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Chi-Kwong Li

Publications and source records attributed to Chi-Kwong Li.

At least 91 records · Page 5Linked to original sources
arXiv2014

Recursive encoding and decoding of the noiseless subsystem for qudits

We give a full explanation of the noiseless subsystem that protects a single-qubit against collective errors and the corresponding recursive scheme described by C.-K. Li et. al. [Phys. Rev. A 84, 044301 (2011)] from a representation theory point of view. Furthermore, we extend the construction to qudits under the influence of collective SU($d$) errors. We find that under this recursive scheme, the asymptotic encoding rate is $1/d$.

quant-ph↗
arXiv2014

Stationary Probability Vectors of Higher-order Markov Chains

We consider the higher-order Markov Chain, and characterize the second order Markov chains admitting every probability distribution vector as a stationary vector. The result is used to construct Markov chains of higher-order with the same property. We also study conditions under which the set of stationary vectors of the Markov chain has a certain affine dimension.

math.PR↗
arXiv2014

Conditions for degradability of tripartite quantum states

Alice, Bob, and Eve share a pure quantum state. We introduce the notion of state degradability by asking whether the joint density of Alice and Eve can be transformed to the joint density of Alice and Bob by processing Eve's part through a quantum channel, in order words, degrading Eve. We prove necessary and sufficient conditions for state degradability and provide an efficient method to quickly rule out degradability for a given state. The problem of determining degradability of states is different from that of quantum channels, although the notion is similar. One application of state degradability is that it can be used to test channel degradability. In particular, the degradability of the output state of a channel obtained from the maximally entangled input state gives information about the degradability of the channel.

quant-ph↗
arXiv2014

Perturbing eigenvalues of non-negative matrices

Let $A$ be an irreducible (entrywise) nonnegative $n\times n$ matrix with eigenvalues $$ρ, b+ic,b-ic, λ_4,\cdots,λ_n,$$ where $ρ$ is the Perron eigenvalue. It is shown that for any $t \in [0, \infty)$ there is a nonnegative matrix with eigenvalues $$ρ+ \tilde t,λ_2+t,λ_3+t, λ_4 \cdots,λ_n,$$ whenever $\tilde t \ge γ_n t$ with $γ_3=1, γ_4 = 2, γ_5=\sqrt 5$ and $γ_n = 2.25$ for $n \ge 6$. The result improves that of Guo et al. Our proof depends on an auxiliary result in geometry asserting that the area of an $n$-sided convex polygon is bounded by $γ_n$ times the maximum area of the triangle lying inside the polygon.

math.SP↗
arXiv2013

Decomposition of quantum gates

A recurrence scheme is presented to decompose an $n$-qubit unitary gate to the product of no more than $N(N-1)/2$ single qubit gates with small number of controls, where $N = 2^n$. Detailed description of the recurrence steps and formulas for the number of $k$-controlled single qubit gates in the decomposition are given. Comparison of the result to a previous scheme is presented, and future research directions are discussed.

quant-ph↗
arXiv2013

Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector

Let $A = \pmatrix A_{11} & A_{12} \cr A_{21} & A_{22}\cr\pmatrix \in M_n$, where $A_{11} \in M_m$ with $m \le n/2$, be such that the numerical range of $A$ lies in the set $\{e^{iφ} z \in \IC: |\Im z| \le (\Re z) \tan α\}$, for some $φ\in [0, 2π)$ and $α\in [0, π/2)$. We obtain the optimal containment region for the generalized eigenvalue $λ$ satisfying $$λ\pmatrix A_{11} & 0 \cr 0 & A_{22}\cr\pmatrix x = \pmatrix 0 & A_{12} \cr A_{21} & 0\cr\pmatrix x \quad \hbox{for some nonzero} x \in \IC^n,$$ and the optimal eigenvalue containment region of the matrix $I_m - A_{11}^{-1}A_{12} A_{22}^{-1}A_{21}$ in case $A_{11}$ and $A_{22}$ are invertible. From this result, one can show $|\det(A)| \le \sec^{2m}(α) |\det(A_{11})\det(A_{22})|$. In particular, if $A$ is a accretive-dissipative matrix, then $|\det(A)| \le 2^m |\det(A_{11})\det(A_{22})|$. These affirm some conjectures of Drury and Lin.

math.NA↗
arXiv2013

Linear maps preserving Ky Fan norms and Schatten norms of tensor products of matrices

For a positive integer $n$, let $M_n$ be the set of $n\times n$ complex matrices. Suppose $\|\cdot\|$ is the Ky Fan $k$-norm with $1 \le k \le mn$ or the Schatten $p$-norm with $1 \le p \le \infty$ ($p\ne 2$) on $M_{mn}$, where $m,n\ge 2$ are positive integers. It is shown that a linear map $ϕ: M_{mn} \rightarrow M_{mn}$ satisfying $$\|A\otimes B\| = \|ϕ(A\otimes B)\| \quad \hbox{for all} A \in M_m \hbox{and} B \in M_n$$ if and only if there are unitary $U, V \in M_{mn}$ such that $ϕ$ has the form $A\otimes B \mapsto U(φ_1(A) \otimes φ_2(B))V$, where $φ_s(X)$ is either the identity map $X \mapsto X$ or the transposition map $X \mapsto X^t$. The results are extended to tensor space $M_{n_1} \otimes \cdots \otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.

math.FA↗
arXiv2013

Criteria and new classes of k-positive maps

We study k-positive maps on operators. Proofs are given to different positivity criteria. Special attention is on positive maps arising in the study of quantum information science. Results of other researchers are extended and improved. New classes of positive maps are constructed. Some open questions are answered.

quant-ph↗
arXiv2013

Decomposition of unitary matrices and quantum gates

A general scheme is presented to decompose a $d$-by-$d$ unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in $d-1$ classes, where each class is isomorphic to the group of $2\times 2$ unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on $n$-qubit registers can be expressed as no more than $2^{n-1}(2^n-1)$ fully controlled single-qubit gates chosen from $2^n-1$ classes, where the quantum gates in each class share the same $n-1$ control qubits. Moreover, it is shown that it is easy to adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.

quant-ph↗
arXiv2012

Perfect partition of some regular bipartite graphs

A graph has a perfect partition if all its perfect matchings can be partitioned so that each part is a 1-factorization of the graph. Let $L_{rm, r}=K_{rm,rm}-mK_{r,r}$. We first give a formula to count the number of perfect matchings of $L_{rm, r}$, then show that $L_{6,1}$ and $L_{8,2}$ have perfect partitions.

math.CO↗
arXiv2012

Physical transformations between quantum states

Given two sets of quantum states {A_1, ..., A_k} and {B_1, ..., B_k}, represented as sets of density matrices, necessary and sufficient conditions are obtained for the existence of a physical transformation T, represented as a trace-preserving completely positive map, such that T(A_i) = B_i for i = 1, ..., k. General completely positive maps without the trace-preserving requirement, and unital completely positive maps transforming the states are also considered.

math-ph↗
arXiv2012

Entanglement transformation between sets of bipartite pure quantum states using local operations

Alice and Bob are given an unknown initial state chosen from a set of pure quantum states. Their task is to transform the initial state to a corresponding final pure state using local operations only. We prove necessary and sufficient conditions on the existence of such a transformation. We also provide efficient algorithms that can quickly rule out the possibility of transforming a set of initial states to a set of final states.

quant-ph↗
arXiv2012

A geometric characterization of invertible quantum measurement maps

A geometric characterization is given for invertible quantum measurement maps. Denote by ${\mathcal S}(H)$ the convex set of all states (i.e., trace-1 positive operators) on Hilbert space $H$ with dim$H\leq \infty$, and $[ρ_1, ρ_2]$ the line segment joining two elements $ρ_1, ρ_2$ in ${\mathcal S}(H)$. It is shown that a bijective map $ϕ:{\mathcal S}(H) \rightarrow {\mathcal S}(H)$ satisfies $ϕ([ρ_1, ρ_2]) \subseteq [ϕ(ρ_1),ϕ(ρ_2)]$ for any $ρ_1, ρ_2 \in {\mathcal S}$ if and only if $ϕ$ has one of the following forms $$ρ\mapsto \frac{MρM^*}{{\rm tr}(MρM^*)}\quad \hbox{or} \quad ρ\mapsto \frac{Mρ^T M^*}{{\rm tr}(Mρ^T M^*)},$$ where $M$ is an invertible bounded linear operator and $ρ^T$ is the transpose of $ρ$ with respect to an arbitrarily fixed orthonormal basis.

quant-ph↗
arXiv2012

Linear preservers and quantum information science

Let $m,n\ge 2$ be positive integers, $M_m$ the set of $m\times m$ complex matrices and $M_n$ the set of $n\times n$ complex matrices. Regard $M_{mn}$ as the tensor space $M_m\otimes M_n$. Suppose $|\cdot|$ is the Ky Fan $k$-norm with $1 \le k \le mn$, or the Schatten $p$-norm with $1 \le p \le \infty$ ($p\ne 2$) on $M_{mn}$. It is shown that a linear map $ϕ: M_{mn} \rightarrow M_{mn}$ satisfying $$|A\otimes B| = |ϕ(A\otimes B)|$$ for all $A \in M_m$ and $B \in M_n$ if and only if there are unitary $U, V \in M_{mn}$ such that $ϕ$ has the form $A\otimes B \mapsto U(φ_1(A) \otimes φ_2(B))V$, where $φ_i(X)$ is either the identity map $X \mapsto X$ or the transposition map $X \mapsto X^t$. The results are extended to tensor space $M_{n_1} \otimes ... \otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.

quant-ph↗
arXiv2011

Recovery in quantum error correction for general noise without measurement

It is known that one can do quantum error correction without syndrome measurement, which is often done in operator quantum error correction (OQEC). However, the physical realization could be challenging, especially when the recovery process involves high-rank projection operators and a superoperator. We use operator theory to improve OQEC so that the implementation can always be done by unitary gates followed by a partial trace operation. Examples are given to show that our error correction scheme outperforms the existing ones in various scenarios.

quant-ph↗
arXiv2011

Efficient Quantum Error Correction for Fully Correlated Noise

We investigate an efficient quantum error correction of a fully correlated noise. Suppose the noise is characterized by a quantum channel whose error operators take fully correlated forms given by $σ_x^{\otimes n}$, $σ_y^{\otimes n}$ and $σ_z^{\otimes n}$, where $n>2$ is the number of qubits encoding the codeword. It is proved that (i) $n$ qubits codeword encodes $(n-1)$ data qubits when $n$ is odd and (ii) $n$ qubits codeword implements a noiseless subsystem encoding $(n-2)$ data qubits when $n$ is even. Quantum circuits implementing these schemes are constructed.

quant-ph↗