 Solution

ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2020/quantum-information/2/d/i/solution

Fix the two subsystem dimensions. For reflexivity,

∣Ψ ⟩ = (I_{d} \otimes I_{d^{'}}) ∣Ψ ⟩

, and both identity operators are unitary.

For symmetry, if

∣Ψ ⟩ = (U \otimes V) ∣Φ ⟩

∣Φ ⟩ = (U^{†} \otimes V^{†}) ∣Ψ ⟩ .

The conjugate transposes of unitary operators are unitary, so the reverse relation holds.

For transitivity, if additionally

∣Φ ⟩ = (R \otimes S) ∣Θ ⟩

∣Ψ ⟩ = (U \otimes V) (R \otimes S) ∣Θ ⟩ = (U R \otimes V S) ∣Θ ⟩ .

Both products are unitary, for example

(U R)^{†} (U R) = R^{†} U^{†} U R = I

. Thus local-unitary equivalence is reflexive, symmetric and transitive, and hence is an equivalence relation.

 New to topics? Read the docs here!