{
  "id": "oai:papers.qnfo.org:prime-states-as-intrinsic-basis-states-entanglement-bounds-preparation-complexit",
  "slug": "prime-states-as-intrinsic-basis-states-entanglement-bounds-preparation-complexit",
  "title": "Prime States as Intrinsic Basis States: Entanglement Bounds, Preparation Complexity, and the Limits of Classical Emulation",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "A recurring proposal in quantum-architecture design is to treat the prime numbers not as parameters of an algorithm but as the fundamental addressable units of a register: a \"Prime state\" that superposes all primes below 2^n. The specific gap this paper addresses is quantitative: no compact derivation exists that a system architect can use to compare the preparation cost, entanglement content, and classical emulability of such a prime-basis register against a standard computational-basis register. Using the Prime-state construction of Latorre and Sierra, in which a Grover oracle implementing the Miller-Rabin test prepares |P_n⟩ = π(2^n)^-1/2∑_p<2^n|p⟩, we derive: (i) the prime-state population π(2^n) ≈ 2^n/(nln 2), giving 75 640 for n=20 under the PNT approximation; (ii) a Grover preparation cost of (π/4)√nln 2 oracle calls, i.e. $5.23$ calls at n=64, against 2⁶4 classical candidates; (iii) an entanglement entropy ceiling of log₂ π(2^n) bits across any bipartition, $16.21$ bits at n=20; and (iv) a coprimality-derived collision floor of 6/π² ≈ 0.6079 for random register pairs. We further show that exact classical emulation of the full n-qubit state vector requires 2^n amplitudes ($17.59$ TB at n=40, double precision) while the sparse prime representation needs only π(2^n), and we identify the regime boundary this creates. The results convert a conjectural architecture idea into checkable cost and entropy bounds.",
  "abstract_markdown": "A recurring proposal in quantum-architecture design is to treat the prime numbers not as parameters of an algorithm but as the fundamental addressable units of a register: a \"Prime state\" that superposes all primes below $2^n$. The specific gap this paper addresses is quantitative: no compact derivation exists that a system architect can use to compare the preparation cost, entanglement content, and classical emulability of such a prime-basis register against a standard computational-basis register. Using the Prime-state construction of Latorre and Sierra, in which a Grover oracle implementing the Miller-Rabin test prepares $|\\mathbb{P}_n\\rangle = \\pi(2^n)^{-1/2}\\sum_{p<2^n}|p\\rangle$, we derive: (i) the prime-state population $\\pi(2^n) \\approx 2^n/(n\\ln 2)$, giving $75\\,640$ for $n=20$ under the PNT approximation; (ii) a Grover preparation cost of $(\\pi/4)\\sqrt{n\\ln 2}$ oracle calls, i.e. $5.23$ calls at $n=64$, against $2^{64}$ classical candidates; (iii) an entanglement entropy ceiling of $\\log_2 \\pi(2^n)$ bits across any bipartition, $16.21$ bits at $n=20$; and (iv) a coprimality-derived collision floor of $6/\\pi^2 \\approx 0.6079$ for random register pairs. We further show that exact classical emulation of the full $n$-qubit state vector requires $2^n$ amplitudes ($17.59$ TB at $n=40$, double precision) while the sparse prime representation needs only $\\pi(2^n)$, and we identify the regime boundary this creates. The results convert a conjectural architecture idea into checkable cost and entropy bounds.",
  "date": "2026-11-02",
  "first_published": "2026-11-02",
  "last_revised": "2026-11-02T00:00:00Z",
  "modified": "2026-11-02T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/prime-states-as-intrinsic-basis-states-entanglement-bounds-preparation-complexit",
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    "creators": [
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Prime States as Intrinsic Basis States: Entanglement Bounds, Preparation Complexity, and the Limits of Classical Emulation"
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        "description": "A recurring proposal in quantum-architecture design is to treat the prime numbers not as parameters of an algorithm but as the fundamental addressable units of a register: a \"Prime state\" that superposes all primes below 2^n. The specific gap this paper addresses is quantitative: no compact derivation exists that a system architect can use to compare the preparation cost, entanglement content, and classical emulability of such a prime-basis register against a standard computational-basis register. Using the Prime-state construction of Latorre and Sierra, in which a Grover oracle implementing the Miller-Rabin test prepares |P_n⟩ = π(2^n)^-1/2∑_p<2^n|p⟩, we derive: (i) the prime-state population π(2^n) ≈ 2^n/(nln 2), giving 75 640 for n=20 under the PNT approximation; (ii) a Grover preparation cost of (π/4)√nln 2 oracle calls, i.e. $5.23$ calls at n=64, against 2⁶4 classical candidates; (iii) an entanglement entropy ceiling of log₂ π(2^n) bits across any bipartition, $16.21$ bits at n=20; and (iv) a coprimality-derived collision floor of 6/π² ≈ 0.6079 for random register pairs. We further show that exact classical emulation of the full n-qubit state vector requires 2^n amplitudes ($17.59$ TB at n=40, double precision) while the sparse prime representation needs only π(2^n), and we identify the regime boundary this creates. The results convert a conjectural architecture idea into checkable cost and entropy bounds.",
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