This document describes a presentation given by Chi-Kwong Li on quantum operations and completely positive linear maps. It begins with an overview of classical and modern computing concepts before introducing quantum computing concepts. It then provides a mathematical formulation of quantum states and operations based on von Neumann's work. It discusses how quantum states of multiple qubits can be represented using tensor products and the exponential complexity this introduces. Finally, it defines quantum operations as completely positive linear maps and provides conditions for complete positivity, unitarity, and trace preservation.

1. Quantum Operations and
Completely Positive Linear Maps
Chi-Kwong Li
Department of Mathematics
The College of William and Mary
Williamsburg, Virginia, USA
Chi-Kwong Li Quantum Operations and Completely Positive Maps

2. Quantum Operations and
Completely Positive Linear Maps
Chi-Kwong Li
Department of Mathematics
The College of William and Mary
Williamsburg, Virginia, USA
Currently visiting GWU and NRL.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

3. Quantum Operations and
Completely Positive Linear Maps
Chi-Kwong Li
Department of Mathematics
The College of William and Mary
Williamsburg, Virginia, USA
Currently visiting GWU and NRL.
Joint work with Yiu-Tung Poon (Iowa State University).
Chi-Kwong Li Quantum Operations and Completely Positive Maps

4. Classical computing
Chi-Kwong Li Quantum Operations and Completely Positive Maps

5. Classical computing
Hardware - Beads and bars.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

6. Classical computing
Hardware - Beads and bars.
Input - Using finger skill to change the states of the device.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

7. Classical computing
Hardware - Beads and bars.
Input - Using finger skill to change the states of the device.
Processor - Mechanical process with algorithms based on elementary
arithmetic rules.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

8. Classical computing
Hardware - Beads and bars.
Input - Using finger skill to change the states of the device.
Processor - Mechanical process with algorithms based on elementary
arithmetic rules.
Output - Beads and bars, then recorded by brush and ink.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

9. Modern Computing
Chi-Kwong Li Quantum Operations and Completely Positive Maps

10. Modern Computing
Hardware - Mechanical/electronic/integrated circuits.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

11. Modern Computing
Hardware - Mechanical/electronic/integrated circuits.
Input - Punch cards, keyboards, scanners, sounds, etc. all converted
to binary bits - (0, 1) sequences.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

12. Modern Computing
Hardware - Mechanical/electronic/integrated circuits.
Input - Punch cards, keyboards, scanners, sounds, etc. all converted
to binary bits - (0, 1) sequences.
Processor - Manipulations of (0, 1) sequences using Boolean logic.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

13. Modern Computing
0∨0=0
0∨1=1
1∨0=1
1∨1=1
Hardware - Mechanical/electronic/integrated circuits.
Input - Punch cards, keyboards, scanners, sounds, etc. all converted
to binary bits - (0, 1) sequences.
Processor - Manipulations of (0, 1) sequences using Boolean logic.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

14. Modern Computing
0∨0=0
0∨1=1
1∨0=1
1∨1=1
Hardware - Mechanical/electronic/integrated circuits.
Input - Punch cards, keyboards, scanners, sounds, etc. all converted
to binary bits - (0, 1) sequences.
Processor - Manipulations of (0, 1) sequences using Boolean logic.
Output - (0, 1) sequences realized as visual images, which can be
viewed or printed.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

15. Quantum computing
Quantum Computing Unit
−→ −→
Optical lattices, NMR
Chi-Kwong Li Quantum Operations and Completely Positive Maps

16. Quantum computing
Quantum Computing Unit
−→ −→
Optical lattices, NMR
Hardware - Super conductor, trapped ions, optical lattices, quantum
dot, MNR, etc.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

17. Quantum computing
Quantum Computing Unit
−→ −→
Optical lattices, NMR
Hardware - Super conductor, trapped ions, optical lattices, quantum
dot, MNR, etc.
Input - Quantum states in a specific form - Quantum bits (Qubits).
Chi-Kwong Li Quantum Operations and Completely Positive Maps

18. Quantum computing
Quantum Computing Unit
−→ −→
Optical lattices, NMR
Hardware - Super conductor, trapped ions, optical lattices, quantum
dot, MNR, etc.
Input - Quantum states in a specific form - Quantum bits (Qubits).
Processor - Provide suitable environment for the quantum system of
qubits to evolve.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

19. Quantum computing
Quantum Computing Unit
−→ −→
Optical lattices, NMR
Hardware - Super conductor, trapped ions, optical lattices, quantum
dot, MNR, etc.
Input - Quantum states in a specific form - Quantum bits (Qubits).
Processor - Provide suitable environment for the quantum system of
qubits to evolve.
Output - Measurement of the resulting quantum states.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

20. Quantum computing
Quantum Computing Unit
−→ −→
Optical lattices, NMR
Hardware - Super conductor, trapped ions, optical lattices, quantum
dot, MNR, etc.
Input - Quantum states in a specific form - Quantum bits (Qubits).
Processor - Provide suitable environment for the quantum system of
qubits to evolve.
Output - Measurement of the resulting quantum states.
All these require the understanding of mathematics, physics,
chemistry, computer sciences, engineering, etc.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

21. Mathematical formulation (by von Neumann)
Suppose a quantum system have
two (discrete) measurable physical
sates, say, up spin and down spin
of a particle represented by
Chi-Kwong Li Quantum Operations and Completely Positive Maps

22. Mathematical formulation (by von Neumann)
Suppose a quantum system have
two (discrete) measurable physical
sates, say, up spin and down spin
of a particle represented by
1 0
|0 = and |1 = .
0 1
Chi-Kwong Li Quantum Operations and Completely Positive Maps

23. Mathematical formulation (by von Neumann)
Suppose a quantum system have
two (discrete) measurable physical
sates, say, up spin and down spin
of a particle represented by
1 0
|0 = and |1 = .
0 1
Before measurement, the vector state may be in superposition state
represented by a complex vector
α
v = |ψ = α|0 + β|1 = ∈ C2 , |α|2 + |β|2 = 1.
β
Chi-Kwong Li Quantum Operations and Completely Positive Maps

24. Mathematical formulation (by von Neumann)
Suppose a quantum system have
two (discrete) measurable physical
sates, say, up spin and down spin
of a particle represented by
1 0
|0 = and |1 = .
0 1
Before measurement, the vector state may be in superposition state
represented by a complex vector
α
v = |ψ = α|0 + β|1 = ∈ C2 , |α|2 + |β|2 = 1.
β
One can apply a quantum operation to a state in superposition.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

25. Mathematical formulation (by von Neumann)
Suppose a quantum system have
two (discrete) measurable physical
sates, say, up spin and down spin
of a particle represented by
1 0
|0 = and |1 = .
0 1
Before measurement, the vector state may be in superposition state
represented by a complex vector
α
v = |ψ = α|0 + β|1 = ∈ C2 , |α|2 + |β|2 = 1.
β
One can apply a quantum operation to a state in superposition.
That is the famous “phenomenon” that one can apply a
transformation to the half alive and half dead Schrödinger cat.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

26. Matrix formulation
It is convenient to represent the quantum state |ψ as a rank one
orthogonal projection:
1 1+z x + iy
Q = |ψ ψ| =
2 x − iy 1−z
with x, y, z ∈ R such that x2 + y 2 + z 2 = 1.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

27. Matrix formulation
It is convenient to represent the quantum state |ψ as a rank one
orthogonal projection:
1 1+z x + iy
Q = |ψ ψ| =
2 x − iy 1−z
with x, y, z ∈ R such that x2 + y 2 + z 2 = 1.
There is a Bloch sphere representation of a qubit
Chi-Kwong Li Quantum Operations and Completely Positive Maps

28. Tensor product and complexity
The state of k qubits is represented as the tensor product of k 2 × 2
matrices
Q1 ⊗ · · · ⊗ Qk .
Chi-Kwong Li Quantum Operations and Completely Positive Maps

29. Tensor product and complexity
The state of k qubits is represented as the tensor product of k 2 × 2
matrices
Q1 ⊗ · · · ⊗ Qk .
Recall that for matrices X = (xij ) and Y (may be of different
sizes),
X ⊗ Y = (xij Y ).
Chi-Kwong Li Quantum Operations and Completely Positive Maps

30. Tensor product and complexity
The state of k qubits is represented as the tensor product of k 2 × 2
matrices
Q1 ⊗ · · · ⊗ Qk .
Recall that for matrices X = (xij ) and Y (may be of different
sizes),
X ⊗ Y = (xij Y ).
General entangled states are represented as 2k × 2k density
matrices, i.e., trace one positive semidefinite matrices.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

31. Tensor product and complexity
The state of k qubits is represented as the tensor product of k 2 × 2
matrices
Q1 ⊗ · · · ⊗ Qk .
Recall that for matrices X = (xij ) and Y (may be of different
sizes),
X ⊗ Y = (xij Y ).
General entangled states are represented as 2k × 2k density
matrices, i.e., trace one positive semidefinite matrices.
For k = 1000, we have 21000 = (210 )100 ≈ 10300 . If a high speed
computer can do 1015 operations per second, to do one operation
on each of the states,
Chi-Kwong Li Quantum Operations and Completely Positive Maps

32. Tensor product and complexity
The state of k qubits is represented as the tensor product of k 2 × 2
matrices
Q1 ⊗ · · · ⊗ Qk .
Recall that for matrices X = (xij ) and Y (may be of different
sizes),
X ⊗ Y = (xij Y ).
General entangled states are represented as 2k × 2k density
matrices, i.e., trace one positive semidefinite matrices.
For k = 1000, we have 21000 = (210 )100 ≈ 10300 . If a high speed
computer can do 1015 operations per second, to do one operation
on each of the states, one needs
1030 /1015 = 1015 seconds > 300, 000 centuries!
Chi-Kwong Li Quantum Operations and Completely Positive Maps

33. Quantum operations
Solving the Schrödinger equation, one sees that quantum operations
(channels) on a closed system are unitary similarity transform on
quatnum states ρ (represented as density matrices), i.e.,
ρ(t) → Ut ρ0 Ut† .
Chi-Kwong Li Quantum Operations and Completely Positive Maps

34. Quantum operations
Solving the Schrödinger equation, one sees that quantum operations
(channels) on a closed system are unitary similarity transform on
quatnum states ρ (represented as density matrices), i.e.,
ρ(t) → Ut ρ0 Ut† .
When the state ρ of principal system interacts (involuntarily) with
the state ρE of the (external) environment, one would trace out the
environment so that
Chi-Kwong Li Quantum Operations and Completely Positive Maps

35. Quantum operations
Solving the Schrödinger equation, one sees that quantum operations
(channels) on a closed system are unitary similarity transform on
quatnum states ρ (represented as density matrices), i.e.,
ρ(t) → Ut ρ0 Ut† .
When the state ρ of principal system interacts (involuntarily) with
the state ρE of the (external) environment, one would trace out the
environment so that
r
ρ(t) → Tr E (Ut (ρ ⊗ ρE )Ut† ) = Fj ρFj†
j=1
Fj† Fj = I.
r
for some m × n matrices F1 , . . . , Fr satisfying j=1
Chi-Kwong Li Quantum Operations and Completely Positive Maps

36. Completely positive linear maps
Let Mn be the set of n × n complex matrices,
Chi-Kwong Li Quantum Operations and Completely Positive Maps

37. Completely positive linear maps
Let Mn be the set of n × n complex matrices,
Hn be the set of n × n complex Hermitian matrices.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

38. Completely positive linear maps
Let Mn be the set of n × n complex matrices,
Hn be the set of n × n complex Hermitian matrices.
A map L : Mn → Mm is completely positive if L admits an operator sum
representation
r
L(A) = Fj AFj† ,
j=1
where F1 , . . . , Fr are m × n complex matrices.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

39. Completely positive linear maps
Let Mn be the set of n × n complex matrices,
Hn be the set of n × n complex Hermitian matrices.
A map L : Mn → Mm is completely positive if L admits an operator sum
representation
r
L(A) = Fj AFj† ,
j=1
where F1 , . . . , Fr are m × n complex matrices.
Fj Fj† = Im ;
r
In addition, L is unital if L(In ) = Im , equivalently, j=1
Chi-Kwong Li Quantum Operations and Completely Positive Maps

40. Completely positive linear maps
Let Mn be the set of n × n complex matrices,
Hn be the set of n × n complex Hermitian matrices.
A map L : Mn → Mm is completely positive if L admits an operator sum
representation
r
L(A) = Fj AFj† ,
j=1
where F1 , . . . , Fr are m × n complex matrices.
Fj Fj† = Im ;
r
In addition, L is unital if L(In ) = Im , equivalently, j=1
Fj† Fj = In .
r
L is trace preserving if Tr A = Tr L(A), equivalently, j=1
Chi-Kwong Li Quantum Operations and Completely Positive Maps

41. General problems
Since every quantum operation / channel is a trace preserving completely
positive linear map, it is interesting to study the following.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

42. General problems
Since every quantum operation / channel is a trace preserving completely
positive linear map, it is interesting to study the following.
Question
Given A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm , is there a (unital/trace
preserving) completely positive linear map satisfying L(Aj ) = Bj for all
j = 1, . . . , k?
Chi-Kwong Li Quantum Operations and Completely Positive Maps

43. General problems
Since every quantum operation / channel is a trace preserving completely
positive linear map, it is interesting to study the following.
Question
Given A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm , is there a (unital/trace
preserving) completely positive linear map satisfying L(Aj ) = Bj for all
j = 1, . . . , k?
It is also interesting to consider the following related problems.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

44. General problems
Since every quantum operation / channel is a trace preserving completely
positive linear map, it is interesting to study the following.
Question
Given A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm , is there a (unital/trace
preserving) completely positive linear map satisfying L(Aj ) = Bj for all
j = 1, . . . , k?
It is also interesting to consider the following related problems.
Understand the duality relation between the trace preserving
completely positive linear maps and the unital preserving completely
positive linear maps.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

45. General problems
Since every quantum operation / channel is a trace preserving completely
positive linear map, it is interesting to study the following.
Question
Given A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm , is there a (unital/trace
preserving) completely positive linear map satisfying L(Aj ) = Bj for all
j = 1, . . . , k?
It is also interesting to consider the following related problems.
Understand the duality relation between the trace preserving
completely positive linear maps and the unital preserving completely
positive linear maps.
Determine / deduce properties of L based on the information of
L(A) for some special matrices A.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

46. Basic results
For A ∈ Hn with eigenvalues λ1 (A) ≥ · · · ≥ λn (A), denote by
λ(A) = (λ1 (A), . . . , λn (A)).
Chi-Kwong Li Quantum Operations and Completely Positive Maps

47. Basic results
For A ∈ Hn with eigenvalues λ1 (A) ≥ · · · ≥ λn (A), denote by
λ(A) = (λ1 (A), . . . , λn (A)).
Definition
For x, y ∈ R1×m , we say that x is majorized by y, denoted by x y, if
the sum of entries of x is the same as that of y, and the sum of the k
largest entries of x is not larger than that of y for k = 1, . . . , k − 1.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

48. Basic results
For A ∈ Hn with eigenvalues λ1 (A) ≥ · · · ≥ λn (A), denote by
λ(A) = (λ1 (A), . . . , λn (A)).
Definition
For x, y ∈ R1×m , we say that x is majorized by y, denoted by x y, if
the sum of entries of x is the same as that of y, and the sum of the k
largest entries of x is not larger than that of y for k = 1, . . . , k − 1.
Example (5, 4, 1) (7, 3, 0), (5, 4, 1) (6, 2, 2), (6, 2, 2) (5, 4, 1).
Chi-Kwong Li Quantum Operations and Completely Positive Maps

49. Basic results
For A ∈ Hn with eigenvalues λ1 (A) ≥ · · · ≥ λn (A), denote by
λ(A) = (λ1 (A), . . . , λn (A)).
Definition
For x, y ∈ R1×m , we say that x is majorized by y, denoted by x y, if
the sum of entries of x is the same as that of y, and the sum of the k
largest entries of x is not larger than that of y for k = 1, . . . , k − 1.
Example (5, 4, 1) (7, 3, 0), (5, 4, 1) (6, 2, 2), (6, 2, 2) (5, 4, 1).
It is known that x y if and only if there is a doubly stochastic matrix
D such that x = yD.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

50. Basic results
For A ∈ Hn with eigenvalues λ1 (A) ≥ · · · ≥ λn (A), denote by
λ(A) = (λ1 (A), . . . , λn (A)).
Definition
For x, y ∈ R1×m , we say that x is majorized by y, denoted by x y, if
the sum of entries of x is the same as that of y, and the sum of the k
largest entries of x is not larger than that of y for k = 1, . . . , k − 1.
Example (5, 4, 1) (7, 3, 0), (5, 4, 1) (6, 2, 2), (6, 2, 2) (5, 4, 1).
It is known that x y if and only if there is a doubly stochastic matrix
D such that x = yD.
Recall that a nonnegative matrix D is row (respectively, column)
stochastic if D have all row (respectively, column) sums equal to one;
Chi-Kwong Li Quantum Operations and Completely Positive Maps

51. Basic results
For A ∈ Hn with eigenvalues λ1 (A) ≥ · · · ≥ λn (A), denote by
λ(A) = (λ1 (A), . . . , λn (A)).
Definition
For x, y ∈ R1×m , we say that x is majorized by y, denoted by x y, if
the sum of entries of x is the same as that of y, and the sum of the k
largest entries of x is not larger than that of y for k = 1, . . . , k − 1.
Example (5, 4, 1) (7, 3, 0), (5, 4, 1) (6, 2, 2), (6, 2, 2) (5, 4, 1).
It is known that x y if and only if there is a doubly stochastic matrix
D such that x = yD.
Recall that a nonnegative matrix D is row (respectively, column)
stochastic if D have all row (respectively, column) sums equal to one;
D is doubly stochastic if all row and column sums equal one.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

52. Theorem
Suppose A ∈ Hn and B ∈ Hm . Let a+ (respectively, a− ) be the sum of
the positive (respectively, negative) eigenvalues of A.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

53. Theorem
Suppose A ∈ Hn and B ∈ Hm . Let a+ (respectively, a− ) be the sum of
the positive (respectively, negative) eigenvalues of A. The following
conditions are equivalent.
There is a trace preserving completely positive linear map
L : Mn → Mm such that L(A) = B.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

54. Theorem
Suppose A ∈ Hn and B ∈ Hm . Let a+ (respectively, a− ) be the sum of
the positive (respectively, negative) eigenvalues of A. The following
conditions are equivalent.
There is a trace preserving completely positive linear map
L : Mn → Mm such that L(A) = B.
λ(B) (a+ , 0, . . . , 0, a− ) in R1×m .
Chi-Kwong Li Quantum Operations and Completely Positive Maps

55. Theorem
Suppose A ∈ Hn and B ∈ Hm . Let a+ (respectively, a− ) be the sum of
the positive (respectively, negative) eigenvalues of A. The following
conditions are equivalent.
There is a trace preserving completely positive linear map
L : Mn → Mm such that L(A) = B.
λ(B) (a+ , 0, . . . , 0, a− ) in R1×m .
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.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

56. Theorem
Suppose A ∈ Hn and B ∈ Hm . Let a+ (respectively, a− ) be the sum of
the positive (respectively, negative) eigenvalues of A. The following
conditions are equivalent.
There is a trace preserving completely positive linear map
L : Mn → Mm such that L(A) = B.
λ(B) (a+ , 0, . . . , 0, a− ) in R1×m .
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.
One can use D to construct m × n matrices F1 , . . . , Fr with
r = max(m, n) such that B = j=1 Fj AFj† and j=1 Fj† Fj = In .
r r
Chi-Kwong Li Quantum Operations and Completely Positive Maps

57. Theorem
Suppose A ∈ Hn and B ∈ Hm . Let a+ (respectively, a− ) be the sum of
the positive (respectively, negative) eigenvalues of A. The following
conditions are equivalent.
There is a trace preserving completely positive linear map
L : Mn → Mm such that L(A) = B.
λ(B) (a+ , 0, . . . , 0, a− ) in R1×m .
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.
One can use D to construct m × n matrices F1 , . . . , Fr with
r = max(m, n) such that B = j=1 Fj AFj† and j=1 Fj† Fj = In .
r r
Remark For density matrices A and B, the condition trivially holds.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

58. Theorem
Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

59. Theorem
Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent.
There is a unital completely positive linear map L such that
L(A) = B.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

60. Theorem
Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent.
There is a unital completely positive linear map L such that
L(A) = B.
λn (A) ≤ λj (B) ≤ λ1 (A) for all j = 1, . . . , m.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

61. Theorem
Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent.
There is a unital completely positive linear map L such that
L(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.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

62. Theorem
Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent.
There is a unital completely positive linear map L such that
L(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.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

63. Theorem
Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent.
There is a unital completely positive linear map L such that
L(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
Can we deduce this result from the previous one using duality of
completely positive linear map?
Chi-Kwong Li Quantum Operations and Completely Positive Maps

64. Question
Assume there is a unital completely positive map sending A to B, and
also a trace preserving completely positive map sending A to B.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

65. Question
Assume there is a unital completely positive map sending A to B, and
also a trace preserving completely positive map sending A to B. Is there
a unital trace preserving completely positive map sending A to B?
Chi-Kwong Li Quantum Operations and Completely Positive Maps

66. Question
Assume there is a unital completely positive map sending A to B, and
also a trace preserving completely positive map sending A to B. Is there
a unital trace preserving completely positive map sending A to B?
The following example shows that the answer is negative.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

67. Question
Assume there is a unital completely positive map sending A to B, and
also a trace preserving completely positive map sending A to B. Is there
a unital trace preserving completely positive map sending A to B?
The following example shows that the answer is negative.
Example
Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is a
trace preserving completely positive map sending A to B, and also a
unital completely positive map sending A to B.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

68. Question
Assume there is a unital completely positive map sending A to B, and
also a trace preserving completely positive map sending A to B. Is there
a unital trace preserving completely positive map sending A to B?
The following example shows that the answer is negative.
Example
Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is a
trace preserving completely positive map sending A to B, and also a
unital completely positive map sending A to B. But there is no trace
preserving completely positive linear map sending A1 to B1 if
A1 = A − I4 = diag (3, 0, 0, −1) and B1 = B − I4 = diag (2, 2, −1, −1).
Chi-Kwong Li Quantum Operations and Completely Positive Maps

69. Question
Assume there is a unital completely positive map sending A to B, and
also a trace preserving completely positive map sending A to B. Is there
a unital trace preserving completely positive map sending A to B?
The following example shows that the answer is negative.
Example
Suppose A = diag (4, 1, 1, 0) and B = diag (3, 3, 0, 0). Then there is a
trace preserving completely positive map sending A to B, and also a
unital completely positive map sending A to B. But there is no trace
preserving completely positive linear map sending A1 to B1 if
A1 = A − I4 = diag (3, 0, 0, −1) and B1 = B − I4 = diag (2, 2, −1, −1).
Hence, there is no unital trace preserving completely positive map
sending A to B.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

70. Theorem
Let A, B ∈ Hn . The following conditions are equivalent.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

71. Theorem
Let A, B ∈ Hn . The following conditions are equivalent.
There exists a unital trace preserving completely positive map L
such that L(A) = B.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

72. Theorem
Let A, B ∈ Hn . The following conditions are equivalent.
There exists a unital trace preserving completely positive map L
such that L(A) = B.
For each t ∈ R, there exists a trace preserving completely positive
map L such that L(A − tI) = B − tI.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

73. Theorem
Let A, B ∈ Hn . The following conditions are equivalent.
There exists a unital trace preserving completely positive map L
such that L(A) = B.
For each t ∈ R, there exists a trace preserving completely positive
map L such that L(A − tI) = B − tI.
λ(B) λ(A). i.e., there is a doubly stochastic matrix D such that
λ(B) = λ(A)D.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

74. Theorem
Let A, B ∈ Hn . The following conditions are equivalent.
There exists a unital trace preserving completely positive map L
such that L(A) = B.
For each t ∈ R, there exists a trace preserving completely positive
map L such that L(A − tI) = B − tI.
λ(B) λ(A). i.e., there is a doubly stochastic matrix D such that
λ(B) = λ(A)D.
There is a unitary U ∈ Mn such that U AU † has diagonal entries
λ1 (B), . . . , λn (B).
Chi-Kwong Li Quantum Operations and Completely Positive Maps

75. Theorem
Let A, B ∈ Hn . The following conditions are equivalent.
There exists a unital trace preserving completely positive map L
such that L(A) = B.
For each t ∈ R, there exists a trace preserving completely positive
map L such that L(A − tI) = B − tI.
λ(B) λ(A). i.e., there is a doubly stochastic matrix D such that
λ(B) = λ(A)D.
There is a unitary U ∈ Mn such that U AU † has diagonal entries
λ1 (B), . . . , λn (B).
There exist unitary matrices Uj , 1 ≤ j ≤ n such that
1 n †
B = n j=1 Uj AUj .
Chi-Kwong Li Quantum Operations and Completely Positive Maps

76. Theorem
Let A, B ∈ Hn . The following conditions are equivalent.
There exists a unital trace preserving completely positive map L
such that L(A) = B.
For each t ∈ R, there exists a trace preserving completely positive
map L such that L(A − tI) = B − tI.
λ(B) λ(A). i.e., there is a doubly stochastic matrix D such that
λ(B) = λ(A)D.
There is a unitary U ∈ Mn such that U AU † has diagonal entries
λ1 (B), . . . , λn (B).
There exist unitary matrices Uj , 1 ≤ j ≤ n such that
1 n †
B = n j=1 Uj AUj .
B is in the convex hull of the unitary orbit U(A) of A:
{U AU † : U unitary}.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

77. Results and questions on multiple matrices
Theorem
Suppose A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm are diagonal matrices.
Then there is a unital / trace preserving / unital and trace preserving
completely positive linear maps L such that
L(Aj ) = Bj for j = 1, . . . , k
Chi-Kwong Li Quantum Operations and Completely Positive Maps

78. Results and questions on multiple matrices
Theorem
Suppose A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm are diagonal matrices.
Then there is a unital / trace preserving / unital and trace preserving
completely positive linear maps L such that
L(Aj ) = Bj for j = 1, . . . , k
if and only if there is an n × m column / row / doubly stochastic matrix
D such that d(Bj ) = d(Aj )D for j = 1, . . . , k.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

79. Results and questions on multiple matrices
Theorem
Suppose A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm are diagonal matrices.
Then there is a unital / trace preserving / unital and trace preserving
completely positive linear maps L such that
L(Aj ) = Bj for j = 1, . . . , k
if and only if there is an n × m column / row / doubly stochastic matrix
D such that d(Bj ) = d(Aj )D for j = 1, . . . , k.
Here d(R) is the vector of diagonal entries of the square matrix R.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

80. Results and questions on multiple matrices
Theorem
Suppose A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm are diagonal matrices.
Then there is a unital / trace preserving / unital and trace preserving
completely positive linear maps L such that
L(Aj ) = Bj for j = 1, . . . , k
if and only if there is an n × m column / row / doubly stochastic matrix
D such that d(Bj ) = d(Aj )D for j = 1, . . . , k.
Here d(R) is the vector of diagonal entries of the square matrix R.
Remark Evidently, the result can be applied to commuting families
{A1 , . . . , Ak } and {B1 , . . . , Bk }.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

81. Results and questions on multiple matrices
Theorem
Suppose A1 , . . . , Ak ∈ Mn and B1 , . . . , Bk ∈ Mm are diagonal matrices.
Then there is a unital / trace preserving / unital and trace preserving
completely positive linear maps L such that
L(Aj ) = Bj for j = 1, . . . , k
if and only if there is an n × m column / row / doubly stochastic matrix
D such that d(Bj ) = d(Aj )D for j = 1, . . . , k.
Here d(R) is the vector of diagonal entries of the square matrix R.
Remark Evidently, the result can be applied to commuting families
{A1 , . . . , Ak } and {B1 , . . . , Bk }.
Question What about non-commuting families?
Chi-Kwong Li Quantum Operations and Completely Positive Maps

82. Further research
It would be interesting to study the interpolation problem for
general families {A1 , . . . , Ak } and {B1 , . . . , Bk }.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

83. Further research
It would be interesting to study the interpolation problem for
general families {A1 , . . . , Ak } and {B1 , . . . , Bk }.
It is closely related to dilation theory and the study of spectral
inequalities relating a Hermitian matrix and its principal
submatrices.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

84. Further research
It would be interesting to study the interpolation problem for
general families {A1 , . . . , Ak } and {B1 , . . . , Bk }.
It is closely related to dilation theory and the study of spectral
inequalities relating a Hermitian matrix and its principal
submatrices.
For given A1 , . . . , Ak ∈ Hn , it is interesting to characterize the
compact convex set CPm (A1 , . . . , Ak ) of k-tuples of matrices
k
(B1 , . . . , Bk ) ∈ Hm such that
(B1 , . . . , Bk ) = (L(A1 ), . . . , L(Ak ))
for some completely positive linear map L : Mn → Mm .
Chi-Kwong Li Quantum Operations and Completely Positive Maps

85. Further research
It would be interesting to study the interpolation problem for
general families {A1 , . . . , Ak } and {B1 , . . . , Bk }.
It is closely related to dilation theory and the study of spectral
inequalities relating a Hermitian matrix and its principal
submatrices.
For given A1 , . . . , Ak ∈ Hn , it is interesting to characterize the
compact convex set CPm (A1 , . . . , Ak ) of k-tuples of matrices
k
(B1 , . . . , Bk ) ∈ Hm such that
(B1 , . . . , Bk ) = (L(A1 ), . . . , L(Ak ))
for some completely positive linear map L : Mn → Mm .
The problem is interesting even when k = 1.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

86. Further research
It would be interesting to study the interpolation problem for
general families {A1 , . . . , Ak } and {B1 , . . . , Bk }.
It is closely related to dilation theory and the study of spectral
inequalities relating a Hermitian matrix and its principal
submatrices.
For given A1 , . . . , Ak ∈ Hn , it is interesting to characterize the
compact convex set CPm (A1 , . . . , Ak ) of k-tuples of matrices
k
(B1 , . . . , Bk ) ∈ Hm such that
(B1 , . . . , Bk ) = (L(A1 ), . . . , L(Ak ))
for some completely positive linear map L : Mn → Mm .
The problem is interesting even when k = 1.
It is also interesting to impose restriction on the Karus (Choi) rank
r of the completely positive linear map L(A) = j=1 Fj AFj† .
r
Chi-Kwong Li Quantum Operations and Completely Positive Maps

87. A computaional approach
Derive numerical scheme (using gradient flow, positive semi-definite
programming, etc.) to solve the following:
Given A1 , . . . , Ak ∈ Hn , B1 , . . . , Bk ∈ Hm , determine L such that
Chi-Kwong Li Quantum Operations and Completely Positive Maps

88. A computaional approach
Derive numerical scheme (using gradient flow, positive semi-definite
programming, etc.) to solve the following:
Given A1 , . . . , Ak ∈ Hn , B1 , . . . , Bk ∈ Hm , determine L such that
L(Aj ) = Bj , j = 1, . . . , k,
Chi-Kwong Li Quantum Operations and Completely Positive Maps

89. A computaional approach
Derive numerical scheme (using gradient flow, positive semi-definite
programming, etc.) to solve the following:
Given A1 , . . . , Ak ∈ Hn , B1 , . . . , Bk ∈ Hm , determine L such that
L(Aj ) = Bj , j = 1, . . . , k,
and
[L(Eij )] ≥ 0.
Chi-Kwong Li Quantum Operations and Completely Positive Maps

90. A computaional approach
Derive numerical scheme (using gradient flow, positive semi-definite
programming, etc.) to solve the following:
Given A1 , . . . , Ak ∈ Hn , B1 , . . . , Bk ∈ Hm , determine L such that
L(Aj ) = Bj , j = 1, . . . , k,
and
[L(Eij )] ≥ 0.
We may impose additional conditons such as:
Chi-Kwong Li Quantum Operations and Completely Positive Maps

91. A computaional approach
Derive numerical scheme (using gradient flow, positive semi-definite
programming, etc.) to solve the following:
Given A1 , . . . , Ak ∈ Hn , B1 , . . . , Bk ∈ Hm , determine L such that
L(Aj ) = Bj , j = 1, . . . , k,
and
[L(Eij )] ≥ 0.
We may impose additional conditons such as:
L(In ) = Im (unital).
Chi-Kwong Li Quantum Operations and Completely Positive Maps

92. A computaional approach
Derive numerical scheme (using gradient flow, positive semi-definite
programming, etc.) to solve the following:
Given A1 , . . . , Ak ∈ Hn , B1 , . . . , Bk ∈ Hm , determine L such that
L(Aj ) = Bj , j = 1, . . . , k,
and
[L(Eij )] ≥ 0.
We may impose additional conditons such as:
L(In ) = Im (unital).
Tr L(Eij ) = δij for 1 ≤ i, j ≤ n (trace preserving).
Chi-Kwong Li Quantum Operations and Completely Positive Maps

93. A computaional approach
Derive numerical scheme (using gradient flow, positive semi-definite
programming, etc.) to solve the following:
Given A1 , . . . , Ak ∈ Hn , B1 , . . . , Bk ∈ Hm , determine L such that
L(Aj ) = Bj , j = 1, . . . , k,
and
[L(Eij )] ≥ 0.
We may impose additional conditons such as:
L(In ) = Im (unital).
Tr L(Eij ) = δij for 1 ≤ i, j ≤ n (trace preserving).
The sum of r × r principal submatrix of L: Sr (L) = 0 for a given r
(L has rank less than r).
Chi-Kwong Li Quantum Operations and Completely Positive Maps

94. Any comments and suggestions are welcome!
Chi-Kwong Li Quantum Operations and Completely Positive Maps

95. Any comments and suggestions are welcome!
Thank you for your attention!
Chi-Kwong Li Quantum Operations and Completely Positive Maps