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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=f9007b104c3add9b55801c7ca7f6e76d" title="Click to view the MathML source">C^⁎class="mathContainer hidden">class="mathCode">$C^{⁎}$-algebras. Among several results, we establish a noncommutative Heisenberg uncertainty relation. More precisely, we show that if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=8a1f7b62f1e129dc47a24a8475ca94a4" title="Click to view the MathML source">Φ:A→Bclass="mathContainer hidden">class="mathCode">$\mathrm{\Phi }:\mathcal{A}\to \mathcal{B}$ is a tracial positive linear map between class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=f9007b104c3add9b55801c7ca7f6e76d" title="Click to view the MathML source">C^⁎class="mathContainer hidden">class="mathCode">$C^{⁎}$-algebras, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_mathId=si3.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=3b4aee1b2704f8ec04f9dbcb8de633be" title="Click to view the MathML source">ρ∈Aclass="mathContainer hidden">class="mathCode">$\rho \in \mathcal{A}$ is a Φ-density element and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_mathId=si4.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=dbed8a1072cb05337894b69eb53bc605" title="Click to view the MathML source">A,Bclass="mathContainer hidden">class="mathCode">$A,B$ are self-adjoint operators of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_mathId=si5.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=1ff0654e956a78c29dce004b7de59916" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode">$\mathcal{A}$ such that class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_mathId=si6.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=25e7c2cb63b6bac84c54935d71534d76">class="imgLazyJSB inlineImage" height="20" width="185" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306230-si6.gif">class="mathContainer hidden">class="mathCode">$\mathrm{sp}(\text{-i}{\rho }^{\frac{1}{2}}[A,B]{\rho }^{\frac{1}{2}})\subseteq [m,M]$ for some scalers class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_mathId=si131.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=050285b7163f300282cd833edf6cda8a" title="Click to view the MathML source">0<m<Mclass="mathContainer hidden">class="mathCode">$0<m<M$, then under some conditions where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_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])class="mathContainer hidden">class="mathCode">$K_{m,M}(\rho [A,B])$ is the Kantorovich constant of the operator class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_mathId=si10.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=61bcfc5c79430e8681c00e84548eedd4">class="imgLazyJSB inlineImage" height="20" width="88" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306230-si10.gif">class="mathContainer hidden">class="mathCode">$\text{-i}{\rho }^{\frac{1}{2}}[A,B]{\rho }^{\frac{1}{2}}$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306230&_mathId=si11.gif&_user=111111111&_pii=S0022247X16306230&_rdoc=1&_issn=0022247X&md5=0ac858277be6d30b24f2d5a03d40cfbb" title="Click to view the MathML source">V_ρ,Φ(X)class="mathContainer hidden">class="mathCode">$V_{\rho ,\mathrm{\Phi }}(X)$ 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.