Abstract We investigate the extendibility problem for Brauer states, focusing on the symmetric two-sided extendibility and the de Finetti extendibility. By employing the representation theory of the unitary and orthogonal groups, we provide a general recipe for determining the set of -extendible and -de Finetti-extendible Brauer states. From the concrete form of the commutant of the diagonal action of the orthogonal group, we explicitly determine the set of parameters for which the Brauer states are -, - and -extendible in any dimension and find that Brauer states extend with a non-trivial trade-off in and . Using the same recipe we also provide an estimate of the set of -extendible Brauer states for any and dimension . Finally, using the branching rules from to , we obtain the set of -de Finetti-extendible Brauer states in any dimension, and also analytically describe the limiting shape which turns out not to be a polygon for odd dimensions.
Abstract We investigate the extendibility problem for Brauer states, focusing on the symmetric two-sided extendibility and the de Finetti extendibility. By employing the representation theory of the unitary and orthogonal groups, we provide a general recipe for determining the set of -extendible and -de Finetti-extendible Brauer states. From the concrete form of the commutant of the diagonal action of the orthogonal group, we explicitly determine the set of parameters for which the Brauer states are -, - and -extendible in any dimension and find that Brauer states extend with a non-trivial trade-off in and . Using the same recipe we also provide an estimate of the set of -extendible Brauer states for any and dimension . Finally, using the branching rules from to , we obtain the set of -de Finetti-extendible Brauer states in any dimension, and also analytically describe the limiting shape which turns out not to be a polygon for odd dimensions.
We investigate the two-sided symmetric extendibility problem of Werner states. The interplay of the unitary symmetry of these states and the inherent bipartite permutation symmetry of the extendibility scenario allows us to map this problem into the ground state problem of a highly symmetric spin-model Hamiltonian. We solve this ground state problem analytically by utilizing the representation theory of SU(d), in particular a result related to the dominance order of Young diagrams in Littlewood-Richarson decompositions. As a result, we obtain necessary and sufficient conditions for the extendibility of Werner states for arbitrary extension size and local dimension. Interestingly, the range of extendible states has a non-trivial trade-off between the extension sizes on the two sides. We compare our result with the two-sided extendibility problem of isotropic states, where there is no such trade-off.
We prove a generalization of the quantum de Finetti theorem when the local space is an infinite-dimensional Fock space. In particular, instead of considering the action of the permutation group on n copies of that space, we consider the action of the unitary group U(n) on the creation operators of the n modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in U(n). Our first result is a complete characterization of this subspace, which turns out to be spanned by a family of generalized coherent states related to the special unitary group SU(p,q) of signature (p,q). More precisely, this construction yields a unitary representation of the noncompact simple real Lie group SU(p,q). We therefore find a dual unitary representation of the pair of groups U(n) and SU(p,q) on an n(p+q)-mode Fock space.The (Gaussian) SU(p,q) coherent states resolve the identity on the symmetric subspace, which implies a Gaussian de Finetti theorem stating that tracing over a few modes of a unitary-invariant state yields a state close to a mixture of Gaussian states. As an application of this de Finetti theorem, we show that the n×n upper-left submatrix of an n×n Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if n3=O(m).
We prove a generalization of the quantum de Finetti theorem when the local space is an infinite-dimensional Fock space. In particular, instead of considering the action of the permutation group on n copies of that space, we consider the action of the unitary group U(n) on the creation operators of the n modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in U(n). Our first result is a complete characterization of this subspace, which turns out to be spanned by a family of generalized coherent states related to the special unitary group SU(p, q) of signature (p, q). More precisely, this construction yields a unitary representation of the noncompact simple real Lie group SU(p, q). We therefore find a dual unitary representation of the pair of groups U(n) and SU(p, q) on an n(p + q)-mode Fock space. The (Gaussian) SU(p, q) coherent states resolve the identity on the symmetric subspace, which implies a Gaussian de Finetti theorem stating that tracing over a few modes of a unitary-invariant state yields a state close to a mixture of Gaussian states. As an application of this de Finetti theorem, we show that the n × n upper-left submatrix of an n × n Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if n3 = O(m).
De Finetti theorems show how sufficiently exchangeable states are well-approximated by convex combinations of independent identically distributed states. Recently, it was shown that in many quantum information applications, a more relaxed de Finetti reduction (i.e., only a matrix inequality between the symmetric state and one of the de Finetti forms) is enough and that it leads to more concise and elegant arguments. Here we show several uses and general flexible applicability of a constrained de Finetti reduction in quantum information theory, which was recently discovered by Duan, Severini, and Winter. In particular, we show that the technique can accommodate other symmetries commuting with the permutation action and permutation-invariant linear constraints. We then demonstrate that, in some cases, it is also fruitful with convex constraints, in particular separability in a bipartite setting. This is a constraint particularly interesting in the context of the complexity class QMA(2) of interactive quantum Merlin-Arthur games with unentangled provers, and our results relate to the soundness gap amplification of QMA(2) protocols by parallel repetition. It is also relevant for the regularization of certain entropic channel parameters. As an aside, we present an alternative way of attacking this problem, relying on an entanglement measure theory rather than the de Finetti approach. Finally, we explore an extension to infinite-dimensional systems, which usually pose inherent problems to de Finetti techniques in the quantum case.
We prove optimal finite quantum de Finetti upper bounds. Given a bosonic state ρN∈D(SymN(Cd)), there is a probability measure ν on the unit sphere such that ρN(2)−∫∣u⟩⟨u∣⊗2dν(u)1≤N−1d−1. By purification, the bosonic theorem also gives the optimal O(d/N) upper bound for arbitrary exchangeable states. These results settle the dimension dependence left open by Christandl, König, Mitchison, and Renner (CMP 2007). The proof casts de Finetti approximation as sum-of-squares rounding and applies the argmax method of Jeronimo, Wu, and Xu (manuscript 2026).
More generally, t-site marginals satisfy O(td/N) bosonic and O(td/N) permutation-invariant bounds. Our proof formulates de Finetti approximation as the integrality gap of a symmetric-extension semidefinite program and rounds an optimum by the argmax principle. The sharp bounds have several consequences. For every fixed ε∈(0,1), we construct a channel with input dimension D=exp(Oε(dlogd))=exp(o(d)) whose outputs are ε-close to separable states of local dimension d and whose image contains every such separable state, thereby refuting Watrous's disentangler conjecture. We also obtain deterministic exp(O(d/ε))-time algorithms for explicit Best Separable State without perfect completeness and for trace-distance separability testing.
Finally, spectral truncation gives the first dimension-free bosonic de Finetti theorem in Hilbert--Schmidt distance, with the optimal rate Θ(N−1/2) when the dimension may grow.
Abstract We study bipartite unitary operators which stay invariant under the local actions of diagonal unitary and orthogonal groups. We investigate structural properties of these operators, arguing that the diagonal symmetry makes them suitable for analytical study. As a first application, we construct large new families of dual unitary gates in arbitrary finite dimensions, which are important toy models for entanglement spreading in quantum circuits. We then analyze the non-local nature of these invariant operators, both in discrete (operator Schmidt rank) and continuous (entangling power) settings. Our scrutiny reveals that these operators can be used to simulate any bipartite unitary gate via stochastic local operations and classical communication. Furthermore, we establish a one-to-one connection between the set of local diagonal unitary invariant dual unitary operators with maximum entangling power and the set of complex Hadamard matrices. Finally, we discuss distinguishability of unitary operators in the setting of the stated diagonal symmetry.