Quantum Computing (AL-802 (D)) - Important Questions

Define a Hilbert space. Give examples relevant to quantum computing and state the axioms that characterize a complex Hilbert space.

State the properties of the inner product on a complex Hilbert space and prove the Cauchy--Schwarz inequality: $$\left|\left\langle \psi \middle| \phi \right\rangle\right| \le \sqrt{\left\langle \psi \middle| \psi \right\rangle}\;\sqrt{\left\langle \phi \middle| \phi \right\rangle}.$$

Explain Dirac bra--ket notation. Define the adjoint (Hermitian conjugate) of an operator $A$ and show how bras and kets transform under the adjoint operation. Illustrate with a generic operator acting on $|\psi\rangle$.

Define an orthonormal basis in a Hilbert space. Show that any state $|\psi\rangle$ can be uniquely expanded as $$|\psi\rangle = \sum_{i} c_{i} \, |e_{i}\rangle$$ where $\{ |e_{i}\rangle \}$ is an orthonormal basis, and express the coefficients $c_{i}$ in terms of inner products.

Define the tensor product of two Hilbert spaces and of two state vectors. If $|a\rangle$ and $|b\rangle$ are single-qubit states, write the two-qubit product state and show how a single-qubit operator $A$ acting on the first qubit is represented as $A\otimes I$ on the two-qubit space.

Define a unitary operator. Prove that if $U$ is unitary then it preserves inner products and norms, i.e. for all $|\psi\rangle,|\phi\rangle$ $$\left\langle U\psi \middle| U\phi \right\rangle = \left\langle \psi \middle| \phi \right\rangle.$$ Show that if $H$ is Hermitian then the time evolution operator $$U(t)=\exp\left( -\dfrac{iHt}{\hbar} \right)$$ is unitary.

Define a projector $P$ and state its properties. For a projective measurement with projectors $\{P_{k}\}$ satisfying $P_{k}P_{l}=\delta_{kl}P_{k}$ and $\sum_{k}P_{k}=I$, derive the probability $p_{k}$ of outcome $k$ for a state $|\psi\rangle$ and the post-measurement state. Express $p_{k}$ and the post-measurement state using $P_{k}$ and $|\psi\rangle$.

State the spectral decomposition theorem for a Hermitian operator $A$ and write $A$ in terms of its eigenvalues and projectors. Explain how this decomposition relates to measurement outcomes and expectation values.

Define the density operator $\rho$ for a quantum system. State and prove its key properties (Hermiticity, unit trace, positivity). Distinguish pure and mixed states using $\rho^{2}$, and show how to obtain a reduced density matrix by taking a partial trace over a subsystem.

Represent a general single-qubit pure state on the Bloch sphere by writing $$|\psi\rangle = \cos\left(\dfrac{\theta}{2}\right) |0\rangle + e^{i\phi}\,\sin\left(\dfrac{\theta}{2}\right) |1\rangle.$$ Derive the Bloch vector components $\left\langle \sigma_{x} \right\rangle$, $\left\langle \sigma_{y} \right\rangle$, $\left\langle \sigma_{z} \right\rangle$ in terms of $\theta$ and $\phi$.