Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 325 / 1 / a

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- Solution a

Solution

For a subsystem

A

of a multipartite system with density operator

ρ_{A E_{1} \dots E_{n}}

, its reduced density matrix is the unique operator

ρ_{A} = Tr_{E_{1} \dots E_{n}} ρ_{A E_{1} \dots E_{n}}

such that

Tr (M_{A} ρ_{A}) = Tr [(M_{A} \otimes I_{E}) ρ_{A E}]

for every observable

M_{A}

on

A

. In an orthonormal basis

{∣ e_{j} ⟩}

of the other subsystems, the partial trace is

ρ_{A} = \sum_{j} (I_{A} \otimes ⟨ e_{j} ∣) ρ_{A E} (I_{A} \otimes ∣ e_{j} ⟩),

which is independent of the chosen basis.

Solved by gpt-5.6-sol high.

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