# Post-quantum cryptography canonical hilbert space

I am reading Post-quantum cryptography and on page 20, the mathematical space that allows the superposition of state is defined. It is defined as follows:
Quantum memory storing one quantum bit, or qubit as we will call it in all that follows, will have to allow for a superposition of the two states 0 and 1. Hence it is two-dimensional and can be modeled by the canonical two-dimensional Hilbert space $$\mathcal{H} = \mathcal{H}_1 = \mathbb{C} \oplus \mathbb{C}$$ We will use the set consisting of $(1, 0)$ and $(0, 1)$ as the standard (computational) basis for $\mathcal{H}$, and denote these vectors by $|0\rangle$, and $|1\rangle$, respectively.

First of all, why choosing the notation $\oplus$? To me, $\oplus$ represents the direct sum between two spaces. Moreover, there is also a proposition that says that the sum $E_1 + E_2$ is direct iff $E_1 \cap E_2 = \{0\}$, which is clearly not our case here, so why this notation? To me $\mathbb{C} \oplus \mathbb{C} = \mathbb{C}$, what am I missing here?

The definition of the basis is not clear either for me. When they say We will use the set consisting of $(1, 0)$ and $(0, 1)$ as the standard (computational) basis for $\mathcal{H}$, and denote these vectors by $|0\rangle$, and $|1\rangle$, respectively, they mean $(1, 0) = e_1$ and $(0, 1) = e_2$ like the usual basis for $\mathbb{R}^2$, right?

But if we compute $\alpha |0\rangle + \beta |1\rangle$ with $\alpha, \beta \in \mathbb{C}$, the result is in $\mathbb{C}^2$ which has dimension $4$. So once again, what am I missing?

Thanks for any help.

## 1 Answer

The (somewhat informal) notation $\mathbb C \oplus \mathbb C$ refers to the direct sum of two isomorphic but different copies of $\mathbb C$.

Take two abstract symbols (here $|0\rangle$ and $|1\rangle$, whatever those are) and consider formal linear combinations $\alpha |0\rangle + \beta |1\rangle$, $\alpha, \beta \in \mathbb C$ (that's just really a fancy way of writing $(\alpha, \beta)$). Formally, you can add them and muliply them by complex scalars. This gives you naturally a complex vector space of dimension 2, of basis $\{|0\rangle, |1\rangle\}$.

Here think of $|0\rangle$ and $|1\rangle$ as just abstract symbols without a particular meaning (think of the construction of complex numbers where we start by looking at formal sums $x+iy$, with $x,y \in \mathbb R$, as a fancy way of writing $(x,y)$.