Several open problems in quantum information theory can be formulated as equipartition problems for positive operators, asking for a decomposition into bounded-rank positive parts under uniform constraints. The existence problems for SIC-POVMs and MUBs are of this type, as is the Ruskai-Audenaert conjecture. We first show that certain problems of this form can be attacked using equivariant cohomology, and then present new results on the Ruskai-Audenaert conjecture. In its weak form, this conjecture asserts that every quantum channel admits a convex decomposition into a minimal number of generalized extreme points; in its strong form, one with equal weights. We prove the strong conjecture in all dimensions for a set of channels of nonzero measure, including all cq- and qc-channels, as well as for all channels with qubit inputs. We also prove the weak conjecture for all qutrit channels, along with further results on convex decompositions of quantum channels.
We develop Delsarte theory for quantum association schemes, allowing both composition and the quantum Schur product to be noncommutative. A MacWilliams identity and positivity of the inner and dual distributions yield conic upper bounds for codes and lower bounds for designs in terms of effective size, which extends su...
We first solve in ZFC a problem that the first author and Weaver posed ten years ago: a state on a von Neumann algebra is countably additive on orthogonal projections if and only if it is sequentially weak* continuous, equivalently, sequentially normal. Such states may be considered the quantum (countably additive) pro...
We establish exact thresholds for the failure of simultaneous quantum typicality: three parties for mixed states and four for pure states. For the four-qubit Higuchi--Sudbery state, every sequence satisfying the expected purity bounds on three overlapping marginals, with sufficiently small fixed positive entropy slack,...
We give an exact translation between fat-point interpolation in complex projective space and a dark-state problem for fixed-particle-number bosons. Under the unitary Fischer--Bargmann realization, homogeneous degree becomes particle number, the Veronese lift becomes a bosonic symmetric product state, and order-$m$ vani...
We prove a generalized quantum Stein lemma for parallel discrimination of an arbitrary finite-dimensional channel against families of alternative channels. The alternatives are compact and convex, closed under tensor products, and contain a faithful replacer. At every fixed type-I error tolerance, the optimal type-II e...
Minbo Gao, Zheng-Feng Ji, Cheng-Hua Liu· 1 citation
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