Quantum Conical Designs

Abstract

Complex projective t-designs, particularly SICs and full sets of MUBs, play an important role in quantum information. We introduce a generalization which we call conical t-designs. They include arbitrary rank symmetric informationally complete measurements (SIMs) and full sets of arbitrary rank mutually unbiased measurements (MUMs). They are deeply implicated in the description of entanglement (as we show in a subsequent paper). Viewed in one way a conical 2-design is a symmetric decomposition of a separable Werner state (up to a normalization factor). Viewed in another way it is a certain kind of polytope in the Bloch body. In the Bloch body picture SIMs and full sets of MUMs form highly symmetric polytopes (a single regular simplex in the one case; the convex hull of a set of orthogonal regular simplices in the other). We give the necessary and sufficient conditions for an arbitrary polytope to be what we call a homogeneous conical 2-design. This suggests a way to search for new kinds of projective 2-design.

12 pages. v3: additional references; title changed to match published version

References in corpus (15)

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• A Decomposition of Separable Werner States
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Cited by in corpus (21)

• The SIC Question: History and State of Play
• Linking Entanglement Detection and State Tomography via Quantum 2-Designs
• Universally Fisher-Symmetric Informationally Complete Measurements
• Tight Frames, Hadamard Matrices and Zauner's Conjecture
• Iso-entangled mutually unbiased bases, symmetric quantum measurements and mixed-state designs
• Sporadic SICs and the Normed Division Algebras
• Designing Quantum Information Processing via Structural Physical Approximation
• What are the minimal conditions required to define a SIC POVM?
• Entanglement properties of multipartite informationally complete quantum measurements
• Entanglement and Designs
• Geometric and Information-Theoretic Properties of the Hoggar Lines
• Equiangular tight frames and unistochastic matrices
• Uncertainty Relations in the Presence of Quantum Memory for Mutually Unbiased Measurements
• Communication capacity of mixed quantum t designs
• Uncertainty relations for quantum measurements from generalized equiangular tight frames
• Generalization of Pauli channels through mutually unbiased measurements
• User-friendly confidence regions for quantum state tomography
• The Decompositions of Werner and Isotropic States
• Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories
• All classes of informationally complete symmetric measurements in finite dimensions
• Mutually unbiased measurements with arbitrary purity