scholarly journals Quantifying the quantumness of ensembles via unitary similarity invariant norms

2018 ◽  
Vol 13 (4) ◽  
Author(s):  
Xian-Fei Qi ◽  
Ting Gao ◽  
Feng-Li Yan
Keyword(s):  
Unitary Similarity ◽  
Similarity Invariant

scholarly journals A complete unitary similarity invariant for unicellular matrices

2011 ◽  
Vol 435 (2) ◽  
pp. 409-419 ◽  
Author(s):  
Douglas Farenick ◽  
Tatiana G. Gerasimova ◽  
Nadya Shvai
Keyword(s):  
Unitary Similarity ◽  
Similarity Invariant

scholarly journals Ranks and eigenvalues of states with prescribed reduced states

2014 ◽  
Vol 27 ◽  
Author(s):  
Chi-Kwong Li ◽  
Yiu-Tung Poon ◽  
Xuefeng Wang
Keyword(s):  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Open Problems ◽  
Unitary Similarity ◽  
Partial Trace ◽  
Invariant Norm ◽  
Compact Convex Set ◽  
Rank One ◽  
Similarity Invariant

For a quantum state represented as an $n\times n$ density matrix $\sigma \in M_n$, let $\cS(\sigma)$ be the compact convex set of quantum states $\rho = (\rho_{ij}) \in M_{m\cdot n}$ with the first partial trace equal to $\sigma$, i.e., $\tr_1(\rho) =\rho_{11} + \cdots + \rho_{mm} = \sigma$. It is known that if $m\ge n$ then there is a rank one matrix $\rho \in \cS(\sigma)$ satisfying $\tr_1(\rho) = \sigma$. If $m < n$, there may not be any rank one matrix in $\cS(\sigma)$. In this paper, we determine the ranks of the elements and ranks of the extreme points of the set $\cS$. We also determine $\rho^* \in \cS(\sigma)$ with rank bounded by $k$ such that $\|\tr_1(\rho^*) - \sigma\|$ is minimum for a given unitary similarity invariant norm $\|\cdot\|$. Furthermore, the relation between the eigenvalues of $\sigma$ and those of $\rho \in \cS(\sigma)$ is analyzed. Extension of the results and open problems will be mentioned.

scholarly journals Mappings on matrices: invariance of functional values of matrix products

2006 ◽  
Vol 81 (2) ◽  
pp. 165-184 ◽  
Author(s):  
Jor-Ting Chan ◽  
Chi-Kwong Li ◽  
Nung-Sing Sze
Keyword(s):  
Standard Form ◽  
Complex Matrices ◽  
Unitary Similarity ◽  
Invariant Functions ◽  
Matrix Products ◽  
Rank One ◽  
Similarity Invariant ◽  
Special Case ◽  
Real Matrices

AbstractLet Mn, be the algebra of all n × n matrices over a field F, where n ≧ 2. Let S be a subset of Mn containing all rank one matrices. We study mappings φ: S → Mn, such that F(φ (A)φ (B)) = F(A B) for various families of functions F including all the unitary similarity invariant functions on real or complex matrices. Very often, these mappings have the form A ↦ μ(A)S(σ (aij))S-1 for all A= (aij) ∈ S for some invertible S ∈ Mn, field monomorphism σ of F, and an F*-valued mapping μ defined on S. For real matrices, σ is often the identity map; for complex matrices, σ is often the identity map or the conjugation map: z ↦ z. A key idea in our study is reducing the problem to the special case when F:Mn → {0, 1} is defined by F(X) = 0, if X = 0, and F(X) = 1 otherwise. In such a case, one needs to characterize φ: S → Mn such that φ(A) φ (B) = 0 if and only if AB = 0. We show that such a map has the standard form described above on rank one matrices in S.

scholarly journals Some unitary similarity invariant sets preservers of skew Lie products

2014 ◽  
Vol 457 ◽  
pp. 76-92 ◽  
Author(s):  
Jianlian Cui ◽  
Qiting Li ◽  
Jinchuan Hou ◽  
Xiaofei Qi
Keyword(s):  
Invariant Sets ◽  
Unitary Similarity ◽  
Similarity Invariant

Similarity-invariant signatures for partially occluded planar shapes

1992 ◽  
Vol 7 (3) ◽  
pp. 271-285 ◽  
Author(s):  
Alfred M. Bruckstein ◽  
Nir Katzir ◽  
Michael Lindenbaum ◽  
Moshe Porat
Keyword(s):  
Invariant Signatures ◽  
Planar Shapes ◽  
Similarity Invariant ◽  
Partially Occluded

scholarly journals Preservers of unitary similarity functions on Lie products of matrices

2016 ◽  
Vol 498 ◽  
pp. 160-180 ◽  
Author(s):  
Jianlian Cui ◽  
Chi-Kwong Li ◽  
Yiu-Tung Poon
Keyword(s):  
Unitary Similarity ◽  
Similarity Functions ◽  
Products Of Matrices
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