Quantum information inequalities via tracial positive linear maps

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• 作者：Ali Dadkhah ^{dadkhah61@yahoo.com" class="auth_mail" title="E-mail the corresponding author} ; Mohammad Sal Moslehian ; ^{moslehian@um.ac.ir" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Tracial positive linear map ; Heisenberg uncertainty relation ; Generalized covariance ; Generalized correlation ; Generalized Wigner&ndash ; Yanase skew information ; C⁎C⁎-algebra
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：447
• 期：1
• 页码：666-680
• 全文大小：350 K

文摘

We present some generalizations of quantum information inequalities involving tracial positive linear maps between 9b55801c7ca7f6e76d" title="Click to view the MathML source">C^⁎$C^{⁎}$-algebras. Among several results, we establish a noncommutative Heisenberg uncertainty relation. More precisely, we show that if 8a1f7b62f1e129dc47a24a8475ca94a4" title="Click to view the MathML source">Φ:A→B$\mathrm{\Phi }:\mathcal{A}\to \mathcal{B}$ is a tracial positive linear map between 9b55801c7ca7f6e76d" title="Click to view the MathML source">C^⁎$C^{⁎}$-algebras, b4aee1b2704f8ec04f9dbcb8de633be" title="Click to view the MathML source">ρ∈A$\rho \in \mathcal{A}$ is a Φ-density element and 8a1072cb05337894b69eb53bc605" title="Click to view the MathML source">A,B$A,B$ are self-adjoint operators of e59916" title="Click to view the MathML source">A$\mathcal{A}$ such that $\mathrm{sp}(\text{-i}{\rho }^{\frac{1}{2}}[A,B]{\rho }^{\frac{1}{2}})\subseteq [m,M]$ for some scalers 8a" title="Click to view the MathML source">0<m<M$0<m<M$, then under some conditions where K_m,M(ρ[A,B])$K_{m,M}(\rho [A,B])$ is the Kantorovich constant of the operator $\text{-i}{\rho }^{\frac{1}{2}}[A,B]{\rho }^{\frac{1}{2}}$ and e6d30b24f2d5a03d40cfbb" title="Click to view the MathML source">V_ρ,Φ(X)$V_{\rho ,\mathrm{\Phi }}(X)$ is the generalized variance of X. In addition, we use some arguments differing from the scalar theory to present some inequalities related to the generalized correlation and the generalized Wigner–Yanase–Dyson skew information.