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Quantum Interference Needs Convention: Overlap-Determinability and Unified No-Superposition Principle

Jeongho Bang, Kyoungho Cho, Ki Hyuk Yee

Abstract

Quantum superposition is often phrased as the ability to add state vectors. In practice, however, the physical quantity is a ray (a rank-one projector), so each input specifies only a projector and leaves a gauge freedom in the phases of its vector representatives. This becomes a real operational barrier when one asks for a device that, given two independently prepared unknown pure states, outputs a coherent state proportional to a prescribed linear combination. We identify the missing ingredient as not probabilistic but phase-like. One needs a physical scenario that fixes a single phase convention on the relevant set of rays, so that the overlaps become well defined complex numbers. Thus, we formalize this through phase conventions and a single notion -- dubbed as "overlap-determinability." Our main theorem gives an exact equivalence: A nonzero completely positive trace-nonincreasing map that probabilistically produces superposition on a domain exists if and only if that domain is overlap-determinable. This unifies modern no-superposition results and characterizes the exceptional yes-go protocols, which succeed precisely when side information supplies the required missing resource. We then show that granting universal access to such convention-fixed overlaps destabilizes the familiar foundational and computational constraints. It enables forbidden transformations akin to quantum cloning and yields super-luminal signaling. It would also permit reflections about unknown states, leading to exponentially fast overlap amplification and a collapse of Grover's search lower bound to a logarithmic query complexity.

Quantum Interference Needs Convention: Overlap-Determinability and Unified No-Superposition Principle

Abstract

Quantum superposition is often phrased as the ability to add state vectors. In practice, however, the physical quantity is a ray (a rank-one projector), so each input specifies only a projector and leaves a gauge freedom in the phases of its vector representatives. This becomes a real operational barrier when one asks for a device that, given two independently prepared unknown pure states, outputs a coherent state proportional to a prescribed linear combination. We identify the missing ingredient as not probabilistic but phase-like. One needs a physical scenario that fixes a single phase convention on the relevant set of rays, so that the overlaps become well defined complex numbers. Thus, we formalize this through phase conventions and a single notion -- dubbed as "overlap-determinability." Our main theorem gives an exact equivalence: A nonzero completely positive trace-nonincreasing map that probabilistically produces superposition on a domain exists if and only if that domain is overlap-determinable. This unifies modern no-superposition results and characterizes the exceptional yes-go protocols, which succeed precisely when side information supplies the required missing resource. We then show that granting universal access to such convention-fixed overlaps destabilizes the familiar foundational and computational constraints. It enables forbidden transformations akin to quantum cloning and yields super-luminal signaling. It would also permit reflections about unknown states, leading to exponentially fast overlap amplification and a collapse of Grover's search lower bound to a logarithmic query complexity.
Paper Structure (39 sections, 11 theorems, 93 equations, 1 figure)

This paper contains 39 sections, 11 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and Formal Problem Statement
  3. Pure states as rays and the phase ambiguity of "superposition"
  4. Quantum operations and postselection
  5. The superposition-generation task
  6. Overlap data and what is physically well-defined
  7. Partial prior information and gauge-fixed superpositions
  8. Formal problem statement for the unified no-superposition principle
  9. Related Work: No-Go Results and Probabilistic Constructions
  10. From quantum adders to the phase-gauge obstruction
  11. Universal no-go theorems for exact superposition under CPTNI maps
  12. Probabilistic superposition with partial prior information
  13. Unified Principle: "Overlap-Determinability" as the Missing Resource
  14. From rays to coherent addition: the obstruction is the interference term
  15. Overlap-determinability and no-superposition principle
  16. ...and 24 more sections

Key Result

Theorem 1

Let $\mathcal{H}$ be a Hilbert space with $\dim\mathcal{H}\ge 2$, and let $\alpha,\beta$ be any two nonzero complex numbers satisfying $|\alpha|^2+|\beta|^2=1$. There exists no nonzero CPTNI map $\Lambda_{\alpha,\beta}:\mathcal{H}^{\otimes 2}\rightarrow\mathcal{H}$ satisfying Eq. (eq:universal_super

Figures (1)

  • Figure 1: Overlap-determinability as a unifying bottleneck. Independently prepared unknown rays do not, by themselves, determine a phase-fixed overlap. If a scenario were to make overlap-determinability freely available on generic domains, it would enable primitives such as universal superposition and unknown-state reflections; these, in turn, trigger a cascade of post-quantum consequences, including probabilistic cloning of linearly dependent families, finite-resource superluminal signaling, and the collapse of Grover's query lower bound via aggressive overlap amplification.

Theorems & Definitions (22)

  • Definition 1: Family of $\alpha$-$\beta$ superpositions of two rays
  • Definition 2: Universal probabilistic superposition map
  • Theorem 1: No-superposition theorem Oszmaniec2016
  • Definition 3: Phase convention and overlap-determinability
  • Theorem 2: Overlap-determinability is necessary for coherent superposition
  • proof : Proof Sketch
  • Theorem 3: Unified No-Superposition Principle
  • proof : Proof sketch
  • Corollary 1: No universal unknown-plus-unknown superposition
  • proof
  • ...and 12 more