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Yiu-Tung Poon

Publications and source records attributed to Yiu-Tung Poon.

At least 37 records · Page 2Linked to original sources
arXiv2016

Inequalities on generalized matrix functions

We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number $r \in \{1\} \cup [2, \infty)$, positive semi-definite matrices $A_i,\ B_i,\ C_i\in M_{n_i}$, $i=1,2$, and generalized matrix functions $d_χ, d_ξ$ such as the determinant and permanent, etc., we have \begin{eqnarray*}&&\left(d_χ(A_1+B_1+C_1)d_ξ(A_2+B_2+C_2)\right)^r \\ &&\hskip 1in + \left(d_χ(A_1)d_ξ(A_2)\right)^r + \left(d_χ(B_1)d_ξ(B_2)\right)^r + \left(d_χ(C_1)d_ξ(C_2)\right)^r \\ & \ge &\left(d_χ(A_1+B_1 )d_ξ(A_2+B_2 )\right)^r + \left(d_χ(A_1+ C_1)d_ξ(A_2+ C_2)\right)^r + \left(d_χ( B_1+C_1)d_ξ( B_2+C_2)\right)^r\,.\end{eqnarray*} A general scheme is introduced to prove more general inequalities involving $m$ positive semi-definite matrices for $m \ge 3$ that extend the results of other authors.

math.FA↗
arXiv2016

Joint product numerical range and geometry of reduced density matrices

The reduced density matrices of a many-body quantum system form a convex set, whose three-dimensional projection $Θ$ is convex in $\mathbb{R}^3$. The boundary $\partialΘ$ of $Θ$ may exhibit nontrivial geometry, in particular ruled surfaces. Two physical mechanisms are known for the origins of ruled surfaces: symmetry breaking and gapless. In this work, we study the emergence of ruled surfaces for systems with local Hamiltonians in infinite spatial dimension, where the reduced density matrices are known to be separable as a consequence of the quantum de Finetti's theorem. This allows us to identify the reduced density matrix geometry with joint product numerical range $Π$ of the Hamiltonian interaction terms. We focus on the case where the interaction terms have certain structures, such that ruled surface emerge naturally when taking a convex hull of $Π$. We show that, a ruled surface on $\partialΘ$ sitting in $Π$ has a gapless origin, otherwise it has a symmetry breaking origin. As an example, we demonstrate that a famous ruled surface, known as the oloid, is a possible shape of $Θ$, with two boundary pieces of symmetry breaking origin separated by two gapless lines.

quant-ph↗
arXiv2015

Maximal noiseless code rates for collective rotation channels on qudits

We study noiseless subsystems on collective rotation channels of qudits, i.e., quantum channels with operators in the set ${\mathcal E}(d,n) = \{ U^{\otimes n}: U \in {\mathrm{SU}}(d)\}.$ This is done by analyzing the decomposition of the algebra ${\mathcal A}(d,n)$ generated by ${\mathcal E}(d,n)$. We summarize the results for the channels on qubits ($d=2$), and obtain the maximum dimension of the noiseless subsystem that can be used as the quantum error correction code for the channel. Then we extend our results to general $d$. In particular, it is shown that the code rate, i.e., the number of protected qudits over the number of physical qudits, always approaches 1 for a suitable noiseless subsystem. Moreover, one can determine the maximum dimension of the noiseless subsystem by solving a non-trivial discrete optimization problem. The maximum dimension of the noiseless subsystem for $d = 3$ (qutrits) is explicitly determined by a combination of mathematical analysis and the symbolic software Mathematica.

quant-ph↗
arXiv2015

Minkowski product of convex sets and product numerical range

Let $K_1, K_2$ be two compact convex sets in $\mathit{C}$. Their Minkowski product is the set $K_1K_2 = \{ab: a \in K_1, b\in K_2\}$. We show that the set $K_1K_2$ is star-shaped if $K_1$ is a line segment or a circular disk. Examples for $K_1$ and $K_2$ are given so that $K_1$ and $K_2$ are triangles (including interior) and $K_1K_2$ is not star-shaped. This gives a negative answer to a conjecture by Puchala et. al concerning the product numerical range in the study of quantum information science. Additional results and open problems are presented.

math.MG↗
arXiv2015

Discontinuity of Maximum Entropy Inference and Quantum Phase Transitions

In this paper, we discuss the connection between two genuinely quantum phenomena --- the discontinuity of quantum maximum entropy inference and quantum phase transitions at zero temperature. It is shown that the discontinuity of the maximum entropy inference of local observable measurements signals the non-local type of transitions, where local density matrices of the ground state change smoothly at the transition point. We then propose to use the quantum conditional mutual information of the ground state as an indicator to detect the discontinuity and the non-local type of quantum phase transitions in the thermodynamic limit.

quant-ph↗
arXiv2014

Ranks and eigenvalues of states with prescribed reduced states

For a quantum state represented as an $n\times n$ density matrix $σ\in M_n$, let ${\cal S}(σ)$ be the compact convex set of quantum states $ρ= (ρ_{ij}) \in M_{m\cdot n}$ with the first partial trace equal to $σ$, i.e., ${\rm tr}_1(ρ) = ρ_{11} + \cdots + ρ_{mm} = σ$. It is known that if $m \ge n$ then there is a rank one matrix $ρ\in {\cal S}(σ)$ satisfying ${\rm tr}_1(ρ) = σ$. If $m < n$, there may not be rank one matrix in ${\cal S}(σ)$. In this paper, we determine the ranks of the elements and ranks of the extreme points of the set ${\cal S}$ We also determine $ρ^* \in {\cal S}(σ)$ with rank bounded by $k$ such that $\|{\rm tr}_1ρ^*) - σ\|$ is minimum for a given unitary similarity invariant norm $\|\cdot\|$. Furthermore, the relation between the eigenvalues of $σ$ and those of $ρ\in {\cal S}(σ)$ is analyzed. Extension of our results and open problems will be mentioned.

quant-ph↗
arXiv2014

Preservers of Unitary Similarity Functions on Lie Products of Matrices

Denote by $M_n$ the set of $n\times n$ complex matrices. Let $f: M_n \rightarrow [0,\infty)$ be a continuous map such that $f(μUAU^*)= f(A)$ for any complex unit $μ$, $A \in M_n$ and unitary $U \in M_n$, $f(X)=0$ if and only if $X=0$ and the induced map $t \mapsto f(tX)$ is monotonic increasing on $[0,\infty)$ for any rank 1 nilpotent $X \in M_n$. Characterizations are given for surjective maps $ϕ$ on $M_n$ satisfying $f(AB-BA) = f(ϕ(A)ϕ(B)-ϕ(B)ϕ(A))$. The general theorem are then used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius, that answers a question of Molnar raised at the 2014 CMS summer meeting.

math.FA↗
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

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

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↗
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↗
arXiv2011

A note on the realignment criterion

For a quantum state in a bipartite system represented as a density matrix, researchers used the realignment matrix and functions on its singular values to study the separability of the quantum state. We obtain bounds for elementary symmetric functions of singular values of realignment matrices. This answers some open problems proposed by Lupo, Aniello, and Scardicchio. As a consequence, we show that the proposed scheme by these authors for testing separability would not work if the two subsystems of the bipartite system have the same dimension.

quant-ph↗
arXiv2011

Recursive Encoding and Decoding of Noiseless Subsystem and Decoherence Free Subspace

When the environmental disturbace to a quantum system has a wavelength much larger than the system size, all qubits localized within a small area are under action of the same error operators. Noiseless subsystem and decoherence free subspace are known to correct such collective errors. We construct simple quantum circuits, which implement these collective error correction codes, for a small number $n$ of physical qubits. A single logical qubit is encoded with $n=3$ and $n=4$, while two logical qubits are encoded with $n=5$. The recursive relations among the subspaces employed in noiseless subsystem and decoherence free subspace play essential rôles in our implementation. The recursive relations also show that the number of gates required to encode $m$ logical qubits increases linearly in $m$.

quant-ph↗
arXiv2011

The automorphism group of separable states in quantum information theory

We show that the linear group of automorphism of Hermitian matrices which preserves the set of separable states is generated by \emph{natural} automorphisms: change of an orthonormal basis in each tensor factor, partial transpose in each tensor factor, and interchanging two tensor factors of the same dimension. We apply our results to preservers of the product numerical range.

quant-ph↗
arXiv2010

Interpolation problems by completely positive maps

Given commuting families of Hermitian matrices {A1, ..., Ak} and {B1, ...., Bk}, conditions for the existence of a completely positive map L, such that L(Aj) = Bj for j = 1, ...,k, are studied. Additional properties such as unital or / and trace preserving on the map ? are also considered. Connections of the study to dilation theory, matrix inequalities, unitary orbits, and quantum information science are mentioned.

math.FA↗
arXiv2010

Higher rank numerical ranges of normal matrices

The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix $A \in M_n$ has eigenvalues $a_1, \..., a_n$, then its higher rank numerical range $Λ_k(A)$ is the intersection of convex polygons with vertices $a_{j_1}, \..., a_{j_{n-k+1}}$, where $1 \le j_1 < \... < j_{n-k+1} \le n$. In this paper, it is shown that the higher rank numerical range of a normal matrix with $m$ distinct eigenvalues can be written as the intersection of no more than $\max\{m,4\}$ closed half planes. In addition, given a convex polygon ${\mathcal P}$ a construction is given for a normal matrix $A \in M_n$ with minimum $n$ such that $Λ_k(A) = {\mathcal P}$. In particular, if ${\mathcal P}$ has $p$ vertices, with $p \ge 3$, there is a normal matrix $A \in M_n$ with $n \le \max\left\{p+k-1, 2k+2 \right\}$ such that $Λ_k(A) = {\mathcal P}$.

math.FA↗