The Exercise

A quantum operation $\mathcal{E}$ on a system with state space $\mathcal{H}$ is given by finitely many operation elements $E_k$ , operators on $\mathcal{H}$ , and acts on a state as $\mathcal{E}(\rho) = \sum_k E_k \rho E_k^\dagger$ . The elements satisfy $\sum_k E_k^\dagger E_k \le I$ , so $\mathcal{E}$ may lower the trace of a state: $\operatorname{tr}\mathcal{E}(\rho)$ is the probability that the process described by $\mathcal{E}$ takes place.

The exercise asks for a model of such an $\mathcal{E}$ with an environment, of the kind an operation that preserves trace has: the environment starts in a pure state, a unitary acts on system and environment together, and the environment is traced out. The elements $E_k$ are completed by one more operator $E_\infty$ , the positive square root of $I - \sum_k E_k^\dagger E_k$ , so that the completed set sums to the identity, and $\mathcal{E}$ is recovered from a unitary on system and environment by projecting the environment off the basis state that belongs to $E_\infty$ before the trace.

The Extra Operation Element

The extra element $E_\infty$ of a family of operators $E_k$ on $\mathcal{H}$ with $\sum_k E_k^\dagger E_k \le I$ is the positive square root of the positive operator $I - \sum_k E_k^\dagger E_k$ : $E_\infty = \sqrt{I - \textstyle\sum_k E_k^\dagger E_k}.$ (1)^✦

The completed family $\{F_j\}$ of operators $E_k$ on $\mathcal{H}$ is the family of operators on $\mathcal{H}$ labelled by the indices $k$ together with one more index $\infty$ , equal to $E_k$ at each $k$ and to the extra element at $\infty$ : $F_k = E_k, \qquad F_\infty = E_\infty.$ (2)^✦

The Environment

The environment has state space $\mathcal{K}$ , with an orthonormal basis $\{|b_j\rangle\}$ labelled like the completed family, by the indices $k$ and $\infty$ . It starts in a pure state $|\omega\rangle$ , a unit vector of $\mathcal{K}$ .

For an orthonormal basis $\{|b_j\rangle\}$ of $\mathcal{K}$ , the partial trace over $\mathcal{K}$ of a linear operator $T$ on $\mathcal{H}\otimes\mathcal{K}$ is the operator on $\mathcal{H}$ $\operatorname{tr}_{\mathcal{K}} T = \sum_j \bigl(I\otimes\langle b_j|\bigr)\, T\, \bigl(I\otimes|b_j\rangle\bigr),$ (3) where the embedding $I\otimes|b_j\rangle$ takes $|\psi\rangle$ to $|\psi\rangle|b_j\rangle$ and the restriction $I\otimes\langle b_j|$ takes $|\psi\rangle|\varphi\rangle$ to $\langle b_j|\varphi\rangle\,|\psi\rangle$ .

The projector $P$ of the basis $\{|b_j\rangle\}$ is the operator on $\mathcal{H}\otimes\mathcal{K}$ that acts as the identity on the system and projects the environment onto the orthogonal complement of $|b_\infty\rangle$ : $P = I \otimes \bigl(I - |b_\infty\rangle\langle b_\infty|\bigr).$ (4)^✦

The Claim

Theorem 1. Let $E_k$ be finitely many operators on $\mathcal{H}$ with $\sum_k E_k^\dagger E_k \le I$ . Their completed family satisfies the completeness relation $\sum_j F_j^\dagger F_j = I.$ (5)^✦

Theorem 2. Let $E_k$ be finitely many operators on $\mathcal{H}$ with $\sum_k E_k^\dagger E_k \le I$ , let $\{|b_j\rangle\}$ be an orthonormal basis of $\mathcal{K}$ labelled by the indices $k$ together with $\infty$ , and let $|\omega\rangle$ be a unit vector of $\mathcal{K}$ . There is a unitary $U$ on $\mathcal{H}\otimes\mathcal{K}$ such that, for every operator $\rho$ on $\mathcal{H}$ , $\operatorname{tr}_{\mathcal{K}}\bigl[P\,U\,(\rho\otimes|\omega\rangle\langle\omega|)\, U^\dagger P\bigr] = \sum_k E_k \rho E_k^\dagger.$ (6)^✦

For a state $\rho$ the right side of (6) is $\mathcal{E}(\rho)$ , and the left side is the model the exercise asks for.

What Is Checked

The unitary model (6) is stated over the projector $P$ of (4) and the partial trace (3), both taken in the environment basis $\{|b_j\rangle\}$ .

What the library proves about the extra element (1) and the completed family (2) is the completeness relation (5).