Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 5 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

The generalized measurement postulate specifies operators

{M_{i}}

satisfying

\sum_{i} M_{i}^{†} M_{i} = I .

On state

ρ

, outcome

i

has probability

p_{i} = Tr (M_{i}^{†} M_{i} ρ),

and, when

p_{i} > 0

, conditional state

ρ_{i} = \frac{M _{i} ρ M _{i}^{†}}{p _{i}} .

The effects

E_{i} = M_{i}^{†} M_{i}

form a positive operator-valued measure.

ii

Solution

Introduce a quantum ancilla with basis

∣ i ⟩

V ∣ ψ ⟩ = \sum_{i} M_{i} ∣ ψ ⟩ \otimes ∣ i ⟩ .

The completeness relation gives

V^{†} V = I

, so

V

is an isometry and extends to a unitary operator on a sufficiently large system-plus-ancilla space. Prepare the ancilla in a fixed state, apply that unitary, and perform the projective measurement

{I \otimes ∣ i ⟩ ⟨ i ∣}

. Outcome

i

has probability

∥ M_{i} ∣ ψ ⟩ ∥^{2}

and leaves the system in the normalized state

M_{i} ∣ ψ ⟩

. By linearity the same holds for mixed states, implementing the generalized measurement.

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