Quantum operations and completely positive linear maps

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


Chi-Kwong Li

Currently visiting GWU and NRL.



Chi-Kwong Li

Currently visiting GWU and NRL.
Joint work with Yiu-Tung Poon (Iowa State University).


Classical computing


Classical computing

Hardware - Beads and bars.


Classical computing

Input - Using ﬁnger skill to change the states of the device.


Classical computing

Processor - Mechanical process with algorithms based on elementary
arithmetic rules.


Classical computing

Processor - Mechanical process with algorithms based on elementary
arithmetic rules.
Output - Beads and bars, then recorded by brush and ink.


Modern Computing


Modern Computing

Hardware - Mechanical/electronic/integrated circuits.


Modern Computing

Input - Punch cards, keyboards, scanners, sounds, etc. all converted
to binary bits - (0, 1) sequences.


Modern Computing

Processor - Manipulations of (0, 1) sequences using Boolean logic.


Modern Computing

0∨0=0
0∨1=1
1∨0=1
1∨1=1



Modern Computing

0∨0=0
0∨1=1
1∨0=1
1∨1=1


Output - (0, 1) sequences realized as visual images, which can be
viewed or printed.


Quantum computing

Quantum Computing Unit
−→ −→
Optical lattices, NMR


Quantum computing

−→ −→

Hardware - Super conductor, trapped ions, optical lattices, quantum
dot, MNR, etc.


Quantum computing

−→ −→

dot, MNR, etc.
Input - Quantum states in a speciﬁc form - Quantum bits (Qubits).


Quantum computing

−→ −→

dot, MNR, etc.
Processor - Provide suitable environment for the quantum system of
qubits to evolve.


Quantum computing

−→ −→

dot, MNR, etc.
qubits to evolve.
Output - Measurement of the resulting quantum states.


Quantum computing

−→ −→

dot, MNR, etc.
qubits to evolve.
Output - Measurement of the resulting quantum states.
All these require the understanding of mathematics, physics,
chemistry, computer sciences, engineering, etc.


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




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




1 0
|0 = and |1 = .
0 1


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.




1 0
|0 = and |1 = .
0 1


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.


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.


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


Tensor product and complexity

The state of k qubits is represented as the tensor product of k 2 × 2
matrices
Q1 ⊗ · · · ⊗ Qk .



matrices
Q1 ⊗ · · · ⊗ Qk .

Recall that for matrices X = (xij ) and Y (may be of diﬀerent
sizes),
X ⊗ Y = (xij Y ).



matrices
Q1 ⊗ · · · ⊗ Qk .

sizes),
X ⊗ Y = (xij Y ).

General entangled states are represented as 2k × 2k density
matrices, i.e., trace one positive semideﬁnite matrices.



matrices
Q1 ⊗ · · · ⊗ Qk .

sizes),
X ⊗ Y = (xij Y ).

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,



matrices
Q1 ⊗ · · · ⊗ Qk .

sizes),
X ⊗ Y = (xij Y ).

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!


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† .


Quantum operations


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


Quantum operations


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


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.



representation
r
L(A) = Fj AFj† ,
j=1

Fj Fj† = Im ;
r
In addition, L is unital if L(In ) = Im , equivalently, j=1



representation
r
L(A) = Fj AFj† ,
j=1

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


General problems

Since every quantum operation / channel is a trace preserving completely
positive linear map, it is interesting to study the following.


General problems

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?


General problems

Question
j = 1, . . . , k?

It is also interesting to consider the following related problems.


General problems

Question
j = 1, . . . , k?


Understand the duality relation between the trace preserving
completely positive linear maps and the unital preserving completely
positive linear maps.


General problems

Question
j = 1, . . . , k?


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.


Basic results

For A ∈ Hn with eigenvalues λ1 (A) ≥ · · · ≥ λn (A), denote by

λ(A) = (λ1 (A), . . . , λn (A)).


Basic results


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.


Basic results


λ(A) = (λ1 (A), . . . , λn (A)).

Deﬁnition

Example (5, 4, 1) (7, 3, 0), (5, 4, 1) (6, 2, 2), (6, 2, 2) (5, 4, 1).


Basic results


λ(A) = (λ1 (A), . . . , λn (A)).

Deﬁnition

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.


Basic results


λ(A) = (λ1 (A), . . . , λn (A)).

Deﬁnition

Example (5, 4, 1) (7, 3, 0), (5, 4, 1) (6, 2, 2), (6, 2, 2) (5, 4, 1).
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;


Basic results


λ(A) = (λ1 (A), . . . , λn (A)).

Deﬁnition

Example (5, 4, 1) (7, 3, 0), (5, 4, 1) (6, 2, 2), (6, 2, 2) (5, 4, 1).
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.


Theorem
Suppose A ∈ Hn and B ∈ Hm . Let a+ (respectively, a− ) be the sum of
the positive (respectively, negative) eigenvalues of A.


Theorem
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.


Theorem
λ(B) (a+ , 0, . . . , 0, a− ) in R1×m .


Theorem
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.


Theorem
λ(B) (a+ , 0, . . . , 0, a− ) in R1×m .
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


Theorem
λ(B) (a+ , 0, . . . , 0, a− ) in R1×m .
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.


Theorem
Let A ∈ Hn and B ∈ Hm . The following conditions are equivalent.


Theorem

There is a unital completely positive linear map L such that
L(A) = B.


Theorem

L(A) = B.
λn (A) ≤ λj (B) ≤ λ1 (A) for all j = 1, . . . , m.


Theorem

L(A) = B.
There is an n × m column stochastic matrix D such that
λ(B) = λ(A)D.


Theorem

L(A) = B.
λ(B) = λ(A)D.

Remark The condition may fail even if A and B are density matrices.


Theorem

L(A) = B.
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?


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.


Question
also a trace preserving completely positive map sending A to B. Is there
a unital trace preserving completely positive map sending A to B?


Question

The following example shows that the answer is negative.


Question

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.


Question

Example
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).


Question

Example
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.


Theorem
Let A, B ∈ Hn . The following conditions are equivalent.


Theorem

There exists a unital trace preserving completely positive map L
such that L(A) = B.


Theorem

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.


Theorem

such that L(A) = B.
λ(B) λ(A). i.e., there is a doubly stochastic matrix D such that
λ(B) = λ(A)D.


Theorem

such that L(A) = B.
λ(B) = λ(A)D.
There is a unitary U ∈ Mn such that U AU † has diagonal entries
λ1 (B), . . . , λn (B).


Theorem

such that L(A) = B.
λ(B) = λ(A)D.
λ1 (B), . . . , λn (B).
There exist unitary matrices Uj , 1 ≤ j ≤ n such that
1 n †
B = n j=1 Uj AUj .


Theorem

such that L(A) = B.
λ(B) = λ(A)D.
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}.


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



Theorem

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.



Theorem

L(Aj ) = Bj for j = 1, . . . , k

Here d(R) is the vector of diagonal entries of the square matrix R.



Theorem

L(Aj ) = Bj for j = 1, . . . , k


Remark Evidently, the result can be applied to commuting families
{A1 , . . . , Ak } and {B1 , . . . , Bk }.



Theorem

L(Aj ) = Bj for j = 1, . . . , k


Remark Evidently, the result can be applied to commuting families
{A1 , . . . , Ak } and {B1 , . . . , Bk }.
Question What about non-commuting families?


Further research

It would be interesting to study the interpolation problem for
general families {A1 , . . . , Ak } and {B1 , . . . , Bk }.


Further research

It is closely related to dilation theory and the study of spectral
inequalities relating a Hermitian matrix and its principal
submatrices.


Further research

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 .


Further research

submatrices.
k
(B1 , . . . , Bk ) ∈ Hm such that

(B1 , . . . , Bk ) = (L(A1 ), . . . , L(Ak ))

The problem is interesting even when k = 1.


Further research

submatrices.
k
(B1 , . . . , Bk ) ∈ Hm such that

(B1 , . . . , Bk ) = (L(A1 ), . . . , L(Ak ))

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


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,




L(Aj ) = Bj , j = 1, . . . , k,

and
[L(Eij )] ≥ 0.




L(Aj ) = Bj , j = 1, . . . , k,

and
[L(Eij )] ≥ 0.

We may impose additional conditons such as:




L(Aj ) = Bj , j = 1, . . . , k,

and
[L(Eij )] ≥ 0.


L(In ) = Im (unital).




L(Aj ) = Bj , j = 1, . . . , k,

and
[L(Eij )] ≥ 0.



Tr L(Eij ) = δij for 1 ≤ i, j ≤ n (trace preserving).




L(Aj ) = Bj , j = 1, . . . , k,

and
[L(Eij )] ≥ 0.



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).


Any comments and suggestions are welcome!


Any comments and suggestions are welcome!
Thank you for your attention!


Quantum operations and completely positive linear maps

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