QuantumCore

Hands-on lab · 1 step

Coordinates in the ± basis

Express v=(3+i)∣0⟩+(1−i)∣1⟩v=(3+i)\lvert0\rangle+(1-i)\lvert1\rangle in the basis {∣+⟩,∣−⟩}\{\lvert+\rangle,\lvert-\rangle\} of C2\mathbb{C}^2.

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Step 1: Find c₊ and c₋

Let ∣±⟩=12(∣0⟩±∣1⟩)\lvert\pm\rangle=\frac{1}{\sqrt2}(\lvert0\rangle\pm\lvert1\rangle). Write the vector v=(3+i)∣0⟩+(1−i)∣1⟩∈C2v=(3+i)\lvert0\rangle+(1-i)\lvert1\rangle\in\mathbb{C}^2 as v=c+∣+⟩+c−∣−⟩v=c_+\lvert+\rangle+c_-\lvert-\rangle and enter the coordinates c+c_+ and c−c_- (as complex numbers). The coordinates are unique because {∣+⟩,∣−⟩}\{\lvert+\rangle,\lvert-\rangle\} is a basis. Answers are accepted within 10−310^{-3}.

Enter exact expressions or decimals, e.g. 1/sqrt(2), pi/3, (1+i)/sqrt(6), exp(i*pi/4). Complex values are allowed.

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