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Convexity and Positivity in Quantum Information: Part II

Andrew Landahl, University of New Mexico

(Session 101 : Thursday from 16:00-18:00)

Abstract. Convex sets and convex cones occur frequently in quantum information theory. For example, normalized density matrices, completely positive maps, separable states, and POVMs all form convex sets. Finding an optimal quantum information processing protocol can often be cast as minimizing a linear function over a compact convex set, a problem for which much is known. This tutorial will cover some of the basic theory of convex sets and optimization over them, and applications to quantum information processing.

The second half of the tutorial will be given by Andrew Landahl and will focus on some concrete applications of semidefinite programming in quantum information:

Outline of Tutorial:

I. Optimal quantum error correction
• Convexity of quantum process distance measures.
• Optimal optimal encoding and recovery as convex optimization problems.
• Semidefinite program (SDP) formulation for some distance measures.
• Comparison to stabilizer coding.
II. Optimal quantum protocols for weak coin flipping
• Definition of weak coin flipping (WCF).
• Maximum cheating bias in a quantum WCF protocol as an SDP.
• Minimum bias over all quantum WCF protocols as an SDP.
• Why incorporating cheat detection makes problem nonconvex.
III. Optimal quantum state discrimination
• Definition of quantum state discrimination problems.
• Optimal minimum-error state discrimination measurement as an SDP.
• Optimal unambiguous state discrimination measurement as an SDP.
• Optimal "hybrid" state discrimination measurement as an SDP.

Andrew Landahl,

(Session 12 : Sunday from )

Abstract.