The Exercise

A system with $d$ levels passes through a depolarizing channel: with probability $p$ its state is replaced by the maximally mixed state $I/d$, and with probability $1-p$ it is left as it was. The exercise asks for a set of operation elements for this channel.

The library’s answer has $2^d\,d! + 1$ elements. One is a multiple of the identity and leaves the state alone; the others are equal multiples of $2^d\,d!$ unitaries, each of which permutes the standard basis of the $d$ levels and reverses the sign of some of its vectors.

A Qudit and Its States

A quantum system is a physical object whose state space is a finite dimensional complex Hilbert space, and each such space $H$ is the state space of a system. The qudit with $d$ levels is the quantum system whose state space is $\mathbb{C}^d$ with its standard inner product.^✦ The computational basis $\{|i\rangle\}$ of the qudit with $d$ levels is the orthonormal basis of $\mathbb{C}^d$ made of the standard unit vectors, $$|i\rangle = e_i, \qquad \langle i | i' \rangle = \delta_{ii'}, \qquad i = 0, \dots, d-1,$$(1)^✦ where $e_i$ has a one in position $i$ and zeros elsewhere.

A state $\rho$ of a quantum system $S$ is a positive operator of unit trace on the state space of $S$.^✦ The maximally mixed state $I/d$ of a system $S$ whose state space is nonzero, of dimension $d$, is the state of $S$ whose operator is the identity divided by $d$.^✦

The Channel

The depolarizing channel $\mathcal{E}$ on the qudit with $d \ge 1$ levels, for a probability $p$ between $0$ and $1$, is the map on the states of the qudit given by $$\mathcal{E}(\rho) = (1-p)\,\rho + p\,\frac{I}{d}.$$(2)^✦

The depolarizing unitaries of the qudit are labelled by pairs $j = (s, \pi)$ of a string $s$ of $d$ bits, one bit for each basis vector, and a permutation $\pi$ of the basis labels, that is, an element of the symmetric group $S_d$ on $d$ letters. The depolarizing unitary $U_j$ is the unitary on $\mathbb{C}^d$ that permutes the computational basis vectors by $\pi$ and reverses the sign of those selected by $s$; they are indexed by the labels, $$U_j, \qquad j = (s, \pi) \in J = \{0,1\}^d \times S_d.$$(3)^✦

The operation elements $E_k$ for the probability $p$ are the operators on $\mathbb{C}^d$ labelled by $k \in \{0\} \cup J$, $$E_0 = \sqrt{1-p}\; I, \qquad E_j = \sqrt{\frac{p}{2^d\,d!}}\; U_j \quad (j \in J).$$(4)^✦

The Claim

What Is Checked

The claim is stated over the depolarizing channel (2) on the states of the qudit and over its operation elements (4), which are built from the identity and the depolarizing unitaries (3) of the computational basis (1).

The library proves that for $d \ge 1$ the qudit has a maximally mixed state, its state space $\mathbb{C}^d$ being nonzero and $I/d$ being positive with unit trace, and that $\mathcal{E}(\rho)$, the convex combination of the states $\rho$ and $I/d$ with weights $1-p$ and $p$, is again a state for every state $\rho$.