Put

c = cos (θ /2)

s = sin (θ /2)

, and

ζ = e^{2 πi /3}

. Averaging the three states cancels the off-diagonal phases:

\overset{ρ}{ˉ} = \frac{1}{3} \sum_{k = 0}^{2} ∣ ψ_{k} ⟩ ⟨ ψ_{k} ∣ = (c^{2} 0 0 s^{2}) .

For

c, s > 0

M_{k} = \frac{1}{3} (1 ζ^{k} ζ^{- k} 1)

because

\overset{ρ}{ˉ}^{- 1/2} ∣ ψ_{k} ⟩ = ∣0 ⟩ + ζ^{k} ∣1 ⟩

. The matrices are positive and

\sum_{k} M_{k} = I

. At the endpoint values of

θ

, the same formula is understood on the support of

\overset{ρ}{ˉ}

and may be completed arbitrarily on its kernel.

Solved by gpt-5.6-sol high.

ii

Solution

The Holevo optimality conditions say a POVM is optimal when

Γ = \sum_{k} p_{k} ρ_{k} M_{k}

is Hermitian and

Γ - p_{k} ρ_{k} \geq 0

for every

k

. Here

Γ = \frac{c + s}{3} (c 0 0 s)

and

Γ - \frac{1}{3} ρ_{k} = \frac{cs}{3} (1 - ζ^{k} - ζ^{- k} 1) \geq 0,

0

and

2 cs /3

. The pretty good measurement is therefore optimal. Its success probability is

P_{succ} = \frac{1}{3} (c + s)^{2} = \frac{1}{3} (1 + sin θ) .

Solved by gpt-5.6-sol high.

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