arXiv subjects

Yiu-Tung Poon

Publications and source records attributed to Yiu-Tung Poon.

40 records · Page 3Linked to original sources

arXiv2008

Quantum error correction and generalized numerical ranges

For a noisy quantum channel, a quantum error correcting code exists if and only if the joint higher rank numerical ranges associated with the error operators of the channel is non-empty. In this paper, geometric properties of the joint higher rank numerical ranges are obtained and their implications to quantum computing are discussed. It is shown that if the dimension of the underlying Hilbert space of the quantum states is sufficiently large, the joint higher rank numerical range of operators is always star-shaped and contains a non-empty convex subset. In case the operators are infinite dimensional, the joint infinite rank numerical range of the operators is a convex set closely related to the joint essential numerical ranges of the operators.

math.FA↗

arXiv2008

Higher rank numerical ranges and low rank perturbations of quantum channels

For a positive integer $k$, the rank-$k$ numerical range $Λ_k(A)$ of an operator $A$ acting on a Hilbert space $\cH$ of dimension at least $k$ is the set of scalars $λ$ such that $PAP = λP$ for some rank $k$ orthogonal projection $P$. In this paper, a close connection between low rank perturbation of an operator $A$ and $Λ_k(A)$ is established. In particular, for $1 \le r < k$ it is shown that $Λ_k(A) \subseteq Λ_{k-r}(A+F)$ for any operator $F$ with $\rank (F) \le r$. In quantum computing, this result implies that a quantum channel with a $k$-dimensional error correcting code under a perturbation of rank $\le r$ will still have a $(k-r)$-dimensional error correcting code. Moreover, it is shown that if $A$ is normal or if the dimension of $A$ is finite, then $Λ_k(A)$ can be obtained as the intersection of $Λ_{k-r}(A+F)$ for a collection of rank $r$ operators $F$. Examples are given to show that the result fails if $A$ is a general operator. The closure and the interior of the convex set $Λ_k(A)$ are completely determined. Analogous results are obtained for $Λ_\infty(A)$ defined as the set of scalars $λ$ such that $PAP = λP$ for an infinite rank orthogonal projection $P$. It is shown that $Λ_\infty(A)$ is the intersection of all $Λ_k(A)$ for $k = 1, 2, >...$. If $A - μI$ is not compact for any $μ\in \IC$, then the closure and the interior of $Λ_\infty(A)$ coincide with those of the essential numerical range of $A$. The situation for the special case when $A-μI$ is compact for some $μ\in \IC$ is also studied.

math.FA↗

arXiv2007

Condition for the higher rank numerical range to be non-empty

It is shown that the rank-$k$ numerical range of every $n$-by-$n$ complex matrix is non-empty if $n \ge 3k - 2$. The proof is based on a recent characterization of the rank-$k$ numerical range by Li and Sze, the Helly's theorem on compact convex sets, and some eigenvalue inequalities. In particular, the result implies that $Λ_2(A)$ is non-empty if $n \ge 4$. This confirms a conjecture of Choi et al. If $3k-2>n>0$, an $n$-by-$n$ complex matrix is given for which the rank-$k$ numerical range is empty. Extension of the result to bounded linear operators acting on an infinite dimensional Hilbert space is also discussed.

math.FA↗

arXiv2003

Eigenvalues, singular values, and Littlewood-Richardson coefficients

We characterize the relationship between the singular values of a complex Hermitian (resp., real symmetric, complex symmetric) matrix and the singular values of its off-diagonal block. We also characterize the eigenvalues of an Hermitian (or real symmetric) matrix C=A+B in terms of the combined list of eigenvalues of A and B. The answers are given by Horn-type linear inequalities. The proofs depend on a new inequality among Littlewood-Richardson coefficients.

math.AG↗