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 bei _ non-zero. A nontrivial linear combination is one such that at least one amplitude is non-zero.

(b) Again, given the standard basis, is a qubit in the state 2​1​(∣+⟩+∣−⟩) in a superposition with respect to the standard basis? where ∣+⟩=(2​1​2​1​​) and ∣−⟩=(2​1​2​−1​​).

Answer 1

A qubit state given by ½(|+> + |->) is indeed in a superposition with respect to the standard basis, as it is a nontrivial combination of basis states with non-zero probability amplitudes.

Step-by-step explanation:

The question pertains to the concept of quantum superposition within quantum mechanics, more specifically in the context of qubit states in a quantum computer. A qubit can exist in a superposition of states, which in the context of the standard basis, involves the states |0> and |1>. Given the standard basis, the superposition state ½(|+> + |->), where |+> = (1/√2, 1/√2)^T and |-> = (1/√2, -1/√2)^T, is indeed a superposition with respect to the standard basis. In this case, the qubit state is a nontrivial linear combination of |0> and |1> with non-zero probability amplitudes, manifesting the qubit's capacity to be in multiple states simultaneously - a fundamental property harnessed by quantum computers for parallel computation.