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

TL;DR

This paper formalizes the concept of phase conventions in quantum superposition, introduces overlap-determinability, and characterizes when superpositions can be probabilistically created, revealing foundational and computational implications.

Contribution

It introduces the notion of overlap-determinability and provides a theorem linking it to superposition protocols, unifying existing results and exploring their foundational consequences.

Findings

Superposition protocols exist iff the domain is overlap-determinable.

Universal access to fixed overlaps enables forbidden quantum transformations.

Such access could lead to superluminal signaling and collapse of quantum complexity bounds.

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.