The quantum condition space

TL;DR

This paper introduces the quantum condition space, a dual to classical outcome space, revealing entangled conditions, duality with quantum states, and implications for quantum measurement, uncertainty, and circuit complexity.

Contribution

It defines the quantum condition space, explores its properties, duality with quantum states, and potential applications in understanding quantum measurements and circuit complexity.

Findings

Quantum condition space is dual to classical outcome space.

Entangled conditions have no classical equivalent.

Fourier transform relates condition space to quantum state space.

Abstract

In this work we first propose to exploit the fundamental properties of quantum physics to evaluate the probability of events with projection measurements. Next, to study what events can be specified by quantum methods, we introduce the concept of the condition space, which is found to be the dual space of the classical outcome space of bit strings. Just like the classical outcome space generates the quantum state space, the condition space generates the quantum condition space that is the central idea of this work. The quantum condition space permits the existence of entangled conditions that have no classical equivalent. In addition, the quantum condition space is related to the quantum state space by a Fourier transform guaranteed by the Pontryagin duality, and therefore an entropic uncertainty principle can be defined. The quantum condition space offers a novel perspective of understanding quantum states with the duality picture. In addition, the quantum conditions have physical meanings and realizations of their own and thus may be studied for purposes beyond the original motivation of characterizing events for probability evaluation. Finally, the relation between the condition space and quantum circuits provides insights into how quantum states are collectively modified by quantum gates, which may lead to deeper understanding of the complexity of quantum circuits.