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 6230&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=f9007b104c3add9b55801c7ca7f6e76d" 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 6230&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=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 6230&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=f9007b104c3add9b55801c7ca7f6e76d" title="Click to view the MathML source">C^⁎$C^{⁎}$-algebras, 6230&_mathId=si3.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=3b4aee1b2704f8ec04f9dbcb8de633be" title="Click to view the MathML source">ρ∈A$\rho \in \mathcal{A}$ is a Φ-density element and 6230&_mathId=si4.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=dbed8a1072cb05337894b69eb53bc605" title="Click to view the MathML source">A,B$A,B$ are self-adjoint operators of 6230&_mathId=si5.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=1ff0654e956a78c29dce004b7de59916" title="Click to view the MathML source">A$\mathcal{A}$ such that 6230&_mathId=si6.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=25e7c2cb63b6bac84c54935d71534d76">6230-si6.gif">$\mathrm{sp}(\text{-i}{\rho }^{\frac{1}{2}}[A,B]{\rho }^{\frac{1}{2}})\subseteq [m,M]$ for some scalers 6230&_mathId=si131.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=050285b7163f300282cd833edf6cda8a" title="Click to view the MathML source">0<m<M$0<m<M$, then under some conditions where 6230&_mathId=si142.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=fadfd7d5195c31aa50f144cfd5c662ae" title="Click to view the MathML source">K_m,M(ρ[A,B])$K_{m,M}(\rho [A,B])$ is the Kantorovich constant of the operator 6230&_mathId=si10.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=61bcfc5c79430e8681c00e84548eedd4">6230-si10.gif">$\text{-i}{\rho }^{\frac{1}{2}}[A,B]{\rho }^{\frac{1}{2}}$ and 6230&_mathId=si11.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=0ac858277be6d30b24f2d5a03d40cfbb" 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.