While many bounds have been proved for partial trace inequalities over the last decades for a large variety of quantities, recent problems in quantum information theory demand sharper bounds. In this work, we study optimal bounds for partial trace quantities in terms of the spectrum; equivalently, we determine the best bounds attainable over unitary orbits of matrices. We solve this question for Schur-convex functionals acting on a single partial trace in terms of eigenvalues for self-adjoint matrices and then we extend these results to singular values of general matrices. We subsequently extend the study to Schur-convex functionals that act on several partial traces simultaneously and present sufficient conditions for sharpness. In cases where closed-form maximizers cannot be identified, we present quadratic programs that yield new computable upper bounds for any Schur-convex functional. We additionally present examples demonstrating improvements over previously known bounds. Finally, we conclude with the study of optimal bounds for an $n$-qubit system and its subsystems of dimension $2$.
Comments
Revised version. Corrected the formulation and proof of Theorem 9.1; improved the sufficient conditions in Section 5; reorganized the presentation and improved clarity throughout
While many bounds have been proved for partial trace inequalities over the last decades for a large variety of quantities, recent problems in quantum information theory demand sharper bounds. In this work, we study optimal bounds for partial trace quantities in terms of the spectrum; equivalently, we determine the best bounds attainable over unitary orbits of matrices. We solve this question for Schur-convex functionals acting on a single partial trace in terms of eigenvalues for self-adjoint matrices and then we extend these results to singular values of general matrices. We subsequently extend the study to Schur-convex functionals that act on several partial traces simultaneously and present sufficient conditions for sharpness. In cases where closed-form maximizers cannot be identified, we present quadratic programs that yield new computable upper bounds for any Schur-convex functional. We additionally present examples demonstrating improvements over previously known bounds. Finally, we conclude with the study of optimal bounds for an n-qubit system and its subsystems of dimension 2.
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.
Convex optimization problems arise naturally in quantum information theory, often in terms of minimizing a convex function over a convex subset of the space of hermitian matrices. In most cases, finding exact solutions to these problems is usually impossible. As inspired by earlier investigations into the relative entropy of entanglement (REE) (Miranowicz and Ishizaka 2008 Phys. Rev. A 78 032310 ), we introduce a general method to solve the converse problem rather than find explicit solutions. That is, given a matrix in a convex set, we determine a family of convex functions that are minimized at this point. This method allows us find explicit formulae for the REE and the Rains bound, two well-known upper bounds on the distillable entanglement, and yields interesting information about these quantities, such as the fact that they coincide in the case where at least one subsystem of a multipartite state is a qubit.
We (the author and the AI systems that did the heavy lifting) show that the quantum query complexity of minimizing a convex function over a convex subset of Rn with evaluation and membership queries is Ω~(n), nearly matching the best known upper bound. In particular, we show this even for quadratic minimization, which is equivalent to inverting an n×n real matrix using matrix-vector queries. We also show linear or nearly linear lower bounds on the quantum query complexity of computing the trace, the sign of the determinant, and the magnitude of the determinant of a real matrix in the matrix-vector query model. We use a novel quantum lower bound technique, the determinantal witness method, based on identifying a witness whose Fourier transform vanishes on low-rank matrices and that correlates well with the function being computed.
Maximal quantum f-divergences, defined via the commutant Radon--Nikodym derivative, form a fundamental class of distinguishability measures for quantum states associated with operator convex functions. In this paper, we study the optimization of maximal quantum f-divergences along unitary orbits of two quantum states. For any operator convex function f:(0,+∞)→R, we determine the exact minimum and maximum of U⟼Sf(ρ∥U∗σU) over the unitary group, and derive explicit spectral formulas for these extremal values together with complete characterizations of the unitaries that attain them. Our approach combines the integral representation of operator convex functions with majorization theory and a unitary-orbit variational method. A key step is to show that any extremizer must commute with the reference state, which reduces the noncommutative optimization problem to a spectral permutation problem. As a consequence, the minimum is achieved by pairing the decreasing eigenvalues of ρ and σ, while the maximum corresponds to pairing the decreasing eigenvalues of ρ with the increasing eigenvalues of σ. Hence, the range of the maximal quantum f-divergence along the unitary orbit is exactly the closed interval determined by these two extremal configurations. Finally, we compare our results with recent unitary-orbit optimization results for quantum f-divergences defined via the quantum hockey-stick divergence, highlighting fundamental structural differences between the two frameworks. Our findings extend earlier extremal results for Umegaki, Rényi, and related quantum divergences, and clarify the distinct operator-theoretic nature of maximal quantum f-divergences.
Motivated by the equivalent partial-trace formulations of Werner-state distillability [P. Costa Rico, Lett. Math. Phys. 115, 47 (2025); S.-Y. Qi et al., Phys. Rev. A 110, 012406 (2024)], we prove a bipartite partial-trace inequality for every matrix of rank at most two. As applications, we prove the two-copy undistillability of NPT Werner states in arbitrary local dimension, thereby resolving this open problem highlighted in [P. Horodecki et al., PRX Quantum 3, 010101 (2022)]. We further prove a two-parameter extension of the matrix inequality and show that two individually one-copy-undistillable NPT Werner states cannot activate each other's one-copy distillability. We also resolve the singular-value maximization problem associated with the two-ququart case.
We describe recent work of Kim [J. Math. Phys. 53, 122204 (2012)] to show that operator convex functions associated with quasi-entropies can be used to prove a large class of new matrix inequalities in the tri-partite and bi-partite setting by taking a judiciously chosen partial trace over all but one of the spaces. We give some additional examples in both settings. Furthermore, we observe that the equality conditions for all the new inequalities are essentially the same as those for strong subadditivity.