A qubit is in state (|0> - |1>)/sqrt(2). What is the probability of measuring 0 in the computational basis?
A0
B1/4
C1/2
D1
The amplitude of |0> is 1/sqrt(2), so the probability of measuring 0 is |1/sqrt(2)|^2 = 1/2. The minus sign on the |1> amplitude is a phase — it affects interference in subsequent gates but does not change |amplitude|^2. Both (|0> + |1>)/sqrt(2) and (|0> - |1>)/sqrt(2) give 50/50 outcomes in the computational basis, even though they are physically distinct states.
Question 2 True / False
After measuring a qubit and obtaining outcome |1>, the qubit is in state |1> regardless of what its pre-measurement state was.
TTrue
FFalse
Answer: True
This is the projection postulate (state collapse). Upon obtaining outcome 1, the qubit's state is projected onto |1> and renormalized. All information about the original superposition is irreversibly lost. The qubit is now deterministically in |1>, and any subsequent measurement in the same basis will yield 1 with certainty.
Question 3 Short Answer
You have a qubit in state |+> = (|0> + |1>)/sqrt(2) and want to distinguish it from |-> = (|0> - |1>)/sqrt(2) with certainty. How can you do this?
Think about your answer, then reveal below.
Model answer: Apply a Hadamard gate before measuring in the computational basis. H maps |+> to |0> and |-> to |1>, so the computational basis measurement after H perfectly distinguishes the two states. Measuring directly in the computational basis gives 50/50 for both states and cannot distinguish them.
This illustrates that the choice of measurement basis matters. The states |+> and |-> are orthogonal in the X basis (Hadamard basis) but have identical computational-basis measurement statistics. By applying H — which is a change-of-basis transformation — you rotate the measurement axis to align with the states you want to distinguish. In general, to distinguish two orthogonal states, you must measure in a basis that includes them.
Question 4 Multiple Choice
Can you determine the full quantum state alpha|0> + beta|1> of a single qubit by performing measurements on it?
AYes — measure in three different bases to reconstruct alpha and beta
BYes — a single measurement in the computational basis reveals the state
CNo — measurement is probabilistic and collapses the state, so a single copy provides at most one bit of information
DNo — quantum states are fundamentally unknowable
A single measurement yields one classical bit (0 or 1) and destroys the state. You cannot determine alpha and beta from this. However, if you have many identical copies of the state, you can estimate the probabilities by repeated measurement, and by measuring in multiple bases (X, Y, Z), you can perform quantum state tomography to reconstruct the full state. The key constraint is that a single copy cannot be fully characterized — this is related to the no-cloning theorem.