In the QECC condition $\langle\psi|E_a^\dagger E_b|\phi\rangle=C_{ab}\langle\psi|\phi\rangle$, what is $C_{ab}$?

Question

In this book, Theorem 2.7 has the QECC conditions. I attach a snippet here

Theorem 2.7 (QECC Conditions). $(Q, \mathcal{E})$ is a $Q E C C$ iff $\forall|\psi\rangle,|\phi\rangle \in Q, \forall E_a, E_b \in \mathcal{E}$, $$ \left\langle\psi\left|E_a^{\dagger} E_b\right| \phi\right\rangle=C_{a b}\langle\psi \mid \phi\rangle . $$

The proof includes the following

Taking the adjoint of the equation and putting in $|\phi\rangle=|\psi\rangle$, we find that $C_{a b}^{\dagger}=C_{b a}^*$, i.e., the matrix $C$ is Hermitian. Therefore $C_{a b}$ is diagonalizable, and by choosing an appropriate spanning set $\{F_a\}$ for $\mathcal{E}$ we can actually diagonalize $C_{a b}$.

I don't understand if $C_{ab}$ is a matrix or a number. The author states multiple times in the proof that $C_{ab}$ is diagonalizable, has eigenvalues etc. but the statement of the theorem seems to suggest it is just a number. How is $C_{ab}$ a matrix in the statement of the theorem and how is it related to $C$ that is referred to in the proof?

Answer 1

The concept of $C_{ab} $ in the context of the Quantum Error-Correcting Code (QECC) conditions as described in Theorem 2.7 can indeed be confusing due to the mathematical notation and the terminology used. I will clarify this for you.

In the statement of Theorem 2.7, $C_{ab}$ appears to represent a scalar quantity based on the equation provided:

$\langle \psi | E_a^\dagger E_b | \phi \rangle = C_{ab} \langle \psi | \phi \rangle $

Here, for any two error operators $E_a $ and $E_b $ from the set $E$, and for any two states $\psi \rangle$ and $\phi \rangle$ from the code space $ Q $, the product $ E_a^\dagger E_b $ behaves in such a way that its matrix element between $ | \psi \rangle $ and $ | \phi \rangle $ is proportional to the inner product $ \langle \psi | \phi \rangle $ with a proportionality constant $ C_{ab} $.

However, the nature of $ C_{ab} $ as discussed in the proof might suggest it is not just a scalar but an element of a matrix $ C $ where $ C $ is composed of these elements $ C_{ab} $ for all combinations of $ a $ and $ b $. This matrix $ C $ is Hermitian, as indicated by the relation $ C_{ab}^\dagger = C_{ba}^* $, which makes it eligible for properties such as being diagonalizable and having eigenvalues.

Here’s how $ C_{ab} $ functions both as a scalar and a matrix element:

1. As a Scalar: In each specific instance of $ a $ and $ b $, $ C_{ab} $ is indeed a scalar, indicating the proportionality between $ \langle \psi | E_a^\dagger E_b | \phi \rangle $ and $ \langle \psi | \phi \rangle $.

2. As a Matrix Element: When considering all possible pairs of $ a $ and $ b $, $ C_{ab} $ forms the elements of the matrix $ C $. The entire matrix $ C $ is what is referred to when discussing properties like diagonalizability and eigenvalues in the proof.

The confusion arises due to the dual role of $ C_{ab} $ as both individual scalars and as components of the matrix $ C $. The matrix $ C $ is important because its properties (like being Hermitian and diagonalizable) are crucial for the QECC to effectively correct errors represented by the error set $ E $. The diagonalization of $ C $ means that you can choose a basis in which $ C $ is diagonal, simplifying the analysis and manipulation of the error correction properties of the code.

Thus, in summary, $ C_{ab} $ in each instance is a scalar, but collectively, these scalars make up the matrix $ C $, which possesses the described properties and is central to the theory and application of QECCs.

Answer 2

I imagine it's supposed to be a matrix $C$ with elements $C_{ab}$, i.e. $a$ indexes the row, and $b$ the column of $C$.