Randomized measurement protocols give experimental access to low-degree polynomial invariants of a quantum state rather than to the state itself, which raises the question of what the best entanglement criterion is that can be built from finitely many such invariants. In this work we answer this question through convex duality.
For each degree we introduce the cone of local unitary trace inequalities that are valid for separable bipartite states in all local dimensions, together with its one-matrix counterparts for Hermitian and positive semidefinite matrices. Each cone is the polar dual of a moment cone; we also determine the minimal combinatorial parametrization of the separable cone. The framework organizes the known moment criteria: the PPT criterion using the first three moments of the partial transposition is the image under an embedding of a Hankel determinant from the positive semidefinite cone to the bipartite cone, which we show to be an extremal ray, and we determine the first nontrivial separable cone completely, finding exactly four extremal rays.
The framework allows us to analyze the realignment criterion and its centered (or enhanced) variant. We prove that the even singular value moments of the (centered) realignment matrix are trace invariants, associated with explicit fixed-point-free involutions. Extracting the optimal trace-norm bound from m such moments thus becomes a dimension-free truncated moment problem, which we solve in closed form for m=2 and relax to a semidefinite program of size O(m) for arbitrary m. The dual solutions are polynomial minorants of x on [0,1], and two explicit families, binomial and L1-weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT entangled states.
With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials Xα1,…,Xαr and their traces tr(Xα1,…,Xαr). Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley–Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.
Symbolic integration over the Haar measure of compact groups is a computational cornerstone in quantum information science and random matrix theory. We present \texttt{IntegrateUnitary.jl}, a comprehensive Julia package for computing exact expectations of polynomial functions over a wide range of compact groups (U(d), O(d), Sp(d), and SU(d) for balanced polynomials), circular and Gaussian ensembles, Ginibre ensembles, permutation groups, random pure states, and unitary t-designs. The package provides a fully open-source implementation of the Weingarten calculus and Wick contractions with broad symbolic-d support for entry-wise and trace-polynomial integrals, while selected workflows currently require concrete integer dimensions (including higher pure trace moments ∣tr(U)∣2k for k>1 and HCIZ with \texttt{SymbolicMatrix} inputs, and direct matrix-valued integration of \texttt{SymbolicMatrix}/\texttt{SymbolicMatrixProduct} expressions), automatic asymptotic expansions, a high-level symbolic trace interface that reconstructs Weingarten graphs from index-free expressions, and a bridge to \texttt{ITensors.jl} for tensor network averaging. We discuss the underlying algorithms, including the Murnaghan-Nakayama rule and symplectic-orthogonal duality, and demonstrate that the package efficiently handles high-degree moments and quantum information metrics.
Quantum state tomography provides a foundation for characterizing state preparation and predicting measurement outcomes. We characterize the minimax expected trace norm risk of quantum state tomography over classes with polynomial spectral decay under randomized nonadaptive single copy measurements. Under an explicit sufficient block size condition for the two layer periodic Clifford ensemble, we establish statewise expected trace norm oracle inequalities for operator norm minimum distance and measurement weighted projected least squares, with bounds that depend on each state's actual spectral tail, while the estimators require no prior knowledge of the state's structural characteristics, including its rank, eigenbasis, or spectral structure. Over these polynomial spectral decay classes, both estimators attain the minimax expected trace norm rates up to constants. As a direct consequence, the same estimators attain the minimax rate over every fixed rank constrained class, while remaining adaptive to the unknown structural characteristics of the state. For admissible logarithmic block sizes, the periodic Clifford measurements attain these guarantees with logarithmic elementary gate depth. The main technical ingredient is a uniform bound on the fourth moment of projectors for the periodic Clifford ensemble. The bound is sharp in dimension and holds for arbitrary test states, including those entangled across blocks. Covariance concentration and positivity yield adaptive oracle bounds, while a shared rotation information argument on spectral tail packings establishes matching lower bounds. We also provide polynomial time reconstruction algorithms, numerical comparisons, and Lean formalizations of all labeled theoretical results.