# Quantum Channel Models

The so called depolarizing channel is the channel model that is mostly used when constructing quantum error correction codes. The action of such channel over a quantum state $\rho$ is

$\rho\rightarrow(1-p_x-p_y-p_z)\rho+p_xX\rho X+p_yY\rho Y+p_zZ\rho Z$

I was wondering which other channel models are considered in quantum communications, and how the construction of error correction codes is affected by considering such other channels.

First let me mention a minor point concerning terminology. The type of channel you are suggesting is often called a Pauli channel; the term depolarizing channel usually refers to the case where $p_x = p_y = p_z$.

Anyway, it is not really correct to say that Pauli channels are the channel model considered for quantum error correction. Standard quantum error correcting codes can protect against arbitrary errors (represented by any quantum channel you might choose) so long as the errors do not affect too many qubits.

As an example, let us consider an arbitrary single-qubit error, represented by a channel $\Phi$ mapping one qubit to one qubit. Such a channel can be expressed in Kraus form as $$\Phi(\rho) = A_1 \rho A_1^{\dagger} + \cdots + A_m \rho A_m^{\dagger}$$ for some choice of Kraus operators $A_1,\ldots,A_m$. (For a qubit channel we can always take $m = 4$ if we want.) You could, for instance, choose these operators so that $\Phi(\rho) = |0\rangle \langle 0|$ for every qubit state $\rho$, you could make the error unitary, or whatever else you choose. The choice can even be adversarial, selected after you know how the code works.

Each of the Kraus operators $A_k$ can be expressed as a linear combination of Pauli operators, because the Pauli operators form a basis for the space of 2 by 2 complex matrices: $$A_k = a_k I + b_k X + c_k Y + d_k Z.$$ If you now expand out the Kraus representation of $\Phi$ above, you will obtain a messy expression where $\Phi(\rho)$ looks like a linear combination of operators of the form $P_i \rho P_j$ where $i,j\in\{1,2,3,4\}$ and $P_1 = I$, $P_2 = X$, $P_3 = Y$, and $P_4 = Z$.

Now imagine that you have a quantum error correcting code that protects against an $X$, $Y$, or $Z$ error on one qubit. The usual way this works is that some extra qubits in the 0 state are tacked on to the encoded data and a unitary operation is performed that reversibly computes into these extra qubits a syndrome describing which error occurred, if any, and which qubit was affected.

Supposing that the arbitrary error $\Phi$ happened on the first qubit for simplicity, after the syndrome computation you will end up with a state that looks like a linear combination of terms like this: $$P_i |\psi\rangle \langle \psi| P_j \otimes |P_i\: \text{syndrome}\rangle\langle P_j\:\text{syndrome}|.$$ The assumption here is that $|\psi\rangle$ represents the encoded data without any noise, $P_i$ and $P_j$ act on the first qubit, and that "$P_i$ syndrome" and "$P_j$ syndrome" refer to the standard basis states that indicate that these errors have occurred on the first qubit. (The situation is similar for the error affecting any other qubit; I'm just trying to keep the notation simple by assuming the error happened to the first qubit.)

Now the key is that you measure the syndrome to see what error occurred, and all of the cross terms disappear because of the measurement. You are left with a probabilistic mixture of states that look like $$P_i |\psi\rangle \langle \psi| P_i \otimes |P_i\: \text{syndrome}\rangle\langle P_i\:\text{syndrome}|.$$ The error is corrected and the original state is recovered. In effect, by measuring the syndrome, you "project" or "collapse" the error to something that looks like a Pauli channel.

This is all described (somewhat briefly) in Section 10.2 of Nielsen and Chuang.

• Thanks for the insight, but I was aware of such effect called "discretization of errors", that causes the fact that correction the Pauli channel for a single qubit does indeed correct arbitrary errors on single qubits due to the collapse of the state after measurement. However, I am interested in what you actually state in "if the errors do not affect too many qubits". What would happen if that is not true? Thanks. – Josu Etxezarreta Martinez Jun 1 '18 at 7:50
• Also just to point out, I have seen that the Pauli channel is referred as "asymmetric Pauli channel" sometimes in literature, so that's why I asked the question using such expression. – Josu Etxezarreta Martinez Jun 1 '18 at 7:52
• My apologies if I've explained something you already know, I've just tried to answer the question that I interpret from what you have written. Also, my comments on terminology merely reflect my view of what is typical and aim only to avoid confusion -- everyone is free to use the terminology they prefer and of course there is not always perfect agreement, both over time and in terms of what people prefer. – John Watrous Jun 1 '18 at 15:14
• I think the answer to the question in your comment is that it depends on both the error and the code. Of course the code might fail to correct an error if it affects too many qubits. On the other hand there are, for example, so-called degenerate codes that correct more errors than they can actually identify, and they can be useful for high noise rates. These are very interesting objects, but I believe that many fundamental questions about them remain unanswered. – John Watrous Jun 1 '18 at 15:15
• @JohnWatrous, I invite you to my question here: quantumcomputing.stackexchange.com/questions/5794/… Kindly help to answer. – Tobias Fritzn Mar 28 at 16:18