What does the inequality $\sum_k A_k^* A_k \leq I$ represent in Kraus' theorem?

Question

Nielsen and Chuang's statement of Kraus' Theorem includes the inequality $$ \sum_k A_k^* A_k \leq I.$$ What does this inequality represent? What quantities are being compared by this symbol?

Answer 1

$\sum_k A_k^\dagger A_k \leq 1$ is basically a statement about how trace-preserving the operator is.

The simplest example of this is a 2 level system that emits energy randomly. It has the Kraus operators $K_0 = \sqrt{1-p}I, K_1 = \sqrt{p}|1\rangle\langle0|$ so $$\sum_i K_i^\dagger K_i = K_0^\dagger K_0 + K_1^\dagger K_1 = (1-p)I+p|0\rangle\langle0|={\rm diag}(1,1-p)$$ Since $\sum_i K_i^\dagger K_i$ has eigenvalues less than 1 (e.g. $\langle1|\sum_i K_i^\dagger K_i|1\rangle = 1-p$) we say $\sum_i K_i^\dagger K_i\leq I.$

Note of course that when applied to some object like say a density matrix, the trace changes. Physically for a density matrix it's trace becomes different to 1. This is because you are 'throwing away' states either intentionally because you post-select a measurement outcome, or unintentionally because your measurement apparatus is imperfect (say you lose 9 out of every 10 photons that go throguh your system).

Note that if all the eigenvalues of $\sum_i K_i^\dagger K_i$ are equal to 1, $\sum_i K_i^\dagger K_i$ is exactly the identity.

To answer the second question, as commented by Navneeth on the question, the inequality is a statement of positive definiteness: $$A \leq B \iff 0 \leq B - A\,,$$ i.e. $B - A$ is positive semi-definite.