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arXiv:2505.20208 · Physical Review A

Estimation of multivariate traces of states given partial classical information

Kyrylo Simonov, Rafael Wagner, Ernesto F. Galvão

Published Nov 14, 2025• 0 citations
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Abstract

Bargmann invariants of order n , defined as multivariate traces of quantum states Tr [ ρ 1 ρ 2 ... ρ n ] , are useful in applications ranging from quantum metrology to certification of nonclassicality. A standard quantum circuit used to estimate Bargmann invariants is the cycle test. In this work, we propose generalizations of the cycle test applicable to a situation where n systems are given and unknown, and classical information on m systems ( m ≤ n ) is available, allowing estimation of invariants of order n + m . Our main result is a generalization of results on fourth-order invariants appearing in “double” weak values from Chiribella [. The use of classical information on some of the states enables circuits on fewer qubits and with fewer gates, decreasing the experimental requirements for their estimation and enabling multiple applications we briefly review.

DOI
10.1103/mrpz-xq28

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Bargmann invariants of order n , defined as multivariate traces of quantum states Tr [ ρ 1 ρ 2 ... ρ n ] , are useful in applications ranging from quantum metrology to certification of nonclassicality. A standard quantum circuit used to estimate Bargmann invariants is the cycle test. In this work, we propose generalizations of the cycle test applicable to a situation where n systems are given and unknown, and classical information on m systems ( m ≤ n ) is available, allowing estimation of invariants of order n + m . Our main result is a generalization of results on fourth-order invariants appearing in “double” weak values from Chiribella [. The use of classical information on some of the states enables circuits on fewer qubits and with fewer gates, decreasing the experimental requirements for their estimation and enabling multiple applications we briefly review.
Certain unitary invariants, known as Bargmann invariants or multivariate traces of quantum states, have recently gained attention due to their applications in quantum information theory. However, determining the boundaries of sets of Bargmann invariants remains a theoretical challenge. In this study, we address the problem by developing a unified, dimension-independent formulation that characterizes the sets of the third and fourth Bargmann invariants. In particular, our result for the set of fourth Bargmann invariants confirms the conjecture given by Fernandes et al. [Phys. Rev. Lett. 133, 190201 (2024)]. Based on the obtained results, we conjecture that the unified, dimension-independent formulation of the boundaries for sets of third- and fourth-order Bargmann invariants may extend to the general case of the nthn\mathrm{th}nth-order Bargmann invariants. These results deepen our understanding of the fundamental physical limits within quantum mechanics and pave the way for novel applications of Bargmann invariants in quantum information processing and related fields.
The Hong-Ou-Mandel test estimates the overlap between spectral functions characterizing the internal degrees of freedom of two single photons. It can be viewed as a photon-native protocol that implements the well-known quantum SWAP test. Here, we propose a native linear-optical protocol that efficiently estimates multivariate traces of quantum states called Bargmann invariants, which are ubiquitous in quantum mechanics. Our protocol may be understood as a photon-native version of the cycle test in the circuit model, which encompasses many-photon multimode quantum states. We show the protocol is sample-efficient and discuss applications, such as generalized suppression laws, efficient quantum kernel estimation for quantum machine learning, eigenspectrum estimation, and the characterization of multiphoton indistinguishability.
Noncommutativity of states and observables is a fundamental signature of quantum theory, and a minimal requirement for nonclassicality. We provide a universal necessary and sufficient condition for pairwise commutativity of quantum states ρ1ρ_1ρ1​ and ρ2ρ_2ρ2​: they commute if and only if tr(ρ12ρ22)=tr(ρ1ρ2ρ1ρ2)\mathrm{tr}(ρ_1^2ρ_2^2) = \mathrm{tr}(ρ_1 ρ_2 ρ_1 ρ_2)tr(ρ12​ρ22​)=tr(ρ1​ρ2​ρ1​ρ2​). For qubits the identity simplifies to an equality between polynomials of purities and of the two-state overlap tr(ρ1ρ2)\mathrm{tr}(ρ_1ρ_2)tr(ρ1​ρ2​). These multivariate traces (known as Bargmann invariants) are directly measurable, allowing commutativity tests that bypass full state tomography. We point out possible applications to the analysis of POVM simulability and partial photonic distinguishability.
Bargmann invariants, also known as multivariate traces of quantum states $\mathrm{Tr}({\ensuremath{\rho}}_{1}{\ensuremath{\rho}}_{2}\ensuremath{\cdots}{\ensuremath{\rho}}_{n})$, are unitary invariant quantities used to characterize weak values, Kirkwood-Dirac quasiprobabilities, out-of-time-order correlators, and geometric phases. Here we give a complete characterization of the set Bn{B}_{n}Bn​ of complex values that nthn\mathrm{th}nth-order invariants can take, resolving some recently proposed conjectures. We show that Bn{B}_{n}Bn​ is equal to the range of invariants arising from pure states described by Gram matrices of circulant form. We show that both ranges are equal to the nthn\mathrm{th}nth power of the complex unit nnn-gon and are therefore convex, which provides a simple geometric intuition. Finally, we show that any Bargmann invariant of order nnn is realizable using either qubit states or circulant qutrit states.
Bargmann invariants, a class of unitary-invariant quantities arising from the overlaps of quantum state vectors, provide a profound and unifying framework for understanding the relative geometry of the projective Hilbert space. This survey offers a comprehensive overview of their theoretical characterization and practical applications, with particular emphasis on recent progress in determining the full structure of their admissible set. The core of this review demonstrates how these invariants serve as a powerful tool for characterizing the intrinsic geometry of the space of quantum states, leading to applications in determining local unitary equivalence and constructing a complete set of polynomial invariants for mixed states. On the operational side, we review the cycle-test quantum circuit for the direct estimation of Bargmann invariants without full state tomography, and demonstrate their utility in witnessing quantum imaginary, discriminating local unitary equivalence, and detecting entanglement via partial-transpose moments---with explicit complete invariant sets provided for two-qubit systems. By connecting fundamental geometric classification with experimentally feasible estimation protocols,this survey establishes Bargmann invariants as indispensable probes of the relational, noncommutative, and geometric structure of quantum states, and identifies key open problems for multipartite high-dimensional systems and for quantum resource theories.
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