Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2601.14204

Native linear-optical protocol for efficient multivariate trace estimation

Leonardo Novo, Marco Robbio, Ernesto F. Galvão, Nicolas J. Cerf

quant-phSubmitted Jan 20, 2026• Announced Jan 21, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

Related papers

Measuring relational information between quantum states, and applications

Michał Oszmaniec, Daniel J. Brod, Ernesto F. Galvão

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.

arXiv:2109.10006•Sep 21, 2021•3 citations

Optimal interferometric certification of multi-photon indistinguishability

Marco Robbio, Michał Oszmaniec, Nicolas J. Cerf +2 more

Multiphoton indistinguishability is a key resource for photonic quantum technologies, yet its characterization typically relies on resource-intensive methods. In this work, we develop two efficient and experimentally friendly protocols to estimate or bound the fidelity $F_{\mathrm{ind}}$ of an $N$-photon state to the closest perfectly indistinguishable state. The first protocol applies to sources preparing separable states, uses a single Fourier interferometer together with photon-number-resolving detection, and yields tight two-sided bounds on $F_{\mathrm{ind}}$. The second protocol combines randomized implementations of linear-optical interferometers with photon counting, enabling direct estimation of $F_{\mathrm{ind}}$ for arbitrary $N$-photon states. Both protocols can certify $F_{\mathrm{ind}} = 1 - \mathcal{O}(ε)$ using provably optimal $\mathcal{O}(1/ε)$ samples, in contrast to previous approaches which required prior assumptions on the model of partial distinguishability. Our methods, based on a multiphoton generalization of the Hong-Ou-Mandel test, bring the rigorous and operationally meaningful certification of multiphoton indistinguishability within reach of current photonic technologies.

quant-pharXiv:2609.37620•Sep 30, 2026•0 citations

Measuring relational information between quantum states, and applications

Michał Oszmaniec, Daniel J. Brod, Ernesto F. Galvão

Abstract 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.

New Journal of Physics•Jan 1, 2024•32 citations

Quantum advantage for single-photon state characterization

S. N. van den Hoven, M. C. Anguita, S. Marzban +1 more

We propose a multiphoton interference protocol that characterizes the pairwise overlaps of the internal modes of single photons more efficiently than pairwise Hong-Ou-Mandel characterization experiments. We experi mentally implement this protocol to characterize three photons. We show that our implementation of the char acterization protocol outperforms the pairwise Hong-Ou-Mandel characterization, even if the Hong-Ou-Mandel characterization would have been performed in a noiseless, perfect experiment. We demonstrate this via the Fisher information matrix. Surprisingly, this advantage persists at arbitrary nonzero transmission, demonstrat ing the viability of this protocol for real-world characterization of single photons.

quant-pharXiv:2512.04903•Physical Review Letters•Dec 5, 2025•0 citations

Commutativity from a single Bargmann invariant equality

Rafael Wagner, Ernesto F. Galvão

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 $ρ_1$ and $ρ_2$: they commute if and only if $\mathrm{tr}(ρ_1^2ρ_2^2) = \mathrm{tr}(ρ_1 ρ_2 ρ_1 ρ_2)$. For qubits the identity simplifies to an equality between polynomials of purities and of the two-state overlap $\mathrm{tr}(ρ_1ρ_2)$. 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.

quant-pharXiv:2605.07405•May 11, 2026•0 citations

Estimation of multivariate traces of states given partial classical information

Kyrylo Simonov, Rafael Wagner, Ernesto F. Galvão

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.

arXiv:2505.20208•Physical Review A•Nov 14, 2025•0 citations
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.
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.
Multiphoton indistinguishability is a key resource for photonic quantum technologies, yet its characterization typically relies on resource-intensive methods. In this work, we develop two efficient and experimentally friendly protocols to estimate or bound the fidelity FindF_{\mathrm{ind}}Find​ of an NNN-photon state to the closest perfectly indistinguishable state. The first protocol applies to sources preparing separable states, uses a single Fourier interferometer together with photon-number-resolving detection, and yields tight two-sided bounds on FindF_{\mathrm{ind}}Find​. The second protocol combines randomized implementations of linear-optical interferometers with photon counting, enabling direct estimation of FindF_{\mathrm{ind}}Find​ for arbitrary NNN-photon states. Both protocols can certify Find=1−O(ε)F_{\mathrm{ind}} = 1 - \mathcal{O}(ε)Find​=1−O(ε) using provably optimal O(1/ε)\mathcal{O}(1/ε)O(1/ε) samples, in contrast to previous approaches which required prior assumptions on the model of partial distinguishability. Our methods, based on a multiphoton generalization of the Hong-Ou-Mandel test, bring the rigorous and operationally meaningful certification of multiphoton indistinguishability within reach of current photonic technologies.
Abstract 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.
We propose a multiphoton interference protocol that characterizes the pairwise overlaps of the internal modes of single photons more efficiently than pairwise Hong-Ou-Mandel characterization experiments. We experi mentally implement this protocol to characterize three photons. We show that our implementation of the char acterization protocol outperforms the pairwise Hong-Ou-Mandel characterization, even if the Hong-Ou-Mandel characterization would have been performed in a noiseless, perfect experiment. We demonstrate this via the Fisher information matrix. Surprisingly, this advantage persists at arbitrary nonzero transmission, demonstrat ing the viability of this protocol for real-world characterization of single photons.
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 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.