The PANORAMA. Numerical Ranges in. Modern Times. Man-Duen Choi 蔡文端. For the 12 th WONRA.

Transcription

1 The PANORAMA of Numerical Ranges in Modern Times Man-Duen Choi 蔡文端 For the 12 th WONRA

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

3 What a great impact if Quantum Channels look like

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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 C 2 } = {one-dimensional complex linear subspaces of C 2 } = {special two-dimensional real subspaces of R 4 }

8 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 R 3. Thus, S 2 need not be a material surface. There are uncountably many 1-qubits, to make S 2 symmetry with continuity for approximation, while 1-bits are just the north pole and the south pole.

9 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 S Million points of codes are not feasible by construction --- Million points are still discrete and independent --- No way of approximation in terms of continuity.

10 However, S 2 is a mathematical simple object. The combinatorial effect of S 2 symmetry is not comparable to 2 n when n > 30. 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 2 n Hilbert space, which can also be identified as a rank-1 projection in the form as a 2 n X 2 n complex matrix.

11 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.

12 Numerical Ranges through Quantum Computers For an n x n matrix T, the numerical range W(T) = {x*tx: xεc n, 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.

13 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.

14 Concrete Example: Consider T = 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

15 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 M 2 n. Def: A density matrix is a positive semidefinite matrix of trace 1. Thus, each 2 x 2 density matrix is expressed as x y iz y 1 iz x with x 2 +y 2 + z 2 < 1; so all density matrices fill up the whole solid sphere with S 2 as boundary. Nevertheless, for the case n >2, there is no geometrical picture for the collection of all n x n density matrices.

16 QUERY: Sanya is a subset of the surface of the earth. What is a random point of Sanya? 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?

17 Numerical Ranges as elliptical discs Let : M 2 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 S 2 is an elliptical disc. So the map also sends all rank-1 projections onto an elliptical disc.

18 Numerical Ranges as related to density matrices For an n x n matrix T, the numerical range W(T) = {x*tx: xεc n, 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

19 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*}.

20 Affine Transforms induces Linear Maps M n = M n+ - M n + + i M + n i M + n M + n = 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}. 20

21 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 : M n M k is completely positive iff [ (E ij )] is positive iff when is of the form (A) = V j *AV j for all A in M n. More details for the structure theory. (This 1975 paper has been cited in more than 1100 research papers, as of 2014 July Google Scholars.)

22 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. 22

23 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

24 Quantum Channels as Trace-preserving completely positive linear maps A linear map : M n M n is a quantum channel when is of the form (A) = V j *AV j for all A in M n with V j V j * = 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.