On Stable Convex Sets Colloquium of the Pure Mathematics Research - PowerPoint PPT Presentation

On Stable Convex Sets Colloquium of the Pure Mathematics Research Centre Queen’s University Belfast, Northern Ireland, UK 17 November 2017 speaker Stephan Weis Université libre de Bruxelles, Belgium 1 / 39

Overview A convex set is stable if the midpoint map p x , y q ÞÑ 1 2 p x ` y q is open. Section 1 and 3 follow the chronological development of the theory of stable compact convex sets during the 1970’s as described by Papadopoulou, Jber. d. Dt. Math.-Verein (1982) 92. The theory includes work by Vesterstrøm, Lima, O’Brien, Clausing, and Papadopoulou, among others. Section 2 reports on a theory of generalized compactness ( µ -compactness) developed by Holevo, Shirokov, and Protasov in the first decade of the 21st century. Density matrices form a stable µ -compact convex set. Applications to the continuity of entanglement monotones and von Neumann entropy are mentioned. Sections 4 and 5 describe problems in finite dimensions related to stability of the set of density matrices: Continuity of inference, ground state problems, geometry of reduced density matrices, and continuity of correlation quantities. 2 / 39

Table of Contents 1. Stability of compact convex sets (4+8) 2. Stability of density matrices and applications (7) 3. The face function (1+2) 4. Continuity of inference (6) 5. Why is continuity of inference interesting? (6) 6. Conclusion (1) 3 / 39

The CE-property (“continuous envelope”) Definition 1. K , Y , A are subsets of a locally convex Hausdorff space; A is closed and bounded, C p A q is the set of bounded continuous real functions on A , and M ` 1 p A q the space of regular Borel probability measures on A (weak topology); if A is convex, then A p A q is the set of continuous affine real functions on A ; K is a compact convex set. 4 / 39

The CE-property (“continuous envelope”) Definition 1. K , Y , A are subsets of a locally convex Hausdorff space; A is closed and bounded, C p A q is the set of bounded continuous real functions on A , and M ` 1 p A q the space of regular Borel probability measures on A (weak topology); if A is convex, then A p A q is the set of continuous affine real functions on A ; K is a compact convex set. if A is convex, then the lower envelope of f P C p A q is ˇ ˇ f : A Ñ R , f p x q “ sup t g p x q : g ď f , g P A p A qu , ş the barycenter of µ P M ` 1 p K q is b p µ q “ K x d µ p x q 4 / 39

The CE-property (“continuous envelope”) Definition 1. K , Y , A are subsets of a locally convex Hausdorff space; A is closed and bounded, C p A q is the set of bounded continuous real functions on A , and M ` 1 p A q the space of regular Borel probability measures on A (weak topology); if A is convex, then A p A q is the set of continuous affine real functions on A ; K is a compact convex set. if A is convex, then the lower envelope of f P C p A q is ˇ ˇ f : A Ñ R , f p x q “ sup t g p x q : g ď f , g P A p A qu , ş the barycenter of µ P M ` 1 p K q is b p µ q “ K x d µ p x q Theorem 1. [Vesterstrøm, J. London Math. Soc. 2 (1973) 289 ] b : M ` 1 p K q Ñ K is open if and only if f P C p K q ñ ˇ f P C p K q . 4 / 39

On the proof of Theorem 1 Reminder. [Alfsen, Compact Convex Sets and Boundary Integrals, Berlin: Springer (1971) ] ˇ f p x q “ min t f p µ q : x “ b p µ q , µ P M ` 1 p K qu , f P C p K q M ` 1 p K q is w ˚ -compact, b : M ` 1 p K q Ñ K is a continuous, affine, and surjective map, C p K q – A p M ` 1 p K qq 5 / 39

On the proof of Theorem 1 Reminder. [Alfsen, Compact Convex Sets and Boundary Integrals, Berlin: Springer (1971) ] ˇ f p x q “ min t f p µ q : x “ b p µ q , µ P M ` 1 p K qu , f P C p K q M ` 1 p K q is w ˚ -compact, b : M ` 1 p K q Ñ K is a continuous, affine, and surjective map, C p K q – A p M ` 1 p K qq abstractly: let Y be a compact convex set, φ : Y Ñ K a continuous, affine, and surjective map, and f P A p Y q ; define f φ : K Ñ R , ˇ ˇ f φ p x q “ min t f p y q : x “ φ p y q , y P Y u 5 / 39

On the proof of Theorem 1 Reminder. [Alfsen, Compact Convex Sets and Boundary Integrals, Berlin: Springer (1971) ] ˇ f p x q “ min t f p µ q : x “ b p µ q , µ P M ` 1 p K qu , f P C p K q M ` 1 p K q is w ˚ -compact, b : M ` 1 p K q Ñ K is a continuous, affine, and surjective map, C p K q – A p M ` 1 p K qq abstractly: let Y be a compact convex set, φ : Y Ñ K a continuous, affine, and surjective map, and f P A p Y q ; define f φ : K Ñ R , ˇ ˇ f φ p x q “ min t f p y q : x “ φ p y q , y P Y u Theorem 2. [Vesterstrøm, ibid ] TFAE a) φ is open f φ P C p K q for all f P A p Y q f b “ ˇ b) ˇ ( ˇ f proves Thm. 1) 5 / 39

On the proof of Theorem 1 Reminder. [Alfsen, Compact Convex Sets and Boundary Integrals, Berlin: Springer (1971) ] ˇ f p x q “ min t f p µ q : x “ b p µ q , µ P M ` 1 p K qu , f P C p K q M ` 1 p K q is w ˚ -compact, b : M ` 1 p K q Ñ K is a continuous, affine, and surjective map, C p K q – A p M ` 1 p K qq abstractly: let Y be a compact convex set, φ : Y Ñ K a continuous, affine, and surjective map, and f P A p Y q ; define f φ : K Ñ R , ˇ ˇ f φ p x q “ min t f p y q : x “ φ p y q , y P Y u Theorem 2. [Vesterstrøm, ibid ] TFAE a) φ is open f φ P C p K q for all f P C p Y q c) ˇ Lima, Proc. London M. Soc. (’72) 5 / 39

Remark (continuity of inference maps) Definition 2. Let φ : Y Ñ K as before. Assume f P C p Y q has for all x P K a unique minimum on φ ´ 1 p x q and define Ψ p x q “ argmin t f p y q : y P φ ´ 1 p x qu . Ψ : K Ñ Y , 6 / 39

Remark (continuity of inference maps) Definition 2. Let φ : Y Ñ K as before. Assume f P C p Y q has for all x P K a unique minimum on φ ´ 1 p x q and define Ψ p x q “ argmin t f p y q : y P φ ´ 1 p x qu . Ψ : K Ñ Y , note: the inference map Ψ choses a point in each fiber of φ which is optimal in the sense of minimizing f , a ranking function; the optimal value is f p Ψ p x qq “ ˇ f φ p x q “ min t f p y q : y P φ ´ 1 p x qu 6 / 39

Remark (continuity of inference maps) Definition 2. Let φ : Y Ñ K as before. Assume f P C p Y q has for all x P K a unique minimum on φ ´ 1 p x q and define Ψ p x q “ argmin t f p y q : y P φ ´ 1 p x qu . Ψ : K Ñ Y , note: the inference map Ψ choses a point in each fiber of φ which is optimal in the sense of minimizing f , a ranking function; the optimal value is f p Ψ p x qq “ ˇ f φ p x q “ min t f p y q : y P φ ´ 1 p x qu Observation 1. [Continuity of inference] If f P C p Y q has a unique minimum in each fiber of φ , then φ : Y Ñ K open ù ñ Ψ : K Ñ Y continuous. 6 / 39

Remark (continuity of inference maps) Definition 2. Let φ : Y Ñ K as before. Assume f P C p Y q has for all x P K a unique minimum on φ ´ 1 p x q and define Ψ p x q “ argmin t f p y q : y P φ ´ 1 p x qu . Ψ : K Ñ Y , note: the inference map Ψ choses a point in each fiber of φ which is optimal in the sense of minimizing f , a ranking function; the optimal value is f p Ψ p x qq “ ˇ f φ p x q “ min t f p y q : y P φ ´ 1 p x qu Observation 1. [Continuity of inference] If f P C p Y q has a unique minimum in each fiber of φ , then φ : Y Ñ K open ù ñ Ψ : K Ñ Y continuous. Proof. use Thm. 2 c) and compactness of Y 6 / 39

Stability of compact convex sets Def. 3. K is stable if K ˆ K Ñ K , p x , y q ÞÑ x ` y is open. 2 note: relative topologies are used on K and K ˆ K 7 / 39

Stability of compact convex sets Def. 3. K is stable if K ˆ K Ñ K , p x , y q ÞÑ x ` y is open. 2 note: relative topologies are used on K and K ˆ K Theorem 3. [O’Brien, Math. Ann. 223 (1976) 207 ] TFAE a) the interior of every convex subset of K is convex b) the convex hull of every open subset of K is open c) K is stable d) @ λ P r 0 , 1 s : K ˆ K Ñ K , p x , y q ÞÑ p 1 ´ λ q x ` λ y is open e) K ˆ K ˆ r 0 , 1 s Ñ K , p x , y , λ q ÞÑ p 1 ´ λ q x ` λ y is open f) the barycenter map b : M ` 1 p K q Ñ K is open a)–e) are equivalent for general convex sets (Clausing and Papadopoulou, Math. Ann. 231 (’78) 193 ) 7 / 39

Standard example of a non-stable convex set let K be the convex hull of the union of the circle tp 0 , y , z q : y 2 ` z 2 “ 1 u and singletons p˘ 1 , 0 , 1 q 8 / 39

Example 1 a) Failure of the CE-property consider f P C p K q f p x , y , z q “ 1 ´ | x | f p‚q “ 0, f p‚q “ 1 9 / 39

Example 1 a) Failure of the CE-property ˇ f p a q “ f p a q for all ex- treme points a of K ˇ f p‚q “ 0, ˇ f p‚q “ 1 ñ ˇ ù f is discontinuous 9 / 39

Example 1 b) Non-convex interior of a convex set consider the cylinder C “ tp x , y , z q : y 2 ` p z ´ 1 2 q 2 ď p 1 2 q 2 u which extends in x -direction, and the convex set K X C (blue) 10 / 39

Example 1 b) Non-convex interior of a convex set the boundary of K X C is the surface tp x , y , z q P K : | x | ď 1 2 , y 2 ` p z ´ 1 2 q 2 “ p 1 2 q 2 u , the interior of K X C is depicted blue region 10 / 39

Example 1 b) Non-convex interior of a convex set the red segment ends on both sides in the interior of K X C (blue), but crosses the boundary of K X C ù ñ the interior of K X C is not convex 10 / 39

Example 1 c) Non-open convex hull of an open set consider the open sets O ˘ “ tp x , y , z q P K : ˘ x ą 1 2 u and their union O “ O ´ Y O ` (blue) 11 / 39

Example 1 c) Non-open convex hull of an open set conv p O q is the union of the interior of K X C (blue) and the red segment ù ñ conv p O q is not open 11 / 39