M. Christandl conjectured that the composition of any trace preserving PPT map with itself is entanglement breaking. We prove that Christandl's conjecture holds asymptotically by showing that the distance between the iterates of any unital or trace preserving PPT map and the set of entanglement breaking maps tends to zero. Finally, for every graph we define a one-parameter family of maps on matrices and determine the least value of the parameter such that the map is variously, positive, completely positive, PPT and entanglement breaking in terms of properties of the graph. Our estimates are sharp enough to conclude that Christandl's conjecture holds for these families.
We establish the dual equivalence of the category of (potentially non-unital) operator systems and the category of pointed compact nc (noncommutative) convex sets, extending a result of Davidson and the first author. We then apply this dual equivalence to establish a number of results about operator systems, some of which are new even in the unital setting.For example, we show that the maximal and minimal C*-covers of an operator system can be realized in terms of the C*-algebra of continuous nc functions on its nc quasistate space, clarifying recent results of Connes and van Suijlekom. We also characterize "C*-simple" operator systems, i.e. operator systems with simple minimal C*-cover, in terms of their nc quasistate spaces.We develop a theory of quotients of operator systems that extends the theory of quotients of unital operator algebras. In addition, we extend results of the first author and Shamovich relating to nc Choquet simplices. We show that an operator system is a C*-algebra if and only if its nc quasistate space is an nc Bauer simplex with zero as an extreme point, and we show that a second countable locally compact group has Kazhdan's property (T) if and only if for every action of the group on a C*-algebra, the set of invariant quasistates is the quasistate space of a C*-algebra. ContentsM. KENNEDY, S.J. KIM, AND N. MANOR 5. Pointed noncommutative functions 23 6. Minimal and maximal C*-covers 24 7. Characterization of unital operator systems 29 8. Quotients of operator systems 31 9. Noncommutative faces 35 10. C*-simplicity 37 11. Characterization of C*-algebras 39 12. Stable equivalence 42 13. Dynamics and Kazhdan's property (T) 43 References 47
We establish the dual equivalence of the category of generalized (i.e., potentially nonunital) operator systems and the category of pointed compact noncommutative (nc) convex sets, extending a result of Davidson and the 1st author. We then apply this dual equivalence to establish a number of results about generalized operator systems, some of which are new even in the unital setting. For example, we show that the maximal and minimal C*-covers of a generalized operator system can be realized in terms of theC*-algebra of continuous nc functions on its nc quasistate space, clarifying recent results of Connes and van Suijlekom. We also characterize “C*-simple” generalized operator systems, that is, generalized operator systems with a simple minimal C*-cover, in terms of their nc quasistate spaces. We develop a theory of quotients of generalized operator systems that extends the theory of quotients of unital operator systems. In addition, we extend results of the 1st author and Shamovich relating to nc Choquet simplices. We show that a generalized operator system is a C*-algebra if and only if its nc quasistate space is an nc Bauer simplex with zero as an extreme point, and we show that a second countable locally compact group has Kazhdan’s property (T) if and only if for every action of the group on a C*-algebra, the set of invariant quasistates is the quasistate space of a C*-algebra.
M. Christandl conjectured that the composition of any trace preserving PPT map with itself is entanglement breaking. We prove that Christandl's conjecture holds asymptotically by showing that the distance between the iterates of any unital or trace preserving PPT map and the set of entanglement breaking maps tends to zero. Finally, for every graph we define a one-parameter family of maps on matrices and determine the least value of the parameter such that the map is variously, positive, completely positive, PPT and entanglement breaking in terms of properties of the graph. Our estimates are sharp enough to conclude that Christandl's conjecture holds for these families.
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