Past exam of the computer science course of the University of Oxford / 2022 / Quantum information / 3 / c

The state is separable. The identity matrix resolves into the four projectors of the computational basis, giving the explicit quantum state ensemble

\frac{I _{4}}{4} = \frac{1}{4} i, j = 0 \sum 1 (∣ i ⟩ ⟨ i ∣)_{A} \otimes (∣ j ⟩ ⟨ j ∣)_{B} = \frac{I _{2}}{2} \otimes \frac{I _{2}}{2} .

Each member is a pure state that is a product state, so this is a separable quantum state by its defining decomposition.

ii

The state is separable. The off-diagonal terms of the two Bell state projectors have opposite signs and cancel:

\frac{1}{2} ∣ ϕ_{+} ⟩ ⟨ ϕ_{+} ∣ + \frac{1}{2} ∣ ϕ_{-} ⟩ ⟨ ϕ_{-} ∣ = \frac{1}{2} ∣00 ⟩ ⟨ 00∣ + \frac{1}{2} ∣11 ⟩ ⟨ 11∣.

1/2

each, establishing a separable quantum state. Its correlations are classical; an ensemble decomposition into entangled states does not itself establish quantum entanglement of the density matrix.

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