0
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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 $(x,y)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.