```
Matrix Problems in Quantum Information Science  - Chi-Kwong Li.
================================================================

One may see http://en.wikipedia.org/wiki/Quantum_computing
for a basic introduction of quantum computing.

There are many interesting matrix problems in quantum information
science. We describe a few of them in the following.

1. Quantum operations and completlely positive linear maps
==========================================================
Quantum states can be represented as density matrices, i.e.,
positive semidefinite matrices with trace one. Every quantum
operation T can be represented as a ompletely positive linear
map on desnity matrices that admits an operator sum representation:

T(A) = F_1AF_1* + .... + F_r A F_r*.

For example, it would be interesting to study the following.

1.a [Interpolating problem] Given nxn density matrices A_1, ..., A_m
and mxm matrices B_1, ..., B_m, determine whether there is a quantum
operation T such that T(A_j) = B_j for all j.

1.b [Tomograpy] Suppose we know that there is a quantum operation T
and some density matrices A_1, ..., A_m such that T(A_j) = B_j
for j = 1, ..., m. From B_1, ..., B_m, what can we say about the
input A_1, ..., A_m, and the quantum operation T?

2. Quantum error correction and generalized numerical ranges.
============================================================
A quantum channel can also be represented as a completely positive linear
map T as described in (1). The operators F_1, ...., F_r, are known as the
error operators of the channel. In the study of quantum error correction
one would like to find a rank k orthogonal projection P so that one can
construct a recovery quantum operation (a completely positive linear map)
R such that
R(T(A)) = A    whenever  PAP = A.

By a result of Knill and Laflamme, P is such an orthogonal projection if and
only if
P(F_i*F_j)P is a multiple of P for each (i,j) pairs.
Thus, if we set
(F_1*F_1, F_1*F_2, ...., F_m*F_m) = (A_1, ..., A_m),

it is of interesting  to study the set of m-tuples (a_1, ..., a_m)
such that there exists a rank k  orthogonal projection P such that

PA_jP = a_j P     for   j = 1, ..., m.

The set of these m-tuples is called the rank-k numerical range of
A = (A_1, ..., A_m), denoted by Lambda_k(A).

2.a. It is important to know when the set is non-empty.
One readily sees that the set is non-empty if k = 1, what is the
maximum k so that Lambda_k(A) is non-empty.

2.b. It is also interesting to study the geometric properties of the set.
For example, if A_1, A_2, A_3 are Hermitian, then can we generate
the set Lambda_k(A_1, A_2, A_3) in R^3.

3. Separability of density matrices
===================================
In quantum information science, it is important to determine whether
an (mn)-by-(mn) density matrix is the convex combination of tensor
states, i.e., density matrices obtained as the tensor product of an
m-by-m density matrix and an n-by-n density matrix. Such sates are
call separable states. Determining whether a quantum state is separable
is an NP hard problem. In any event, it is of interest to find

3.a efficient necessary conditions for separability,
3.b efficient sufficient conditions for separability,
3.c conditions for special classes of density matrices to be separable.

4. Preserver problems related to quantum information science
============================================================
Presever problems concerns the chracterization of maps on
matrices or operators that leave invariant certain properties,
important subsets, or relations on matrices. For example,
a linear map L on square matrices leaving invariant the
determinant function has the form
L(A) = MAN    for all A, or
L(A) = MA^tN  for all A,
where M, N are matrices satisfying det(MN) = 1.

There are many natural preserver problems related to
quantum information science. For example, characterize
the preservers of
separable states, the higher rank numerical ranges,
the fidelity of quantum states, etc.

References
==========
1. arXiv:1102.1618
Title: Quantum error correction without measurement and an efficient recovery operation
Authors: Chi-Kwong Li, Mikio Nakahara, Yiu-Tung Poon, Nung-Sing Sze, Hiroyuki Tomita

2. arXiv:1012.4221
Title: The automorphism group of separable states in quantum information theory
Authors: Shmuel Friedland, Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze

3. arXiv:1012.1675
Title: Interpolation problems by completely positive maps
Authors: Chi-Kwong Li, Yiu-Tung Poon

4. arXiv:0902.4869
Title: Higher rank numerical ranges of normal matrices
Authors: Hwa-Long Gau, Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze
Journal-ref: SIAM J. Matrix Analysis Appl, 32:23-43, 2011

5. arXiv:0812.4772
Title: Quantum error correction and generalized numerical ranges
Authors: Chi-Kwong Li, Yiu-Tung Poon

6. arXiv:0710.2898 [pdf, ps, other]
Title: Higher rank numerical ranges and low rank perturbations of quantum channels
Authors: Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze
Journal-ref: J. Math. Anal. Appl, 348:843-855, 2008

7. arXiv:0706.1540
Title: Condition for the higher rank numerical range to be non-empty
Authors: Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze
Journal-ref: Linear and Multilinear Algebra, 57:365-368, 2009

8. arXiv:0706.1536
Title: Canonical forms, higher rank numerical range, convexity, totally isotropic subspace,
matrix equations
Authors: Chi-Kwong Li, Nung-Sing Sze
Journal-ref: Proc. Amer. Math. Soc., 136:3013-3023. 2008

9. Title: Linear preservers of higher rank numerical ranges and radii,
Authors: Sean Clark, Chi-Kwong Li, Jennifer Mahle and Leiba Rodman,
Journal-ref: Linear and Multilinear Algebra 57 (2009), 503-521.

10. Title: The joint essential numerical range of operators: Convexity and related results
Authros: Chi-Kwong Li and Yiu-Tung Poon
Journal-ref:  Studia Math. 194 (2009), 91-104.

==========
Prerequisities: Math 211 Linear Algebra.

Suggested prerequisities: Programming skills in Matlab and Maple will be very helpful. One can
write computer program to test the non-emptyness of the joint rank-k numerical range.

Students with more advanced mathematics courses such as Math 405, 408 may be able to study some
theoretical aspects of the problem.

Contact: Chi-Kwong Li

[Li is visiting Hong Kong under a Fulbright Fellowship in year 2011.
You may contact him by e-mail in the Spring 2011, and meet him in
Hong Kong and China in the summer of 2011 to conduct research if
interest.]
```