Academic Journal
A treatment of a determinant inequality of Fiedler and Markham
| العنوان: | A treatment of a determinant inequality of Fiedler and Markham |
|---|---|
| المؤلفون: | Lin, Minghua |
| بيانات النشر: | Institute of Mathematics, Academy of Sciences of the Czech Republic Matematický ústav AV ČR |
| سنة النشر: | 2016 |
| المجموعة: | DML-CZ (Czech Digital Mathematics Library) |
| مصطلحات موضوعية: | keyword:determinant inequality, keyword:partial trace, msc:15A45 |
| الوصف: | summary:Fiedler and Markham (1994) proved $$ \Big (\frac {\mathop {\rm det } \widehat {H}}{k}\Big )^{ k}\ge \mathop {\rm det } H, $$ where $H=(H_{ij})_{i,j=1}^n$ is a positive semidefinite matrix partitioned into $n\times n$ blocks with each block $k\times k$ and $\widehat {H}=(\mathop {\rm tr} H_{ij})_{i,j=1}^n$. We revisit this inequality mainly using some terminology from quantum information theory. Analogous results are included. For example, under the same condition, we prove $$ \mathop {\rm det }(I_n+\widehat {H}) \ge \mathop {\rm det }(I_{nk}+kH)^{{1}/{k}}.$$ |
| نوع الوثيقة: | text |
| وصف الملف: | application/pdf |
| اللغة: | English |
| تدمد: | 0011-4642 1572-9141 |
| Relation: | mr:MR3556864; zbl:Zbl 06644030; reference:[1] Bhatia, R.: Positive Definite Matrices.Texts and Readings in Mathematics 44. New Delhi: Hindustan Book Agency, Princeton Series in Applied Mathematics Princeton University Press, Princeton (2007). Zbl 1125.15300, MR 2284176; reference:[2] Bourin, J.-C., Lee, E.-Y., Lin, M.: Positive matrices partitioned into a small number of Hermitian blocks.Linear Algebra Appl. 438 (2013), 2591-2598. Zbl 1262.15037, MR 3005316; reference:[3] Pillis, J. de: Transformations on partitioned matrices.Duke Math. J. 36 (1969), 511-515. Zbl 0186.33703, MR 0325649, 10.1215/S0012-7094-69-03661-8; reference:[4] Fiedler, M., Markham, T. L.: On a theorem of Everitt, Thompson, and de Pillis.Math. Slovaca 44 (1994), 441-444. Zbl 0828.15023, MR 1301952; reference:[5] Hiroshima, T.: Majorization criterion for distillability of a bipartite quantum state.Phys. Rev. Lett. 91 (2003), no. 057902, 4 pages. http://dx.doi.org/10.1103/PhysRevLett.91.057902. 10.1103/PhysRevLett.91.057902; reference:[6] Horn, R. A., Johnson, C. R.: Matrix Analysis.Cambridge University Press, Cambridge (2013). Zbl 1267.15001, MR 2978290; reference:[7] Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions.Phys. Lett. A 223 (1996), 1-8. Zbl 1037.81501, MR 1421501, 10.1016/S0375-9601(96)00706-2; reference:[8] Jenčová, A., Ruskai, M. B.: A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality.Rev. Math. Phys. 22 (2010), 1099-1121. Zbl 1218.81025, MR 2733251, 10.1142/S0129055X10004144; reference:[9] Lin, M.: Some applications of a majorization inequality due to Bapat and Sunder.Linear Algebra Appl. 469 (2015), 510-517. Zbl 1310.15033, MR 3299075; reference:[10] Petz, D.: Quantum Information Theory and Quantum Statistics.Theoretical and Mathematical Physics Springer, Berlin (2008). Zbl 1145.81002, MR 2363070; reference:[11] Rastegin, A. E.: Relations for symmetric norms and anti-norms before and after partial trace.J. Stat. Phys. 148 (2012), 1040-1053. MR 2975521, 10.1007/s10955-012-0569-8 |
| الاتاحة: | http://hdl.handle.net/10338.dmlcz/145868 |
| Rights: | access:Unrestricted ; rights:DML-CZ Czech Digital Mathematics Library, http://dml.cz/ ; rights:Institute of Mathematics AS CR, http://www.math.cas.cz/ ; conditionOfUse:http://dml.cz/use |
| رقم الانضمام: | edsbas.B6DF64EE |
| قاعدة البيانات: | BASE |
Full Text Finder | تدمد: | 00114642 15729141 |
|---|