Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 10D (Quantum Information and Computation)

A pure bipartite state is entangled when it is not a product state $|u{⟩}_A\otimes |v{⟩}_B$.

Sincewe haveIts coefficient matrix is proportional towhose determinant is $-4\ne 0$. By the two-qubit product-state determinant criterion, a product state would have determinant zero. Therefore $|\alpha ⟩$ is entangled.

This is superdense coding. To encode $00,01,10,11$, Alice applies respectively $I,X,Z,XZ$ to her half of $|{\Phi }^{+}⟩$. These local Pauli operators transform the shared state into the four mutually orthogonal Bell states $|{\Phi }^{+}⟩,|{\Psi }^{+}⟩,|{\Phi }^{-}⟩,|{\Psi }^{-}⟩$, up to irrelevant global phases. Alice sends her one qubit to Bob. Bob now holds both qubits and performs a Bell-basis measurement, whose outcome identifies Alice's two-bit message with certainty. One transmitted qubit communicates two classical bits because one shared Bell pair was available beforehand.