When discussing error correction, what are the objects in the expression $PE_i^\dagger E_j P=\alpha_{ij} P$?

I've started reading the book "Quantum Computation and Quantum Information" by Michael A. Nielsen and Issac L. Chuang, specifically chapter 10 (about quantum error correction), and I'm finding myself a bit confused as to the notations used.

In page 436, theorem 10.1, (10.16), I'd like to ask what is the "type" of each "object"? Is everything a $$2^n \times 2^n$$ matrix? Or is $$E_i$$ a vector, and $${E_i}^\dagger E_j$$ the outer-product of two vectors?

An example for such $$\epsilon$$ with $$\{E_i\}$$ and $$\alpha_{ij}$$ would be appreciated.

The theorem: Let $$C$$ be a quantum code, and let $$P$$ be the projector onto $$C$$. Suppose $$\epsilon$$ is a quantum operation with operation elements $$\{E_i\}$$. A necessary and sufficient condition for the existence of an error-correction operation $$R$$ correcting $$\epsilon$$ on $$C$$ is that $$P E_i^\dagger E_j P = \alpha_{ij} P$$ for some Hermitian matrix $$\alpha$$ of complex numbers. We call the operation elements $$\{E_i\}$$ for the noise $$\epsilon$$ errors, and if such an $$R$$ exists we say that $$\{E_i\}$$ constitutes a correctable set of errors.

Thank you very much :)

Generally speaking, if a quantum channel $$\Phi$$ sends operators in $$\mathcal X$$ into operators in $$\mathcal Y$$, and its Kraus representation reads $$\Phi(X)=\sum_a A_a X A_a^\dagger$$, then the Kraus operators are linear operators of the form $$A_a\in\mathrm{Lin}(\mathcal X,\mathcal Y)$$, that is, $$A_a:\mathcal X\to\mathcal Y$$. It follows that $$A_a^\dagger\in\mathrm{Lin}(\mathcal Y,\mathcal X)$$.
In your case, it seems you have a channel with $$\mathcal X=\mathcal Y\simeq\mathbb C^N$$ with $$N=2^n$$. Thus your projection operator is $$P\in\mathrm{Lin}(\mathbb C^N)$$, and $$PE_i^\dagger E_j P$$ is just a standard matrix multiplication, which results in some operator in $$\mathrm{Lin}(\mathbb C^N)$$.
• Can the projection operator $P$ be seen as the identity matrix in the code space $C$? Nov 22, 2021 at 11:39