A computer stores the state of a three-qubit register as an array of eight complex…
PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum states, measurement and entanglement · Quantum states, measurement and entanglement
Quantum Information and Technology, Ch Qubits, registers and the Bloch sphere
Problem
A computer stores the state of a three-qubit register as an array of eight complex amplitudes. Entry k (k=0,…,7) holds the amplitude of ∣abc⟩, where the bit string abc read as a binary number equals k, qubit 1 being the most significant; so ∣101⟩ is entry 5. The register starts in ∣000⟩, the array (1,0,0,0,0,0,0,0).
((a)) Apply H⊗I⊗I. Give the resulting array and the ket it represents.
((b)) Next apply a CNOT with qubit 1 as control and qubit 3 as target: it flips qubit 3 exactly when qubit 1 is ∣1⟩ and leaves qubit 2 alone. Write this gate in the same form as the chapter's CNOT, as a sum of two Kronecker products, say which pairs of array entries it swaps, and give the final array and ket.
((c)) Qubit 1 of the final state is measured in the computational basis. Find the probability of each outcome and the state left on qubits 2 and 3.
((d)) Suppose qubit 2 is measured instead. Find the probability of each outcome and the state left on qubits 1 and 3. The register is entangled, yet this measurement changes nothing: explain why.
((e)) Check parts (a) and (b) by an array calculation, by hand or in any numerical language: build the two 8×8 matrices from Kronecker products and multiply them into the starting array.