# What does it mean “less than identity” in the operator sum representation?

In Quantum Computation and Quantum Information by Isaac Chuang and Michael Nielsen, 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?

Thanks

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 or equal than 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.)