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

Theorem: Completed POVM with a bin-independent error bound

Compressed oscillator position measurements › Arithmetic-aware completion to a genuine POVM · lines 1350–1425 · proof included or linked

Em04-A@2#thm:completion, Em04-A@2#eq:arithrepair, Em04-A@2#eq:completion, Em04-A@2#eq:TVcompletion

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Partition $[- L, L]$ into $m$ interval bins $B_{k}$ , and let

E_{k} = E_{d} (B_{k}), Q_{k} = Q_{d}^{trap} (B_{k}; L) .

Suppose $A_{k} = A_{k}^{†}$ and certified nonnegative bounds satisfy

∥ A_{k} - Q_{k} ∥_{\infty} \leq α_{k}, ∥ Q_{k} - E_{k} ∥_{\infty} \leq η_{k} .

Define

P_{k} = A_{k} + α_{k} I_{d}, ε_{k} = η_{k} + 2 α_{k}, η = k \sum ε_{k},

and

F_{k} = \frac{P _{k}}{1 + η}, F_{out} = I_{d} - k \sum F_{k} .

These are the effects of a valid POVM. For each interior bin,

∥ F_{k} - E_{k} ∥_{\infty} \leq \frac{ε _{k} + η}{1 + η} .

For every finite input state, the total variation distance from the exact compressed-position law, including the outside bin, is at most

d_{TV} \leq min {1, \frac{2 η}{1 + η}} .

For a full-space state $ρ$ and finite preparation $σ_{d}$ , comparison with the full exact position law for the same bins adds at most $d_{tr} (ρ, σ_{d})$ .

Proof. The arithmetic bound gives $A_{k} \geq Q_{k} - α_{k} I_{d}$ , so $P_{k} \geq Q_{k} \geq 0$ . Also $∥ P_{k} - E_{k} ∥_{\infty} \leq ε_{k}$ . Writing $P = \sum_{k} P_{k}$ and $E = \sum_{k} E_{k} \leq I_{d}$ ,

P \leq E + η I_{d} \leq (1 + η) I_{d} .

Thus $F_{out} \geq 0$ , and the effects sum to the identity. The individual norm estimate follows from

F_{k} - E_{k} = \frac{P _{k} - E _{k} - η E _{k}}{1 + η} .

For a finite input state, put $p_{k} = Tr (σ_{d} E_{k})$ and $u_{k} = Tr (σ_{d} P_{k})$ . Then

∣ u_{k} - p_{k} ∣ \leq ε_{k}, k \sum p_{k} \leq 1.

The completed interior probability difference is

d_{k} = \frac{u _{k} - p _{k} - η p _{k}}{1 + η} .

Consequently

k \sum ∣ d_{k} ∣ \leq \frac{\sum _{k} ε _{k} + η \sum _{k} p _{k}}{1 + η} \leq \frac{2 η}{1 + η} .

The outside difference is $- \sum_{k} d_{k}$ , so total variation is no greater than $\sum_{k} ∣ d_{k} ∣$ . It is also at most one. Finally, applying any exact POVM contracts state trace distance to outcome total variation; the triangle inequality proves the last assertion.

Displayed formulas in this passage

Em04-A@2#eq:arithrepair · line 1362
$P_{k} = A_{k} + α_{k} I_{d}, ε_{k} = η_{k} + 2 α_{k}, η = k \sum ε_{k},$
Em04-A@2#eq:completion · line 1369
$F_{k} = \frac{P _{k}}{1 + η}, F_{out} = I_{d} - k \sum F_{k} .$
Em04-A@2#eq:TVcompletion · line 1381
$d_{TV} \leq min {1, \frac{2 η}{1 + η}} .$
unlabeled · line 1353
$E_{k} = E_{d} (B_{k}), Q_{k} = Q_{d}^{trap} (B_{k}; L) .$
unlabeled · line 1357
$∥ A_{k} - Q_{k} ∥_{\infty} \leq α_{k}, ∥ Q_{k} - E_{k} ∥_{\infty} \leq η_{k} .$
unlabeled · line 1375
$∥ F_{k} - E_{k} ∥_{\infty} \leq \frac{ε _{k} + η}{1 + η} .$
unlabeled · line 1395
$P \leq E + η I_{d} \leq (1 + η) I_{d} .$
unlabeled · line 1400
$F_{k} - E_{k} = \frac{P _{k} - E _{k} - η E _{k}}{1 + η} .$
unlabeled · line 1407
$∣ u_{k} - p_{k} ∣ \leq ε_{k}, k \sum p_{k} \leq 1.$
unlabeled · line 1411
$d_{k} = \frac{u _{k} - p _{k} - η p _{k}}{1 + η} .$
unlabeled · line 1415
$k \sum ∣ d_{k} ∣ \leq \frac{\sum _{k} ε _{k} + η \sum _{k} p _{k}}{1 + η} \leq \frac{2 η}{1 + η} .$

Nearby in this section

remark: Remark: What is and is not completed