General qutrit effect monitoring under a conditional drift contract

General qutrit effect monitoring under a conditional drift contract

Operational contract and units

The input space is span. It may include physical vacuum. The nominated terminal labels are. At clock slot, conditional on the preceding declared classical history, there is a common linear measurement

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E+(t) ≥0,
E−(t) ≥0,
E∅(t) = I −E+(t) −E−(t) ≥0.

It is independent of the current probe’s classical preparation tag; the probe influences it through its input quantum state only. All conditional source/controller, ready-stock, preparation and independence of the joint input and inert reference from the ready stock conditions needed for such an effect representation are admitted. Correlation of an unknown input with an uncontrolled device stock cannot be cured by tomography. The device may otherwise depend on the entire past. A known pathwise bound is required for each nominated conclusive effect:

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∥Ey(t) −Ey(s)∥∞≤ν|t −s|,
y = +, −.
(52)

Here effects and are dimensionless probabilities; has units inverse clock slot. If an external clock measures seconds, replace by the corresponding rate per second and use actual elapsed times. Calibration uses, production uses, delays, controller changes and finite reference depletion all count toward elapsed time. A bound for an averaged master equation does not prove Eq. (52) for actual conditional instruments. A known deterministic drift envelope) can replace in the confidence construction. Every +, time, probe setting and deployment decision is retained. Neither missed clicks nor accepted labels are selected away. This section certifies the declared labels on inert-reference extensions and the explicit pointer/reference distance stated below. It does not identify the postmeasurement source, coherent field, bath or complete quantum instrument. Those require the operational completion and separate retained-output proofs.

Nine probes and a general confidence body

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Use the three basis projectors and, for each i < j, the two projectors onto (|i⟩+|j⟩)/
√
2 and (|i⟩+i|j⟩)/
√

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2.
Number these nine trusted states σj, j = 0, . . . , 8. In every trial their actual conditional preparation
has trace distance at most δprep from its nominated state, and the conditional joint input/reference is
independent of the device’s ready stock. This trusted preparation condition is not inferred from the
same unknown detector.
When |v⟩is physical vacuum and the other modes contain one particle, the four vacuum/occupied
superpositions require a phase reference and compatible preparation access. The finite-reference
resource model applies when appropriate. If number superselection and block-diagonal source/effects
are separately admitted, five probes suffice for the five-dimensional real algebra C ⊕M2; unrestricted
vacuum coherences cannot be certified with that reduced access.
Let Xy,j,k be the indicator of label y on the kth use of probe j. Its true conditional mean py,j,k
is predictable and may change with every preceding record. Define the retained empirical frequency
bpy,j,n = n−1 Pn
k=1 Xy,j,k. Suppose at most K monitoring epochs will be opened. Within each epoch
use any predictable probe schedule, with a finite total budget; our example uses round-robin cycles. Set

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s
δ =
α
2JK ,
J = 9,
u(n) =
log(π2n2/(3δ))
2n
(n ≥1).
(53)

A probe with no observations gives no constraint and cannot trigger a complete certificate.

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M1: finite-budget, all-prefix coverage.
With probability at least 1 −α, simultaneously for every
opened epoch, both conclusive labels, every probe, and every observed prefix n,
bpy,j,n −1
n
X
k=1
py,j,k
 ≤u(n).

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n

Consequently, at any data-selected calibration completion time, for probe times, the actual next-slot effect lies in the physical confidence body

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PT = {(E+, E−) : E± ≥0, E+ + E−≤I,
(54)
| tr(Eyσj) −bpy,j,nj| ≤bj(T),
y = ±, j = 0, . . . , 8},
(55)
bj(T) = u(nj) + δprep + ν
X
k
(T −sj,k).
(56)
nj

If the body is empty, decline deployment; do not call it a vacuous success. Future time may use the same formula with every radius enlarged by, or the operator-ball construction below.

Proof. For a chosen stream, is a martingale difference of conditional range one. The conditional Hoeffding lemma gives the exponential supermartingale with compensator8, also when sampling times are predictable and chosen from the preceding record. At a fixed observed sample index optional stopping, or extension of a stopped stream, gives

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
> nb

X
≤2e−2nb2.
Pr
k≤n
(Xk −pk)


At) the right side is 6), summing to over all1. Union over 2 streams proves the claim, conditional on every history at which an epoch is opened and then averaged over those histories. No stationarity or fresh statistical independence has been used. Conditional probe-preparation error changes an event probability by at most; pathwise drift changes it by at most). Average these inequalities to obtain the body. The same simultaneous event permits arbitrary inspection and stopping within the stated epoch budget. The construction remains conservative if only a predetermined finite set of prefixes is inspected. If just one fixed terminal sample size per epoch is used, the simpler width

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ufixed(n) =
q
log(4JK/α)/(2n)

is sufficient; it must not be substituted after selecting a favorable sample size. Tighter established confidence sequences may replace Eq. (53); the displayed one was chosen for a self-contained proof and transparent implementation.

An exact finite reconstruction constant

For any Hermitian, write its nine probe expectations as. The inverse map is

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Hii = vi,
(57)
Re Hij = vij,R −(vi + vj)/2,
(58)
Im Hij = (vi + vj)/2 −vij,I.
(59)

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This fixes the sign convention for the +i probe. The exact rectangular uncertainty radius is the finite
computation
C(b) =
max
s∈{−1,+1}9 ∥R(s0b0, . . . , s8b8)∥∞.
(60)

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Convexity gives equality: the norm of every point of a box is bounded by its largest vertex value, and
the vertices are included. Thus ∥Ey(T) −R(bpy)∥∞≤C(b) on M1. This is sharp for an unconstrained
Hermitian box; enforcing the physical joint confidence body can make it smaller.

M2: exact constant for a uniform box. For

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C(b) = (1 + 2
√
5)b.
(61)

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A short reproducible algebraic proof accompanies the code.
For every one of 512 sign vertices,
form H = R(s). Its entries are Gaussian integers. All principal minors of (1 + 2
√

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5)I ± H are
nonnegative, checked in exact integer arithmetic in Z[
√

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5]. For clarity, with diagonal entries di, t = tr H,
c2 = P
i<j(didj −|Hij|2), and d = det H, the determinant is

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(61 −21t + c2 −d) + (46 −4t + 2c2)
√
5.

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A 2 × 2 minor for i, j is
21 −di −dj + didj −|Hij|2 + (4 −2di −2dj)
√
5.

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One-dimensional minors are immediate. Their signs are decided exactly by integer squares for A+B
√

5; no floating-point tolerance is used in these 7168 checks. The vertex gives

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



1
−2 + 2i
−2
−2 −2i
1
−2 + 2i
−2
−2 −2i
1
H =

,

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whose characteristic polynomial (λ −1)[(λ −1)2 −20] attains the claimed radius. The finite sign checks
plus this witness prove equality. They do not claim that every sign vertex is itself a physical effect.
Set bE± = R(bp±) and bE∅= I −bE+ −bE−. If these are not physical, define

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λ = max(0, −min
y
λmin( bEy)),
Fy =
bEy + λI
1 + 3λ .

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Then F is a physical center. On M1, for y = ±,

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∥Ey(T + a) −Fy∥∞≤ry(T) + νa,
ry(T) = C(b) + ∥Fy −bEy∥∞.
(62)

This explicit repair need not be an optimal fit; it costs the computed shift. Using max gives a physical operator-ball outer class, denoted). It is not the same class as the full body so sharpness for that outer class is not a sharp multinomial-data claim.

Sharp conditional error for general asymmetric effects

Fix known normalized source marginalsand conditional label priors, summing to one. Let ̄

Wrong outcomes remain a subset of conclusive outcomes, including dark-count labels if the input is vacuum. Put ̄ ̄

M3: operational support certificate and its sharp regime. For every

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WE −εQE ≤B(ε) := WF −εQF + r(∥A+(ε)∥1 + ∥A−(ε)∥1),
QE ≥QF −2r.
(63)

If, any 1] with0 certifies conditional error at most. No source/excitation postselection is used. These statements hold on every inert-reference extension having the nominated marginals.

Both support and success bounds are sharp over the physical operator-ball class whenever

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r ≤rint := min{λmin(F+), λmin(F−), 1
2λmin(F∅)}.
(64)

If the radius exceeds this sufficient condition, the bounds remain valid; their physical attainment is no longer asserted.

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Proof.
Write E± = F± +H±. Then WE −εQE = WF −εQF + P
y tr(HyAy). The duality of operator
and trace norms bounds each term by r∥Ay∥1, with equality at Hy = r sign(Ay). Condition (64) makes
these effects positive and keeps F∅−H+ −H−≥0, so they are a real physical POVM. For success use
H+ = H−= −rI, which is also physical in that regime. The denominator is positive when QF > 2r;
the first inequality then implies the conditional-error statement. References do not alter nominated
label probabilities.
□
There is also a complete pointer/reference guarantee. For the channel ME(ρ) = P
y tr(Eyρ)|y⟩⟨y|,

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1
2∥ME −MF ∥⋄≤2r.

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For every joint input, each partial-trace block for a Hermitian effect perturbation H has trace norm
at most ∥H∥∞, by trace-norm duality. The two conclusive blocks and their negative sum give the
bound. In the interior regime it is sharp: choose H+ = H−= rI, with null perturbation −2rI. All
three blocks are scalar multiples of the reference marginal, giving half trace distance exactly 2r on
every input. This is a bound for the declared pointer plus an arbitrary inert reference; it makes no
statement about retained output source or bath.
For unequal known radii, replace r∥Ay∥1 by ry∥Ay∥1 and 2r by r+ + r−; the analogous margin
condition applies. The proof supplies a finite implementation V = P
y
pEy ⊗|y⟩, retaining the output
source and arbitrary inert reference. It demonstrates attainability within ordinary quantum instruments,
not a derivation of Born statistics or identity with the physically nominated detector’s full bath.

Balanced pure hypotheses: a quadratic. Let, and 1]. In their two-dimensional span,

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∥A+(ε)∥1 + ∥A−(ε)∥1 =
q
1 −4ε(1 −ε)s2.

This follows from tr2 and det4. A zero eigenvalue in unused qutrit directions causes no difficulty. Thus the sharp error in the interior regime is the root in [0

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WF −εQF + r
q
1 −4ε(1 −ε)s2 = 0.
(65)

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Equivalently,
(Q2
F −4r2s2)ε2 + (4r2s2 −2WF QF )ε + W 2
F −r2 = 0,

retaining the unsquared sign0. Since 0 for every member, the physical support function is strictly decreasing in, and the root is the exact worst ratio. Orthogonal hypotheses give ; identical hypotheses have no informative balanced-label advantage. Outside the interior regime use the support inequality as a conservative certificate, not a minimax claim.

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Source uncertainty and arbitrary priors.
The example fixes source marginals. If each actual
conditional ρ′
h has trace distance at most δS from ρh, the valid repair is B(ε) + δS ≤0 and QE ≥
QF −2r −δS. Indeed Ewrong −ε(E+ + E−) lies between −εI and (1 −ε)I; its spectral oscillation is
at most one. This term bounds source error without mistaking a measured detector for a measured
source. The combined radius/source certificate need not be sharp.

For priors that may vary arbitrarily with history, certify each label separately. If tr , then

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εh ≤min{1, (wh + r)/qh},
Qh ≥qh −2r.
(66)

The support is, attained by in the same interior regime. Using max min covers arbitrary label priors. The sharper balanced expression requires balanced conditional priors; neither calibration nor a reset establishes them.

Finite-budget deployment and certificate expiry

A complete protocol is now explicit. Predetermine, a maximum number of probe trials per epoch, trusted preparations, the drift contract, target error ̄, and target conclusive probability ̄ 0. Open an epoch, retain all outcomes, and inspect M1 at any prefixes. Construct; deploy only if ̄ and ̄ (or their source-uncertain/labelwise versions). Continue only while the aged radius in Eq. (62) passes the same gate. Otherwise take more probes, open the next budgeted epoch, or stop. A different physical pulse requires its own calibration or a separately established transformation bound; an online choice among uncalibrated pulse settings is not licensed. For balanced pure inputs the largest admitted error radius at a chosen threshold is

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rerr =
 ̄εQF −WF
p
1 −4 ̄ε(1 − ̄ε)s2 ,
(67)

when numerator and denominator are positive. Also impose ̄2. The allowed monitoring gap follows by subtracting) and dividing by 0; if sharpness is desired rather than just a valid certificate, additionally impose. This is an operational expiry time, not a detector change-point localization theorem. There is a real statistical/drift tradeoff even before deployment. For complete round-robin cycles with probe slots 8, and next-slot anchor, the largest mean age is (9 A transparent sufficient radius is

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runif(n) = (1 + 2
√
5)[u(n) + δprep + ν(9n + 9)/2].
(68)

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For a fixed terminal sample size set u(n) = A/√n, A =
p

log(4. The continuous optimum of this displayed design bound is

Check neighboring integers and the declared finite budget. Thus blindly collecting older data eventually worsens a current detector certificate. This is an exact optimum of the stated bound, not of every tomography schedule or confidence method. For the anytime width the optimum is an elementary one-dimensional minimization, provided in the code.

Online calibration events and adaptive record guarantees

The simultaneous M1 event spans future calibration records; it need not be measurable at the initial time. Conditioning on that future event and reusing the symmetric-family initial-event proof would be invalid. A stopped-process repair gives the correct sequential conclusion. For analysis let be the first completed calibration prefix at which the true current effects leave the certified body. It is a stopping time for the full mathematical filtration, even though an operator is unknown to the experimenter. M1 gives Pr. At each nominated production opportunity

let be the predictable deploy gate, its conclusive and wrong indicators, and its reported bounds. A skipped opportunity has all three indicators zero. Prior to, drift plus the gate gives

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E[Wt −εtCt | Ft−1] ≤0,
E[qtDt −Ct | Ft−1] ≤0.

Use zero increments after the first failed prefix in the analysis. The stopped increments have nonpositive conditional mean and range length at most one. For any fixed 0, a nonnegative exponential supermartingale and its first-crossing inequality yield, with failure at most

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t≤n
(Wt −εtCt) < θn
8 + log(1/β)
X
θ
for every n ≥1.
(69)

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The identical construction applies to P
t≤n(qtDt −Ct). For simultaneous two-sided objectives split β
between them. No conditional law on the future success event was used; on {τ = ∞} the stopped and
actual production records agree. The fee α is paid once for the entire finite monitoring budget, not
once per production use. This is standard stopped-supermartingale reasoning newly supplied for the
online calibration interface.
Here the proof filtration includes completed-trial scoring truths and outcomes, as in Section 7; the
controller sees only the permitted accessible-record subfiltration. The admitted conditional source,
ready-stock and drift promises must hold on that augmented past. Such a promise is not implied by
calibration on a smaller observable history. An inert retained quantum reference does not change the
label bounds; measuring that reference or a bath and feeding it back changes the conditional contract
unless separately included. Physical resets and fresh source independence are not inferred.