What does it mean "less than identity" in the operator sum representation?

Question

In Quantum Computation and Quantum Information by Nielsen and Chuang, Section 8.2.3, $\mathcal{E}=\sum_{k}E_k\rho E_k^{\dagger}$ gives the operator-sum representation. In general, it requires $\sum_k E_k E_k^{\dagger}\leq I$. But, what does it mean by the inequation here? Does it mean every entry of the matrix is a nonnegative real value up to 1?

Answer 1

Matrix inequalities of the form $A\ge B$ should be read as $$ A-B\ge 0\ , $$ which in turn means that all eigenvalues of $A-B$ are larger than or equal to zero.

In the given case, $M\le I$ means that all eigenvalues of $M$ are smaller or equal than one.

(Note that this convention for $\ge$ used on matrices depends on the field. In other fields, "$\ge0$" might refer to a component-wise property, and $\succeq0$ might be used for positivity of the eigenvalues.)