Completely positive maps in quantum information

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Completely positive maps in quantum information

  1. 1. Completely Positive Maps in Quantum Information Yiu Tung Poon Department of Mathematics Iowa State University Ames, Iowa, USA INFAS November 14, 2009 Joint work with Chi-Kwong Li.This research was partially supported by an NSF grant. Yiu Tung Poon Completely Positive Maps in Quantum Information
  2. 2. Basic notation and definitions H, K : Hilbert space. Yiu Tung Poon Completely Positive Maps in Quantum Information
  3. 3. Basic notation and definitions H, K : Hilbert space. B(H) (B(H, K)) : Bounded Linear operators on H (from H to K). Yiu Tung Poon Completely Positive Maps in Quantum Information
  4. 4. Basic notation and definitions H, K : Hilbert space. B(H) (B(H, K)) : Bounded Linear operators on H (from H to K). When dim H = n, take H = Cn , and B(H) = Mn , Yiu Tung Poon Completely Positive Maps in Quantum Information
  5. 5. Basic notation and definitions H, K : Hilbert space. B(H) (B(H, K)) : Bounded Linear operators on H (from H to K). When dim H = n, take H = Cn , and B(H) = Mn , the set of n × n complex matrices. Yiu Tung Poon Completely Positive Maps in Quantum Information
  6. 6. Basic notation and definitions H, K : Hilbert space. B(H) (B(H, K)) : Bounded Linear operators on H (from H to K). When dim H = n, take H = Cn , and B(H) = Mn , the set of n × n complex matrices. Similarly, take (B(H, K)) = Mm, n when dim K = m. Yiu Tung Poon Completely Positive Maps in Quantum Information
  7. 7. Basic notation and definitions H, K : Hilbert space. B(H) (B(H, K)) : Bounded Linear operators on H (from H to K). When dim H = n, take H = Cn , and B(H) = Mn , the set of n × n complex matrices. Similarly, take (B(H, K)) = Mm, n when dim K = m. B(H)+ : Positive semi-definite operators in B(H). For A ∈ B(H)+ , write A ≥ 0. Yiu Tung Poon Completely Positive Maps in Quantum Information
  8. 8. Basic notation and definitions H, K : Hilbert space. B(H) (B(H, K)) : Bounded Linear operators on H (from H to K). When dim H = n, take H = Cn , and B(H) = Mn , the set of n × n complex matrices. Similarly, take (B(H, K)) = Mm, n when dim K = m. B(H)+ : Positive semi-definite operators in B(H). For A ∈ B(H)+ , write A ≥ 0. A C∗ -algebra is a normed closed ∗-subalgebra of some B(H). Yiu Tung Poon Completely Positive Maps in Quantum Information
  9. 9. Basic notation and definitions H, K : Hilbert space. B(H) (B(H, K)) : Bounded Linear operators on H (from H to K). When dim H = n, take H = Cn , and B(H) = Mn , the set of n × n complex matrices. Similarly, take (B(H, K)) = Mm, n when dim K = m. B(H)+ : Positive semi-definite operators in B(H). For A ∈ B(H)+ , write A ≥ 0. A C∗ -algebra is a normed closed ∗-subalgebra of some B(H). A linear map Φ : A → B(K) is said to be positive if Φ(A) ≥ 0 for every A ≥ 0. (Φ ≥ 0) Yiu Tung Poon Completely Positive Maps in Quantum Information
  10. 10. Completely positive maps Let A be a C∗ -algebra. Suppose we have a linear map Φ : A → B(H). Yiu Tung Poon Completely Positive Maps in Quantum Information
  11. 11. Completely positive maps Let A be a C∗ -algebra. Suppose we have a linear map Φ : A → B(H). Then for every k ≥ 1, Yiu Tung Poon Completely Positive Maps in Quantum Information
  12. 12. Completely positive maps Let A be a C∗ -algebra. Suppose we have a linear map Φ : A → B(H). Then for every k ≥ 1, we have Φk : Mk (A) → Mk (B(H)) = B(Hk ) such that Φk ([aij ]) = [Φ (aij )]. Yiu Tung Poon Completely Positive Maps in Quantum Information
  13. 13. Completely positive maps Let A be a C∗ -algebra. Suppose we have a linear map Φ : A → B(H). Then for every k ≥ 1, we have Φk : Mk (A) → Mk (B(H)) = B(Hk ) such that Φk ([aij ]) = [Φ (aij )]. Φ is completely positive (CP) Yiu Tung Poon Completely Positive Maps in Quantum Information
  14. 14. Completely positive maps Let A be a C∗ -algebra. Suppose we have a linear map Φ : A → B(H). Then for every k ≥ 1, we have Φk : Mk (A) → Mk (B(H)) = B(Hk ) such that Φk ([aij ]) = [Φ (aij )]. Φ is completely positive (CP) if Φk ≥ 0 for all k ≥ 1. Yiu Tung Poon Completely Positive Maps in Quantum Information
  15. 15. Completely positive maps Let A be a C∗ -algebra. Suppose we have a linear map Φ : A → B(H). Then for every k ≥ 1, we have Φk : Mk (A) → Mk (B(H)) = B(Hk ) such that Φk ([aij ]) = [Φ (aij )]. Φ is completely positive (CP) if Φk ≥ 0 for all k ≥ 1. Define Φ : M2 → M2 by Φ(A) = At . Then Φ1 ≥ 0 but Φ2 ≥ 0 Yiu Tung Poon Completely Positive Maps in Quantum Information
  16. 16. Completely positive maps Let A be a C∗ -algebra. Suppose we have a linear map Φ : A → B(H). Then for every k ≥ 1, we have Φk : Mk (A) → Mk (B(H)) = B(Hk ) such that Φk ([aij ]) = [Φ (aij )]. Φ is completely positive (CP) if Φk ≥ 0 for all k ≥ 1. Define Φ : M2 → M2 by Φ(A) = At . Then Φ1 ≥ 0 but Φ2 ≥ 0     1 0 0 1 1 0 0 0   0 0 0 0     0 0 1 0   Φ2    =       0 0 0 0   0 1 0 0  1 0 0 1 0 0 0 1 Yiu Tung Poon Completely Positive Maps in Quantum Information
  17. 17. Stinespring’s Theorem Stinespring’s Theorem 1955 Let A be a C∗ -algebra with a unit, let H be a Hilbert space, and let Φ be a linear function from A to B(H). Then a necessary and sufficient condition for Φ to be completely positive is that Φ have the form Φ(A) = V ∗ π(A)V for all A ∈ A, where V ∈ B(H, K); and π is a ∗-representation of A into B(K). Yiu Tung Poon Completely Positive Maps in Quantum Information
  18. 18. Stinespring’s Theorem Stinespring’s Theorem 1955 Let A be a C∗ -algebra with a unit, let H be a Hilbert space, and let Φ be a linear function from A to B(H). Then a necessary and sufficient condition for Φ to be completely positive is that Φ have the form Φ(A) = V ∗ π(A)V for all A ∈ A, where V ∈ B(H, K); and π is a ∗-representation of A into B(K). We may assume that π(I) = IK . Yiu Tung Poon Completely Positive Maps in Quantum Information
  19. 19. Stinespring’s Theorem Stinespring’s Theorem 1955 Let A be a C∗ -algebra with a unit, let H be a Hilbert space, and let Φ be a linear function from A to B(H). Then a necessary and sufficient condition for Φ to be completely positive is that Φ have the form Φ(A) = V ∗ π(A)V for all A ∈ A, where V ∈ B(H, K); and π is a ∗-representation of A into B(K). We may assume that π(I) = IK . Φ is unital if and only if V is an isometry, Yiu Tung Poon Completely Positive Maps in Quantum Information
  20. 20. Stinespring’s Theorem Stinespring’s Theorem 1955 Let A be a C∗ -algebra with a unit, let H be a Hilbert space, and let Φ be a linear function from A to B(H). Then a necessary and sufficient condition for Φ to be completely positive is that Φ have the form Φ(A) = V ∗ π(A)V for all A ∈ A, where V ∈ B(H, K); and π is a ∗-representation of A into B(K). We may assume that π(I) = IK . Φ is unital if and only if V is an isometry, i.e. V ∗ V = IH . Yiu Tung Poon Completely Positive Maps in Quantum Information
  21. 21. Stinespring’s Theorem Stinespring’s Theorem 1955 Let A be a C∗ -algebra with a unit, let H be a Hilbert space, and let Φ be a linear function from A to B(H). Then a necessary and sufficient condition for Φ to be completely positive is that Φ have the form Φ(A) = V ∗ π(A)V for all A ∈ A, where V ∈ B(H, K); and π is a ∗-representation of A into B(K). We may assume that π(I) = IK . Φ is unital if and only if V is an isometry, i.e. V ∗ V = IH . In this case, we say that Φ(A) is a compression of π(A), Yiu Tung Poon Completely Positive Maps in Quantum Information
  22. 22. Stinespring’s Theorem Stinespring’s Theorem 1955 Let A be a C∗ -algebra with a unit, let H be a Hilbert space, and let Φ be a linear function from A to B(H). Then a necessary and sufficient condition for Φ to be completely positive is that Φ have the form Φ(A) = V ∗ π(A)V for all A ∈ A, where V ∈ B(H, K); and π is a ∗-representation of A into B(K). We may assume that π(I) = IK . Φ is unital if and only if V is an isometry, i.e. V ∗ V = IH . In this case, we say that Φ(A) is a compression of π(A), or π(A) is a dilation of Φ(A). Yiu Tung Poon Completely Positive Maps in Quantum Information
  23. 23. Choi’s Theorems Choi’s Theorem 1975 Let Φ : Mn → Mm . Then Φ is completely positive if and only if Φ is of the form r Φ(A) = Fj∗ AFj , (1) j=1 where F1 , . . . , Fr ∈ Mn, m . Yiu Tung Poon Completely Positive Maps in Quantum Information
  24. 24. Choi’s Theorems Choi’s Theorem 1975 Let Φ : Mn → Mm . Then Φ is completely positive if and only if Φ is of the form r Φ(A) = Fj∗ AFj , (1) j=1 where F1 , . . . , Fr ∈ Mn, m . r Φ is unital if and only if j=1 Fj∗ Fj = Im . Yiu Tung Poon Completely Positive Maps in Quantum Information
  25. 25. Choi’s Theorems Choi’s Theorem 1975 Let Φ : Mn → Mm . Then Φ is completely positive if and only if Φ is of the form r Φ(A) = Fj∗ AFj , (1) j=1 where F1 , . . . , Fr ∈ Mn, m . r Φ is unital if and only if j=1 Fj∗ Fj = Im . In quantum information, quantum channels are trace preserving completely positive maps, Yiu Tung Poon Completely Positive Maps in Quantum Information
  26. 26. Choi’s Theorems Choi’s Theorem 1975 Let Φ : Mn → Mm . Then Φ is completely positive if and only if Φ is of the form r Φ(A) = Fj∗ AFj , (1) j=1 where F1 , . . . , Fr ∈ Mn, m . r Φ is unital if and only if j=1 Fj∗ Fj = Im . In quantum information, quantum channels are trace preserving completely positive maps, i.e. tr Φ(A) = tr A. Yiu Tung Poon Completely Positive Maps in Quantum Information
  27. 27. Choi’s Theorems Choi’s Theorem 1975 Let Φ : Mn → Mm . Then Φ is completely positive if and only if Φ is of the form r Φ(A) = Fj∗ AFj , (1) j=1 where F1 , . . . , Fr ∈ Mn, m . r Φ is unital if and only if j=1 Fj∗ Fj = Im . In quantum information, quantum channels are trace preserving completely positive maps, i.e. tr Φ(A) = tr A. tr (ai j ) = i ai i . Yiu Tung Poon Completely Positive Maps in Quantum Information
  28. 28. Choi’s Theorems Choi’s Theorem 1975 Let Φ : Mn → Mm . Then Φ is completely positive if and only if Φ is of the form r Φ(A) = Fj∗ AFj , (1) j=1 where F1 , . . . , Fr ∈ Mn, m . r Φ is unital if and only if j=1 Fj∗ Fj = Im . In quantum information, quantum channels are trace preserving completely positive maps, i.e. tr Φ(A) = tr A. tr (ai j ) = i ai i . r Φ is trace preserving if and only if j=1 Fj Fj∗ = In . Yiu Tung Poon Completely Positive Maps in Quantum Information
  29. 29. Choi’s Theorems Choi’s Theorem 1975 Let Φ : Mn → Mm . Then Φ is completely positive if and only if Φ is of the form r Φ(A) = Fj∗ AFj , (1) j=1 where F1 , . . . , Fr ∈ Mn, m . r Φ is unital if and only if j=1 Fj∗ Fj = Im . In quantum information, quantum channels are trace preserving completely positive maps, i.e. tr Φ(A) = tr A. tr (ai j ) = i ai i . r Φ is trace preserving if and only if j=1 Fj Fj∗ = In . Φ is said to have Choi’s rank ≤ r if it can be represented in (1). Yiu Tung Poon Completely Positive Maps in Quantum Information
  30. 30. Choi’s Theorems Choi’s Theorem 1975 Let Φ : Mn → Mm . Then Φ is completely positive if and only if Φ is of the form r Φ(A) = Fj∗ AFj , (1) j=1 where F1 , . . . , Fr ∈ Mn, m . r Φ is unital if and only if j=1 Fj∗ Fj = Im . In quantum information, quantum channels are trace preserving completely positive maps, i.e. tr Φ(A) = tr A. tr (ai j ) = i ai i . r Φ is trace preserving if and only if j=1 Fj Fj∗ = In . Φ is said to have Choi’s rank ≤ r if it can be represented in (1). r can be interpreted as the dimension of the external environment E Yiu Tung Poon Completely Positive Maps in Quantum Information
  31. 31. Choi’s Theorems Choi’s Theorem 1975 Let Φ : Mn → Mm . Then Φ is completely positive if and only if Φ is of the form r Φ(A) = Fj∗ AFj , (1) j=1 where F1 , . . . , Fr ∈ Mn, m . r Φ is unital if and only if j=1 Fj∗ Fj = Im . In quantum information, quantum channels are trace preserving completely positive maps, i.e. tr Φ(A) = tr A. tr (ai j ) = i ai i . r Φ is trace preserving if and only if j=1 Fj Fj∗ = In . Φ is said to have Choi’s rank ≤ r if it can be represented in (1). r can be interpreted as the dimension of the external environment E required to couple with the principal system Yiu Tung Poon Completely Positive Maps in Quantum Information
  32. 32. Choi’s Theorems Choi’s Theorem 1975 Let Φ : Mn → Mm . Then Φ is completely positive if and only if Φ is of the form r Φ(A) = Fj∗ AFj , (1) j=1 where F1 , . . . , Fr ∈ Mn, m . r Φ is unital if and only if j=1 Fj∗ Fj = Im . In quantum information, quantum channels are trace preserving completely positive maps, i.e. tr Φ(A) = tr A. tr (ai j ) = i ai i . r Φ is trace preserving if and only if j=1 Fj Fj∗ = In . Φ is said to have Choi’s rank ≤ r if it can be represented in (1). r can be interpreted as the dimension of the external environment E required to couple with the principal system so that the CP map can be evaluated from the composite system. Yiu Tung Poon Completely Positive Maps in Quantum Information
  33. 33. Duality A, B = tr AB ∗ defines an inner product on Mn, m . Yiu Tung Poon Completely Positive Maps in Quantum Information
  34. 34. Duality A, B = tr AB ∗ defines an inner product on Mn, m . Suppose Φ : Mn → Mm is linear. Yiu Tung Poon Completely Positive Maps in Quantum Information
  35. 35. Duality A, B = tr AB ∗ defines an inner product on Mn, m . Suppose Φ : Mn → Mm is linear. Then the dual map Φ∗ : Mm → Mn is defined by Φ(A), B = A, Φ∗ (B) for all A ∈ Mn , B ∈ Mm . Yiu Tung Poon Completely Positive Maps in Quantum Information
  36. 36. Duality A, B = tr AB ∗ defines an inner product on Mn, m . Suppose Φ : Mn → Mm is linear. Then the dual map Φ∗ : Mm → Mn is defined by Φ(A), B = A, Φ∗ (B) for all A ∈ Mn , B ∈ Mm . If r Φ(A) = Fj∗ AFj , j=1 Yiu Tung Poon Completely Positive Maps in Quantum Information
  37. 37. Duality A, B = tr AB ∗ defines an inner product on Mn, m . Suppose Φ : Mn → Mm is linear. Then the dual map Φ∗ : Mm → Mn is defined by Φ(A), B = A, Φ∗ (B) for all A ∈ Mn , B ∈ Mm . If r Φ(A) = Fj∗ AFj , j=1 then r Φ∗ (B) = Fj BFj∗ . j=1 Yiu Tung Poon Completely Positive Maps in Quantum Information
  38. 38. Duality A, B = tr AB ∗ defines an inner product on Mn, m . Suppose Φ : Mn → Mm is linear. Then the dual map Φ∗ : Mm → Mn is defined by Φ(A), B = A, Φ∗ (B) for all A ∈ Mn , B ∈ Mm . If r Φ(A) = Fj∗ AFj , j=1 then r Φ∗ (B) = Fj BFj∗ . j=1 A completely positive map Φ : Mn → Mm is unital if and only if Φ∗ is trace preserving. Yiu Tung Poon Completely Positive Maps in Quantum Information
  39. 39. Duality A, B = tr AB ∗ defines an inner product on Mn, m . Suppose Φ : Mn → Mm is linear. Then the dual map Φ∗ : Mm → Mn is defined by Φ(A), B = A, Φ∗ (B) for all A ∈ Mn , B ∈ Mm . If r Φ(A) = Fj∗ AFj , j=1 then r Φ∗ (B) = Fj BFj∗ . j=1 A completely positive map Φ : Mn → Mm is unital if and only if Φ∗ is trace preserving. Can we use this duality to deduce results in trace preserving completely positive maps from those in unital completely positive maps and vice versa? Yiu Tung Poon Completely Positive Maps in Quantum Information
  40. 40. General problems 1 Given A ∈ Mn and B ∈ Mm , when is there a completely positive map Φ(A) = B? Yiu Tung Poon Completely Positive Maps in Quantum Information
  41. 41. General problems 1 Given A ∈ Mn and B ∈ Mm , when is there a completely positive map Φ(A) = B? 2 What about Φ(Ai ) = Bi for A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm ? Yiu Tung Poon Completely Positive Maps in Quantum Information
  42. 42. General problems 1 Given A ∈ Mn and B ∈ Mm , when is there a completely positive map Φ(A) = B? 2 What about Φ(Ai ) = Bi for A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm ? 3 Deduce properties of Φ based on the information of Φ(A) for some special matrices A. Yiu Tung Poon Completely Positive Maps in Quantum Information
  43. 43. General problems 1 Given A ∈ Mn and B ∈ Mm , when is there a completely positive map Φ(A) = B? 2 What about Φ(Ai ) = Bi for A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm ? 3 Deduce properties of Φ based on the information of Φ(A) for some special matrices A. 4 Let Hn = {A ∈ Mn : A = A∗ } be the set of n × n Hermitian matrices. Yiu Tung Poon Completely Positive Maps in Quantum Information
  44. 44. General problems 1 Given A ∈ Mn and B ∈ Mm , when is there a completely positive map Φ(A) = B? 2 What about Φ(Ai ) = Bi for A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm ? 3 Deduce properties of Φ based on the information of Φ(A) for some special matrices A. 4 Let Hn = {A ∈ Mn : A = A∗ } be the set of n × n Hermitian matrices. Φ is determined by its action on Hn . Yiu Tung Poon Completely Positive Maps in Quantum Information
  45. 45. General problems 1 Given A ∈ Mn and B ∈ Mm , when is there a completely positive map Φ(A) = B? 2 What about Φ(Ai ) = Bi for A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm ? 3 Deduce properties of Φ based on the information of Φ(A) for some special matrices A. 4 Let Hn = {A ∈ Mn : A = A∗ } be the set of n × n Hermitian matrices. Φ is determined by its action on Hn . 5 Given A1 , . . . , Ak ∈ Hn and B1 , . . . , Bk ∈ Hm , under what condition can we have a completely positive map Φ such that Φ(Ai ) = Bi ? Yiu Tung Poon Completely Positive Maps in Quantum Information
  46. 46. General problems 1 Given A ∈ Mn and B ∈ Mm , when is there a completely positive map Φ(A) = B? 2 What about Φ(Ai ) = Bi for A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm ? 3 Deduce properties of Φ based on the information of Φ(A) for some special matrices A. 4 Let Hn = {A ∈ Mn : A = A∗ } be the set of n × n Hermitian matrices. Φ is determined by its action on Hn . 5 Given A1 , . . . , Ak ∈ Hn and B1 , . . . , Bk ∈ Hm , under what condition can we have a completely positive map Φ such that Φ(Ai ) = Bi ? If Φ exists, can we choose Φ with additional properties? Yiu Tung Poon Completely Positive Maps in Quantum Information
  47. 47. General problems 1 Given A ∈ Mn and B ∈ Mm , when is there a completely positive map Φ(A) = B? 2 What about Φ(Ai ) = Bi for A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm ? 3 Deduce properties of Φ based on the information of Φ(A) for some special matrices A. 4 Let Hn = {A ∈ Mn : A = A∗ } be the set of n × n Hermitian matrices. Φ is determined by its action on Hn . 5 Given A1 , . . . , Ak ∈ Hn and B1 , . . . , Bk ∈ Hm , under what condition can we have a completely positive map Φ such that Φ(Ai ) = Bi ? If Φ exists, can we choose Φ with additional properties? E.g. unital, Yiu Tung Poon Completely Positive Maps in Quantum Information
  48. 48. General problems 1 Given A ∈ Mn and B ∈ Mm , when is there a completely positive map Φ(A) = B? 2 What about Φ(Ai ) = Bi for A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm ? 3 Deduce properties of Φ based on the information of Φ(A) for some special matrices A. 4 Let Hn = {A ∈ Mn : A = A∗ } be the set of n × n Hermitian matrices. Φ is determined by its action on Hn . 5 Given A1 , . . . , Ak ∈ Hn and B1 , . . . , Bk ∈ Hm , under what condition can we have a completely positive map Φ such that Φ(Ai ) = Bi ? If Φ exists, can we choose Φ with additional properties? E.g. unital, trace preserving, Yiu Tung Poon Completely Positive Maps in Quantum Information
  49. 49. General problems 1 Given A ∈ Mn and B ∈ Mm , when is there a completely positive map Φ(A) = B? 2 What about Φ(Ai ) = Bi for A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm ? 3 Deduce properties of Φ based on the information of Φ(A) for some special matrices A. 4 Let Hn = {A ∈ Mn : A = A∗ } be the set of n × n Hermitian matrices. Φ is determined by its action on Hn . 5 Given A1 , . . . , Ak ∈ Hn and B1 , . . . , Bk ∈ Hm , under what condition can we have a completely positive map Φ such that Φ(Ai ) = Bi ? If Φ exists, can we choose Φ with additional properties? E.g. unital, trace preserving, Choi’s rank ≤ r. Yiu Tung Poon Completely Positive Maps in Quantum Information
  50. 50. Why study? Yiu Tung Poon Completely Positive Maps in Quantum Information
  51. 51. Why study? It is a general interpolation problem for completely positive maps. Yiu Tung Poon Completely Positive Maps in Quantum Information
  52. 52. Why study? It is a general interpolation problem for completely positive maps. It is related to dilation/compression problems of operators. Yiu Tung Poon Completely Positive Maps in Quantum Information
  53. 53. Why study? It is a general interpolation problem for completely positive maps. It is related to dilation/compression problems of operators. In quantum information, quantum operations and channels are completely positive maps. Yiu Tung Poon Completely Positive Maps in Quantum Information
  54. 54. Why study? It is a general interpolation problem for completely positive maps. It is related to dilation/compression problems of operators. In quantum information, quantum operations and channels are completely positive maps. The study is relevant to problems in quantum tomography, quantum control and quantum error correction. Yiu Tung Poon Completely Positive Maps in Quantum Information
  55. 55. Why study? It is a general interpolation problem for completely positive maps. It is related to dilation/compression problems of operators. In quantum information, quantum operations and channels are completely positive maps. The study is relevant to problems in quantum tomography, quantum control and quantum error correction. One often tries to deduce the properties of Φ through the study of Φ(A) for some special A. Yiu Tung Poon Completely Positive Maps in Quantum Information
  56. 56. Why study? It is a general interpolation problem for completely positive maps. It is related to dilation/compression problems of operators. In quantum information, quantum operations and channels are completely positive maps. The study is relevant to problems in quantum tomography, quantum control and quantum error correction. One often tries to deduce the properties of Φ through the study of Φ(A) for some special A. The study is related to other topics such as matrix inequalities (majorization), unitary orbits (algebraic, analytic and geometric properties), algebraic combinatorics etc. Yiu Tung Poon Completely Positive Maps in Quantum Information
  57. 57. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . Then there is a completely positive map Φ : Mn → Mm such that Φ(A) = B if and only if there are real numbers γ1 , γ2 ≥ 0 such that γ1 λ1 (A) ≥ λ1 (B) and λm (B) ≥ γ2 λn (A). (2) Yiu Tung Poon Completely Positive Maps in Quantum Information
  58. 58. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . Then there is a completely positive map Φ : Mn → Mm such that Φ(A) = B if and only if there are real numbers γ1 , γ2 ≥ 0 such that γ1 λ1 (A) ≥ λ1 (B) and λm (B) ≥ γ2 λn (A). (2) Moreover, there is γ = γ1 = γ2 > 0 satisfies (2) Yiu Tung Poon Completely Positive Maps in Quantum Information
  59. 59. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . Then there is a completely positive map Φ : Mn → Mm such that Φ(A) = B if and only if there are real numbers γ1 , γ2 ≥ 0 such that γ1 λ1 (A) ≥ λ1 (B) and λm (B) ≥ γ2 λn (A). (2) Moreover, there is γ = γ1 = γ2 > 0 satisfies (2) if and only if there is a completely positive map Φ : Mn → Mm such that Φ(In ) = γIm and Φ(A) = B. Yiu Tung Poon Completely Positive Maps in Quantum Information
  60. 60. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . Then there is a completely positive map Φ : Mn → Mm such that Φ(A) = B if and only if there are real numbers γ1 , γ2 ≥ 0 such that γ1 λ1 (A) ≥ λ1 (B) and λm (B) ≥ γ2 λn (A). (2) Moreover, there is γ = γ1 = γ2 > 0 satisfies (2) if and only if there is a completely positive map Φ : Mn → Mm such that Φ(In ) = γIm and Φ(A) = B. Here, λ1 (A) ≥ · · · ≥ λn (A) are the eigenvalues of A. Yiu Tung Poon Completely Positive Maps in Quantum Information
  61. 61. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . Then there is a completely positive map Φ : Mn → Mm such that Φ(A) = B if and only if there are real numbers γ1 , γ2 ≥ 0 such that γ1 λ1 (A) ≥ λ1 (B) and λm (B) ≥ γ2 λn (A). (2) Moreover, there is γ = γ1 = γ2 > 0 satisfies (2) if and only if there is a completely positive map Φ : Mn → Mm such that Φ(In ) = γIm and Φ(A) = B. Here, λ1 (A) ≥ · · · ≥ λn (A) are the eigenvalues of A. Let λ(A) = (λ1 (A) ≥ · · · ≥ λn (A)). Yiu Tung Poon Completely Positive Maps in Quantum Information
  62. 62. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . Then there is a completely positive map Φ : Mn → Mm such that Φ(A) = B if and only if there are real numbers γ1 , γ2 ≥ 0 such that γ1 λ1 (A) ≥ λ1 (B) and λm (B) ≥ γ2 λn (A). (2) Moreover, there is γ = γ1 = γ2 > 0 satisfies (2) if and only if there is a completely positive map Φ : Mn → Mm such that Φ(In ) = γIm and Φ(A) = B. Here, λ1 (A) ≥ · · · ≥ λn (A) are the eigenvalues of A. Let λ(A) = (λ1 (A) ≥ · · · ≥ λn (A)). Majorization Let a = (a1 , . . . , an ) and b = (b1 , . . . , bn ), with ai ≥ ai+1 and bi ≥ bi+1 for 1 ≤ i ≤ n − 1. Yiu Tung Poon Completely Positive Maps in Quantum Information
  63. 63. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . Then there is a completely positive map Φ : Mn → Mm such that Φ(A) = B if and only if there are real numbers γ1 , γ2 ≥ 0 such that γ1 λ1 (A) ≥ λ1 (B) and λm (B) ≥ γ2 λn (A). (2) Moreover, there is γ = γ1 = γ2 > 0 satisfies (2) if and only if there is a completely positive map Φ : Mn → Mm such that Φ(In ) = γIm and Φ(A) = B. Here, λ1 (A) ≥ · · · ≥ λn (A) are the eigenvalues of A. Let λ(A) = (λ1 (A) ≥ · · · ≥ λn (A)). Majorization Let a = (a1 , . . . , an ) and b = (b1 , . . . , bn ), with ai ≥ ai+1 and bi ≥ bi+1 for 1 ≤ i ≤ n − 1. b is said to be majorized by a, b a if Yiu Tung Poon Completely Positive Maps in Quantum Information
  64. 64. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . Then there is a completely positive map Φ : Mn → Mm such that Φ(A) = B if and only if there are real numbers γ1 , γ2 ≥ 0 such that γ1 λ1 (A) ≥ λ1 (B) and λm (B) ≥ γ2 λn (A). (2) Moreover, there is γ = γ1 = γ2 > 0 satisfies (2) if and only if there is a completely positive map Φ : Mn → Mm such that Φ(In ) = γIm and Φ(A) = B. Here, λ1 (A) ≥ · · · ≥ λn (A) are the eigenvalues of A. Let λ(A) = (λ1 (A) ≥ · · · ≥ λn (A)). Majorization Let a = (a1 , . . . , an ) and b = (b1 , . . . , bn ), with ai ≥ ai+1 and bi ≥ bi+1 for 1 ≤ i ≤ n − 1. b is said to be majorized by a, b a if k k n n i=1 bi ≤ i=1 ai for all 1 ≤ k ≤ n − 1 and i=1 bi = i=1 ai . Yiu Tung Poon Completely Positive Maps in Quantum Information
  65. 65. Some results for k = 1 Theorem Suppose A ∈ Hn has k non-negative eigenvalues and n − k negative eigenvalues, and B ∈ Hm . The following conditions are equivalent. Yiu Tung Poon Completely Positive Maps in Quantum Information
  66. 66. Some results for k = 1 Theorem Suppose A ∈ Hn has k non-negative eigenvalues and n − k negative eigenvalues, and B ∈ Hm . The following conditions are equivalent. (a) There is a trace preserving completely positive linear map Φ : Mn → Mm such that φ(A) = B. Yiu Tung Poon Completely Positive Maps in Quantum Information
  67. 67. Some results for k = 1 Theorem Suppose A ∈ Hn has k non-negative eigenvalues and n − k negative eigenvalues, and B ∈ Hm . The following conditions are equivalent. (a) There is a trace preserving completely positive linear map Φ : Mn → Mm such that φ(A) = B. (b) λ(B) (a+ , 0, . . . , 0, a− ) ∈ Rm where k n a+ = j=1 λj (A) and a− = j=k+1 λj (A). Yiu Tung Poon Completely Positive Maps in Quantum Information
  68. 68. Some results for k = 1 Theorem Suppose A ∈ Hn has k non-negative eigenvalues and n − k negative eigenvalues, and B ∈ Hm . The following conditions are equivalent. (a) There is a trace preserving completely positive linear map Φ : Mn → Mm such that φ(A) = B. (b) λ(B) (a+ , 0, . . . , 0, a− ) ∈ Rm where k n a+ = j=1 λj (A) and a− = j=k+1 λj (A). (c) There is an n × m row stochastic matrix D such that λ(B) = λ(A)D. Yiu Tung Poon Completely Positive Maps in Quantum Information
  69. 69. Some results for k = 1 Theorem Suppose A ∈ Hn has k non-negative eigenvalues and n − k negative eigenvalues, and B ∈ Hm . The following conditions are equivalent. (a) There is a trace preserving completely positive linear map Φ : Mn → Mm such that φ(A) = B. (b) λ(B) (a+ , 0, . . . , 0, a− ) ∈ Rm where k n a+ = j=1 λj (A) and a− = j=k+1 λj (A). (c) There is an n × m row stochastic matrix D such that λ(B) = λ(A)D. D = (di j ) with di j ≥ 0, Yiu Tung Poon Completely Positive Maps in Quantum Information
  70. 70. Some results for k = 1 Theorem Suppose A ∈ Hn has k non-negative eigenvalues and n − k negative eigenvalues, and B ∈ Hm . The following conditions are equivalent. (a) There is a trace preserving completely positive linear map Φ : Mn → Mm such that φ(A) = B. (b) λ(B) (a+ , 0, . . . , 0, a− ) ∈ Rm where k n a+ = j=1 λj (A) and a− = j=k+1 λj (A). (c) There is an n × m row stochastic matrix D such that n λ(B) = λ(A)D. D = (di j ) with di j ≥ 0, j=1 di j = 1 for all 1 ≤ i ≤ n. Yiu Tung Poon Completely Positive Maps in Quantum Information
  71. 71. Some results for k = 1 Theorem Suppose A ∈ Hn has k non-negative eigenvalues and n − k negative eigenvalues, and B ∈ Hm . The following conditions are equivalent. (a) There is a trace preserving completely positive linear map Φ : Mn → Mm such that φ(A) = B. (b) λ(B) (a+ , 0, . . . , 0, a− ) ∈ Rm where k n a+ = j=1 λj (A) and a− = j=k+1 λj (A). (c) There is an n × m row stochastic matrix D such that n λ(B) = λ(A)D. D = (di j ) with di j ≥ 0, j=1 di j = 1 for all 1 ≤ i ≤ n. (d) There is an n × m row stochastic matrix D with the first k rows all equal and the last n − k rows all equal such that λ(B) = λ(A)D. Yiu Tung Poon Completely Positive Maps in Quantum Information
  72. 72. Some results for k = 1 Theorem Suppose A ∈ Hn has k non-negative eigenvalues and n − k negative eigenvalues, and B ∈ Hm . The following conditions are equivalent. (a) There is a trace preserving completely positive linear map Φ : Mn → Mm such that φ(A) = B. (b) λ(B) (a+ , 0, . . . , 0, a− ) ∈ Rm where k n a+ = j=1 λj (A) and a− = j=k+1 λj (A). (c) There is an n × m row stochastic matrix D such that n λ(B) = λ(A)D. D = (di j ) with di j ≥ 0, j=1 di j = 1 for all 1 ≤ i ≤ n. (d) There is an n × m row stochastic matrix D with the first k rows all equal and the last n − k rows all equal such that λ(B) = λ(A)D. Remark For density matrices, (A, B ≥ 0, tr A = tr B = 1), we have λ(B) (1, 0, . . . , 0) and condition (b) holds. Yiu Tung Poon Completely Positive Maps in Quantum Information
  73. 73. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent. Yiu Tung Poon Completely Positive Maps in Quantum Information
  74. 74. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent. There is a unital completely positive linear map Φ such that Φ(A) = B. Yiu Tung Poon Completely Positive Maps in Quantum Information
  75. 75. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent. There is a unital completely positive linear map Φ such that Φ(A) = B. λn (A) ≤ λj (B) ≤ λ1 (A) for all j = 1, . . . , m. Yiu Tung Poon Completely Positive Maps in Quantum Information
  76. 76. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent. There is a unital completely positive linear map Φ such that Φ(A) = B. λn (A) ≤ λj (B) ≤ λ1 (A) for all j = 1, . . . , m. There is an n × m column stochastic matrix D such that λ(B) = λ(A)D. Yiu Tung Poon Completely Positive Maps in Quantum Information
  77. 77. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent. There is a unital completely positive linear map Φ such that Φ(A) = B. λn (A) ≤ λj (B) ≤ λ1 (A) for all j = 1, . . . , m. There is an n × m column stochastic matrix D such that λ(B) = λ(A)D. Remark The condition may fail even if A and B are density matrices. Yiu Tung Poon Completely Positive Maps in Quantum Information
  78. 78. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent. There is a unital completely positive linear map Φ such that Φ(A) = B. λn (A) ≤ λj (B) ≤ λ1 (A) for all j = 1, . . . , m. There is an n × m column stochastic matrix D such that λ(B) = λ(A)D. Remark The condition may fail even if A and B are density matrices. Question Suppose there is a unital completely positive map sending A to B, Yiu Tung Poon Completely Positive Maps in Quantum Information
  79. 79. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent. There is a unital completely positive linear map Φ such that Φ(A) = B. λn (A) ≤ λj (B) ≤ λ1 (A) for all j = 1, . . . , m. There is an n × m column stochastic matrix D such that λ(B) = λ(A)D. Remark The condition may fail even if A and B are density matrices. Question Suppose there is a unital completely positive map sending A to B, and a trace preserving completely positive map sending A to B. Yiu Tung Poon Completely Positive Maps in Quantum Information
  80. 80. Some results for k = 1 Theorem Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent. There is a unital completely positive linear map Φ such that Φ(A) = B. λn (A) ≤ λj (B) ≤ λ1 (A) for all j = 1, . . . , m. There is an n × m column stochastic matrix D such that λ(B) = λ(A)D. Remark The condition may fail even if A and B are density matrices. Question Suppose there is a unital completely positive map sending A to B, and a trace preserving completely positive map sending A to B. Is there a unital and trace preserving completely positive map sending A to B? Yiu Tung Poon Completely Positive Maps in Quantum Information
  81. 81. Some results for k = 1 Example Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Yiu Tung Poon Completely Positive Maps in Quantum Information
  82. 82. Some results for k = 1 Example Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is a trace preserving CP map sending A to B, Yiu Tung Poon Completely Positive Maps in Quantum Information
  83. 83. Some results for k = 1 Example Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is a trace preserving CP map sending A to B, and also a unital CP map sending A to B. Yiu Tung Poon Completely Positive Maps in Quantum Information
  84. 84. Some results for k = 1 Example Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is a trace preserving CP map sending A to B, and also a unital CP map sending A to B. Let A1 = A − I4 = diag (3, 0, 0, −1) and B1 = B − I4 = diag (2, 2, −1, −1). Yiu Tung Poon Completely Positive Maps in Quantum Information
  85. 85. Some results for k = 1 Example Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is a trace preserving CP map sending A to B, and also a unital CP map sending A to B. Let A1 = A − I4 = diag (3, 0, 0, −1) and B1 = B − I4 = diag (2, 2, −1, −1). Then there is no trace preserving CP map sending A1 to B1 . Yiu Tung Poon Completely Positive Maps in Quantum Information
  86. 86. Some results for k = 1 Example Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is a trace preserving CP map sending A to B, and also a unital CP map sending A to B. Let A1 = A − I4 = diag (3, 0, 0, −1) and B1 = B − I4 = diag (2, 2, −1, −1). Then there is no trace preserving CP map sending A1 to B1 . Hence, there is no unital trace preserving CP map sending A to B. Yiu Tung Poon Completely Positive Maps in Quantum Information
  87. 87. Some results for k = 1 Example Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is a trace preserving CP map sending A to B, and also a unital CP map sending A to B. Let A1 = A − I4 = diag (3, 0, 0, −1) and B1 = B − I4 = diag (2, 2, −1, −1). Then there is no trace preserving CP map sending A1 to B1 . Hence, there is no unital trace preserving CP map sending A to B. Randomized unitary channel r ∗ Suppose Φ : Mn → Mn is given by Φ(A) = j=1 ti Uj AUj , where r U1 , . . . , Ur are unitaries and ti ≥ 0, with j=1 ti = 1. Yiu Tung Poon Completely Positive Maps in Quantum Information
  88. 88. Some results for k = 1 Example Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is a trace preserving CP map sending A to B, and also a unital CP map sending A to B. Let A1 = A − I4 = diag (3, 0, 0, −1) and B1 = B − I4 = diag (2, 2, −1, −1). Then there is no trace preserving CP map sending A1 to B1 . Hence, there is no unital trace preserving CP map sending A to B. Randomized unitary channel r ∗ Suppose Φ : Mn → Mn is given by Φ(A) = j=1 ti Uj AUj , where r U1 , . . . , Ur are unitaries and ti ≥ 0, with j=1 ti = 1. Then Φ is known as a randomized unitary channel in quantum information. Yiu Tung Poon Completely Positive Maps in Quantum Information
  89. 89. Some results for k = 1 Example Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is a trace preserving CP map sending A to B, and also a unital CP map sending A to B. Let A1 = A − I4 = diag (3, 0, 0, −1) and B1 = B − I4 = diag (2, 2, −1, −1). Then there is no trace preserving CP map sending A1 to B1 . Hence, there is no unital trace preserving CP map sending A to B. Randomized unitary channel r ∗ Suppose Φ : Mn → Mn is given by Φ(A) = j=1 ti Uj AUj , where r U1 , . . . , Ur are unitaries and ti ≥ 0, with j=1 ti = 1. Then Φ is known as a randomized unitary channel in quantum information. In this case, Φ is a unital and trace preserving CP map. Yiu Tung Poon Completely Positive Maps in Quantum Information
  90. 90. Some results for k = 1 Example Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is a trace preserving CP map sending A to B, and also a unital CP map sending A to B. Let A1 = A − I4 = diag (3, 0, 0, −1) and B1 = B − I4 = diag (2, 2, −1, −1). Then there is no trace preserving CP map sending A1 to B1 . Hence, there is no unital trace preserving CP map sending A to B. Randomized unitary channel r ∗ Suppose Φ : Mn → Mn is given by Φ(A) = j=1 ti Uj AUj , where r U1 , . . . , Ur are unitaries and ti ≥ 0, with j=1 ti = 1. Then Φ is known as a randomized unitary channel in quantum information. In this case, Φ is a unital and trace preserving CP map. For n ≥ 3, there exists a unital and trace preserving CP map Φ : Mn → Mn which is not a randomized unitary channel. Yiu Tung Poon Completely Positive Maps in Quantum Information
  91. 91. TheoremLet A, B ∈ Hn . The following conditions are equivalent. Yiu Tung Poon Completely Positive Maps in Quantum Information
  92. 92. TheoremLet A, B ∈ Hn . The following conditions are equivalent. Φ(A) = B for a unital trace preserving CP map Φ. Yiu Tung Poon Completely Positive Maps in Quantum Information
  93. 93. TheoremLet A, B ∈ Hn . The following conditions are equivalent. Φ(A) = B for a unital trace preserving CP map Φ. Φ(A) = B for a randomized unitary channel Φ. Yiu Tung Poon Completely Positive Maps in Quantum Information
  94. 94. TheoremLet A, B ∈ Hn . The following conditions are equivalent. Φ(A) = B for a unital trace preserving CP map Φ. Φ(A) = B for a randomized unitary channel Φ. Φ(A) = B for a randomized unitary channel Φ, with all ti = 1/n. Yiu Tung Poon Completely Positive Maps in Quantum Information
  95. 95. TheoremLet A, B ∈ Hn . The following conditions are equivalent. Φ(A) = B for a unital trace preserving CP map Φ. Φ(A) = B for a randomized unitary channel Φ. Φ(A) = B for a randomized unitary channel Φ, with all ti = 1/n. For each t ∈ R, there exists a trace preserving CP map Φt such that Φt (A − tI) = B − tI. Yiu Tung Poon Completely Positive Maps in Quantum Information
  96. 96. TheoremLet A, B ∈ Hn . The following conditions are equivalent. Φ(A) = B for a unital trace preserving CP map Φ. Φ(A) = B for a randomized unitary channel Φ. Φ(A) = B for a randomized unitary channel Φ, with all ti = 1/n. For each t ∈ R, there exists a trace preserving CP map Φt such that Φt (A − tI) = B − tI. λ(B) = diag (U ∗ AU ) for some unitary U . Yiu Tung Poon Completely Positive Maps in Quantum Information
  97. 97. TheoremLet A, B ∈ Hn . The following conditions are equivalent. Φ(A) = B for a unital trace preserving CP map Φ. Φ(A) = B for a randomized unitary channel Φ. Φ(A) = B for a randomized unitary channel Φ, with all ti = 1/n. For each t ∈ R, there exists a trace preserving CP map Φt such that Φt (A − tI) = B − tI. λ(B) = diag (U ∗ AU ) for some unitary U . λ(B) λ(A) Yiu Tung Poon Completely Positive Maps in Quantum Information
  98. 98. TheoremLet A, B ∈ Hn . The following conditions are equivalent. Φ(A) = B for a unital trace preserving CP map Φ. Φ(A) = B for a randomized unitary channel Φ. Φ(A) = B for a randomized unitary channel Φ, with all ti = 1/n. For each t ∈ R, there exists a trace preserving CP map Φt such that Φt (A − tI) = B − tI. λ(B) = diag (U ∗ AU ) for some unitary U . λ(B) λ(A) There is an n × n doubly stochastic (d.s.) matrix D such that λ(B) = λ(A)D. Yiu Tung Poon Completely Positive Maps in Quantum Information
  99. 99. TheoremLet A, B ∈ Hn . The following conditions are equivalent. Φ(A) = B for a unital trace preserving CP map Φ. Φ(A) = B for a randomized unitary channel Φ. Φ(A) = B for a randomized unitary channel Φ, with all ti = 1/n. For each t ∈ R, there exists a trace preserving CP map Φt such that Φt (A − tI) = B − tI. λ(B) = diag (U ∗ AU ) for some unitary U . λ(B) λ(A) There is an n × n doubly stochastic (d.s.) matrix D such that λ(B) = λ(A)D.D is d.s. if it is both row stochastic and column stochastic. Yiu Tung Poon Completely Positive Maps in Quantum Information
  100. 100. Restricted rank Yiu Tung Poon Completely Positive Maps in Quantum Information
  101. 101. Restricted rank We will use CPr (n, m) to denote CP maps with Choi’s rank ≤ r Yiu Tung Poon Completely Positive Maps in Quantum Information
  102. 102. Restricted rank We will use CPr (n, m) to denote CP maps with Choi’s rank ≤ r and CP (n, m) = r≥1 CPr (n, m). Yiu Tung Poon Completely Positive Maps in Quantum Information
  103. 103. Restricted rank We will use CPr (n, m) to denote CP maps with Choi’s rank ≤ r and CP (n, m) = r≥1 CPr (n, m). Theorem Let A ∈ Hn and B ∈ Hm . Suppose nr ≥ m. The following conditions are equivalent. Yiu Tung Poon Completely Positive Maps in Quantum Information
  104. 104. Restricted rank We will use CPr (n, m) to denote CP maps with Choi’s rank ≤ r and CP (n, m) = r≥1 CPr (n, m). Theorem Let A ∈ Hn and B ∈ Hm . Suppose nr ≥ m. The following conditions are equivalent. There exists Φ ∈ CPr (n, m) such that Φ(A) = B. Yiu Tung Poon Completely Positive Maps in Quantum Information
  105. 105. Restricted rank We will use CPr (n, m) to denote CP maps with Choi’s rank ≤ r and CP (n, m) = r≥1 CPr (n, m). Theorem Let A ∈ Hn and B ∈ Hm . Suppose nr ≥ m. The following conditions are equivalent. There exists Φ ∈ CPr (n, m) such that Φ(A) = B. There exists an (nr) × m matrix F such that B = F ∗ (A ⊗ Ir )F . Yiu Tung Poon Completely Positive Maps in Quantum Information
  106. 106. Restricted rank We will use CPr (n, m) to denote CP maps with Choi’s rank ≤ r and CP (n, m) = r≥1 CPr (n, m). Theorem Let A ∈ Hn and B ∈ Hm . Suppose nr ≥ m. The following conditions are equivalent. There exists Φ ∈ CPr (n, m) such that Φ(A) = B. There exists an (nr) × m matrix F such that B = F ∗ (A ⊗ Ir )F . The number of positive (negative) eigenvalues of A ⊗ Ir is not less than the number of positive (negative) eigenvalues of B. Yiu Tung Poon Completely Positive Maps in Quantum Information
  107. 107. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm . Suppose nr ≥ m. The following conditions are equivalent. Yiu Tung Poon Completely Positive Maps in Quantum Information
  108. 108. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm . Suppose nr ≥ m. The following conditions are equivalent. There exists a unital Φ ∈ CPr (n, m) such that Φ(A) = B. Yiu Tung Poon Completely Positive Maps in Quantum Information
  109. 109. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm . Suppose nr ≥ m. The following conditions are equivalent. There exists a unital Φ ∈ CPr (n, m) such that Φ(A) = B. B is a compression of A ⊗ Ir , i.e. B = F ∗ (A ⊗ Ir )F for some (nr) × m F such that F ∗ F = Im . Yiu Tung Poon Completely Positive Maps in Quantum Information
  110. 110. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm . Suppose nr ≥ m. The following conditions are equivalent. There exists a unital Φ ∈ CPr (n, m) such that Φ(A) = B. B is a compression of A ⊗ Ir , i.e. B = F ∗ (A ⊗ Ir )F for some (nr) × m F such that F ∗ F = Im . The eigenvalues of A ⊗ Ir interlace those of B, i.e., λj (A ⊗ Ir ) ≥ λj (B) and λm+1−j (B) ≥ λn+1−j (A ⊗ Ir ) for 1 ≤ j ≤ m. Yiu Tung Poon Completely Positive Maps in Quantum Information
  111. 111. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm . Suppose nr ≥ m. The following conditions are equivalent. There exists a unital Φ ∈ CPr (n, m) such that Φ(A) = B. B is a compression of A ⊗ Ir , i.e. B = F ∗ (A ⊗ Ir )F for some (nr) × m F such that F ∗ F = Im . The eigenvalues of A ⊗ Ir interlace those of B, i.e., λj (A ⊗ Ir ) ≥ λj (B) and λm+1−j (B) ≥ λn+1−j (A ⊗ Ir ) for 1 ≤ j ≤ m. Remark Even if A, B are density matrices, the roles of A and B are not symmetric in the last two theorems. Yiu Tung Poon Completely Positive Maps in Quantum Information
  112. 112. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm be positive semidefinite, with mr ≥ n. The following conditions are equivalent. Yiu Tung Poon Completely Positive Maps in Quantum Information
  113. 113. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm be positive semidefinite, with mr ≥ n. The following conditions are equivalent. (a) There exists a trace preserving Φ ∈ CPr (n, m) such that Φ(A) = B. Yiu Tung Poon Completely Positive Maps in Quantum Information
  114. 114. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm be positive semidefinite, with mr ≥ n. The following conditions are equivalent. (a) There exists a trace preserving Φ ∈ CPr (n, m) such that Φ(A) = B. (b) There exists a unitary U ∈ Mmr such that U ∗ (A ⊕ Omr−n )U = (Ci j ), Ci j ∈ Mm Yiu Tung Poon Completely Positive Maps in Quantum Information
  115. 115. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm be positive semidefinite, with mr ≥ n. The following conditions are equivalent. (a) There exists a trace preserving Φ ∈ CPr (n, m) such that Φ(A) = B. (b) There exists a unitary U ∈ Mmr such that r U ∗ (A ⊕ Omr−n )U = (Ci j ), Ci j ∈ Mm with B = j=1 Cjj . Yiu Tung Poon Completely Positive Maps in Quantum Information
  116. 116. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm be positive semidefinite, with mr ≥ n. The following conditions are equivalent. (a) There exists a trace preserving Φ ∈ CPr (n, m) such that Φ(A) = B. (b) There exists a unitary U ∈ Mmr such that r U ∗ (A ⊕ Omr−n )U = (Ci j ), Ci j ∈ Mm with B = j=1 Cjj . (c) There exist positive semidefinite matrices C1 , . . . , Cr ∈ Hm , Yiu Tung Poon Completely Positive Maps in Quantum Information
  117. 117. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm be positive semidefinite, with mr ≥ n. The following conditions are equivalent. (a) There exists a trace preserving Φ ∈ CPr (n, m) such that Φ(A) = B. (b) There exists a unitary U ∈ Mmr such that r U ∗ (A ⊕ Omr−n )U = (Ci j ), Ci j ∈ Mm with B = j=1 Cjj . (c) There exist positive semidefinite matrices C1 , . . . , Cr ∈ Hm , and unitary matrices U1 , . . . , Ur ∈ Mn Yiu Tung Poon Completely Positive Maps in Quantum Information
  118. 118. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm be positive semidefinite, with mr ≥ n. The following conditions are equivalent. (a) There exists a trace preserving Φ ∈ CPr (n, m) such that Φ(A) = B. (b) There exists a unitary U ∈ Mmr such that r U ∗ (A ⊕ Omr−n )U = (Ci j ), Ci j ∈ Mm with B = j=1 Cjj . (c) There exist positive semidefinite matrices C1 , . . . , Cr ∈ Hm , and unitary matrices U1 , . . . , Ur ∈ Mn such that r ∗ r A = j=1 Uj (Cj ⊕ On−m ) Uj and B = j=1 Cj . Yiu Tung Poon Completely Positive Maps in Quantum Information
  119. 119. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm be positive semidefinite, with mr ≥ n. The following conditions are equivalent. (a) There exists a trace preserving Φ ∈ CPr (n, m) such that Φ(A) = B. (b) There exists a unitary U ∈ Mmr such that r U ∗ (A ⊕ Omr−n )U = (Ci j ), Ci j ∈ Mm with B = j=1 Cjj . (c) There exist positive semidefinite matrices C1 , . . . , Cr ∈ Hm , and unitary matrices U1 , . . . , Ur ∈ Mn such that r ∗ r A = j=1 Uj (Cj ⊕ On−m ) Uj and B = j=1 Cj . (d) There exists a trace preserving Ψ ∈ CPr (m, n) such that Ψ(B) = A. Yiu Tung Poon Completely Positive Maps in Quantum Information
  120. 120. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm be positive semidefinite, with mr ≥ n. The following conditions are equivalent. (a) There exists a trace preserving Φ ∈ CPr (n, m) such that Φ(A) = B. (b) There exists a unitary U ∈ Mmr such that r U ∗ (A ⊕ Omr−n )U = (Ci j ), Ci j ∈ Mm with B = j=1 Cjj . (c) There exist positive semidefinite matrices C1 , . . . , Cr ∈ Hm , and unitary matrices U1 , . . . , Ur ∈ Mn such that r ∗ r A = j=1 Uj (Cj ⊕ On−m ) Uj and B = j=1 Cj . (d) There exists a trace preserving Ψ ∈ CPr (m, n) such that Ψ(B) = A. Cj can be choose in the unitary orbit of some C ∈ Hm . Yiu Tung Poon Completely Positive Maps in Quantum Information
  121. 121. Restricted rank Theorem Let A ∈ Hn and B ∈ Hm be positive semidefinite, with mr ≥ n. The following conditions are equivalent. (a) There exists a trace preserving Φ ∈ CPr (n, m) such that Φ(A) = B. (b) There exists a unitary U ∈ Mmr such that r U ∗ (A ⊕ Omr−n )U = (Ci j ), Ci j ∈ Mm with B = j=1 Cjj . (c) There exist positive semidefinite matrices C1 , . . . , Cr ∈ Hm , and unitary matrices U1 , . . . , Ur ∈ Mn such that r ∗ r A = j=1 Uj (Cj ⊕ On−m ) Uj and B = j=1 Cj . (d) There exists a trace preserving Ψ ∈ CPr (m, n) such that Ψ(B) = A. Cj can be choose in the unitary orbit of some C ∈ Hm . Remark Conditions (c) can be expressed in terms of eigenvalue inequalities using Littlewood-Richardson rules in Schubert calculus. Yiu Tung Poon Completely Positive Maps in Quantum Information
  122. 122. Results and questions on multiple matrices Theorem Suppose {A1 , . . . , Ak } and {B1 , . . . , Bk } are commuting families of matrices in Mn and Mm . Then there is a unital / trace preserving / unital and trace preserving completely positive linear maps Φ such that Φ(Aj ) = Bj for j = 1, . . . , k Yiu Tung Poon Completely Positive Maps in Quantum Information
  123. 123. Results and questions on multiple matrices Theorem Suppose {A1 , . . . , Ak } and {B1 , . . . , Bk } are commuting families of matrices in Mn and Mm . Then there is a unital / trace preserving / unital and trace preserving completely positive linear maps Φ such that Φ(Aj ) = Bj for j = 1, . . . , k if and only if there is an n × m column / row / doubly stochastic matrix D such that λ(Bj ) = λ(Aj )D for j = 1, . . . , k. Yiu Tung Poon Completely Positive Maps in Quantum Information
  124. 124. Results and questions on multiple matrices Theorem Suppose {A1 , . . . , Ak } and {B1 , . . . , Bk } are commuting families of matrices in Mn and Mm . Then there is a unital / trace preserving / unital and trace preserving completely positive linear maps Φ such that Φ(Aj ) = Bj for j = 1, . . . , k if and only if there is an n × m column / row / doubly stochastic matrix D such that λ(Bj ) = λ(Aj )D for j = 1, . . . , k. Question What about non-commuting families? Yiu Tung Poon Completely Positive Maps in Quantum Information
  125. 125. Results and questions on multiple matrices Theorem Suppose {A1 , . . . , Ak } and {B1 , . . . , Bk } are commuting families of matrices in Mn and Mm . Then there is a unital / trace preserving / unital and trace preserving completely positive linear maps Φ such that Φ(Aj ) = Bj for j = 1, . . . , k if and only if there is an n × m column / row / doubly stochastic matrix D such that λ(Bj ) = λ(Aj )D for j = 1, . . . , k. Question What about non-commuting families? Restricted Choi’s rank? Yiu Tung Poon Completely Positive Maps in Quantum Information
  126. 126. Results and questions on multiple matrices Theorem Suppose {A1 , . . . , Ak } and {B1 , . . . , Bk } are commuting families of matrices in Mn and Mm . Then there is a unital / trace preserving / unital and trace preserving completely positive linear maps Φ such that Φ(Aj ) = Bj for j = 1, . . . , k if and only if there is an n × m column / row / doubly stochastic matrix D such that λ(Bj ) = λ(Aj )D for j = 1, . . . , k. Question What about non-commuting families? Restricted Choi’s rank? Consider the case when k = 2. Suppose A1 , A2 ∈ Hn and B1 , B2 ∈ Hm . Yiu Tung Poon Completely Positive Maps in Quantum Information
  127. 127. Results and questions on multiple matrices Theorem Suppose {A1 , . . . , Ak } and {B1 , . . . , Bk } are commuting families of matrices in Mn and Mm . Then there is a unital / trace preserving / unital and trace preserving completely positive linear maps Φ such that Φ(Aj ) = Bj for j = 1, . . . , k if and only if there is an n × m column / row / doubly stochastic matrix D such that λ(Bj ) = λ(Aj )D for j = 1, . . . , k. Question What about non-commuting families? Restricted Choi’s rank? Consider the case when k = 2. Suppose A1 , A2 ∈ Hn and B1 , B2 ∈ Hm . Let A = A1 + iA2 ∈ Mn and B = B1 + iB2 . Yiu Tung Poon Completely Positive Maps in Quantum Information
  128. 128. Results and questions on multiple matrices Theorem Suppose {A1 , . . . , Ak } and {B1 , . . . , Bk } are commuting families of matrices in Mn and Mm . Then there is a unital / trace preserving / unital and trace preserving completely positive linear maps Φ such that Φ(Aj ) = Bj for j = 1, . . . , k if and only if there is an n × m column / row / doubly stochastic matrix D such that λ(Bj ) = λ(Aj )D for j = 1, . . . , k. Question What about non-commuting families? Restricted Choi’s rank? Consider the case when k = 2. Suppose A1 , A2 ∈ Hn and B1 , B2 ∈ Hm . Let A = A1 + iA2 ∈ Mn and B = B1 + iB2 . We will study B = Φ(A). Yiu Tung Poon Completely Positive Maps in Quantum Information
  129. 129. Dilation and compression Let T ∈ B(H). Yiu Tung Poon Completely Positive Maps in Quantum Information
  130. 130. Dilation and compression Let T ∈ B(H). The numerical range of A is W (T ) = { T x, x : x = 1}. Yiu Tung Poon Completely Positive Maps in Quantum Information
  131. 131. Dilation and compression Let T ∈ B(H). The numerical range of A is W (T ) = { T x, x : x = 1}. (Mirman, 1968) Let A ∈ M3 be normal. Then B = Φ(A) for a unital CP map Φ if and only if Yiu Tung Poon Completely Positive Maps in Quantum Information
  132. 132. Dilation and compression Let T ∈ B(H). The numerical range of A is W (T ) = { T x, x : x = 1}. (Mirman, 1968) Let A ∈ M3 be normal. Then B = Φ(A) for a unital CP map Φ if and only if W (B) ⊆ W (A). Yiu Tung Poon Completely Positive Maps in Quantum Information
  133. 133. Dilation and compression Let T ∈ B(H). The numerical range of A is W (T ) = { T x, x : x = 1}. (Mirman, 1968) Let A ∈ M3 be normal. Then B = Φ(A) for a unital CP map Φ if and only if W (B) ⊆ W (A). 0 2 (Ando, 1973) Let A = . 0 0 Yiu Tung Poon Completely Positive Maps in Quantum Information
  134. 134. Dilation and compression Let T ∈ B(H). The numerical range of A is W (T ) = { T x, x : x = 1}. (Mirman, 1968) Let A ∈ M3 be normal. Then B = Φ(A) for a unital CP map Φ if and only if W (B) ⊆ W (A). 0 2 (Ando, 1973) Let A = . Then B = Φ(A) for a unital CP 0 0 map Φ if and only if W (B) ⊆ W (A). Yiu Tung Poon Completely Positive Maps in Quantum Information
  135. 135. Dilation and compression Let T ∈ B(H). The numerical range of A is W (T ) = { T x, x : x = 1}. (Mirman, 1968) Let A ∈ M3 be normal. Then B = Φ(A) for a unital CP map Φ if and only if W (B) ⊆ W (A). 0 2 (Ando, 1973) Let A = . Then B = Φ(A) for a unital CP 0 0 map Φ if and only if W (B) ⊆ W (A). (Choi, Li, 2000, 2001) Let A1 ∈ M2 and A = A1 or A = A1 ⊕ [a]. Yiu Tung Poon Completely Positive Maps in Quantum Information
  136. 136. Dilation and compression Let T ∈ B(H). The numerical range of A is W (T ) = { T x, x : x = 1}. (Mirman, 1968) Let A ∈ M3 be normal. Then B = Φ(A) for a unital CP map Φ if and only if W (B) ⊆ W (A). 0 2 (Ando, 1973) Let A = . Then B = Φ(A) for a unital CP 0 0 map Φ if and only if W (B) ⊆ W (A). (Choi, Li, 2000, 2001) Let A1 ∈ M2 and A = A1 or A = A1 ⊕ [a]. Then B = Φ(A) for a unital CP map Φ if and only if Yiu Tung Poon Completely Positive Maps in Quantum Information
  137. 137. Dilation and compression Let T ∈ B(H). The numerical range of A is W (T ) = { T x, x : x = 1}. (Mirman, 1968) Let A ∈ M3 be normal. Then B = Φ(A) for a unital CP map Φ if and only if W (B) ⊆ W (A). 0 2 (Ando, 1973) Let A = . Then B = Φ(A) for a unital CP 0 0 map Φ if and only if W (B) ⊆ W (A). (Choi, Li, 2000, 2001) Let A1 ∈ M2 and A = A1 or A = A1 ⊕ [a]. Then B = Φ(A) for a unital CP map Φ if and only if W (B) ⊆ W (A). Yiu Tung Poon Completely Positive Maps in Quantum Information
  138. 138. Dilation and compression Let T ∈ B(H). The numerical range of A is W (T ) = { T x, x : x = 1}. (Mirman, 1968) Let A ∈ M3 be normal. Then B = Φ(A) for a unital CP map Φ if and only if W (B) ⊆ W (A). 0 2 (Ando, 1973) Let A = . Then B = Φ(A) for a unital CP 0 0 map Φ if and only if W (B) ⊆ W (A). (Choi, Li, 2000, 2001) Let A1 ∈ M2 and A = A1 or A = A1 ⊕ [a]. Then B = Φ(A) for a unital CP map Φ if and only if W (B) ⊆ W (A).   0 1 0 (Choi, Li, 2000) Let A = diag (1, −1, i, −i) or  0 0 1  and 0 0 0 √ 0 2 B= . 0 0 Yiu Tung Poon Completely Positive Maps in Quantum Information
  139. 139. Dilation and compression Let T ∈ B(H). The numerical range of A is W (T ) = { T x, x : x = 1}. (Mirman, 1968) Let A ∈ M3 be normal. Then B = Φ(A) for a unital CP map Φ if and only if W (B) ⊆ W (A). 0 2 (Ando, 1973) Let A = . Then B = Φ(A) for a unital CP 0 0 map Φ if and only if W (B) ⊆ W (A). (Choi, Li, 2000, 2001) Let A1 ∈ M2 and A = A1 or A = A1 ⊕ [a]. Then B = Φ(A) for a unital CP map Φ if and only if W (B) ⊆ W (A).   0 1 0 (Choi, Li, 2000) Let A = diag (1, −1, i, −i) or  0 0 1  and 0 0 0 √ 0 2 B= . Then W (B) ⊆ W (A) but there is no unital CP 0 0 map Φ such that B = Φ(A). Yiu Tung Poon Completely Positive Maps in Quantum Information
  140. 140. Choi’s rank = 1 Suppose A1 , A2 ∈ Hn and B1 , B2 ∈ Hm . Yiu Tung Poon Completely Positive Maps in Quantum Information
  141. 141. Choi’s rank = 1 Suppose A1 , A2 ∈ Hn and B1 , B2 ∈ Hm . Let A = A1 + iA2 and B = B1 + iB2 . Yiu Tung Poon Completely Positive Maps in Quantum Information
  142. 142. Choi’s rank = 1 Suppose A1 , A2 ∈ Hn and B1 , B2 ∈ Hm . Let A = A1 + iA2 and B = B1 + iB2 . There exists a trace preserving Φ ∈ CP1 (n, m) such that Φ(Aj ) = Bj for j = 1, 2 if and only if the B is unitary similar to A ⊕ Om−n . Yiu Tung Poon Completely Positive Maps in Quantum Information
  143. 143. Choi’s rank = 1 Suppose A1 , A2 ∈ Hn and B1 , B2 ∈ Hm . Let A = A1 + iA2 and B = B1 + iB2 . There exists a trace preserving Φ ∈ CP1 (n, m) such that Φ(Aj ) = Bj for j = 1, 2 if and only if the B is unitary similar to A ⊕ Om−n . There exists a unital Φ ∈ CP1 (n, m) such that Φ(Aj ) = Bj for j = 1, 2 if and only if B is a compression of A, Yiu Tung Poon Completely Positive Maps in Quantum Information
  144. 144. Choi’s rank = 1 Suppose A1 , A2 ∈ Hn and B1 , B2 ∈ Hm . Let A = A1 + iA2 and B = B1 + iB2 . There exists a trace preserving Φ ∈ CP1 (n, m) such that Φ(Aj ) = Bj for j = 1, 2 if and only if the B is unitary similar to A ⊕ Om−n . There exists a unital Φ ∈ CP1 (n, m) such that Φ(Aj ) = Bj for j = 1, 2 if and only if B is a compression of A, i.e. there is a unitary U such that B is a principal submatrix of U ∗ AU . Yiu Tung Poon Completely Positive Maps in Quantum Information
  145. 145. Choi’s rank = 1 Suppose A1 , A2 ∈ Hn and B1 , B2 ∈ Hm . Let A = A1 + iA2 and B = B1 + iB2 . There exists a trace preserving Φ ∈ CP1 (n, m) such that Φ(Aj ) = Bj for j = 1, 2 if and only if the B is unitary similar to A ⊕ Om−n . There exists a unital Φ ∈ CP1 (n, m) such that Φ(Aj ) = Bj for j = 1, 2 if and only if B is a compression of A, i.e. there is a unitary U such that B is a principal submatrix of U ∗ AU . The problem is unsolved for n = 4, m = 2 Yiu Tung Poon Completely Positive Maps in Quantum Information
  146. 146. Choi’s rank = 1 Suppose A1 , A2 ∈ Hn and B1 , B2 ∈ Hm . Let A = A1 + iA2 and B = B1 + iB2 . There exists a trace preserving Φ ∈ CP1 (n, m) such that Φ(Aj ) = Bj for j = 1, 2 if and only if the B is unitary similar to A ⊕ Om−n . There exists a unital Φ ∈ CP1 (n, m) such that Φ(Aj ) = Bj for j = 1, 2 if and only if B is a compression of A, i.e. there is a unitary U such that B is a principal submatrix of U ∗ AU . The problem is unsolved for n = 4, m = 2 even when A1 and A2 commute. Yiu Tung Poon Completely Positive Maps in Quantum Information
  147. 147. Restricted rank on multiple matrices Theorem [Poon 1992] Let Φ ∈ CP (n, m) be unital and A1 , . . . , Ak ∈ Hn . Yiu Tung Poon Completely Positive Maps in Quantum Information
  148. 148. Restricted rank on multiple matrices Theorem [Poon 1992] Let Φ ∈ CP (n, m) be unital and A1 , . . . , Ak ∈ Hn . If 1 ≤ r ≤ mn − 1 and m2 (k + 1) − 1 < (r + 1)2 − δmn, r+1 , Yiu Tung Poon Completely Positive Maps in Quantum Information
  149. 149. Restricted rank on multiple matrices Theorem [Poon 1992] Let Φ ∈ CP (n, m) be unital and A1 , . . . , Ak ∈ Hn . If 1 ≤ r ≤ mn − 1 and m2 (k + 1) − 1 < (r + 1)2 − δmn, r+1 , then there exists a unital Ψ ∈ CPr (n m) such that for all 1 ≤ i ≤ k, Ψ(Ai ) = Φ(Ai ) Yiu Tung Poon Completely Positive Maps in Quantum Information
  150. 150. Restricted rank on multiple matrices Theorem [Poon 1992] Let Φ ∈ CP (n, m) be unital and A1 , . . . , Ak ∈ Hn . If 1 ≤ r ≤ mn − 1 and m2 (k + 1) − 1 < (r + 1)2 − δmn, r+1 , then there exists a unital Ψ ∈ CPr (n m) such that for all 1 ≤ i ≤ k, Ψ(Ai ) = Φ(Ai ) For example, if n > 2, k = 3 and r = 2m − 1 , then the condition is satisfied. Yiu Tung Poon Completely Positive Maps in Quantum Information
  151. 151. Restricted rank on multiple matrices Theorem [Poon 1992] Let Φ ∈ CP (n, m) be unital and A1 , . . . , Ak ∈ Hn . If 1 ≤ r ≤ mn − 1 and m2 (k + 1) − 1 < (r + 1)2 − δmn, r+1 , then there exists a unital Ψ ∈ CPr (n m) such that for all 1 ≤ i ≤ k, Ψ(Ai ) = Φ(Ai ) For example, if n > 2, k = 3 and r = 2m − 1 , then the condition is satisfied. What about trace preserving completely positive maps? Yiu Tung Poon Completely Positive Maps in Quantum Information
  152. 152. Thank you for your attention! Yiu Tung Poon Completely Positive Maps in Quantum Information

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