Multi-state imaginarity and coherence in qubit systems

TL;DR

This paper develops a unitary-invariant framework to analyze imaginarity and coherence in multi-qubit states, providing exact tests, bounds, and invariants that deepen understanding of quantum resources beyond individual states.

Contribution

It introduces a novel framework for characterizing imaginarity and coherence in multi-qubit states using invariants and overlaps, extending resource analysis beyond single states.

Findings

Bloch vectors must be coplanar for imaginarity-free states.

Exact rank-based tests for coherence and imaginarity are derived.

Third-order invariants fully characterize multi-state imaginarity in qubits.

Abstract

Traditionally, the characterization of quantum resources has focused on individual quantum states. Recent literature, however, has increasingly explored the characterization of resources in multi-states (ordered collections of states indexed by a varying parameter). In this work, we provide a unitary-invariant framework to pinpoint imaginarity and coherence in sets of qubit states: we prove that Bloch vectors must be coplanar to be imaginarity-free and colinear to be incoherent, yielding exact rank-based tests of coherence and imaginarity, and closed-form bounds for existing robustness quantifiers, all based on two-state overlaps only. We also show that the set of imaginarity-free multi-states is not convex, and that third-order invariants completely characterize multi-state imaginarity of single-qubits but not of higher-dimensional systems. As our main technical result, we show that every Bargmann invariant of single-qubit states is determined (up to conjugation) by two-state overlaps. Beyond qubits, we give purity and system-agnostic coherence witnesses from equality constraints on higher-order invariants and connect our results to practical protocols: characterization of partial distinguishability, spin-chirality detection, and subchannel discrimination.