Let A, B be positive operators on a Hilbert space with 0 < m ≤ A, B ≤ M. Then for every unital positive linear map Φ,
Φ²((A + B)/2) ≤ K²(h)Φ²(A ♯ B),
and
Φ²((A+B)/2) ≤ K²(h)(Φ(A) ♯ Φ(B))²,
where A ♯ B is the geometric mean and K(h) = (h+1)²/(4h) with h = M/m.

Fiedler and Markham (1994) proved $${\left(\frac{\mathrm{det}\widehat{H}}{k}\right)}^{k}\ge \mathrm{det}H,$$
where $H={\left({H}_{ij}\right)}_{i,j=1}^{n}$ is a positive semidefinite matrix partitioned into $n\times n$ blocks with each block $k\times k$ and $\widehat{H}={\left(\mathrm{tr}{H}_{ij}\right)}_{i,j=1}^{n}$. We revisit this inequality mainly using some terminology from quantum information theory. Analogous results are included. For example, under the same condition, we prove $$\mathrm{det}({I}_{n}+\widehat{H})\ge \mathrm{det}{({I}_{nk}+kH)}^{1/k}.$$

Companion matrices of the second type are characterized by properties that involve bilinear maps.

Download Results (CSV)