The Exercise

A quantum system $Q$ in a state $\sigma$ is measured, and the outcome $y$, one of $0, \dots, r-1$, is written into a second system $A$: a register with $r$ levels, prepared in $|0\rangle$. The exercise asks for one quantum operation $\mathcal{E}$ on $Q$ and $A$ together that measures and records in a single step. For each outcome it leaves $Q$ in the state that outcome produces and the register in $|y\rangle$, the outcomes mixed according to their probabilities, which is the state the book writes as Eq. (12.8). The operation must be trace preserving, and it must act on every state of the pair, not only on those with the register in $|0\rangle$.

The library builds $\mathcal{E}$ from one operation element $K_y$ for each outcome $y$, defined below from the measurement operators on $Q$ and a shift of the register. The resulting operation is trace preserving and acts on every state of the pair.

Systems and States

A quantum system $S$ is given by its state space, a finite dimensional complex Hilbert space.^✦ Two systems $S$ and $T$ form the composite system $S \otimes T$, whose state space is the tensor product of theirs.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$, pure or mixed.^✦

A pure state $\psi$ of $S$ is a unit vector of its state space^✦; it determines the state $|\psi\rangle\langle\psi|$ of $S$, the projector onto that vector.^✦ For a state $\rho$ of $S$ and a state $\tau$ of $T$, the product state $\rho \otimes \tau$ is the state of $S \otimes T$ whose density operator is the tensor product of those of $\rho$ and $\tau$.^✦

The Register

The qudit with $d$ levels is the system whose state space is $\mathbb{C}^d$ itself, with its standard inner product.^✦ Its computational basis $\{|i\rangle\}$, for $i = 0, \dots, d-1$, is the orthonormal basis of standard unit vectors, $|i\rangle$ having a one in position $i$ and zeros elsewhere^✦, and each $|i\rangle$ is a pure state of the qudit.^✦ The register $A$ is the qudit with $r$ levels.

A unitary $U$ of a system $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = 1$.^✦ For a permutation $\pi$ of the labels $0, \dots, d-1$, the basis permutation gate $G_\pi$ of the qudit is the unitary with $G_\pi |x\rangle = |\pi(x)\rangle$ for each basis vector $|x\rangle$. The shift $U_y$ of the register is the gate of the cyclic permutation $x \mapsto x + y$, the sum taken modulo $r$: $$U_y |x\rangle = |x + y\rangle, \qquad x, y = 0, \dots, r-1 .$$(1)^✦

The Measurement and the Operation

A measurement on $Q$ with outcomes $y = 0, \dots, r-1$ is a family of measurement operators $M_y$ on the state space of $Q$, one for each outcome, satisfying the completeness relation $$\sum_{y=0}^{r-1} M_y^\dagger M_y = 1 .$$(2)^✦ Write $E_y = M_y^\dagger M_y$ and $\sqrt{E_y}$ for its positive square root.

For each outcome $y$, the operation element $K_y$ on $Q \otimes A$ is the operator that acts as $\sqrt{E_y}$ on $Q$ and as the shift $U_y$ on the register, $$K_y = \sqrt{E_y} \otimes U_y, \qquad y = 0, \dots, r-1 ,$$(3) and the operation $\mathcal{E}$ takes a state $\rho$ of $Q \otimes A$ to the state $$\mathcal{E}(\rho) = \sum_{y=0}^{r-1} K_y \, \rho \, K_y^\dagger .$$(4)^✦

A map $f$ on the states of a system $S$ is a quantum channel when it is the restriction to states of a linear map $\Phi$ on the operators of the state space of $S$, so that $f(\rho) = \Phi(\rho)$ for every state $\rho$, with $\Phi$ completely positive and trace preserving: $\operatorname{tr} \Phi(X) = \operatorname{tr} X$ for every operator $X$.^✦

The Claim

What Is Checked

The claim is stated over a quantum system $Q$ of any finite dimension with a measurement $\{M_y\}$ of $r$ outcomes, a state $\sigma$ of $Q$, the register $A$ with its basis state $|0\rangle$ and the product state $\sigma \otimes |0\rangle\langle 0|$, and the operation $\mathcal{E}$ of (4), built from the operation elements (3) and the shifts (1).

The operation elements satisfy the completeness relation $\sum_y K_y^\dagger K_y = 1$, so that $\mathcal{E}(\rho)$ is a state of $Q \otimes A$ for every state $\rho$ of $Q \otimes A$.