Past exam of the theoretical physics II course of the University of Cambridge / 2016 / Paper 1 / 2 / a

Assume the composite pure state has unit norm. Put its amplitudes into the matrix

C = (a_{1} a_{3} a_{2} a_{4}), ∣ χ ⟩ = \sum_{s, t} C_{s t} ∣ s ⟩_{A} ∣ t ⟩_{B} .

(1)

The partial trace contracts the same

B

index between ket and bra:

(ρ_{A})_{s s^{'}} = \sum_{t} C_{s t} C_{s^{'} t}^{*} .

(2)

Thus

ρ_{A} = C C^{†} = (∣ a_{1} ∣^{2} + ∣ a_{2} ∣^{2} a_{1}^{*} a_{3} + a_{2}^{*} a_{4} a_{1} a_{3}^{*} + a_{2} a_{4}^{*} ∣ a_{3} ∣^{2} + ∣ a_{4} ∣^{2}) .

(3)

It is Hermitian, positive semidefinite and has trace one. Its determinant is particularly useful:

det ρ_{A} = ∣ det C ∣^{2} = ∣ a_{1} a_{4} - a_{2} a_{3} ∣^{2} .

(4)

This follows from multiplicativity of the determinant, or by expanding the two-by-two expression. If the supplied amplitudes were not normalized, the physical reduced state would be

C C^{†} / tr (C C^{†})

instead.

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Past exam of the theoretical physics II course of the University of Cambridge / 2016 / Paper 1 / 2 / a

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