Past exam of the computer science course of the University of Oxford / 2023 / Quantum information / 1

A density matrix is a complex $d\times d$ Hermitian matrix that is positive semidefinite and has unit trace:Equivalently, its eigenvalues are nonnegative and sum to one, with an orthonormal basis of eigenvectors.^[ref]

A basic measurement is specified by an orthonormal basis $\{|b_j⟩{\}}_{j=0}^{d-1}$. The Born rule for the density matrix $\rho$ gives the outcome probabilitiesThese probabilities are nonnegative by positive semidefiniteness and sum to one by the resolution of the identity and unit trace. In the computational basis, the answers are simply the diagonal entries ${\rho }_{jj}$.^[ref]

Expand the density matrix of the Bell state and take the partial trace over Alice's qubit:The orthonormality of the computational basis removes the off-diagonal terms. Thus Bob's reduced density matrix is the maximally mixed state

Write the proposed quantum state ensemble as $\{(|u⟩,p),(|v⟩,1-p)\}$ with normalized pure states, so thatTaking the expectation values in $|u⟩$ and $|v⟩$, and setting $c=|⟨u|v⟩{|}^2$, givesConsequently $p\le 1/2$ and $p\ge 1/2$. Substitution then gives $c=0$. The two pure states are therefore orthogonal, with equal probabilities:Conversely, any orthonormal basis of a qubit has projectors summing to $I$, so such a quantum state ensemble does give $I/2$. This proves the characterization of two-state ensembles of a maximally mixed qubit.

Alice uses conjugate-basis steering of a Bell pair. Write the desired orthonormal basis for Bob as $|\psi ⟩=a|0⟩+b|1⟩$ and $|{\psi }^{\perp }⟩=c|0⟩+d|1⟩$. Entrywise complex conjugation in the computational basis gives Alice's orthonormal basisFor either $|u⟩$ in Bob's chosen orthonormal basis, the corresponding unnormalized conditional quantum state isIts squared norm is $1/2$. Thus Alice measures in the complex-conjugate basis, obtaining the required Bob pure states with probability $1/2$ each. Averaging over Alice's undisclosed outcome leaves Bob's density matrix equal to $I/2$, consistent with the no-communication theorem.

Use deterministic two-qubit entanglement dilution, retaining the specified complex probability amplitudes. Alice implements a quantum instrument on $A$ with Kraus operatorsThe Kraus representation completeness condition holds even for complex $\alpha ,\beta$:To realize this using the permitted quantum gates and a basic measurement, Alice initializes an ancilla qubit $C$ to $|0⟩$. Conditional on her computational-basis state being $j$, she applies to $C$ the unitary matrixEach has orthonormal columns. Therefore $W=|0⟩⟨0{|}_A\otimes V_0+|1⟩⟨1{|}_A\otimes V_1$ is a unitary gate, withA computational-basis measurement of $C$ implements the desired quantum instrument. Its two unnormalized branches on $AB$ areEach has probability $1/2$. Alice sends her outcome $m$ as one classical bit. For $m=0$, both parties keep their qubits unchanged. For $m=1$, both apply a Pauli X gate, givingThus both outcomes produce the exact target using one classical bit and local operations. The construction includes $\alpha =0$ or $\beta =0$ and arbitrary relative quantum phases; it requires no quantum channel between the parties.