Geodesic
A treatment of a determinant inequality of Fiedler and Markham
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 737-742
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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 $$ \mathop {\rm det }(I_n+\widehat {H}) \ge \mathop {\rm det }(I_{nk}+kH)^{{1}/{k}}.$$
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 $$ \mathop {\rm det }(I_n+\widehat {H}) \ge \mathop {\rm det }(I_{nk}+kH)^{{1}/{k}}.$$
DOI : 10.1007/s10587-016-0289-3
Classification : 15A45
Keywords: determinant inequality; partial trace 
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Lin, Minghua. A treatment of a determinant inequality of Fiedler and Markham. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 737-742. doi: 10.1007/s10587-016-0289-3

[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). | MR | Zbl

[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. | MR | Zbl

[3] Pillis, J. de: Transformations on partitioned matrices. Duke Math. J. 36 (1969), 511-515. | DOI | MR | Zbl

[4] Fiedler, M., Markham, T. L.: On a theorem of Everitt, Thompson, and de Pillis. Math. Slovaca 44 (1994), 441-444. | MR | Zbl

[5] Hiroshima, T.: Majorization criterion for distillability of a bipartite quantum state. Phys. Rev. Lett. 91 (2003), no. 057902, 4 pages. | DOI

[6] Horn, R. A., Johnson, C. R.: Matrix Analysis. Cambridge University Press, Cambridge (2013). | MR | Zbl

[7] Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223 (1996), 1-8. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[9] Lin, M.: Some applications of a majorization inequality due to Bapat and Sunder. Linear Algebra Appl. 469 (2015), 510-517. | MR | Zbl

[10] Petz, D.: Quantum Information Theory and Quantum Statistics. Theoretical and Mathematical Physics Springer, Berlin (2008). | MR | Zbl

[11] Rastegin, A. E.: Relations for symmetric norms and anti-norms before and after partial trace. J. Stat. Phys. 148 (2012), 1040-1053. | DOI | MR

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