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Abstract This paper addresses the overdetermined problem of inverting the n -dimensional cone (or Compton) transform that integrates a function over conical surfaces in R n . The study of the cone transform originates from Compton camera imaging, a nuclear imaging method for the passive detection of gamma-ray sources. We present a new identity relating the n -dimensional cone and Radon transforms through spherical convolutions with arbitrary weight functions. This relationship, which generalizes a previously obtained identity, leads to various inversion formulas in n -dimensions under a mild assumption on the geometry of detectors. We present two such formulas along with the results of their numerical implementation using synthetic phantoms. Compared to our previously discovered inversion techniques, the new formulas are more stable and simpler to implement numerically.
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Monotonicity Under Local Operations: Linear Entropic Formulas
All correlation measures, classical and quantum, must be monotonic under local operations. In this paper, we characterize monotonic formulas that are linear combinations of the von Neumann entropies associated with the quantum state of a physical system that has n parts. We show that these formulas form a polyhedral convex cone, which we call the monotonicity cone, and enumerate its facets. We illustrate its structure and prove that it is equivalent to the cone of monotonic formulas implied by strong subadditivity. We explicitly compute its extremal rays for n ≤ 5. We also consider the symmetric monotonicity cone, in which the formulas are required to be invariant under subsystem permutations. We describe this cone fully for all n. We also show that these results hold when states and operations are constrained to be classical.
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- Award ID(s):
- 1652560
- PAR ID:
- 10169433
- Date Published:
- Journal Name:
- IEEE Transactions on Information Theory
- ISSN:
- 0018-9448
- Page Range / eLocation ID:
- 1 to 1
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TIT.2020.2986728
