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arXiv:2403.15066 · Physical Review Letters

Unitary-Invariant Witnesses of Quantum Imaginarity

Carlos Fernandes, Rafael Wagner, Leonardo Novo, Ernesto F. Galvão

Published Nov 8, 2024• 43 citations
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Abstract

Quantum theory is traditionally formulated using complex numbers. This imaginarity of quantum theory has been quantified as a resource with applications in discrimination tasks, pseudorandomness generation, and quantum metrology. In the standard formulation, a quantum state is said to have "imaginarity" if the associated density matrix is not real-valued in a given, fixed basis. If instead we consider a set of states, it is possible to devise tests that guarantee imaginarity of some state in the set, independently of the basis chosen. Here we propose such basis-independent witnesses for imaginarity that rely on measurements of unitary-invariant properties of sets of states. For three pure states, we completely characterize the invariant values attainable by quantum theory, and give a partial characterization for four pure states. We show that simple pairwise overlap measurements suffice to witness imaginarity of sets of four states, but not for sets of three. Our witnesses are experimentally friendly, opening up a new path for measuring and using imaginarity as a resource.

DOI
10.1103/physrevlett.133.190201

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Quantum theory is traditionally formulated using complex numbers. This imaginarity of quantum theory has been quantified as a resource with applications in discrimination tasks, pseudorandomness generation, and quantum metrology. In the standard formulation, a quantum state is said to have "imaginarity" if the associated density matrix is not real-valued in a given, fixed basis. If instead we consider a set of states, it is possible to devise tests that guarantee imaginarity of some state in the set, independently of the basis chosen. Here we propose such basis-independent witnesses for imaginarity that rely on measurements of unitary-invariant properties of sets of states. For three pure states, we completely characterize the invariant values attainable by quantum theory, and give a partial characterization for four pure states. We show that simple pairwise overlap measurements suffice to witness imaginarity of sets of four states, but not for sets of three. Our witnesses are experimentally friendly, opening up a new path for measuring and using imaginarity as a resource.
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 nnn quantum states. Then, we study the properties of the relevant Bargmann invariant set Bn{\mathcal{B}}_{n}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.
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.
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. [C. Fernandes, R. Wagner, L. Novo, and E. F. Galvão, 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 nnn quantum states. Then, we study the properties of the relevant Bargmann invariant set Bn\mathcal{B}_nBn​ 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.
Quantum imaginarity is an essential physical resource that underpins key functionalities of modern quantum technologies. We improve imaginarity witnessing via the prior knowledge of imaginarity-witness operators. To this end, we derive a rigorous upper bound on the maximum expectation value of an imaginarity witness operator over the set of all free (real) quantum states, which is given by the spectral norm of the real component of the corresponding witness operator. We demonstrate via detailed examples that this bound substantially improves imaginarity detection. We further classify all imaginarity-witness operators into four distinct families based on this bound. For these four witness classes, we perform a comprehensive analysis of their completeness and finite completeness, the joint detection of shared imaginary quantum states by different witnesses, and the conditions for distinct witnesses to identify identical imaginary states. Our results advance the fundamental understanding of imaginarity detection and offer useful insights for both theoretical studies and experimental implementations of quantum imaginarity.
The geometrical arrangement of a set of quantum states can be completely characterized using relational information only. This information is encoded in the pairwise state overlaps, as well as in Bargmann invariants of higher degree written as traces of products of density matrices. We describe how to measure Bargmann invariants using suitable generalizations of the SWAP test. This allows for a complete and robust characterization of the projective-unitary invariant properties of any set of pure or mixed states. As applications, we describe basis-independent tests for linear independence, coherence, and imaginarity. We also show that Bargmann invariants can be used to characterize multi-photon indistinguishability.
Imaginarity has shown to be an important resource in quantum information. The witness theory of quantum resource, such as entanglement witness, coherence witness, and imaginarity witness, has been established, in particular entanglement witness and coherence witness have been extensively explored. We present here another class of imaginarity witnesses beyond the one in [{Phys. Lett. A} \textbf{530}, 130135 (2025)]. Within our framework, any nonreal Hermitian operator under the reference basis is an imaginarity witness and only finite of them can detect all the imaginarity states. As the common approaches that were explored in the witness theories of entanglement and coherence, we then explore the relations between these imaginarity witnesses in different cases: (i) when they can detect common imaginary states, (ii) when they can detect the same sets of imaginary states, and (ii) when they obey the finer relation. Finally, we define an imaginarity measure in terms of witnesses, termed witnessed imaginarity, and prove that it coincides with both the trace norm of imaginarity and the robustness of imaginarity.