Factorizable completely positive maps and the Connes embedding - - PowerPoint PPT Presentation

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

1

Factorizable maps and the Asymptotic Quantum Birkhoff Conjecture

2

Extreme points and factorizability

3

Remarks on the semigroup case

4

Asymptotic properties of factorizable maps and the Connes embedding problem

5

Holevo–Werner channels

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. M

T

• α
• N

P

β∗=β−1◦Eβ(N)

• ◮ 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(Mn(C))) ⊆ F(Mn(C)) ⊆ UCPTn .

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. M

T

• α
• N

P

β∗=β−1◦Eβ(N)

• ◮ 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(Mn(C))) ⊆ F(Mn(C)) ⊆ UCPTn .

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. ◮ K¨ ummerer (1983): UCPT2 = conv(Aut(M2(C))). ◮ For n ≥ 3: UCPTn conv(Aut(Mn(C))) K¨ ummerer (1986): n = 3, K¨ ummerer-Maasen (1987): n ≥ 4, Landau-Streater (1993): another counterexample for n = 3. Conjecture (J. A. Smolin, F. Verstraete, A. Winter, 2005) Let T ∈ UCPTn, n ≥ 3 . Then T satisfies the following asymptotic quantum Birkhoff property (AQBP): lim

k→∞ dcb

k

• i=1

T , conv(Aut(

k

• i=1

Mn(C)))

• = 0 .

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. ◮ K¨ ummerer (1983): UCPT2 = conv(Aut(M2(C))). ◮ For n ≥ 3: UCPTn conv(Aut(Mn(C))) K¨ ummerer (1986): n = 3, K¨ ummerer-Maasen (1987): n ≥ 4, Landau-Streater (1993): another counterexample for n = 3. Conjecture (J. A. Smolin, F. Verstraete, A. Winter, 2005) Let T ∈ UCPTn, n ≥ 3 . Then T satisfies the following asymptotic quantum Birkhoff property (AQBP): lim

k→∞ dcb

k

• i=1

T , conv(Aut(

k

• i=1

Mn(C)))

• = 0 .

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. ◮ K¨ ummerer (1983): UCPT2 = conv(Aut(M2(C))). ◮ For n ≥ 3: UCPTn conv(Aut(Mn(C))) K¨ ummerer (1986): n = 3, K¨ ummerer-Maasen (1987): n ≥ 4, Landau-Streater (1993): another counterexample for n = 3. Conjecture (J. A. Smolin, F. Verstraete, A. Winter, 2005) Let T ∈ UCPTn, n ≥ 3 . Then T satisfies the following asymptotic quantum Birkhoff property (AQBP): lim

k→∞ dcb

k

• i=1

T , conv(Aut(

k

• i=1

Mn(C)))

• = 0 .

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. ◮ K¨ ummerer (1983): UCPT2 = conv(Aut(M2(C))). ◮ For n ≥ 3: UCPTn conv(Aut(Mn(C))) K¨ ummerer (1986): n = 3, K¨ ummerer-Maasen (1987): n ≥ 4, Landau-Streater (1993): another counterexample for n = 3. Conjecture (J. A. Smolin, F. Verstraete, A. Winter, 2005) Let T ∈ UCPTn, n ≥ 3 . Then T satisfies the following asymptotic quantum Birkhoff property (AQBP): lim

k→∞ dcb

k

• i=1

T , conv(Aut(

k

• i=1

Mn(C)))

• = 0 .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Non-factorizable maps and the AQBP

Mendl-Wolf (2009): ∃ T ∈ UCPT3 s.t. T / ∈ conv(Aut(M3(C))), but T ⊗ T ∈ conv(Aut

• M3(C) ⊗ M3(C))
• .

Theorem (Haagerup-M, 2011) Let T ∈ UCPTn, where n ≥ 3. Then, for all k ≥ 1, dcb k

• i=1

T, F k

• i=1

Mn(C)

• ≥ dcb(T, F(Mn(C))) .

◮ If T is not factorizable, then dcb(T , F(Mn(C))) > 0, as F(Mn(C)) is norm-closed. Since conv(Aut(Mn(C))) ⊂ F(Mn(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

Mendl-Wolf (2009): ∃ T ∈ UCPT3 s.t. T / ∈ conv(Aut(M3(C))), but T ⊗ T ∈ conv(Aut

• M3(C) ⊗ M3(C))
• .

Theorem (Haagerup-M, 2011) Let T ∈ UCPTn, where n ≥ 3. Then, for all k ≥ 1, dcb k

• i=1

T, F k

• i=1

Mn(C)

• ≥ dcb(T, F(Mn(C))) .

◮ If T is not factorizable, then dcb(T , F(Mn(C))) > 0, as F(Mn(C)) is norm-closed. Since conv(Aut(Mn(C))) ⊂ F(Mn(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

Mendl-Wolf (2009): ∃ T ∈ UCPT3 s.t. T / ∈ conv(Aut(M3(C))), but T ⊗ T ∈ conv(Aut

• M3(C) ⊗ M3(C))
• .

Theorem (Haagerup-M, 2011) Let T ∈ UCPTn, where n ≥ 3. Then, for all k ≥ 1, dcb k

• i=1

T, F k

• i=1

Mn(C)

• ≥ dcb(T, F(Mn(C))) .

◮ If T is not factorizable, then dcb(T , F(Mn(C))) > 0, as F(Mn(C)) is norm-closed. Since conv(Aut(Mn(C))) ⊂ F(Mn(C)), any non-factorizable unital channel T fails the AQBP.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Theorem (Haagerup-M., 2011) TFAE for T ∈ UCPTn , n ≥ 3, written in Choi canonical form Tx =

d

• i=1

a∗

i xai ,

x ∈ Mn(C) . 1) T is factorizable 2) ∃ vN algebra N with nf tracial state τN and u ∈ U(Mn(N)) st Tx = (idMn(C) ⊗ τN)(u∗(x ⊗ 1N)u) , x ∈ Mn(C) . We say that T has an exact factorization through Mn(C) ⊗ N. 3) ∃ vN algebra N with nf tracial state τN and v1 , . . . , vd ∈ N st u : =

d

• i=1

ai ⊗ vi ∈ U(Mn(N)), τN(v∗

i vj) = δij , 1 ≤ i, j ≤ d

Interpretation in Quantum Information Theory (R. Werner): Factorizable maps are obtained by coupling the input system to a maximally mixed ancillary one, executing a unitary rotation on the combined system, and tracing out the ancilla.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Theorem (Haagerup-M., 2011) TFAE for T ∈ UCPTn , n ≥ 3, written in Choi canonical form Tx =

d

• i=1

a∗

i xai ,

x ∈ Mn(C) . 1) T is factorizable 2) ∃ vN algebra N with nf tracial state τN and u ∈ U(Mn(N)) st Tx = (idMn(C) ⊗ τN)(u∗(x ⊗ 1N)u) , x ∈ Mn(C) . We say that T has an exact factorization through Mn(C) ⊗ N. 3) ∃ vN algebra N with nf tracial state τN and v1 , . . . , vd ∈ N st u : =

d

• i=1

ai ⊗ vi ∈ U(Mn(N)), τN(v∗

i vj) = δij , 1 ≤ i, j ≤ d

Interpretation in Quantum Information Theory (R. Werner): Factorizable maps are obtained by coupling the input system to a maximally mixed ancillary one, executing a unitary rotation on the combined system, and tracing out the ancilla.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Examples of non-factorizable maps

Corollary Let T ∈ UCPTn, with canonical form T = d

i=1 a∗ i xai. If d ≥ 2

and the set {a∗

i aj : 1 ≤ i, j ≤ d}

is linearly independent, then T is not factorizable. Proof: Assume T factorizable. Then ∃(N, τN), ∃v1 , . . . , vd ∈ N st u : =

• ai ⊗ vi ∈ U(Mn(C) ⊗ N),

τN(v∗

i vj) = δij.

= ⇒

d

• i,j=1

a∗

i aj ⊗ (v∗ i vj − δij1N) = u∗u −

d

• i=1

a∗

i ai

• ⊗ 1N = 0 .

By the linear independence of the set {a∗

i aj : 1 ≤ i, j ≤ d} ,

v∗

i vj − δij1N = 0N ,

1 ≤ i, j ≤ d . In particular (since d ≥ 2), v∗

1 v1 = v∗ 2 v2 = 1N and v∗ 1 v2 = 0N .

Impossible, since N is finite.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Examples of non-factorizable maps

Corollary Let T ∈ UCPTn, with canonical form T = d

i=1 a∗ i xai. If d ≥ 2

and the set {a∗

i aj : 1 ≤ i, j ≤ d}

is linearly independent, then T is not factorizable. Proof: Assume T factorizable. Then ∃(N, τN), ∃v1 , . . . , vd ∈ N st u : =

• ai ⊗ vi ∈ U(Mn(C) ⊗ N),

τN(v∗

i vj) = δij.

= ⇒

d

• i,j=1

a∗

i aj ⊗ (v∗ i vj − δij1N) = u∗u −

d

• i=1

a∗

i ai

• ⊗ 1N = 0 .

By the linear independence of the set {a∗

i aj : 1 ≤ i, j ≤ d} ,

v∗

i vj − δij1N = 0N ,

1 ≤ i, j ≤ d . In particular (since d ≥ 2), v∗

1 v1 = v∗ 2 v2 = 1N and v∗ 1 v2 = 0N .

Impossible, since N is finite.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Examples of non-factorizable maps

Corollary Let T ∈ UCPTn, with canonical form T = d

i=1 a∗ i xai. If d ≥ 2

and the set {a∗

i aj : 1 ≤ i, j ≤ d}

is linearly independent, then T is not factorizable. Proof: Assume T factorizable. Then ∃(N, τN), ∃v1 , . . . , vd ∈ N st u : =

• ai ⊗ vi ∈ U(Mn(C) ⊗ N),

τN(v∗

i vj) = δij.

= ⇒

d

• i,j=1

a∗

i aj ⊗ (v∗ i vj − δij1N) = u∗u −

d

• i=1

a∗

i ai

• ⊗ 1N = 0 .

By the linear independence of the set {a∗

i aj : 1 ≤ i, j ≤ d} ,

v∗

i vj − δij1N = 0N ,

1 ≤ i, j ≤ d . In particular (since d ≥ 2), v∗

1 v1 = v∗ 2 v2 = 1N and v∗ 1 v2 = 0N .

Impossible, since N is finite.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Example As an application of corollary, by letting a1 = 1 √ 2   −1 1   , a2 = 1 √ 2   1 −1   , a3 = 1 √ 2   −1 1   we obtained a (first example of a) non-factorizable unital channel. Turned out to be the Holevo-Werner channel, W −

3 , in dimension

n = 3.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Outline

1

Factorizable maps and the Asymptotic Quantum Birkhoff Conjecture

2

Extreme points and factorizability

3

Remarks on the semigroup case

4

Asymptotic properties of factorizable maps and the Connes embedding problem

5

Holevo–Werner channels

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Choi (1975): T ∈ ∂e(UCPn) if and only if {a∗

i aj : 1 ≤ i, j ≤ d} is

a linearly independent set. By the corollary, if T ∈ ∂e(UCPn) ∩ UCPTn then T is not factorizable. (Hence T does not satisfy AQBP.) Crann-Neufang (2012): A class of unital quantum channels arising from abstract harmonic analysis failing the AQBP: Given G finite group, φ pos. definite function on G with φ(e) = 1 Bφ = (φ(t−1s))t,s∈G ∈ M|G|(C) TBφ unital Schur channel. Conditions ensuring that TBφ is an extreme point are discussed.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Choi (1975): T ∈ ∂e(UCPn) if and only if {a∗

i aj : 1 ≤ i, j ≤ d} is

a linearly independent set. By the corollary, if T ∈ ∂e(UCPn) ∩ UCPTn then T is not factorizable. (Hence T does not satisfy AQBP.) Crann-Neufang (2012): A class of unital quantum channels arising from abstract harmonic analysis failing the AQBP: Given G finite group, φ pos. definite function on G with φ(e) = 1 Bφ = (φ(t−1s))t,s∈G ∈ M|G|(C) TBφ unital Schur channel. Conditions ensuring that TBφ is an extreme point are discussed.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Theorem (Haagerup-M.-Ruskai) Let n ≥ 3 and let S : Cn → Cn be the cyclic shift. (1) Let U1, . . . , Un ∈ U(n − 1) and set ai = 1 √n − 1 Si Ui

• S−i ,

1 ≤ i ≤ n . Set Tx = n

i=1 a∗ i xai, x ∈ Mn(C). Then T ∈ UCPTn and

with probability one (w.r.t. Haar measure on n

i=1 U(n − 1)),

T ∈ ∂e(UCPn) ∩ ∂e(CPTn) . In particular, T is not factorizable. (2) Same conclusion holds for ai = 1 √ n − 1 + t2 Si Ui t

• S−i ,

1 ≤ i ≤ n , where t > 0, t = 1 (fixed).

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Landau and Streater (1993): T ∈ ∂e(UCPTn) if and only if {a∗

i aj ⊕ aja∗ i : 1 ≤ i, j ≤ d}

is a linearly independent set. Hence ∂e(UCPTn) ⊇

• ∂e(UCPn) ∪ ∂e(CPTn)
• ∩ UCPTn.

Mendl-Wolf (2009): above inclusion is strict for n = 3 . Ohno (2010): concrete examples for n = 3 , n = 4. Further examples (motivated by a question of Farenick, 2010): Haagerup-M.-Ruskai: A family (Tt)t∈[0,1] ⊂ UCPT3 s.t. Tt ∈ ∂e(UCPT3) \

• ∂e(UCP3) ∪ ∂e(CPT3)
• ,

t ∈ (0, 1) \ {1/2}. Moreover, Tt is factorizable, 0 ≤ t ≤ 1, (through M3(C) ⊗ M2(C)).

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Landau and Streater (1993): T ∈ ∂e(UCPTn) if and only if {a∗

i aj ⊕ aja∗ i : 1 ≤ i, j ≤ d}

is a linearly independent set. Hence ∂e(UCPTn) ⊇

• ∂e(UCPn) ∪ ∂e(CPTn)
• ∩ UCPTn.

Mendl-Wolf (2009): above inclusion is strict for n = 3 . Ohno (2010): concrete examples for n = 3 , n = 4. Further examples (motivated by a question of Farenick, 2010): Haagerup-M.-Ruskai: A family (Tt)t∈[0,1] ⊂ UCPT3 s.t. Tt ∈ ∂e(UCPT3) \

• ∂e(UCP3) ∪ ∂e(CPT3)
• ,

t ∈ (0, 1) \ {1/2}. Moreover, Tt is factorizable, 0 ≤ t ≤ 1, (through M3(C) ⊗ M2(C)).

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Landau and Streater (1993): T ∈ ∂e(UCPTn) if and only if {a∗

i aj ⊕ aja∗ i : 1 ≤ i, j ≤ d}

is a linearly independent set. Hence ∂e(UCPTn) ⊇

• ∂e(UCPn) ∪ ∂e(CPTn)
• ∩ UCPTn.

Mendl-Wolf (2009): above inclusion is strict for n = 3 . Ohno (2010): concrete examples for n = 3 , n = 4. Further examples (motivated by a question of Farenick, 2010): Haagerup-M.-Ruskai: A family (Tt)t∈[0,1] ⊂ UCPT3 s.t. Tt ∈ ∂e(UCPT3) \

• ∂e(UCP3) ∪ ∂e(CPT3)
• ,

t ∈ (0, 1) \ {1/2}. Moreover, Tt is factorizable, 0 ≤ t ≤ 1, (through M3(C) ⊗ M2(C)).

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Outline

1

Factorizable maps and the Asymptotic Quantum Birkhoff Conjecture

2

Extreme points and factorizability

3

Remarks on the semigroup case

4

Asymptotic properties of factorizable maps and the Connes embedding problem

5

Holevo–Werner channels

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Remarks on the semigroup case

K¨ ummerer-Maassen (1987): For n ≥ 3, if (Tt)t≥0 is a

• ne-parameter semigroup of self-adjoint UCPTn maps, then

Tt ∈ conv(Aut(Mn(C))) , t ≥ 0 . In particular, Tt is factorizable, t ≥ 0 . Haagerup-M. (2011): Example of a semigroup (Tt)t≥0 of non self-adjoint UCPT4 maps for which ∃ t0 > 0 s.t. Tt is not factorizable, for any 0 < t < t0 . Junge-Ricard-Shlyakhtenko (independently, Dabrowsky) (2012): If (Tt)t≥0 is a strongly continuous semigroup of self-adjoint UCPT maps on a finite von Neumann algebra, then Tt is factorizable, t ≥ 0 .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Remarks on the semigroup case

K¨ ummerer-Maassen (1987): For n ≥ 3, if (Tt)t≥0 is a

• ne-parameter semigroup of self-adjoint UCPTn maps, then

Tt ∈ conv(Aut(Mn(C))) , t ≥ 0 . In particular, Tt is factorizable, t ≥ 0 . Haagerup-M. (2011): Example of a semigroup (Tt)t≥0 of non self-adjoint UCPT4 maps for which ∃ t0 > 0 s.t. Tt is not factorizable, for any 0 < t < t0 . Junge-Ricard-Shlyakhtenko (independently, Dabrowsky) (2012): If (Tt)t≥0 is a strongly continuous semigroup of self-adjoint UCPT maps on a finite von Neumann algebra, then Tt is factorizable, t ≥ 0 .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Remarks on the semigroup case

K¨ ummerer-Maassen (1987): For n ≥ 3, if (Tt)t≥0 is a

• ne-parameter semigroup of self-adjoint UCPTn maps, then

Tt ∈ conv(Aut(Mn(C))) , t ≥ 0 . In particular, Tt is factorizable, t ≥ 0 . Haagerup-M. (2011): Example of a semigroup (Tt)t≥0 of non self-adjoint UCPT4 maps for which ∃ t0 > 0 s.t. Tt is not factorizable, for any 0 < t < t0 . Junge-Ricard-Shlyakhtenko (independently, Dabrowsky) (2012): If (Tt)t≥0 is a strongly continuous semigroup of self-adjoint UCPT maps on a finite von Neumann algebra, then Tt is factorizable, t ≥ 0 .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Outline

1

Factorizable maps and the Asymptotic Quantum Birkhoff Conjecture

2

Extreme points and factorizability

3

Remarks on the semigroup case

4

Asymptotic properties of factorizable maps and the Connes embedding problem

5

Holevo–Werner channels

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Natural question: Does every factorizable unital quantum channel satisfy the AQBP? This question (that arose at a 2010 Banff meeting) gained a lot of interest due to the following connection to the Connes embedding problem, whether every II1-factor (on a separable Hilbert space) embeds in an ultrapower Rω of the hyperfinite II1 factor R: Theorem (Haagerup-M, 2011) If for any n ≥ 3 , every factorizable UCPTn map satisfies the AQBP, then the Connes embedding problem has a positive answer.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Natural question: Does every factorizable unital quantum channel satisfy the AQBP? This question (that arose at a 2010 Banff meeting) gained a lot of interest due to the following connection to the Connes embedding problem, whether every II1-factor (on a separable Hilbert space) embeds in an ultrapower Rω of the hyperfinite II1 factor R: Theorem (Haagerup-M, 2011) If for any n ≥ 3 , every factorizable UCPTn map satisfies the AQBP, then the Connes embedding problem has a positive answer.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Example Let β = 1/ √ 5 and set B : =         1 β β β β β β 1 β −β −β β β β 1 β −β −β β −β β 1 β −β β −β −β β 1 β β β −β −β β 1         . The unital Schur channel TB is factorizable, but TB / ∈ conv(Aut(M6(C))) Question Does TB satisfy the AQBP?

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Example Let β = 1/ √ 5 and set B : =         1 β β β β β β 1 β −β −β β β β 1 β −β −β β −β β 1 β −β β −β −β β 1 β β β −β −β β 1         . The unital Schur channel TB is factorizable, but TB / ∈ conv(Aut(M6(C))) Question Does TB satisfy the AQBP?

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Theorem (Haagerup-M) Let T be a unital Schur channel on Mn(C) and S be a unital Schur channel on Mk(C) , k, n ≥ 2. Then dcb

• T ⊗S, conv
• Aut(Mnk(C))
• ≥ 1

2 dcb

• T, conv
• Aut(Mn(C))
• In particular, if T /

∈ conv(Aut(Mn(C))) , then T fails the AQBP. Conclusion: The factorizable map TB above fails the AQBP.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Theorem (Haagerup-M) Let T be a unital Schur channel on Mn(C) and S be a unital Schur channel on Mk(C) , k, n ≥ 2. Then dcb

• T ⊗S, conv
• Aut(Mnk(C))
• ≥ 1

2 dcb

• T, conv
• Aut(Mn(C))
• In particular, if T /

∈ conv(Aut(Mn(C))) , then T fails the AQBP. Conclusion: The factorizable map TB above fails the AQBP.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Connection to the Connes embedding problem

Theorem (Haagerup-M) Let T ∈ UCPTn be factorizable. TFAE: (1) T has an exact factorization through a finite von Neumann algebra which embeds into Rω, i.e., ∃ (N, τN) ֒ → Rω , ∃ u ∈ U(Mn(N)) s.t. Tx = (idMn(C) ⊗ τN)

• u∗(x ⊗ 1N)u
• ,

x ∈ Mn(C) . (2) T admits an approximate factorization through matrix algebras. (3) lim

k→∞ dcb

• T ⊗ Sk , conv
• Aut(Mn(C) ⊗ Mk(C))
• = 0,

where Sk is the completely depolarizing channel: Sk(y) = τk(y)1k, y ∈ Mk(C).

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Theorem (Haagerup-M) The Connes embedding problem has a positive answer if and only if every factorizable UCPTn map satisfies one of the equivalent conditions in previous theorem, for all n ≥ 3 . Ideas of proof of previous theorem: If T has an exact factorization through Mn(C) ⊗ Mk(C), then T ⊗ Sk ∈ conv(Aut

• Mn(C) ⊗ Mk(C))
• .

If T ⊗ Sk ∈ conv

• Aut(Mn(C) ⊗ Mk(C))
• , then T has an

exact factorization through Mn(C) ⊗

• Mk(C) ⊗ L∞([0, 1], dx)
• .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Theorem (Haagerup-M) The Connes embedding problem has a positive answer if and only if every factorizable UCPTn map satisfies one of the equivalent conditions in previous theorem, for all n ≥ 3 . Ideas of proof of previous theorem: If T has an exact factorization through Mn(C) ⊗ Mk(C), then T ⊗ Sk ∈ conv(Aut

• Mn(C) ⊗ Mk(C))
• .

If T ⊗ Sk ∈ conv

• Aut(Mn(C) ⊗ Mk(C))
• , then T has an

exact factorization through Mn(C) ⊗

• Mk(C) ⊗ L∞([0, 1], dx)
• .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Theorem (Haagerup-M) The Connes embedding problem has a positive answer if and only if every factorizable UCPTn map satisfies one of the equivalent conditions in above theorem, for all n ≥ 3. Idea of proof: (⇐) Dykema-Jushenko (2009): Fn : =

• k≥1
• B = (bij) ∈ Mn(C) : bij = τk(uiu∗

j ) , uj ∈ U(Mk(C))

• Gn :

=

• B = (bij) ∈ Mn(C) : bij = τM(uiu∗

j ) , uj ∈ U(M) , for

some vN algebra (M, τM) with n.f. tracial state

• = {B ∈ Mn(C) : Schur multiplier TB is factorizable }.

By Kirchberg (1993): The Connes embedding problem has a positive answer iff Fn = Gn , for all n ≥ 1.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Theorem (Haagerup-M) The Connes embedding problem has a positive answer if and only if every factorizable UCPTn map satisfies one of the equivalent conditions in above theorem, for all n ≥ 3. Idea of proof: (⇐) Dykema-Jushenko (2009): Fn : =

• k≥1
• B = (bij) ∈ Mn(C) : bij = τk(uiu∗

j ) , uj ∈ U(Mk(C))

• Gn :

=

• B = (bij) ∈ Mn(C) : bij = τM(uiu∗

j ) , uj ∈ U(M) , for

some vN algebra (M, τM) with n.f. tracial state

• = {B ∈ Mn(C) : Schur multiplier TB is factorizable }.

By Kirchberg (1993): The Connes embedding problem has a positive answer iff Fn = Gn , for all n ≥ 1.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Theorem (Haagerup-M) The Connes embedding problem has a positive answer if and only if every factorizable UCPTn map satisfies one of the equivalent conditions in above theorem, for all n ≥ 3. Idea of proof: (⇐) Dykema-Jushenko (2009): Fn : =

• k≥1
• B = (bij) ∈ Mn(C) : bij = τk(uiu∗

j ) , uj ∈ U(Mk(C))

• Gn :

=

• B = (bij) ∈ Mn(C) : bij = τM(uiu∗

j ) , uj ∈ U(M) , for

some vN algebra (M, τM) with n.f. tracial state

• = {B ∈ Mn(C) : Schur multiplier TB is factorizable }.

By Kirchberg (1993): The Connes embedding problem has a positive answer iff Fn = Gn , for all n ≥ 1.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Idea of proof — continued: Assume that the Connes embedding problem has a negative

• answer. Then Gn \ Fn = ∅ , for some n ≥ 1. Choose

B = (bij)n

i,j=1 ∈ Gn \ Fn .

Then the Schur multiplier TB has an exact factorization through a finite vN algebra embeddable into Rω, so ∃u1, . . . un ∈ U(Rω) s.t. bij = τRω(u∗

i uj),

1 ≤ i, j ≤ n . Approximate each bij by τR(v∗

i vj), where vi ∈ U(R), and further by

unitary matrices (via Kaplansky). Hence B can be approximated by a sequence Bk whose Schur multiplier TBk admits an exact factorization through a matrix algebra. This implies B ∈ Fn .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Idea of proof — continued: Assume that the Connes embedding problem has a negative

• answer. Then Gn \ Fn = ∅ , for some n ≥ 1. Choose

B = (bij)n

i,j=1 ∈ Gn \ Fn .

Then the Schur multiplier TB has an exact factorization through a finite vN algebra embeddable into Rω, so ∃u1, . . . un ∈ U(Rω) s.t. bij = τRω(u∗

i uj),

1 ≤ i, j ≤ n . Approximate each bij by τR(v∗

i vj), where vi ∈ U(R), and further by

unitary matrices (via Kaplansky). Hence B can be approximated by a sequence Bk whose Schur multiplier TBk admits an exact factorization through a matrix algebra. This implies B ∈ Fn .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Outline

1

Factorizable maps and the Asymptotic Quantum Birkhoff Conjecture

2

Extreme points and factorizability

3

Remarks on the semigroup case

4

Asymptotic properties of factorizable maps and the Connes embedding problem

5

Holevo–Werner channels

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Some concrete examples

Let n ≥ 2. Consider the Holevo-Werner channel in dimension n: W −

n (x) =

1 n − 1( Tr(x)1n − xt ) , x ∈ Mn(C) . It has an analogue W +

n (x) =

1 n + 1( Tr(x)1n + xt ) , x ∈ Mn(C) . ◮ W −

n , W + n ∈ UCPTn.

◮ Sn ∈ conv{W −

n , W + n }, since

Sn(x) = 1 nTr(x)1n = n − 1 2n W −

n (x) + n + 1

2n W +

n (x) , x ∈ Mn(C) .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Some concrete examples

Let n ≥ 2. Consider the Holevo-Werner channel in dimension n: W −

n (x) =

1 n − 1( Tr(x)1n − xt ) , x ∈ Mn(C) . It has an analogue W +

n (x) =

1 n + 1( Tr(x)1n + xt ) , x ∈ Mn(C) . ◮ W −

n , W + n ∈ UCPTn.

◮ Sn ∈ conv{W −

n , W + n }, since

Sn(x) = 1 nTr(x)1n = n − 1 2n W −

n (x) + n + 1

2n W +

n (x) , x ∈ Mn(C) .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Some concrete examples

Let n ≥ 2. Consider the Holevo-Werner channel in dimension n: W −

n (x) =

1 n − 1( Tr(x)1n − xt ) , x ∈ Mn(C) . It has an analogue W +

n (x) =

1 n + 1( Tr(x)1n + xt ) , x ∈ Mn(C) . ◮ W −

n , W + n ∈ UCPTn.

◮ Sn ∈ conv{W −

n , W + n }, since

Sn(x) = 1 nTr(x)1n = n − 1 2n W −

n (x) + n + 1

2n W +

n (x) , x ∈ Mn(C) .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Recall: W −

n (x) =

1 n − 1( Tr(x)1n − xt ) , x ∈ Mn(C) . W +

n (x) =

1 n + 1( Tr(x)1n + xt ) , x ∈ Mn(C) . Denote aij = eij − eji, bij = eij + eji , 1 ≤ i, j ≤ n ◮ For all x ∈ Mn(C) we have (Choi canonical forms): W −

n (x)

= 1 n − 1

• i<j

aij x a∗

ij

W +

n (x)

= 1 n + 1

i<j

bij x b∗

ij + 2

• i

eii x eii

• .

◮ W −

3 is not factorizable.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Theorem (Mendl-Wolf, 2009) (1) W +

n ∈ conv(Aut(Mn(C))), for all n ≥ 2.

(2) W −

n ∈ conv(Aut(Mn(C))), for all n even.

(3) For n odd and 0 ≤ λ ≤ 1, λ W +

n + (1 − λ) W − n ∈ conv

• Aut(Mn(C))
• ⇐

⇒ λ ≥ 1 n. In particular, W −

n /

∈ conv

• Aut(Mn(C))
• .

Theorem (Haagerup-M) (1) dcb(W −

3 , F

• M3(C))
• = 4

27.

(2) For n odd, n = 3, W −

n has an exact factorization through Mn(C) ⊗ M4(C).

(3) λ W +

3 + (1 − λ) W − 3 ∈ F(M3(C)) ⇐

⇒

2 27 ≤ λ ≤ 1.

In each such case, λ W +

3 + (1 − λ) W − 3 has an exact

factorization through M3(C) ⊗ M3(C).

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Theorem (Mendl-Wolf, 2009) (1) W +

n ∈ conv(Aut(Mn(C))), for all n ≥ 2.

(2) W −

n ∈ conv(Aut(Mn(C))), for all n even.

(3) For n odd and 0 ≤ λ ≤ 1, λ W +

n + (1 − λ) W − n ∈ conv

• Aut(Mn(C))
• ⇐

⇒ λ ≥ 1 n. In particular, W −

n /

∈ conv

• Aut(Mn(C))
• .

Theorem (Haagerup-M) (1) dcb(W −

3 , F

• M3(C))
• = 4

27.

(2) For n odd, n = 3, W −

n has an exact factorization through Mn(C) ⊗ M4(C).

(3) λ W +

3 + (1 − λ) W − 3 ∈ F(M3(C)) ⇐

⇒

2 27 ≤ λ ≤ 1.

In each such case, λ W +

3 + (1 − λ) W − 3 has an exact

factorization through M3(C) ⊗ M3(C).

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

For 0 ≤ λ ≤ 1, set Tλ : = λ W +

3 + (1 − λ) W − 3 . By Mendl-Wolf:

Tλ ∈ conv(Aut

• M3(C))
• ⇐

⇒ λ ≥ 1/3. Theorem (Mendl-Wolf, 2009) There exists λ0 ∈

• 0, 1

3

• such that for all λ ≥ λ0,

Tλ ⊗ Tλ ∈ conv

• Aut(M3(C) ⊗ M3(C))
• .

Theorem (Haagerup-M) There exists λ1 ∈

• 0, 1

3

• such that for all λ ≥ λ1,

Tλ ⊗ Tλ ⊗ Tλ ∈ conv

• Aut(M3(C) ⊗ M3(C) ⊗ M3(C))
• .

Hence Tλ satisfies the AQBP, for all 1 ≥ λ ≥ max{λ0, λ1} ≥ 2/27.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

For 0 ≤ λ ≤ 1, set Tλ : = λ W +

3 + (1 − λ) W − 3 . By Mendl-Wolf:

Tλ ∈ conv(Aut

• M3(C))
• ⇐

⇒ λ ≥ 1/3. Theorem (Mendl-Wolf, 2009) There exists λ0 ∈

• 0, 1

3

• such that for all λ ≥ λ0,

Tλ ⊗ Tλ ∈ conv

• Aut(M3(C) ⊗ M3(C))
• .

Theorem (Haagerup-M) There exists λ1 ∈

• 0, 1

3

• such that for all λ ≥ λ1,

Tλ ⊗ Tλ ⊗ Tλ ∈ conv

• Aut(M3(C) ⊗ M3(C) ⊗ M3(C))
• .

Hence Tλ satisfies the AQBP, for all 1 ≥ λ ≥ max{λ0, λ1} ≥ 2/27.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

For 0 ≤ λ ≤ 1, set Tλ : = λ W +

3 + (1 − λ) W − 3 . By Mendl-Wolf:

Tλ ∈ conv(Aut

• M3(C))
• ⇐

⇒ λ ≥ 1/3. Theorem (Mendl-Wolf, 2009) There exists λ0 ∈

• 0, 1

3

• such that for all λ ≥ λ0,

Tλ ⊗ Tλ ∈ conv

• Aut(M3(C) ⊗ M3(C))
• .

Theorem (Haagerup-M) There exists λ1 ∈

• 0, 1

3

• such that for all λ ≥ λ1,

Tλ ⊗ Tλ ⊗ Tλ ∈ conv

• Aut(M3(C) ⊗ M3(C) ⊗ M3(C))
• .

Hence Tλ satisfies the AQBP, for all 1 ≥ λ ≥ max{λ0, λ1} ≥ 2/27.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

For 0 ≤ λ ≤ 1, set Tλ : = λ W +

3 + (1 − λ) W − 3 . By Mendl-Wolf:

Tλ ∈ conv(Aut

• M3(C))
• ⇐

⇒ λ ≥ 1/3. Theorem (Mendl-Wolf, 2009) There exists λ0 ∈

• 0, 1

3

• such that for all λ ≥ λ0,

Tλ ⊗ Tλ ∈ conv

• Aut(M3(C) ⊗ M3(C))
• .

Theorem (Haagerup-M) There exists λ1 ∈

• 0, 1

3

• such that for all λ ≥ λ1,

Tλ ⊗ Tλ ⊗ Tλ ∈ conv

• Aut(M3(C) ⊗ M3(C) ⊗ M3(C))
• .

Hence Tλ satisfies the AQBP, for all 1 ≥ λ ≥ max{λ0, λ1} ≥ 2/27.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

About some of the proofs

Proof of the fact that W −

5 is factorizable.

W −

5 = 1

4

• i<j

aijxa∗

ij,

aij = eij − eji , W −1

5

is factorizable iff ∃ (N, τN) and u = (uij)i,j=1,5 ∈ U(Mn(N)) s.t. uij = −uji, ∀i, j and {uij}i,j=1,5 is orthonormal w.r.t. inner product given by τN. This can be achieved with (N, τN) = (M4(C), tr4). Key: ∃v1, . . . , v5 ∈ U(M4(C)) self-adjoint anti-commuting which form an orthonormal set w.r.t. inner product given by tr4.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

About some of the proofs

Proof of the fact that W −

5 is factorizable.

W −

5 = 1

4

• i<j

aijxa∗

ij,

aij = eij − eji , W −1

5

is factorizable iff ∃ (N, τN) and u = (uij)i,j=1,5 ∈ U(Mn(N)) s.t. uij = −uji, ∀i, j and {uij}i,j=1,5 is orthonormal w.r.t. inner product given by τN. This can be achieved with (N, τN) = (M4(C), tr4). Key: ∃v1, . . . , v5 ∈ U(M4(C)) self-adjoint anti-commuting which form an orthonormal set w.r.t. inner product given by tr4.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Let a = − 1

2 + i √ 3 2 , b = − 1 2 − i √ 3 2 . Then

σ =       a b b a a a b b b a a b b b a a a b b a       ∈ U(M5(C)) . Set u : =    v1 ... v5    (σ⊗14)    v1 ... v5    = (σijvivj)i,j=1,5 . Then u ∈ U(M5(C) ⊗ M4(C)) does the trick.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Proof of the fact that T = 2

27 W + 3 + 25 27 W − 3 is factorizable.

One can check that for x ∈ M3(C), 2 27 W +

3 + 25

27 W −

3

• (x) = (idM3(C) ⊗ tr3)(u(x ⊗ 13)u∗),

where u =              

1 3

− 2

3

− 2

3

1 1 1 − 2

3 1 3

− 2

3

1 1 1 − 2

3

− 2

3 1 3

              ∈ U(M9(C)) .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Then dcb(W −

3 , F(M3(C)))

≤

• W −

3 −

2 27 W +

3 + 25

27 W −

3

• cb

= 2 27W +

3 − W − 3 cb = 4

27 , wherein we have used the fact that W +

n − W − n cb = 2,

n ≥ 2 . The statement that λ W +

3 + (1 − λ) W − 3 is factorizable if and only if 2 27 ≤ λ ≤ 1

follows once we show that dcb(W −

3 , F(M3(C))) ≥ 4/27, since

2λ = W −

3 −(λ W + 3 +(1−λ) W − 3 )cb ≥ dcb(W − 3 , F(M3(C))) = 4/27 .

Note that the (‘if’) part follows right-away by convexity.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Then dcb(W −

3 , F(M3(C)))

≤

• W −

3 −

2 27 W +

3 + 25

27 W −

3

• cb

= 2 27W +

3 − W − 3 cb = 4

27 , wherein we have used the fact that W +

n − W − n cb = 2,

n ≥ 2 . The statement that λ W +

3 + (1 − λ) W − 3 is factorizable if and only if 2 27 ≤ λ ≤ 1

follows once we show that dcb(W −

3 , F(M3(C))) ≥ 4/27, since

2λ = W −

3 −(λ W + 3 +(1−λ) W − 3 )cb ≥ dcb(W − 3 , F(M3(C))) = 4/27 .

Note that the (‘if’) part follows right-away by convexity.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Then dcb(W −

3 , F(M3(C)))

≤

• W −

3 −

2 27 W +

3 + 25

27 W −

3

• cb

= 2 27W +

3 − W − 3 cb = 4

27 , wherein we have used the fact that W +

n − W − n cb = 2,

n ≥ 2 . The statement that λ W +

3 + (1 − λ) W − 3 is factorizable if and only if 2 27 ≤ λ ≤ 1

follows once we show that dcb(W −

3 , F(M3(C))) ≥ 4/27, since

2λ = W −

3 −(λ W + 3 +(1−λ) W − 3 )cb ≥ dcb(W − 3 , F(M3(C))) = 4/27 .

Note that the (‘if’) part follows right-away by convexity.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

An averaging technique (twirl)

For T ∈ B(Mn(C)) set F(T): =

• U(n)

ad(u) T ad(ut) du , where du is the Haar measure on U(n). Properties: If T ∈ UCPTn then F(T) ∈ UCPTn. Moreover, F(T)cb ≤ Tcb . F(conv(Aut(Mn(C)))) ⊆ conv(Aut(Mn(C))). F(F(Mn(C))) ⊆ F(Mn(C)).

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

An averaging technique (twirl)

For T ∈ B(Mn(C)) set F(T): =

• U(n)

ad(u) T ad(ut) du , where du is the Haar measure on U(n). Properties: If T ∈ UCPTn then F(T) ∈ UCPTn. Moreover, F(T)cb ≤ Tcb . F(conv(Aut(Mn(C)))) ⊆ conv(Aut(Mn(C))). F(F(Mn(C))) ⊆ F(Mn(C)).

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Choi (1975): T ∈ B(Mn(C)) is CP ⇐ ⇒ ˆ T is positive, where ˆ T : = 1 n

n

• i,j=1

T(eij) ⊗ eij ∈ Mn(C) ⊗ Mn(C) . Vollbrecht-Werner (2001): F(T) = E( ˆ T), where E(x) =

• U(n)

(u ⊗ u)x(u∗ ⊗ u∗)du, x ∈ Mn(C) ⊗ Mn(C), is the trace-preserving cond. expectation of Mn(C) ⊗ Mn(C) onto span{P+, P−}, where P+, P− are the orthogonal projections onto (Cn ⊗ Cn)sym and (Cn ⊗ Cn)antisym , respectively.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

◮ F(W +

n ) = W + n and F(W − n ) = W − n .

Theorem (Haagerup-M) If T ∈ UCPTn, then F(T) ∈ conv{W +

n , W − n }.

More precisely, if T = d

i=1 aixa∗ i (Choi canonical form), then

F(T) = c+(T)W +

n + c−(T)W − n ,

where c+(T) = 1

4 d

• i=1

ai + at

i 2 2 , c−(T) = 1 4 d

• i=1

ai − at

i 2 2 .

Corollary: (1) If at

i = ai for all i, then F(T) = W + n .

(2) If at

i = −ai for all i, then F(T) = W − n .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

◮ F(W +

n ) = W + n and F(W − n ) = W − n .

Theorem (Haagerup-M) If T ∈ UCPTn, then F(T) ∈ conv{W +

n , W − n }.

More precisely, if T = d

i=1 aixa∗ i (Choi canonical form), then

F(T) = c+(T)W +

n + c−(T)W − n ,

where c+(T) = 1

4 d

• i=1

ai + at

i 2 2 , c−(T) = 1 4 d

• i=1

ai − at

i 2 2 .

Corollary: (1) If at

i = ai for all i, then F(T) = W + n .

(2) If at

i = −ai for all i, then F(T) = W − n .

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Proof of the fact that W −

n is factorizable, for all n ≥ 7, n odd.

Let k = n−5

2

∈ N. Define S ∈ UCPTn by S(x): = W −

5 (x11)

W −

2k(x22)

• ,

where x = x11 x12 x21 x22

• ∈ Cn = C5 ⊕ C2k.

W −

5 ∈ F(M5(C)) , W − 2k ∈ conv(Aut(M2k(C))) ⇒ S ∈ F(Mn(C)).

An application of above corollary shows that F(S) = W −

n . The

conclusion follows.

Factorizable maps and AQBC Extreme points Semigroup case Connes embedding Holevo–Werner channels

Proof of the fact that dcb(W −

3 , F(M3(C))) ≥ 4/27.

Let T ∈ F(M3(C)). Then ∃(N, τN) and u ∈ U(M3(N)) s.t. Tx = (idM3(C) ⊗ τN)(u∗(x ⊗ 1N)u) , x ∈ M3(C) . A refinement of previous theorem shows that F(T) = λW +

3 + (1 − λ)W − 3 ,

where λ = 1

4u + ut2

• 2. Moreover, we can prove that λ ≥ 2
• 27. Then

W −

3 −Tcb ≥ W − 3 −F(T)cb = λW − 3 −W + 3 cb = 2λ ≥ 4/27 .