Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Factorizable completely positive maps and the Connes embedding problem Joint work with Uffe Haagerup, and part in collaboration with M. B. Ruskai, Tufts University Magdalena Musat University of Copenhagen Banach Algebras and Applications Gothenburg, July 31, 2013
Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Outline Factorizable maps and the Asymptotic Quantum Birkhoff 1 Conjecture Extreme points and factorizability 2 Remarks on the semigroup case 3 Asymptotic properties of factorizable maps and the Connes 4 embedding problem Holevo–Werner channels 5
� � � Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Definition (Anantharaman-Delaroche, 2005) ◮ Let ( M , φ ) , ( N , ψ ) be vN algebras with n.f. tracial states. A trace-preserving UCP map T : M → N is called factorizable if ∃ vN algebra P with n.f. tracial state χ and injective trace-preserving unital ∗ -homs α : M → P , β : N → P s.t. T = β ∗ ◦ α . T M N � � � � � � � � � � α � � β ∗ = β − 1 ◦ E β ( N ) � � � P ◮ Markov maps between abelian vN algebras are factorizable. • The set of factorizable maps F ( M , N ) is convex and closed under composition and taking adjoints. In particular, for n ≥ 2, conv(Aut( M n ( C ))) ⊆ F ( M n ( C )) ⊆ UCPT n .
� � � Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Definition (Anantharaman-Delaroche, 2005) ◮ Let ( M , φ ) , ( N , ψ ) be vN algebras with n.f. tracial states. A trace-preserving UCP map T : M → N is called factorizable if ∃ vN algebra P with n.f. tracial state χ and injective trace-preserving unital ∗ -homs α : M → P , β : N → P s.t. T = β ∗ ◦ α . T M N � � � � � � � � � � α � � β ∗ = β − 1 ◦ E β ( N ) � � � P ◮ Markov maps between abelian vN algebras are factorizable. • The set of factorizable maps F ( M , N ) is convex and closed under composition and taking adjoints. In particular, for n ≥ 2, conv(Aut( M n ( C ))) ⊆ F ( M n ( C )) ⊆ UCPT n .
Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Definition (Anantharaman-Delaroche, 2005) ◮ Let ( M , φ ) , ( N , ψ ) be vN algebras with n.f. tracial states. A trace-preserving UCP map T : M → N is called factorizable if ∃ vN algebra P with n.f. tracial state χ and injective trace-preserving unital ∗ -homs α : M → P , β : N → P s.t. T = β ∗ ◦ α . Theorem (Anantharaman-Delaroche, 2005) T : M → M factorizable with T = T ∗ = ⇒ T 2 has a Rota dilation. Problem (Anantharaman-Delaroche) Is every UCP trace-preserving map factorizable? ◮ Ricard (2008): Schur multipliers associated to positive-definite real matrices having diagonal entries equal to 1 are factorizable.
Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Definition (Anantharaman-Delaroche, 2005) ◮ Let ( M , φ ) , ( N , ψ ) be vN algebras with n.f. tracial states. A trace-preserving UCP map T : M → N is called factorizable if ∃ vN algebra P with n.f. tracial state χ and injective trace-preserving unital ∗ -homs α : M → P , β : N → P s.t. T = β ∗ ◦ α . Theorem (Anantharaman-Delaroche, 2005) T : M → M factorizable with T = T ∗ = ⇒ T 2 has a Rota dilation. Problem (Anantharaman-Delaroche) Is every UCP trace-preserving map factorizable? ◮ Ricard (2008): Schur multipliers associated to positive-definite real matrices having diagonal entries equal to 1 are factorizable.
Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Connections to quantum information ◮ Gregoratti-Werner (2003): Channels which are convex combinations of unitarily implemented ones allow for complete error correction, given suitable feedback of classical information from the environment. ummerer (1983): UCPT 2 = conv(Aut( M 2 ( C ))). ◮ K¨ UCPT n � conv(Aut( M n ( C ))) For n ≥ 3: ◮ ummerer (1986): n = 3, K¨ ummerer-Maasen (1987): n ≥ 4, K¨ Landau-Streater (1993): another counterexample for n = 3. Conjecture (J. A. Smolin, F. Verstraete, A. Winter, 2005) Let T ∈ UCPT n , n ≥ 3 . Then T satisfies the following asymptotic quantum Birkhoff property ( AQBP ): � k k � � � M n ( C ))) k →∞ d cb lim T , conv ( Aut ( = 0 . i =1 i =1
Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Connections to quantum information ◮ Gregoratti-Werner (2003): Channels which are convex combinations of unitarily implemented ones allow for complete error correction, given suitable feedback of classical information from the environment. ummerer (1983): UCPT 2 = conv(Aut( M 2 ( C ))). ◮ K¨ UCPT n � conv(Aut( M n ( C ))) For n ≥ 3: ◮ ummerer (1986): n = 3, K¨ ummerer-Maasen (1987): n ≥ 4, K¨ Landau-Streater (1993): another counterexample for n = 3. Conjecture (J. A. Smolin, F. Verstraete, A. Winter, 2005) Let T ∈ UCPT n , n ≥ 3 . Then T satisfies the following asymptotic quantum Birkhoff property ( AQBP ): � k k � � � M n ( C ))) k →∞ d cb lim T , conv ( Aut ( = 0 . i =1 i =1
Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Connections to quantum information ◮ Gregoratti-Werner (2003): Channels which are convex combinations of unitarily implemented ones allow for complete error correction, given suitable feedback of classical information from the environment. ummerer (1983): UCPT 2 = conv(Aut( M 2 ( C ))). ◮ K¨ UCPT n � conv(Aut( M n ( C ))) For n ≥ 3: ◮ ummerer (1986): n = 3, K¨ ummerer-Maasen (1987): n ≥ 4, K¨ Landau-Streater (1993): another counterexample for n = 3. Conjecture (J. A. Smolin, F. Verstraete, A. Winter, 2005) Let T ∈ UCPT n , n ≥ 3 . Then T satisfies the following asymptotic quantum Birkhoff property ( AQBP ): � k k � � � M n ( C ))) k →∞ d cb lim T , conv ( Aut ( = 0 . i =1 i =1
Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Connections to quantum information ◮ Gregoratti-Werner (2003): Channels which are convex combinations of unitarily implemented ones allow for complete error correction, given suitable feedback of classical information from the environment. ummerer (1983): UCPT 2 = conv(Aut( M 2 ( C ))). ◮ K¨ UCPT n � conv(Aut( M n ( C ))) For n ≥ 3: ◮ ummerer (1986): n = 3, K¨ ummerer-Maasen (1987): n ≥ 4, K¨ Landau-Streater (1993): another counterexample for n = 3. Conjecture (J. A. Smolin, F. Verstraete, A. Winter, 2005) Let T ∈ UCPT n , n ≥ 3 . Then T satisfies the following asymptotic quantum Birkhoff property ( AQBP ): � k k � � � M n ( C ))) k →∞ d cb lim T , conv ( Aut ( = 0 . i =1 i =1
Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Non-factorizable maps and the AQBP ∈ conv(Aut( M 3 ( C ))), Mendl-Wolf (2009): ∃ T ∈ UCPT 3 s.t. T / but � M 3 ( C ) ⊗ M 3 ( C )) � T ⊗ T ∈ conv(Aut . Theorem (Haagerup-M, 2011) Let T ∈ UCPT n , where n ≥ 3 . Then, for all k ≥ 1 , � k � k �� � � M n ( C ) ≥ d cb ( T , F ( M n ( C ))) . d cb T , F i =1 i =1 ◮ If T is not factorizable, then d cb ( T , F ( M n ( C ))) > 0, as F ( M n ( C )) is norm-closed. Since conv(Aut( M n ( C ))) ⊂ F ( M n ( C )) , any non-factorizable unital channel T fails the AQBP.
Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Non-factorizable maps and the AQBP ∈ conv(Aut( M 3 ( C ))), Mendl-Wolf (2009): ∃ T ∈ UCPT 3 s.t. T / but � M 3 ( C ) ⊗ M 3 ( C )) � T ⊗ T ∈ conv(Aut . Theorem (Haagerup-M, 2011) Let T ∈ UCPT n , where n ≥ 3 . Then, for all k ≥ 1 , � k � k �� � � M n ( C ) ≥ d cb ( T , F ( M n ( C ))) . d cb T , F i =1 i =1 ◮ If T is not factorizable, then d cb ( T , F ( M n ( C ))) > 0, as F ( M n ( C )) is norm-closed. Since conv(Aut( M n ( C ))) ⊂ F ( M n ( C )) , any non-factorizable unital channel T fails the AQBP.
Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels Non-factorizable maps and the AQBP ∈ conv(Aut( M 3 ( C ))), Mendl-Wolf (2009): ∃ T ∈ UCPT 3 s.t. T / but � M 3 ( C ) ⊗ M 3 ( C )) � T ⊗ T ∈ conv(Aut . Theorem (Haagerup-M, 2011) Let T ∈ UCPT n , where n ≥ 3 . Then, for all k ≥ 1 , � k � k �� � � M n ( C ) ≥ d cb ( T , F ( M n ( C ))) . d cb T , F i =1 i =1 ◮ If T is not factorizable, then d cb ( T , F ( M n ( C ))) > 0, as F ( M n ( C )) is norm-closed. Since conv(Aut( M n ( C ))) ⊂ F ( M n ( C )) , any non-factorizable unital channel T fails the AQBP.
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