Technical Companion: Detector Interfaces and Retained Outputs

Preparation cost of a product-response reconstruction

Jeremy RodgersSection 5 of 12Original PDF ↗

Preparation cost of a product-response reconstruction

Take a finite mode register or a finite spatial bin, a normalized state ρ, a dimensionless intensity effect
0 ≤A ≤I, and the normalized template projector B = |R⟩⟨R|. For a pure field Ψ =
√ — Technical passage, source page 14.

$Nψ$ with fixed known total intensity $N >$ 0, the normalized template coherence used in [7, 8 $] is ⟨ψ|B|ψ⟩$ and the bin intensity is a fixed intensity scale times $N⟨ψ|A|ψ⟩$ . Absorb these fixed scales and the susceptibility into $κ ≥$ 0 with units s $−1$ . This gives the precise proposed quantum-state interpretation

x = Tr(Aρ),
y = Tr(Bρ),
Γ(ρ) = κxy,
psrc(ρ) = 1 −e−ηxy,
η = κ∆t. — Source page 14.

A state-dependent total intensity or susceptibility must be treated as a further source/controller variable; it cannot be silently absorbed into a state-independent detector. Point evaluation on an unrestricted continuum is not a bounded effect, so the finite bin is also an explicit addition.

Proposition 4.1 (Single-copy obstruction and two-copy effect). The product Tr $(Aρ)$ Tr $(Bρ)$ is affine on the full density-matrix state space if and only if at least one of $A, B$ is scalar. In general it is not a single-copy click effect. Nevertheless, for $0 ≤η ≤1$ the two-copy effect

Ec = ηA ⊗B,
E0 = I −Ec — Source page 14.

is positive and complete, and on independently prepared copies it gives pc = ηxy. Its difference from
the source frame rule obeys
0 ≤pc −psrc ≤1
2η2x2y2 ≤1
2η2. — Technical passage, source page 14.

Proof. Write $ρ = I/d+εX$ for arbitrary traceless Hermitian $X$ and sufficiently small $ε$ . Affinity requires Tr $(AX)$ Tr $(BX$ ) = 0 for every $X$ . Two nonzero linear functionals cannot have their kernels cover the whole vector space: choose a vector off the first kernel and, if necessary, add a multiple of a vector off the second. Thus one functional vanishes on all traceless matrices, equivalently its effect is scalar. The converse is immediate. For an explicit obstruction even to an ensemble-independent extension of pure-state rates, take $A = B = |0⟩⟨0|$ . The equally weighted ${|0⟩, |1⟩}$ and ${|+⟩, |−⟩}$ ensembles both prepare $I/$ 2, but their averaged pure-state product rates are 1 $/$ 2 and 1 $/$ 4. Positivity and completeness of the two-copy effects follow from 0 $≤A ⊗B ≤I$ . The bound uses 0 $≤z −(1 −e−z) ≤z2/$ 2 for $z ≥0.$

An explicit retained-reference instrument is Ic(ω) = √Ec ω√Ec and I0(ω) = √E0 ω√E0, with the
identity on an arbitrary reference, and orthogonal recorded flags. Its probabilities on a correlated
two-register input are Tr(Ecω12); they factor as ηxy only under the declared product preparation.
This needs two preparations, not a machine cloning an unknown input.
The standard isometry
V = |c⟩√Ec + |0⟩√E0 provides a finite apparatus dilation, but a material coupling remains to be
supplied. The square roots specify a state update not determined by the scalar rate alone. — Technical passage, source page 14.

Proposition 4.2 (Exact exponential needs an unbounded stock or more input). For $η > 0, no$ fixed finite number $m$ of identically prepared copies and a fixed quantum effect implements $1 −e−ηp2$

for every unknown qubit pure state with $p = |⟨0|ψ⟩|2$ . An exact protocol using a random number of independent two-copy tests exists: sample $K ∼$ Poisson $(η)$ independently, perform the effect $A ⊗B$ on each independently prepared pair, and report a click if any test clicks, performing all $K$ tests. Then $pc = 1 −e−ηxy$ and the expected number of consumed copies is $2η$ (early stopping can reduce consumption but is not used in this count). If the protocol instead has at most $m$ pairs and retains a separate stock-loss flag when $K > m$ , its classical report law is within Pr ${K > m}$ in total variation of the exact law (padded with a zero-probability loss flag). For $m + 1 > η,$

m+1
.
Pr{K > m} ≤e−η

eη
m + 1 — Source page 14.

Proof. Apply a putative m-copy effect to |ψp,φ⟩= √p |0⟩+ eiφ√1 −p |1⟩. Since the desired result is
independent of φ, average the effect over the common phase. On these symmetric inputs its expectation
becomes Pm
k=0
�m
k
pk(1 −p)m−kek, with fixed 0 ≤ek ≤1. This is a polynomial in p, whereas 1 −e−ηp2 — Technical passage, source page 15.

is not. Extra state-independent ancillas or randomization do not change the polynomial obstruction. For the random-stock construction the no-click probability is $E(1 −xy)K$ = exp $(−ηxy$ ). Couple the exact and truncated protocols using the same $K$ and tests; their classical reports differ only on $K > m.$ The tail bound is the usual Poisson exponential bound, optimized at $et$ $= (m + 1)/η.$

For $η = 0.8, xy = 0.$ 24 and at most six tests (twelve copies), the exact click probabil-ity is 0 $.$ 1746931315; the truncated value is 0 $.$ 1746753459, with retained stock-loss probability $2.0746838 × 10−5.$ The analytic tail certificate is 1 $.2547853 × 10−4.$ Standard-library code supplement/operational/10_verify_product_reader.py reproduces these values. This is an exact preparation-cost interface for one interpretation of the source, not a claim that a classical driven field needs duplicated unknown quantum states. It supplies neither fresh independence from a reset nor a primitive Born-event selection. The comparison bound concerns the declared classical reports; it does not identify quantum residual outputs of two incompatible apparatuses. All loss flags, controller draws, input stock, clock and retained references remain part of the complete model.