Complete retained-output comparisons and microscopic detector closure

Complete retained-output comparisons and microscopic detector closure

A complete one-transition model and a reference-uniform certificate

Fix a finite input Hilbert space, horizon, and common terminal output spaces. All changes of basis used for comparison are declared in advance; an arbitrary environment unitary chosen afterwards would change the meaning of retaining the physical output. For detector 1, let

and a jump-amplitude map be bounded, measurable coefficients. The post-transition propagation and every retained mark are included in; this propagation must be an isometry, or its lost channels must be included. Set

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QX(t) = BX(t)∗BX(t),
GX(t) = −iHX(t)/ħ−QX(t)/2,
 ̇NX(t) = GX(t)NX(t),
NX(0) = I.

Here is in joules, and in s in s, and is dimensionless. This is an absorbing, one-transition model: there are no later transitions back into the untriggered sector. A finite resonance or continuum response may be used, but its amplitude map must actually be supplied. Define the complete amplitude isometry

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VXψ = NX(T)ψ ⊕
t 7→BX(t)NX(t)ψ

in
H0 ⊕L2([0, T]; O).
(8)

The first summand is the retained null/no-transition outcome, not a discarded population. The corresponding time-resolved instrument has Kraus density) and null operator ). It may retain the output in, coarsen time into bins, or apply a fixed common decoder. Its normalized law is the ordinary quantum law of this specified instrument. It is not a derivation of that law from a source theory. For a registered exact time, cq outputs are integrable trace-class conditional kernels over the null atom and time interval, with integrated trace norm. This classical output space is distinct from the coherent temporal field in (8); no trace-class diagonal density operator on that field is assumed. Coarsening to finite time bins gives ordinary cq blocks.

Proposition 2.1 (Full-output amplitude and kernel mismatch certificate). Let

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e(t) =
Z t
0
∥(G1(s) −G0(s))N0(s)∥ds,
A2
T = e(T)2 +
Z T
0
(∥(B1(t) −B0(t))N0(t)∥+ ∥B1(t)∥e(t))2 dt.
(9)

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Then V ∗
XVX = I and ∥V1 −V0∥≤AT . For every input state on H0 ⊗R, with arbitrary finite inert
reference R,
1
2∥(V1 ⊗IR)ρ(V ∗
1 ⊗IR) −(V0 ⊗IR)ρ(V ∗
0 ⊗IR)∥1 ≤min{1, AT }.
(10)

The same bound applies to the complete time/null/mark cq instrument after a common readout. With a fixed input-independent original stock this is also a half-diamond upper bound for the two declared output channels.

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Proof. G∗
X + GX = −QX gives
d
dt(N∗
XNX) = −N∗
XQXNX.

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Integration proves NX(T)∗NX(T) +
R KX(t)∗KX(t)dt = I, hence isometry and conservation of the null
population. The two-parameter propagator generated by G1 is a contraction. Variation of constants
gives

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N1(t) −N0(t) =
Z t
0
N1(t, s)(G1(s) −G0(s))N0(s)ds,
∥N1(t) −N0(t)∥≤e(t).

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Since
K1 −K0 = (B1 −B0)N0 + B1(N1 −N0),

the direct-sum norm in (8), followed by the triangle inequality, proves (9). For normalized pure inputs, the trace distance of two output vectors is at most their vector distance. Tensoring an operator with does not change its operator norm; mixtures follow by convexity or purification and partial trace. Any common time measurement, mark decoder or CPTP readout contracts trace distance. This proves the reference-uniform statement without dividing by a branch probability or charging a reference dimension. The fixed-stock condition supplies actual channels on an unknown input; arbitrary input-correlated preparations supply only their stated statewise comparisons.

This certificate includes a term absent from a rate-only comparison: (. Even does not make it vanish. Conversely, matching the declared complex amplitude maps, null dynamics and terminal propagation makes the full output equal. Under exact matching, the computable cross-Gram operator

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S = V ∗
0 V1 = N(T)∗N(T) +
Z T
0
N(t)∗B0(t)∗B1(t)N(t)dt
(11)

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retains the omitted phase and mark data. For the coherent complete channels their exact half-diamond
distance is
q

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1 −ν(S)2,
ν(S) =
min
ρ≥0, Tr ρ=1 | Tr(ρS)|.
(12)

Indeed the overlap of the two pure outputs of any reference-extended input is Tr), where is its input marginal; every such has a finite purification. Mixed inputs cannot improve the optimum, by purification and contraction. Equation (12) is the standard numerical-range description of isometric-channel discrimination, here made explicit for the proposed detector lift. Time measurement can reduce this distance, so a time-record comparison cannot be substituted for it.

Exact scalar kernel formula: what intensity matching pays for

For a prepared, untriggered scalar detector write

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ΛX(t) =
Z t
0
ΓX(s)ds,
pX(∅) = e−ΛX(T),
pX(t) = ΓX(t)e−ΛX(t).

Letbe the normalized retained quantum mark following a transition at time; assume the null ray is common. On the common time/null space the positive output kernel is

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σX(∅) = pX(∅)|0⟩⟨0|,
σX(t) = pX(t)|bX(t)⟩⟨bX(t)|.

The mark may include outgoing electron energy, polarization, vibrational state, source recoil and other specified outputs; it is not merely a rendered color. Put

Proposition 2.2 (Exact full time/mark discrepancy). The retained cq half trace distance is exactly

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Dcq = 1
2|p1(∅) −p0(∅)|
Z T
q
[p1(t) −p0(t)]2 + 4p1(t)p0(t)[1 −c(t)2] dt.
(13)
+ 1
2
0

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For equal rates it reduces to
Dcq =
Z T
0
Γ(t)e−Λ(t)q
1 −c(t)2 dt.
(14)

For unequal rates one has the constructive calibration budget

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Dcq ≤TV(P1, P0) +
Z T
0
min(p1(t), p0(t))
q
1 −c(t)2 dt,
(15)

where is the complete time-plus-null probability law.

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Proof. For unit vectors u, v and nonnegative p, q, the nonzero eigenvalues of p|u⟩⟨u|−q|v⟩⟨v| lie in their
span, have sum p−q and product −pq(1−|⟨u, v⟩|2). Their absolute sum is
p

Integrate this identity on the time/null space. For the last bound split off times the larger-weight ray and compare the remaining equal weights. Common unchanged output factors, including a product inert reference in this scalar-input comparison, may be tensored on without altering the result. For input-correlated quantum outputs the corresponding full mark states must instead be included explicitly in the compared kernels.

Thus exact intensity/rate matching really does fix all waiting-time statistics in this one-transition model, not only one terminal probability. Its extension cost is determined by output-state overlaps. For constant Γ, orthogonal marks give

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Dbit = 0,
Dcq = 1 −e−ΓT .

At Γ = 1, this is 06321205588. The reduced detector equivalence in [7, pp. 10–15] concerns a smaller output interface. The calculation here quantifies the additional condition required for complete retained-output comparison.

A coherent-field discriminator despite identical count records

A time-resolved measurement has already discarded coherence between different emission times. Retaining the unmeasured complete field is a stronger promise. A declared, broadband absorbing jump coupling can carry a time-dependent phase while leaving its reduced dissipator unchanged. With constant rate Γ 0, compare the two normalized complete vectors

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|ΨX⟩= e−ΓT/2|0, vac⟩+
Z T
√
Γe−Γt/2eiθX(t)|1, 1t⟩dt.
(16)
0

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Here |1t⟩denotes the continuum single-quantum temporal-mode basis; the integral is an L2 wavepacket,
not a distribution-valued state evaluated at a point. This is a signal-marker dilation of the abstract
absorbing detector, not a microscopic model of photoelectron emission or a microtubule. The symbols
0, 1 label ready/triggered states, not their energy ordering. An explicit Markov dilation uses the
coupling LX(t) =
√

Γ with vacuum input; once triggered it cannot jump again. The phase schedule and its source are externally prescribed resources. An autonomous realization would have to include and retain that controller and clock and reprove the comparison. For

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zT = ⟨Ψ0|Ψ1⟩= e−ΓT +
Γ
Γ −i∆ω[1 −e−(Γ−i∆ω)T ],
Dcoherent =
q
1 −|zT |2.
(17)

The reduced detector evolution and the entire count-time distribution agree exactly. Nevertheless, at 5551876958. In the long-time limit,

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Dcoherent =
|∆ω|
√
Γ2 + ∆ω2 .
(18)

Consequently a claimed complete coherent-output error at most 1 requires, for this family,

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|∆ω| ≤Γε/
p
1 −ε2.
(19)

For one percent error the permissible detuning is approximately 00100005Γ. This is a usable tolerance, not a proposed new physical law. Interference of the outgoing temporal mode with a calibrated reference mode, or a phase-sensitive field measurement, can probe the missing phase; ideal equal-prior Helstrom discrimination succeeds with probability (1+2. That optimal measurement is an in-principle statement. Its apparatus is not included for free in the detector model. Counting and coherent-field measurements are alternative output experiments; one cannot first measure an exact click time and then claim its erased temporal coherence was retained.

When is the binary reduction microscopically closed?

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Let the true finite detector have an untriggered subspace P0H, a triggered subspace P1H, a block-
preserving Hamiltonian, and absorbing jumps Ja = P1JaP0.
Define Q = P
a J∗
aJa.
The binary
population readout is C(ρ) = (Tr P0ρ, Tr P1ρ).

Proposition 2.3 (Exact closure and finite spectral repair). For a time-independent generator, the binary populations obey the same absorbing scalar rate law ̇Γ for every microscopic input if and only if (20)

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This statement concerns populations; fixed output marks and a two-dimensional quantum-coherence law
require additional assumptions.
If instead Q = Pk
i=1 γi|i⟩⟨i| on P0H, [H, Q] = 0, consider the declared controlled-rate model whose
complete jump preserves the label and is

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k
X
√γi |1, i⟩⟨0, i|
(21)
J =
i=1

A-label register is an exact finite repair of this specified model, including label coherence and its inert purifying reference; additional output marks or phases, if present, must be retained in the repaired jump as well. The register must be prepared once as part of the original stock, not redrawn after each use.

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Proof. The dual generator gives L∗(P0) = −Q and L∗(P1) = Q. Equality of the scalar population
derivative for all states forces Q = ΓP0 by polarization; this condition also makes the two-dimensional
population observable space invariant under the dual generator, proving sufficiency for all times.
Under the second hypotheses, each label has no-jump amplitude e−γit/2, with its specified commuting
Hamiltonian phase, and jump amplitude √γie−γit/2. Equation (21) and its common no-jump dynamics
give exactly these amplitudes, also when the label is entangled with an inert reference. Full comparison
still retains any additional microscopic channels; none are deleted by naming a label.

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For a diagonal original label stock with probabilities wi > 0, survival is S(t) = P
i wie−γit, not
generally an exponential. Its no-transition posterior is wi(t) = wie−γit/S(t). Therefore a reset wave or a
reset binary flag does not reset the actual label stock. An arbitrarily prescribed time-dependent hazard
−S′(t)/S(t) can imitate the survival of this one specified mixture; it does not establish preparation-
independent microscopic closure, restore the retained label correlations, or license independent successive
samples.
For k distinct rates, the number of labels in a time-homogeneous linear classical realization cannot
in general be reduced below k. Indeed its k × k derivative Hankel matrix is

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i<j
(γi −γj)2 > 0.
Hab = (−1)a+bS(a+b)(0) =
X
i
wiγa+b
i
,
0 ≤a, b < k,
det H = (
Y
i
wi)
Y

Any-state time-homogeneous linear realization has Hankel rank at most, by factoring each derivative as. Thus. This is a standard realization-rank argument applied constructively here, not a lower bound on every possible quantum or time-dependent surrogate. An approximate repair can cluster rates while retaining their original label. For a rate replaced by 0, the exact affinity of its complete time/null law is

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aT (γ, κ) = 2√γκ
γ + κ +

1 −2√γκ

e−(γ+κ)T/2.
(22)
γ + κ

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This follows by adding the null overlap e−(γ+κ)T/2 to the integral of √γκe−(γ+κ)t/2. Cauchy–Schwarz

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1 −a2
T . The same bound holds for their
coherent positive-amplitude wavepackets by the pure-state identity. With original label weights wi the
cq cost is at most P
i wi
q
applied to |p −q| = |√p −√q|(√p + √q) gives TV(Pγ, Pκ) ≤
q

. For an unknown label distribution use the maximum. More strongly, for the fully specified controlled model with the same retained labels, Hamiltonian phases and marks on both sides, the coherent cross-Gram matrix is diagonal with entries Equation (12) then gives the exact reference-uniform coherent-channel distance

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q
1 −aT (γi, κg(i))2.
max
i

This bound admits label superpositions and arbitrary inert references; it does not replace their amplitudes by a classical mixture. Different retained marks or phase schedules must instead be charged with the full amplitude certificate. Thus a finite approximate repair has an explicit worst-case tolerance, beyond a match to one mean rate.