Manuscript I@2 · I.tex · section · 85631fa7

D1: three distinct measurement interfaces

After Enter: records, instruments, decoders, and cutoffs › D1: three distinct measurement interfaces · lines 2200–2255

I@2#eq:destructive-interface, I@2#eq:coherent-interface, I@2#eq:nondestructive-interface

After characterization, write $L = S ⊔ B$ . A classical register is represented below by a diagonal matrix algebra. It is not an additional live quantum carrier in the term language merely because Hilbert-space notation is used for its labels.

The three interfaces distinguished in Cheung's En03-A CheungEn03A (definitions :enter, :recording, and :nondestructive) are:

Enter. Its complete classical/quantum channel is

M_{S} (ρ) = z \sum ∣ z ⟩ ⟨ z ∣_{R} \otimes M^{S}_{z} ρ (M^{S}_{z})^{†} .

The output residual is $B$ ; the factor $S$ is absent.

recording. The copying isometry is

A_{S} = z \sum ∣ z ⟩_{R} \otimes P^{S}_{z}, A_{S}^{†} A_{S} = I,

and the coherent channel is

C_{S} (ρ) = A_{S} ρ A_{S}^{†} .

Both $S$ and the quantum record carrier $R$ remain. Off-diagonal coherences in $R$ are retained in the joint state. This is not yet a classical readout.

nondestructive projective instrument. Copying and then destructively reading the record carrier implements

I^{nd}_{S} (ρ) = z \sum ∣ z ⟩ ⟨ z ∣_{R} \otimes P^{S}_{z} ρ P^{S}_{z} .

Its classical record is $R$ , and its quantum residual interface is $S \otimes B$ . It is represented by allocation, CNOT, and Enter; it is not the destructive primitive.

These formulas follow from the characterized semantics. They do not constitute another derivation or axiom of the Born rule.

Displayed formulas in this passage

I@2#eq:destructive-interface · line 2216
$M_{S} (ρ) = z \sum ∣ z ⟩ ⟨ z ∣_{R} \otimes M^{S}_{z} ρ (M^{S}_{z})^{†} .$
I@2#eq:coherent-interface · line 2232
$C_{S} (ρ) = A_{S} ρ A_{S}^{†} .$
I@2#eq:nondestructive-interface · line 2242
$I^{nd}_{S} (ρ) = z \sum ∣ z ⟩ ⟨ z ∣_{R} \otimes P^{S}_{z} ρ P^{S}_{z} .$
unlabeled · line 2226
$A_{S} = z \sum ∣ z ⟩_{R} \otimes P^{S}_{z}, A_{S}^{†} A_{S} = I,$

Nearby in this section

definition: Definition: Record and continuation