Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 4 / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

i

Solution

For the binary POVM

{E, I - E}

and equal priors,

p_{succ} = \frac{1}{2} Tr (E ρ_{1}) + \frac{1}{2} Tr [(I - E) ρ_{2}] = \frac{1}{2} [1 + Tr E (ρ_{1} - ρ_{2})] .

Maximizing with the variational characterization of trace distance gives the equal-prior Holevo–Helstrom theorem

p_{succ}^{m a x} = \frac{1}{2} [1 + D (ρ_{1}, ρ_{2})] .

ii

Solution

For priors

p_{1}, p_{2}

,

p_{succ} = p_{2} + Tr [E (p_{1} ρ_{1} - p_{2} ρ_{2})] .

If

Δ = p_{1} ρ_{1} - p_{2} ρ_{2}

, the maximum is

p_{2} + Tr Δ_{+}

. Since

Tr Δ_{+} = \frac{1}{2} (∥Δ ∥_{1} + Tr Δ) = \frac{1}{2} (∥Δ ∥_{1} + p_{1} - p_{2}),

one obtains

p_{succ}^{m a x} = \frac{1}{2} (1 + ∥ p_{1} ρ_{1} - p_{2} ρ_{2} ∥_{1}) .

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