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 mathmlsrc">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^⁎mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll">C⁎math>-algebras. Among several results, we establish a noncommutative Heisenberg uncertainty relation. More precisely, we show that if mathmlsrc">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→BmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll">mathvariant="normal">Φ:mathvariant="script">A→mathvariant="script">Bmath> is a tracial positive linear map between mathmlsrc">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^⁎mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll">C⁎math>-algebras, mathmlsrc">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∈AmathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll">ρ∈mathvariant="script">Amath> is a Φ-density element and mathmlsrc">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,BmathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll">A,Bmath> are self-adjoint operators of mathmlsrc">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">AmathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll">mathvariant="script">Amath> such that mathmlsrc">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">lineImage" 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">mathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll">mathvariant="normal">sp(-iρ12[A,B]ρ12)⊆[m,M]math> for some scalers mathmlsrc">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<MmathContainer hidden">mathCode"><math altimg="si131.gif" overflow="scroll">0<m<Mmath>, then under some conditions where mathmlsrc">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])mathContainer hidden">mathCode"><math altimg="si142.gif" overflow="scroll">Km,M(ρ[A,B])math> is the Kantorovich constant of the operator mathmlsrc">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">lineImage" 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">mathContainer hidden">mathCode"><math altimg="si10.gif" overflow="scroll">-iρ12[A,B]ρ12math> and mathmlsrc">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)mathContainer hidden">mathCode"><math altimg="si11.gif" overflow="scroll">Vρ,mathvariant="normal">Φ(X)math> 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.