Manuscript Em04-A@2 · Em04-A-v2.tex · lemma · f5b28abc

Lemma: Normalized projection error

Compressed oscillator position measurements › Exact objects and state approximation · lines 1074–1105 · proof included or linked

Em04-A@2#lem:projection, Em04-A@2#eq:stateerror

Back to manuscript

For any oscillator state $ρ$ , suppose

δ_{d} = Tr ((I - P_{d}) ρ) < 1, ρ_{d} = \frac{P _{d} ρ P _{d}}{1 - δ _{d}} .

Embedding $ρ_{d}$ into the full space,

d_{tr} (ρ, ρ_{d}) \leq δ_{d} .

For any finite state $σ_{d}$ ,

d_{tr} (ρ, σ_{d}) \leq δ_{d} + d_{tr} (ρ_{d}, σ_{d}) .

Every effect-probability difference is bounded by the corresponding trace distance.

Proof. A purification $∣Ψ ⟩$ of $ρ$ and its normalized projection have overlap $1 - δ_{d}$ . Their pure-state trace distance is $δ_{d}$ , by diagonalizing their difference on their two-dimensional span. Partial trace is contractive, proving the first assertion. The triangle inequality proves (eq:stateerror). For an effect $0 \leq F \leq I$ , the Jordan decomposition of a trace-zero state difference gives $∣ Tr (F (ρ - σ)) ∣ \leq d_{tr} (ρ, σ)$ .

Displayed formulas in this passage

Em04-A@2#eq:stateerror · line 1086
$d_{tr} (ρ, σ_{d}) \leq δ_{d} + d_{tr} (ρ_{d}, σ_{d}) .$
unlabeled · line 1077
$δ_{d} = Tr ((I - P_{d}) ρ) < 1, ρ_{d} = \frac{P _{d} ρ P _{d}}{1 - δ _{d}} .$
unlabeled · line 1082
$d_{tr} (ρ, ρ_{d}) \leq δ_{d} .$

Lemma: Normalized projection error

Displayed formulas in this passage

Nearby in this section