Quantum operations#

We will start with density matrices and CPTP maps.

Given the Hilbert space $\vecs{H}$, let $L(\vecs{H})$ be the set of automorphisms on this space. Then a density operator is such a linear operator $\rho \in L(\vecs{H})$, such that $\rho$ is

  • Hermitian: $\rho = \rho^\dagger$,
  • Normalized: $\text{Tr}(\rho)=1$,
  • Positive semidefinite $\bra{\psi}\rho\ket{\psi} \ge 0$ (written $\rho \ge 0)$).

A density operator is the most complete description of a quantum system. The most general evolution of density operators, including unitary operations, transient interactions with other systems and measurements, is described by quantum operations, more formally referred to as Completely-Positive Trace-Preserving (CPTP) maps.

A quantum operation is a CPTP map $\oper{S}: L(\vecs{H}) \to L(\vecs{H})$ that maps density operators to density operators such that $\oper{S}$ is

  • Positive: if $\rho \ge 0$, then $\oper{S}(\rho) \ge 0$,
  • Completely-positive: if $\sigma \in L(\vecs{H}^{AB})$ such that $\sigma \ge 0$ and $\text{Tr}_B(\sigma) = \rho$, then $(I \otimes S)(\sigma) \ge 0$.
  • Trace-preserving: $\text{Tr}(S(\rho)) = \text{Tr}(\rho)$.

Any such operation has a representation, called the Kraus representation.

For any quantum operation $\oper{S}$, there exist a non-unique set of operators $\set{A_k}_k$, where $A_k\in L(\vecs{H})$, such that \begin{equation} S(\rho) = \sum_k A_k\rho A_k^\dagger, \end{equation} and $\sum_k A_k^\dagger A_k = I$. This is called the Kraus representation.

Example: The dephasing channel...

Example: The depolarizing channel...

The noise channel model#

We are now ready to define the noise that any qubits sent through the channel experience. Suppose, Alice sends $n$-qubits to Bob via the channel. Then, the noise model is the same single-qubit channel $S$ applied to each qubit. So, to describe the noise the $n$-qubit quantum operation is \begin{equation} \oper{N} = \oper{S}^{\otimes n}. \end{equation}

Given the description of noise model an the $n$-qubit operator $\oper{N}$, we can identify a Krauss decomposition \begin{equation} \oper{N}(\rho) = \sum_i A_i\rho A_i^\dagger, \end{equation} where there are some $A_i$ with weight equal to $n$. However, recall that in the classical case if the probability of error $p$ is small, then large errors are unlikely. There is a similar theorem for noise in the quantum systems.

A CPTP map on $n$-qubits that can be written with Krauss operators each of which is a sum of operators with weight less than $t$ is a $t$-qubit error.

Theorem: Let $\oper{S}$ be a single-qubit quantum channel close to the identity $||\oper{S} - I||_\diamond < \epsilon$ for some $\epsilon$. Then there exists a $t$-qubit error channel $\oper{\tilde E}$ such that \begin{equation} ||\oper{S}^{\otimes n} - \oper{\tilde E}||_\diamond = \mathcal{O}\left(\binom{n}{t} \epsilon^t\right). \end{equation}

In simple language the above theorem states that if each of the $n$-qubits has only a very small chance of incurring an error, then one is very likely to observe errors on not very many qubits.

Suppose now that we have a $t$-qubit error channel $\oper{E}$, with a Krauss representation given by the operators $\set{E_i}_i$. From now on, we will refer to the $E_i$ as errors, and $\set{E_i}_i$ as the set of errors. For instance, if $S$ is the dephasing channel, and $n = 2$, then \begin{equation} \st{E} = \set{I, Z_1, Z_2, Z_1Z_2}, \end{equation} where the $I$ is included as the no-error possibility.

Our goal is to find quantum codes that are able to correct a given set of errors. Though often the research process proceeds in the opposite direction, and first a quantum code is defined and then its set of correctable errors is determined. We are now in the position to define a quantum code.