Extreme Points of Unital Quantum Channels Mary Beth Ruskai - PowerPoint PPT Presentation

Extreme Points of Unital Quantum Channels Mary Beth Ruskai University of Vermont ruskai@member.ams.org joint work with U. Haagerup and M. Musat QMATH13 October, 2016 1 M. B. Ruskai Extreme Points of UCPT maps

Overview • Review: Stinespring, extreme points conds, etc. • Family of factorizable extreme UCPT maps extreme mixed states with max mixed quant marginals • Extreme points of CPT and UCP with Choi-rank d Kraus ops are partial isometries and generalization • Example for d = 2 ν + 1 odd • Universal example Reformulate linear independence as eigenvalue problem Associate eigenvectors (lin dep) with irreps of S n 2 M. B. Ruskai Extreme Points of UCPT maps

Complete positivity Def: Φ : M d A �→ M d B is completely positive (CP) if Φ ⊗ I d E preserves positivity ∀ d E . Suffices to consider d E = min { d A , d B } 3 M. B. Ruskai Extreme Points of UCPT maps

Complete positivity Def: Φ : M d A �→ M d B is completely positive (CP) if Φ ⊗ I d E preserves positivity ∀ d E . Suffices to consider d E = min { d A , d B } � � Φ is CP ⇔ J Φ = � Thm: (Choi) jk | e j �� e k | ⊗ Φ | e j �� e k | ≥ 0 Quantum Channel: Φ is CP and trace-preserving (CPT) TP means Tr Φ( A ) = Tr A ∀ A ∈ M d A Φ UCP if unital, i.e., Φ( I d A ) = I d B and CP 3 M. B. Ruskai Extreme Points of UCPT maps

Complete positivity Def: Φ : M d A �→ M d B is completely positive (CP) if Φ ⊗ I d E preserves positivity ∀ d E . Suffices to consider d E = min { d A , d B } � � Φ is CP ⇔ J Φ = � Thm: (Choi) jk | e j �� e k | ⊗ Φ | e j �� e k | ≥ 0 Quantum Channel: Φ is CP and trace-preserving (CPT) TP means Tr Φ( A ) = Tr A ∀ A ∈ M d A Φ UCP if unital, i.e., Φ( I d A ) = I d B and CP Φ : M d A �→ M d B is TP ⇔ � Φ : M d B �→ M d A is unital Φ adjoint wrt Hilb-Schmidt inner prod. Tr [ � � Φ( A )] ∗ B = Tr A ∗ Φ( B ) 3 M. B. Ruskai Extreme Points of UCPT maps

Choi condition for extremeality Φ( A ) = � k F k AF ∗ Choi-Kraus CP F k not unique but k 4 M. B. Ruskai Extreme Points of UCPT maps

Choi condition for extremeality Φ( A ) = � k F k AF ∗ Choi-Kraus CP F k not unique but k Choi obtained F k by “stacking” e-vec of J Φ with non-zero evals Thm: (Choi) Φ is extreme in set of CP maps with � k F ∗ k F k = X ⇔ { F ∗ j F k } is linearly independent. ⇒ Φ = � k F k AF ∗ k extreme CPT map ⇔ { F ∗ j F k } is lin indep. ⇒ Φ = � k F k AF ∗ k extreme UCP map ⇔ { F j F ∗ k } is lin indep. Cor: extreme CPT ⇒ d E ≤ d B extreme UCP ⇒ d E ≤ d A 4 M. B. Ruskai Extreme Points of UCPT maps

Factorizable maps on matrix algebras � � ρ ⊗ 1 U ∗ Factorizable: ∃ unitary U such that Φ( ρ ) = Tr E U d I d 5 M. B. Ruskai Extreme Points of UCPT maps

Factorizable maps on matrix algebras � � U ∗ ρ ⊗ | φ �� φ | Recall Stinespring Φ( ρ ) = Tr E U � � ρ ⊗ 1 U ∗ Factorizable: ∃ unitary U such that Φ( ρ ) = Tr E U d I d � � Φ( ρ ) = � 1 U ∗ ⇒ ρ ⊗ | e k �� e k | d Tr E U k Factorizable ⇒ Not Extreme Extreme ⇒ Not Factorizable 5 M. B. Ruskai Extreme Points of UCPT maps

Factorizable maps on matrix algebras � � U ∗ ρ ⊗ | φ �� φ | Recall Stinespring Φ( ρ ) = Tr E U � � ρ ⊗ 1 U ∗ Factorizable: ∃ unitary U such that Φ( ρ ) = Tr E U d I d � � Φ( ρ ) = � 1 U ∗ ⇒ ρ ⊗ | e k �� e k | d Tr E U k Factorizable ⇒ Not Extreme Extreme ⇒ Not Factorizable Question: “small environment” For Φ : M d �→ M d can one make environment d E ≤ d if replace � � U ∗ | φ �� φ | by DM γ s. t. Φ( ρ ) = Tr E U ρ ⊗ γ More general: arbitrary γ rather than max mixed 1 d I d More restrictive: ρ ∈ M d rather than higher dim environment 5 M. B. Ruskai Extreme Points of UCPT maps

Factorizable maps on matrix algebras � � U ∗ ρ ⊗ | φ �� φ | Recall Stinespring Φ( ρ ) = Tr E U � � ρ ⊗ 1 U ∗ Factorizable: ∃ unitary U such that Φ( ρ ) = Tr E U d I d � � Φ( ρ ) = � 1 U ∗ ⇒ ρ ⊗ | e k �� e k | d Tr E U k Factorizable ⇒ Not Extreme Extreme ⇒ Not Factorizable Question: “small environment” two groups showed false ≈ 1999 For Φ : M d �→ M d can one make environment d E ≤ d if replace � � U ∗ | φ �� φ | by DM γ s. t. Φ( ρ ) = Tr E U ρ ⊗ γ More general: arbitrary γ rather than max mixed 1 d I d More restrictive: ρ ∈ M d rather than higher dim environment 5 M. B. Ruskai Extreme Points of UCPT maps

UCPT maps Question: Are there UCPT maps Φ : M d �→ M d not extreme in either UCP or CPT maps, but are extreme in UCPT maps. Thm: (Landau-Streater) Φ : M d �→ M d is extreme in set of UCPT j } linearly independent Φ( ρ ) = � maps ⇔ { A ∗ j A k ⊕ A k A ∗ k A k ρ A ∗ k 6 M. B. Ruskai Extreme Points of UCPT maps

UCPT maps Question: Are there UCPT maps Φ : M d �→ M d not extreme in either UCP or CPT maps, but are extreme in UCPT maps. Thm: (Landau-Streater) Φ : M d �→ M d is extreme in set of UCPT j } linearly independent Φ( ρ ) = � maps ⇔ { A ∗ j A k ⊕ A k A ∗ k A k ρ A ∗ k By C-J isomorphism convex set of UCPT maps isomorphic to bipartite states with maximally mixed quantum marginals Equiv: Are there extreme points in convex set of bipartite states ρ AB with ρ A = ρ B = 1 d I d which are not max entang pure states? 6 M. B. Ruskai Extreme Points of UCPT maps

UCPT maps Question: Are there UCPT maps Φ : M d �→ M d not extreme in either UCP or CPT maps, but are extreme in UCPT maps. Thm: (Landau-Streater) Φ : M d �→ M d is extreme in set of UCPT j } linearly independent Φ( ρ ) = � maps ⇔ { A ∗ j A k ⊕ A k A ∗ k A k ρ A ∗ k By C-J isomorphism convex set of UCPT maps isomorphic to bipartite states with maximally mixed quantum marginals Equiv: Are there extreme points in convex set of bipartite states ρ AB with ρ A = ρ B = 1 d I d which are not max entang pure states? Def: Entanglement of Formation � � � k x k E ( ψ k ) : � EoF ( ρ AB ) = inf k x k | ψ k �� ψ k | = ρ AB E ( ψ AB ) = S ( ρ A ) , ρ A = Tr B | ψ AB �� ψ AB | , S ( ρ ) = − Tr ρ log ρ 6 M. B. Ruskai Extreme Points of UCPT maps

Known results about extreme points of CPT maps • Qubit channels Φ : M 2 �→ M 2 * Ruskai, Szarek Werner (2002) all extreme points * UCPT much earlier, essent conj with I 2 or Pauli matrix correspond to max entangled Bells states – tetrahedron • Parthsarathy ρ AB state on C 2 ⊗ C d extreme ⇔ max entang 7 M. B. Ruskai Extreme Points of UCPT maps

Known results about extreme points of CPT maps • Qubit channels Φ : M 2 �→ M 2 * Ruskai, Szarek Werner (2002) all extreme points * UCPT much earlier, essent conj with I 2 or Pauli matrix correspond to max entangled Bells states – tetrahedron • Parthsarathy ρ AB state on C 2 ⊗ C d extreme ⇔ max entang • General UCPT Φ : M d �→ M d unitary conj are extreme • Few other results — very special * d = 3 Werner-Holevo channel and symmetric variant ext. not true for Werner-Holevo when d > 3 * Arveson-Ohno examples – few high rank in low dims one low rank family using partial isometries 7 M. B. Ruskai Extreme Points of UCPT maps

Family of high rank extreme points of UCPT maps 4 � A ∗ Φ α,β ( ρ ) = k ρ A k k =1 Def: For | α | 2 + | β | 2 = 1 let A 1 = α | e 1 �� e 1 | + | e 2 �� e 3 | A 2 = β | e 1 �� e 3 | + | e 3 �� e 2 | A 3 = | e 1 �� e 2 | + β | e 3 �� e 1 | A 4 = | e 2 �� e 1 | + α | e 3 �� e 3 | 8 M. B. Ruskai Extreme Points of UCPT maps

Family of high rank extreme points of UCPT maps 4 � A ∗ Φ α,β ( ρ ) = k ρ A k 1-1 correspond with qubit pure states k =1 Def: For | α | 2 + | β | 2 = 1 let or, equiv., vectors in R 3 A 1 = α | e 1 �� e 1 | + | e 2 �� e 3 | A 2 = β | e 1 �� e 3 | + | e 3 �� e 2 | A 3 = | e 1 �� e 2 | + β | e 3 �� e 1 | A 4 = | e 2 �� e 1 | + α | e 3 �� e 3 | 8 M. B. Ruskai Extreme Points of UCPT maps

Family of high rank extreme points of UCPT maps 4 � A ∗ Φ α,β ( ρ ) = k ρ A k 1-1 correspond with qubit pure states k =1 Def: For | α | 2 + | β | 2 = 1 let or, equiv., vectors in R 3 A 1 = α | e 1 �� e 1 | + | e 2 �� e 3 | A 2 = β | e 1 �� e 3 | + | e 3 �� e 2 | A 3 = | e 1 �� e 2 | + β | e 3 �� e 1 | A 4 = | e 2 �� e 1 | + α | e 3 �� e 3 | � A 1 � A 2 Observe U = is unitary ∈ M 3 ⊗ M 2 − A 3 A 4 4 � ( I 3 ⊗ Tr )( U ∗ � � ρ ⊗ 1 ⇒ Φ α,β ( ρ ) = 2 I 2 ) U factorizable k =1 8 M. B. Ruskai Extreme Points of UCPT maps

Family of high rank extreme points of UCPT maps 4 � A ∗ Φ α,β ( ρ ) = k ρ A k 1-1 correspond with qubit pure states k =1 Def: For | α | 2 + | β | 2 = 1 let or, equiv., vectors in R 3 A 1 = α | e 1 �� e 1 | + | e 2 �� e 3 | A 2 = β | e 1 �� e 3 | + | e 3 �� e 2 | A 3 = | e 1 �� e 2 | + β | e 3 �� e 1 | A 4 = | e 2 �� e 1 | + α | e 3 �� e 3 | � A 1 � A 2 Observe U = is unitary ∈ M 3 ⊗ M 2 − A 3 A 4 4 � ( I 3 ⊗ Tr )( U ∗ � � ρ ⊗ 1 ⇒ Φ α,β ( ρ ) = 2 I 2 ) U factorizable k =1 Thm: Φ α,β is an extreme UCPT map for α, β � = 0 , 1 2 , 1 corresponds to N and S poles and equator on Bloch sphere 8 M. B. Ruskai Extreme Points of UCPT maps