Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 323 / 2 / ii / a / Solution

Memorylessness means that the ensemble's average density operator is

ρ^{(n)} = π^{\otimes n}

. Its spectral decomposition is

ρ^{(n)} = \sum_{i^{n}} q_{i_{1}} \dots q_{i_{n}} ∣ ϕ_{i^{n}} ⟩ ⟨ ϕ_{i^{n}} ∣, ∣ ϕ_{i^{n}} ⟩ = ∣ ϕ_{i_{1}} ⟩ \otimes \dots \otimes ∣ ϕ_{i_{n}} ⟩ .

The product eigenvalues give

S (ρ^{(n)}) = - \sum_{i^{n}} (\prod_{j} q_{i_{j}}) lo g_{2} (\prod_{j} q_{i_{j}}) = n H (q) = n S (π) .

Zero eigenvalues contribute zero to the Von Neumann entropy and are excluded from logarithmic typicality tests.

The quantum typical subspace is the span of eigenvectors whose eigenvalues satisfy

- \frac{1}{n} lo g_{2} (q_{i_{1}} \dots q_{i_{n}}) - S (π) \leq ε .

Its orthogonal projection

P_{ε}^{(n)}

selects precisely the classical typical set for the eigenvalue distribution

q

. The typical-set cardinality bounds and the weak law of large numbers therefore give

dim T_{ε}^{(n)} \leq 2^{n (S (π) + ε)}, Tr (ρ^{(n)} P_{ε}^{(n)}) \geq 1 - δ

for any fixed

δ > 0

and all sufficiently large

n

.

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