Set coherence and set imaginarity are two important quantum resources defined by sets of states. We investigate tuples of qubit states under vanishing higher-order Bargmann invariants and show that their set coherence and set imaginarity are each related to whether they contain a pair of orthogonal states. Specifically, when the relevant higher-order Bargmann invariants vanish, the tuple contains a pair of orthogonal states if and only if removing some state makes the remaining tuple incoherent. Moreover, when maximally mixed states are excluded, the same vanishing conditions imply that the tuple is imaginarity-free if and only if it contains a pair of orthogonal states. As a consequence, under these conditions, a tuple of single-qubit states has imaginarity precisely when every tuple obtained by deleting one of its states is coherent.
Set coherence and set imaginarity are two important quantum resources defined by sets of states. We investigate tuples of qubit states under vanishing higher-order Bargmann invariants and show that their set coherence and set imaginarity are each related to whether they contain a pair of orthogonal states. Specifically, when the relevant higher-order Bargmann invariants vanish, the tuple contains a pair of orthogonal states if and only if removing some state makes the remaining tuple incoherent. Moreover, when maximally mixed states are excluded, the same vanishing conditions imply that the tuple is imaginarity-free if and only if it contains a pair of orthogonal states. As a consequence, under these conditions, a tuple of single-qubit states has imaginarity precisely when every tuple obtained by deleting one of its states is coherent.
Traditionally, the characterization of quantum resources has focused on individual quantum states. Recent literature, however, has increasingly explored the characterization of resources in 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 collinear 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 multistates is not convex, and that third-order invariants completely characterize multistate imaginarity of single-qubits but not of higher-dimensional systems. As our main technical result, we show that 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.
Recent studies have shown that Bargmann invariants provide effective detectors of set imaginarity. In this paper, we investigate set imaginarity as a quantum resource in qubit systems. By exploiting the structure of Bargmann invariants, we show that the free operations for qubit set imaginarity consist precisely of common unital operations and common planarized operations. Based on this characterization, we introduce an axiomatic framework for set-imaginarity measures (SIMs). In particular, we propose two refined notions, namely unified SIMs and complete SIMs, which allow a more fine-grained quantification of set imaginarity. To make these notions concrete, we construct two qubit SIMs from the Bargmann invariants of three-state subsets. We prove that one of them is a unified SIM, while the other satisfies the stronger requirements of a complete SIM. Furthermore, we revisit the robustness of set imaginarity previously introduced in the literature. We show that, although this robustness is a valid SIM for qubit systems, it is neither a unified SIM nor a complete SIM. To overcome this limitation, we propose an improved robustness-type measure and rigorously prove that it defines a complete qubit SIM.
The imaginary in quantum theory plays a crucial role in describing quantum coherence and is widely applied in quantum information tasks such as state discrimination, pseudorandomness generation, and quantum metrology. A recent paper by Fernandes et al. [Phys. Rev. Lett. 133, 190201 (2024)] showed how to use the Bargmann invariant to witness the imaginarity of a set of quantum states. In this work, we delve into the structure of Bargmann invariants and their quantum realization in qubit systems. First, we present a characterization of special sets of Bargmann invariants (also studied by Fernandes et al. for a set of four states) for a general set of n quantum states. Then, we study the properties of the relevant Bargmann invariant set Bn and its quantum realization in qubit systems. Our results provide new insights into the structure of Bargmann invariants, contributing to the advancement of quantum information techniques, particularly within qubit systems.
Considerable effort has been devoted to developing techniques for witnessing and characterizing quantum resources that emerge from collective properties of a set of states. In this context, Bargmann invariants play a central role: they witness coherence and related resources, and underpin important applications. In this work, we introduce a unified formalism that fully characterizes and organizes the capability of Bargmann invariants to witness different manifestations of coherence in sets of states. It is formulated around the construction of Bargmann scenarios, which specify relevant tuples of Bargmann invariants, and Bargmann polytopes, which bound the values that said invariants can have when the states are incoherent. We study their basic geometry, connect them to existing formalisms, and illustrate their physical relevance. Our construction opens new opportunities for the certification of quantum devices and lays the path toward a full quantum resource theory based entirely on multivariate traces of states.
Abstract Since both coherence and imaginarity have proven to be fundamental resources in quantum information processing and their definitions are both relative to a given basis, it is natural to investigate the conversion between them. In this work, we prove that a quantum state can be transformed into a state with nonvanishing imaginarity resource by incoherent operations if and only if it has nonvanishing coherence resource. On the other hand, we demonstrate that a quantum state can be transformed into a state with vanishing coherence resource by orthogonal operations if and only if it has vanishing imaginarity resource. Based on these results, we show that for any imaginarity measure, a corresponding coherence measure can be defined as the maximal amount of imaginarity generated by incoherent operations, and that for any coherence quantifier, a corresponding imaginarity quantifier can be defined as the minimal coherence generated by orthogonal operations.
Set coherence is a basis-independent relational form of quantum coherence: a finite family of quantum states is set incoherent exactly when all its members are diagonal in one common basis. We determine how much low-order Bargmann data are needed to decide this property. For two states, second-order data are complete for qubits but fail for qutrits, while complete third-order data are sufficient for qutrits but fail already in dimension four. We then show that fourth-order, ordering-sensitive Bargmann invariants give the first universal pairwise criterion for set coherence. Applied to all unordered pairs, this criterion yields a complete test for arbitrary finite families. The result provides a low-order hierarchy connecting cyclic trace invariants with the noncommutativity that prevents a common incoherent basis.