Given a set of ordered quantum states, described by density operators {ρj}j=1n, the Bargmann invariant of {ρj}j=1n is defined as tr(ρ1ρ2, …, ρn). Bargmann invariant serves as a fundamental concept for quantum mechanics and has diverse applications in quantum information science. Bosonic Gaussian states are a class of quantum states on infinite-dimensional Hilbert space, widely used in quantum optics and quantum information science. Bosonic Gaussian states are conveniently and conventionally characterized by their means and covariance matrices. In this work, we provide the expression of Bargmann invariant tr(ρ1ρ2, …, ρn) for any m-mode bosonic Gaussian states {ρj}j=1n in terms of the means and covariance matrices of {ρj}j=1n. We also use this expression to explore the permissible values of Bargmann invariants for bosonic Gaussian states.
Given a set of ordered quantum states, described by density operators {ρj}j=1n, the Bargmann invariant of {ρj}j=1n is defined as tr(ρ1ρ2, …, ρn). Bargmann invariant serves as a fundamental concept for quantum mechanics and has diverse applications in quantum information science. Bosonic Gaussian states are a class of quantum states on infinite-dimensional Hilbert space, widely used in quantum optics and quantum information science. Bosonic Gaussian states are conveniently and conventionally characterized by their means and covariance matrices. In this work, we provide the expression of Bargmann invariant tr(ρ1ρ2, …, ρn) for any m-mode bosonic Gaussian states {ρj}j=1n in terms of the means and covariance matrices of {ρj}j=1n. We also use this expression to explore the permissible values of Bargmann invariants for bosonic Gaussian states.
Let Γ(H) be the boson Fock space over a finite dimensional Hilbert space H. It is shown that every Gaussian symmetry admits a Klauder–Bargmann integral representation in terms of coherent states. Furthermore, Gaussian states, Gaussian symmetries, and second quantization contractions belong to a weakly closed self-adjoint semigroup E2(H) of bounded operators in Γ(H). This yields a common parametrization for these operators. It is shown that the new parametrization for Gaussian states is a fruitful alternative to the customary parametrization by position–momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries: (i) every Gaussian state ρ admits a factorization ρ=Z1†Z1, where Z1 is an element of E2(H) and has the form Z1=cΓ(P)exp∑r=1nλrar+∑r,s=1nαrsaras on the dense linear manifold generated by all exponential vectors, where c is a positive scalar, Γ(P) is the second quantization of a positive contractive operator P in H, ar, 1 ≤ r ≤ n, are the annihilation operators corresponding to the n different modes in Γ(H), λr∈C, and [αrs] is a symmetric matrix in Mn(C); (ii) an explicit particle basis expansion of an arbitrary mean zero pure Gaussian state vector along with a density matrix formula for a general Gaussian state in terms of its E2(H)-parameters; (iii) a class of examples of pure n-mode Gaussian states that are completely entangled; (iv) tomography of an unknown Gaussian state in Γ(Cn) by the estimation of its E2(Cn) parameters using O(n2) measurements with a finite number of outcomes.
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.
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 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.
Bosonic Gaussian thermal states form a fundamental class of states in quantum information science. This paper explores the information geometry of these states, focusing on characterizing the distance between two nearby states and the geometry induced by a parameterization in terms of their mean vectors and Hamiltonian matrices. In particular, for the family of bosonic Gaussian thermal states, we derive expressions for their Fisher-Bures, Kubo-Mori, and α-z information matrices with respect to their mean vectors and Hamiltonian matrices. An important application of our formulas consists of fundamental limits on how well one can estimate these parameters. We additionally establish formulas for the derivatives and the symmetric logarithmic derivatives of bosonic Gaussian thermal states. The former could have applications in gradient descent algorithms for quantum machine learning when using bosonic Gaussian thermal states as an ansatz, and the latter in formulating optimal strategies for single parameter estimation of bosonic Gaussian thermal states. Finally, the expressions for the aforementioned information matrices could have additional applications in natural gradient descent algorithms when using bosonic Gaussian thermal states as an ansatz.
Let Γ(H) be the boson Fock space over a finite dimensional Hilbert space H. It is shown that every gaussian symmetry admits a Klauder-Bargmann integral representation in terms of coherent states. Furthermore, gaussian symmetries, gaussian states and second quantization contractions, all of these operators belong to a weakly closed, selfadjoint semigroup E2(H) of bounded operators in Γ(H). This yields, a new parametrization of gaussian states, which is a very fruitful alternative to the customary parametrization by position-momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries:
(i) every gaussian state ρ admits a factorization ρ=Z1†Z1, where Z1 is an element of E2(H) and has the form Z1=cΓ(Λ)exp∑r=1nλrar+∑r,s=1nαrsaras on the dense linear manifold generated by all exponential vectors, Λ being a positive operator in H, ar,1≤r≤n are the annihilation operators corresponding to the n different modes in Γ(H), λr∈C and [αrs] is a symmetric matrix in Mn(C);
(ii) an explicit particle basis expansion of an arbitrary mean zero pure gaussian state vector along with a density matrix formula for a general gaussian state in terms of its E2(H)-parameters;
(iii) an easy test for the entanglement of pure gaussian states and a class of examples of pure n-mode gaussian states which are completely entangled;
(iv) tomography of an unknown gaussian state in Γ(Cn) by the estimation of its E2(Cn)-parameters using O(n2) measurements with a finite number of outcomes.
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 of complex values that nth-order invariants can take, resolving some recently proposed conjectures. We show that 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 nth power of the complex unit n-gon and are therefore convex, which provides a simple geometric intuition. Finally, we show that any Bargmann invariant of order n is realizable using either qubit states or circulant qutrit states.