1. Numerical Ranges in Modern Times 14th WONRA at Man-Duen Choi
14th WONRA at

2. In the last century,
Toeplitz-Hausdorff Theorem, …
Ando-Arveson Dilation Theorem, …
In the 21st century of Quantum Computer

3. What a great impact if Quantum Channels look like

7. Quantum Computer with 2x2 matrices
A qubit (quantum bit) is a unit of quantum computer memory.
Mathematically, each 1-qubit is regarded as an element in
= {all 2 X 2 rank-1 projection matrices}
= {all vector states acting on C2 }
= {one-dimensional complex linear subspaces of C2 }
= {special two-dimensional real subspaces of R4}

8. with misleading pictures?

9. Thus, S2 need not be a material surface.
Physically, a 1-qubit is a superposition of the spherical surface (called the Bloch sphere), because an “electron” can move freely to any direction from the origin of R3.
Thus, S2 need not be a material surface.
There are uncountably many 1-qubits, to make S2 symmetry with continuity for approximation, while 1-bits are just the north pole and the south pole.

10. --- Million points of codes are not feasible by construction
May try to achieve the effect of 1-qubits with conventional computer, using million sensitive codes on a spherical surface, but
---How to put millions points uniformly on S2
--- Million points of codes are not feasible by construction
--- Million points are still discrete and independent
--- No way of approximation in terms of continuity.

11. However, S2 is a mathematical simple object
However, S2 is a mathematical simple object. The combinatorial effect of S2 symmetry is not comparable to 2n when n > So, a 1-qubit computer cannot replace the conventional computer, as used in digital photo and music.
But, we should look further in n-qubit computers, with physical meanings and mathematical ideas in terms of continuity and dimensions and analogs.
Def. An n-qubit (= a vector state) is regarded as a 1-dimensional complex linear subspace of the dim 2n Hilbert space, which can also be identified as a rank-1 projection in the form as a 2n X 2n complex matrix.

12. Usual computer produces a finite 0-1 sequences.
1-qubit computer produces 2 x 2
projections, as photons (instead
of electrons that cannot be isolated)
realized by laser.
A more general quantum computer
can produces larger-size matrices.

13. Numerical Ranges through Quantum Computers
For an n x n matrix T, the numerical range
W(T) = {x*Tx: xεCn , ‖x ‖ = 1}
={ trace Txx*: ||x|| =1}
={trace TP: for all rank-1 projections P}.
Thus the numerical range W(T) is the BASIC feature of the matrix T, as needed/provided for quantum information.

14. What On earth Does it mean?
How could laser be shoot
to produce the numerical range of a matrix
yields
What On earth Does it mean?
It means to be quantum information, as a direction to shoot laser, or a 2x2 rank-1 projection .

15. → Concrete Example: Consider T = 2 3 0 −2
The numerical range of T is the effect of laser that yields an elliptical disc of major axis of length 5 and minor axis of length 3, with foci at 2 and -2. →
This is a two-to-one mapping except from the equator.
The equator → the boundary of W(T).
Two poles → the center of the elliptical disc
Four eigen projections → two foci

16. deep featureS of 2 x 2 matrices:
W(T) determines T up to unitary equivalence.
In other words, each ellipse as W(T) determines the norm of T, and the eigenvalues of T .
The norm structure is built in hard meaning.

17. Density Matrices
In the formal setting of non-commutative probability, the random position of an n-qubit can be regarded as a density matrix, to be defined as a convex combination of rank-1 projections in M2n.
Def: A density matrix is a positive semidefinite matrix of trace 1.
Thus, each 2 x 2 density matrix is expressed as
with x2 +y2 + z2 < 1; so all density matrices fill up the whole solid sphere with S2 as boundary.
Nevertheless, for the case n >2, there is no geometrical picture for the collection of all n x n density matrices.

18. QUERY: Munich is a subset of the surface of the earth
QUERY: Munich is a subset of the surface of the earth. What is a random point of Munich?
Answer: A point in the solid sphere, not just a point on the surface.
Similarly, I went to 23 places on the surface of earth. Then my average position is a point of the solid earth. Could this point of the solid earth provide you some useful information about my whole locus?
Most geographers /computer scientists / experimental physicists could never understand the meaning of random position.
How to define the average of projective spaces?

19. Numerical Ranges as elliptical discs
Let  : M2  C be a linear map. Then  restricted to {2 x 2 density matrices} is an affine map (preserving the convex structure).
As {2 x 2 density matrices} is same as the solid ball as a convex set, each affine map will send the solid ball onto an elliptical disc of the complex plane.
But, the affine image of S2 is an elliptical disc. So the map  also sends all rank-1 projections onto an elliptical disc.

20. Numerical Ranges as related to density matrices
For an n x n matrix T, the numerical range
W(T) = {x*Tx: xεCn , ‖x ‖ = 1}
={ trace Txx*: ||x|| =1}
={trace TP: for all rank-1 projections P}.
=(trace TR: for all density matrices}.
Thus the numerical range W(T) , as the BASIC feature of the matrix T, justifies the need of density matrices for qubits,
the need of state space for pure states

21. Unital Positive Linear Maps vs Numerical Ranges
Usually, no way to describe of positive linear maps between 3 x 3 matrices, but there is a graphical way: .
Key Fact: If W(T) is a subset of W(S), then the unital linear map sending I to I and S to T is a positive linear map from the span of {I, S, S*} onto the span of {I,T, T*}.

22. Affine Transforms induces Linear Maps
Mn = Mn+ - Mn i Mn+ – i Mn+
Mn+ = R+ x { density matrices}
{affine transforms on density matrices}
≈ {trace-preserving positive linear maps}.
{affine transforms on density matrices, fixing scalar matrices }
≈ {unital trace-preserving positive linear maps}.
{feasible affine transforms on density matrices}
≈ {trace-preserving completely positive linear maps}. 22

23. Completely positive linear maps
Completely positive linear maps serves better than positive linear maps in structure theory as shown in
Thm: (Choi, 1975) A linear map
: Mn  Mk is completely positive
iff [(Eij)] is positive
iff when  is of the form
(A) =  Vj*AVj for all A in Mn .
More details for the structure theory.
(This paper has been cited in more than 1650 research papers, as of June Google Scholars.)

24. Completely positive linear maps in circuit theory
Each transformer defines a positive linear map A --> V*AV . Thus several transformers in series define a completely positive linear maps.
Main question in circuit theory: Besides transformers, what else can make a positive linear map?
Concrete linear maps of mathematical expressions are not implementable.

25. Non-normal Dilations in the easiest setting of the Stinespring Theorem
Thm: (Ando,Arveson) If the numerical range of T is a subset of the closed unit disc, then T can be dilated to
The proof is non-constructive (not enough for the need of quantum information).
Major Problem: Given T, how to get concrete A, B, C so that
is unitarily equivalent to

26. Quantum Channels as Trace-preserving completely positive linear maps
A linear map  : Mn  Mn is a quantum channel when  is of the form
(A) =  Vj*AVj
for all A in Mn with VjVj* = I .
Open Question: Let T be an n x n matrix.
What are (T) for all possible quantum channels ?
Partial answer: Suppose T is positive. Then All
(T) are precisely all positive n x n matrices of the same trace as T.