classical entanglement brings metrological advantages previously thought to require quantum entanglement. The Forbes group's own 2026 *Nature Communications* paper demonstrates "fundamental equivalence between pure topological quantum states and their classical counterparts" under turbulence, with identical skyrmion number preservation. This classical-quantum topological equivalence is perhaps the strongest existing evidence for the proposed thesis—though the Forbes group frames it as parallel behavior rather than as evidence for classical primacy. Spreeuw, *PRA* 63, 062302 (2001), directly concluded that **"quantum nonlocality is the essential ingredient of a quantum computer, even more so than entanglement"**—a provocative statement distinguishing nonlocality from nonseparability that the proposed paper should engage with. --- ## Spekkens' toy model and epistemic frameworks provide philosophical support **Spekkens, *Physical Review A* 75, 032110 (2007)** constructed a purely classical toy theory with an epistemic restriction ("knowledge balance principle") that reproduces entanglement, no-cloning, teleportation, dense coding, interference, and monogamy—failing only at Bell nonlocality and contextuality. Catani & Browne, *New Journal of Physics* 19, 073035 (2017) proved this toy theory is operationally equivalent to stabilizer quantum mechanics in odd prime dimensions, identifying **contextuality** as the genuine quantum resource. This directly supports the proposed thesis's underlying logic: entanglement-like structures can emerge from classical information under access restrictions. The ψ-epistemic program, formalized by Harrigan & Spekkens, *Foundations of Physics* 40, 125 (2010), treats quantum states as encoding classical information about underlying ontic states. Fuchs, Mermin & Schack's QBism (*AJP* 82, 749, 2014) goes further, treating quantum states as subjective Bayesian probabilities—the Bloch vector becomes entirely a classical encoding of expectations. The PBR theorem (*Nature Physics* 8, 476, 2012) constrains but does not eliminate ψ-epistemic models under preparation independence assumptions. --- ## Topological invariants sit at the classical-quantum boundary The computational complexity of topological invariants reveals a nuanced picture. Freedman, Kitaev, Larsen & Wang, *Bull. AMS* 40, 31 (2003) proved that simulating TQFT is computationally equivalent to quantum computation, suggesting topological invariants are "genuinely quantum." However, Aharonov, Jones & Landau, *Algorithmica* 55, 395 (2009) showed that the Jones polynomial—a classical discrete knot invariant—is BQP-complete to approximate, meaning a classical object becomes the hardest problem quantum computers solve. **Atiyah's foundational paper, *Publ. Math. IHÉS* 68, 175 (1988), contains a remarkably prescient observation**: "fundamental topological aspects of such a quantum field theory should be independent of the parameters and it is therefore reasonable to expect them to be computable (in some sense) by examining the classical limit... such topological information is essentially robust and should be independent of the fine analytical details (and difficulties) of the full quantum theory." This directly supports characterizing topological invariants as classical discrete objects emerging from quantum substrates. Chen et al., *Nature Communications* 10, 1557 (2019) demonstrated a **classical microwave analog of topological entanglement entropy**, reproducing the discrete invariant γ = ln 2 for Z₂ order using entirely classical states. Jiang, Wang & Balents, *Nature Physics* 8, 902 (2012) showed classical DMRG algorithms efficiently compute topological entanglement entropy. Colón Vargas et al. (arXiv:2512.19028, 2024) found that WRT invariants of torus bundles are classically computable in polynomial time, while extracting individual coefficients is #P-complete—the invariant is partly classical, partly quantum. --- ## Novelty assessment and positioning recommendations **The proposed thesis is novel.** No published paper or preprint argues that entanglement serves as "access control" for classical discrete information structures. The closest work—the Forbes group's "digitization of quantum information"—frames topology as a robust encoding of quantum information generated by entanglement, not as evidence that entanglement merely gates pre-existing classical structures. However, the novelty space is tightly bounded. The proposed paper should engage with these five distinct bodies of evidence: - **Forbes group topological program** (2024–2026): The empirical foundation. The *d*² − 1 dimensional topological manifold, the sector decomposition via Gell-Mann triplets, and the "digitization" concept are all directly relevant. The proposed reinterpretation reframes their results. - **Classical simulation costs** (Toner-Bacon, Vidal, Jozsa-Linden): The modest classical communication cost of simulating entanglement supports the access-control interpretation. - **Classical nonseparability** (Spreeuw, Aiello, Forbes): The mathematical equivalence of classical and quantum entanglement structures, with nonlocality as the distinguishing feature. - **Epistemic/toy model frameworks** (Spekkens, Catani-Browne, QBism): The philosophical underpinning that entanglement-like phenomena emerge from classical information under epistemic restrictions, with contextuality as the true quantum resource. - **Topological invariant complexity** (Freedman-Kitaev-Larsen-Wang, Aharonov-Jones-Landau, Atiyah): The nuanced classical-quantum boundary of topological invariants. The strongest version of the proposed paper would likely argue that the Forbes group's topological spectrum constitutes a complete discrete classical skeleton of OAM entangled states, that entanglement determines *which* skeleton elements are physically realized (the "access control" function), and that contextuality—not entanglement—marks the boundary where this classical picture fails. The Toner-Bacon 1-bit simulation result and Spekkens' toy model provide the information-theoretic and foundational support, while the *d*² − 1 coincidence between topological manifold dimension and Bloch-space dimension provides the quantitative anchor.. https://doi.org/10.5281/zenodo.19306509