Imaginarity as a Resource within Quantum Coherence: Geometric Decomposition and Operational Conversion

Research Overview

This paper investigates the structural relationship between two critical quantum resources: quantum coherence (superposition in a preferred basis) and imaginarity (the necessity of complex numbers in quantum state representation). While both are essential for quantum information processing, their precise hierarchical and operational connection has remained elusive. The authors introduce a geometric framework to partition total coherence into two distinct parts: imaginarity and residual coherence.

Geometric Decomposition

The authors define residual coherence as the metric distance between the closest real state and the closest incoherent state. By applying the triangle inequality to these distances, they derive a fundamental trade-off relation: total coherence is bounded by the sum of imaginarity and residual coherence. This result provides a rigorous geometric explanation for why coherence cannot always be fully attributed to imaginarity, identifying residual coherence as an inherent overhead caused by reference-frame misalignment.

Operational Conversion

To move beyond theoretical abstraction, the researchers developed an explicit operational protocol to convert imaginarity in one subsystem into coherence in another. Using a bipartite system where one subsystem is initialized in an incoherent state, they demonstrate that by applying a controlled-Z gate, a Hadamard gate, and a Y-basis measurement, the imaginarity of the second subsystem can be mapped directly to the coherence of the first. This protocol proves that imaginarity is not just a theoretical construct but a convertible resource that can be managed in distributed quantum networks.

Dynamical Constraints

The study also characterizes the dynamical evolution of these resources under diagonal Hamiltonians. The findings reveal a conservation law for total coherence, while imaginarity and residual coherence exhibit complementary oscillations. This suggests that in distributed architectures, the efficiency of resource utilization is dictated by the geometric composition of the state, rather than just the total amount of coherence available.