A Prime-Log Orbit Is Dense Across an Equal-Amplitude Phase Family

Can a family of quantum states become easier to clone in the worst case just because it is described by one real parameter? For one constructed family, a bounded derivation gives a precise answer: over all real times, the parameter traces a dense subset of the full equal-amplitude phase family. The all-time worst-case global-fidelity problem therefore has the same value as the full-family problem for each fixed channel.

The phase-state cloning framework comes from Herasymenko, Lin, Verduyn Lunel and Grinko's Approximate cloning of structured pure states. The result developed here is the prime-log application: unique prime factorisation gives rationally independent relative frequencies, and density carries the fixed-channel worst-case fidelity value from the full phase family to this orbit. This article is not a review of the source paper.

The phase family

Choose any finite set of $d \geq 2$ distinct primes $p_0,\ldots,p_{d-1}$. The associated equal-amplitude states are

A common phase has no effect on the state. Removing the phase from $p_0$ leaves the relative phases $\theta_j(t)=t\log(p_j/p_0)\pmod{2\pi}$ for $1\leq j<d$. These relative phases move on a $(d-1)$-dimensional torus.

Why the orbit is dense

Suppose integers $m_1,\ldots,m_{d-1}$ satisfy

Exponentiating turns this into an equality between products of integer powers of distinct primes. Moving negative powers to the other side and applying unique prime factorisation forces every $m_j$ to be zero. So the relative frequencies have no nonzero integer relation.

For a continuous-time linear flow on a torus, this rational independence makes the trajectory dense. Thus the prime-log orbit comes arbitrarily close to every equal-amplitude phase state. Density is a statement about the closure of the orbit; it does not say that every state is reached at a finite time.

The cloning consequence

Fix positive integers $N$ and $K$, and a deterministic quantum channel $C$ that maps $N$ input copies to $N+K$ output copies. Its squared global fidelity is a continuous function of the phase vector on the compact torus. A continuous function has the same infimum over a dense subset as its minimum over the full compact space. Therefore, for this fixed channel, the infimum along the prime-log orbit equals the minimum over the full equal-amplitude phase family. Taking the optimum over the same class of channels preserves that equality.

In this all-time, worst-case global-fidelity sense, reducing the description to one real parameter does not reduce the phase family that a fixed channel must handle. The orbit infimum need not be attained at a finite time. The conclusion assumes a fixed channel with no classical label revealing $t$; it concerns joint-output global fidelity, not a trace-distance or single-copy optimum.

What this does not establish

The argument gives no coverage guarantee or convergence rate for a bounded time interval. It makes no claim about prime prediction, physical spectra, a physical encoding mechanism, or transfer to Infinity Research. The density follows from rational independence and is not unique to primes. No literature-priority or field-level novelty claim is made.

The derivation and its source map are recorded in the Research Artifact Ledger at commit 55dd367dca43e40bc4ece1cfa49a7b0f1d630093 and the claim-to-source map. The Ledger reports an independent high-effort model challenge that found the deduction valid at its stated scope. That is not human specialist review, peer review or formal verification.

AI assistance disclosure

AI assisted with source and claim-boundary review, drafting and editorial development of this article. The featured image is an AI-generated conceptual illustration, not data or a dimension-specific model. The Ledger records a separate high-effort model challenge; neither that check nor the assistance here constitutes human specialist review or peer review. The author is responsible for the work.