Quantum Information HT24, Qubits
Flashcards
Describe every possible state of a qubit, also taking into account phase.
Consider all vectors of the form
\[\vert \psi\rangle = \alpha \vert 0 \rangle + \beta \vert 1\rangle\]where $\alpha, \beta \in \mathbb C$ and $ \vert \alpha \vert ^2 + \vert \beta \vert ^2 = 1$. Define an equivalence relation
\[\psi \sim \psi'\]if there exists $\gamma \in [0, 2\pi]$ (called global phase) such that $ \vert \psi’\rangle = e^{i\gamma} \vert \psi\rangle$. Then the states of the qubit are the equivalence classes of this relation.
Can you give a necessary and sufficient condition for two generic vectors $ \vert \psi\rangle, \vert \psi’\rangle$ to represent the same quantum state?
\[\vert \psi\rangle, \vert \psi'\rangle \text{ represent same state} \iff \vert \langle \psi \vert \psi'\rangle \vert = 1\]
Suppose we have the qubit in the state $\alpha \vert 0\rangle + \beta \vert 1\rangle$. Can you describe a qubit in a state orthogonal to this?
\[\overline \beta \vert 0\rangle - \overline \alpha \vert 1\rangle\]