Question

2. A single qubit state is a superposition with respect to a given basis if it can be written as a nontrivial linear combination of the basis with all probability amplitudes being non-zero. A nontrivial linear combination is one such that at least one amplitude is non-zero.

(a) Given the standard basis ∣0⟩,∣1⟩ show that a qubit in the state ∣+⟩ is in a state of superposition with respect to the standard basis

Answer 1

A qubit in the state |+\wrangle is in a state of superposition with respect to the standard basis |0\wrangle, |1\wrangle because it can be expressed as a nontrivial linear combination of these basis states with equal, nonzero probability amplitudes.

Step-by-step explanation:

To show that a qubit in the state |+\wrangle is in a state of superposition with respect to the standard basis |0\wrangle, |1\wrangle, we first need to express |+\wrangle in terms of the standard basis. The state |+\wrangle is defined as (|0\wrangle + |1\wrangle) / \sqrt{2}, which represents an equal superposition of the states |0\wrangle and |1\wrangle because it is a nontrivial linear combination of these two basis states where both coefficients (probability amplitudes) are nonzero.

The nontrivial linear combination demonstrates that the qubit is neither just in the state |0\wrangle nor |1\wrangle, but simultaneously in both to some extent. The coefficients in front of |0\wrangle and |1\wrangle give the probabilities of measuring the qubit in each state upon observation, which in this case are equal. Since both coefficients are nonzero, it confirms that the state |+\wrangle is a superposition of |0\wrangle and |1\wrangle as per the definition of a superposition state in quantum mechanics.