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

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?

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.

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 $$C$$.