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Homework: Tensor products and the partial trace

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Homework: Tensor products and the partial trace

Georg Maringer

Thursday 19

th

April, 2018

Let|Ψi = √1

2|0i ⊗ |2i+√1

6|1i ⊗(|0i+i|1i+|2i) ∈ C2⊗C3

where {|0i,|1i}and{|0i,|1i,|2i}are ONBs ofC2andC3, respectively.

1) Expressρ12 =|ΨihΨ| as a matrix in a suitable orthonormal basis ofC2⊗C3.

2) Compute the reduced density operators (partial traces) of the first system (qubit) and the second system (qutrit).

3) Compute the eigenvalues of the reduced density operators.

4) Compute the Schmidt-decomposition of the of the state

ψ= √1

3(|0i ⊗ |1i+|1i ⊗ |0i − |1i ⊗ |1i) (0.1) Hint: The existence of the Schmidt decomposition |ψi = ∑ri=1√

λi|eii ⊗ |fji was shown in the presentation. It was also shown thatspec(tr2|ψihψ|) = {λ1. . .λr} = spec(tr1|ψihψ|) = {λ1, . . .λr}.

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