Results for “positive semidefinite matrix completion”

DOE OSTI2022

A non-commutative Bayes' theorem

Using a diagrammatic reformulation of Bayes' theorem, we provide a necessary and sufficient condition for the existence of Bayesian inference in the setting of finite-dimensional C* -algebras. In other words, we prove an analogue of Bayes' theorem in the joint classical and quantum context. Our analogue is justified by recent advances in categorical probability theory, which have provided an abstract formulation of the classical Bayes' theorem. In the process, we further develop non-commutative almost everywhere equivalence and illustrate its important role in non-commutative Bayesian inversion. The construction of such Bayesian inverses, when they exist, involves solving a positive semidefinite matrix completion problem for the Choi matrix. This gives a solution to the open problem of constructing Bayesian inversion for completely positive unital maps acting on density matrices that do not have full support. In conclusion, we illustrate how the procedure works for several examples relevant to quantum information theory.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Pareto-optimal target definition for multi-axis random vibration testing

In random vibration testing with multiple control channels, existing control laws require specification of a complete spectral density matrix at each control frequency. Spectral density matrices include autospectral densities on the diagonal and cross-spectral densities on the off-diagonal. In practice, the off-diagonal terms are often unknown, and recent vibration testing research has focused on fixing the diagonal and specifying the off-diagonal to minimize the required control energy, subject to a constraint that the target matrix is positive semidefinite. This paper shows that, even with a fixed diagonal, off-diagonal terms strongly affect control residuals. This overlooked effect occurs in both square and rectangular systems. By jointly considering input energy and control residuals, open-loop inputs are derived directly from the diagonal without specifying the off-diagonal terms. Vibration targets that can be used in closed-loop control are then derived using the optimal inputs, with positive semidefinite constraints applied during the derivation. The result is a set of Pareto-optimal control solutions. For each solution in the set, any other possible solution produces greater control error, greater input energy, or both. A balanced solution is selected automatically, though others can be chosen based on test needs. Simulations and experiments show that the proposed method outperforms state-of-the-art energy-minimizing approaches, achieving significant reductions in both control error and input energy.

Autospectral density↗