Linear maps preserving determinant of tensor products of Hermitian matrices

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• 作者：Yuting Ding^a ; ^{yuting840810@163.com" class="auth_mail" title="E-mail the corresponding author} ; Ajda Fo&scaron ; ner^b ; ^{ajda.fosner@fm-kp.si" class="auth_mail" title="E-mail the corresponding author} ; Jinli Xu^a ; ^{jclixv@qq.com" class="auth_mail" title="E-mail the corresponding author} ; Baodong Zheng^c ; ^{zbd@hit.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Linear preserver ; Tensor product ; Determinant ; Hermitian matrix ; Quantum information science
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1139-1153
• 全文大小：359 K

文摘

Let pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304462&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304462&_rdoc=1&_issn=0022247X&md5=216722b88a835a1cb2985b8b14701e3e" title="Click to view the MathML source">m,n≥2pan>pan class="mathContainer hidden">pan class="mathCode">$m,n\ge 2$pan>pan>pan> be positive integers and let pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304462&_mathId=si2.gif&_user=111111111&_pii=S0022247X16304462&_rdoc=1&_issn=0022247X&md5=219d3e94864619b2e8261a1089ed5d09" title="Click to view the MathML source">H_npan>pan class="mathContainer hidden">pan class="mathCode">$H_n$pan>pan>pan> be the set of pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304462&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304462&_rdoc=1&_issn=0022247X&md5=611dc687009b172d410fb7678924c7ff" title="Click to view the MathML source">n×npan>pan class="mathContainer hidden">pan class="mathCode">$n\times n$pan>pan>pan> complex Hermitian matrices. We study linear maps pan id="mmlsi219" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304462&_mathId=si219.gif&_user=111111111&_pii=S0022247X16304462&_rdoc=1&_issn=0022247X&md5=3e455dd8bfe1c10cff3effdd09850711" title="Click to view the MathML source">ϕ:H_mn→H_mnpan>pan class="mathContainer hidden">pan class="mathCode">$\varphi :H_{mn}\to H_{mn}$pan>pan>pan> satisfying pan id="mmlsi48" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304462&_mathId=si48.gif&_user=111111111&_pii=S0022247X16304462&_rdoc=1&_issn=0022247X&md5=2d63517a9d1b82009e2b0401df3e61d8" title="Click to view the MathML source">det⁡(A⊗B)=det⁡(ϕ(A⊗B))pan>pan class="mathContainer hidden">pan class="mathCode">$\det (A\otimes B)=\det \left(\varphi (A\otimes B)\right)$pan>pan>pan> for all pan id="mmlsi220" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304462&_mathId=si220.gif&_user=111111111&_pii=S0022247X16304462&_rdoc=1&_issn=0022247X&md5=925319a30ff3b9fd41318dbd148e5ef0" title="Click to view the MathML source">A∈H_mpan>pan class="mathContainer hidden">pan class="mathCode">$A\in H_m$pan>pan>pan>, pan id="mmlsi131" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304462&_mathId=si131.gif&_user=111111111&_pii=S0022247X16304462&_rdoc=1&_issn=0022247X&md5=94b83c8260aa7bce28877f2d1f051d39" title="Click to view the MathML source">B∈H_npan>pan class="mathContainer hidden">pan class="mathCode">$B\in H_n$pan>pan>pan>. The connection of the problem to quantum information science is mentioned.