Finite charge references and complete heralded phase measurement

Finite charge references and complete heralded phase measurement

Physical encoding and admitted experiment

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Let the source S have basis |x⟩, x = 0, . . . , N −1, N ≥2, and dimensionless charge QS = P
x x|x⟩⟨x|.
For an energy implementation take HS = ħωQS, with ω > 0 in radians per second. A fixed, known
chirp setting a defines the orthonormal basis

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N−1
X
|a, b⟩=
1
√
x=0
exp[2πi(ax2 + bx)/N]|x⟩,
b = 0, . . . , N −1.
(23)
N

No primality assumption is required for this single basis. Its diagonal quadratic mask commutes with ; the noncommuting resource is the Fourier mixing/readout, not the diagonal phase notation. The active reference has integer charges and a fixed prepared state, independent of the unknown source and its arbitrary inert reference. At most charge sectors are populated. The ideal pure reference is

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|uM⟩= M−1/2
M−1
X
n=0
|n⟩.

Admissible readout effects on commute with. All other accessible apparatus states are invariant under the same(1) symmetry and pointer records carry zero charge. Equivalently one may use charge-preserving joint unitaries with invariant ancillas and a neutral pointer satisfying the Yanase condition. A neutral-pointer Naimark dilation realizes every invariant POVM used here. This is an explicit conditional resource model. It applies, for example, to coherent mixing of equally spaced energy modes when no additional phase-bearing drive is admitted. It does not impose a phase-reference cost on arbitrary degenerate spatial modes: if is proportional to, their basis

measurement commutes with charge and can be exact with no such token. An accessible macroscopic oscillator, pump, hidden phase controller or purifying asymmetry register must be added to; its resource cannot be silently treated as free. A clock and fixed setting record are common to the compared experiments; their allowed couplings must obey the declared symmetry. No speed, bandwidth or interaction-strength bound is inferred from this resource count. The task is an exact state-independent heralded measurement: outcomes have induced effects , and the retained null effect is (1. The theorem below optimizes in this class, not every possible approximate measurement with errors.

Optimal exact heralding from finite reference support

Sharp support/loss theorem. For any reference supported on at most integer charge sectors, the largest possible exact heralding probability is

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ηmax(N, M) = (M −N + 1)+
M
.
(24)

A uniform pure reference on consecutive charges attains it. For, equality among pure reference states requires this uniform probability profile on a consecutive interval, up to charge translation and arbitrary known phases compensated by the readout. For, every reference has zero exact success; no uniqueness assertion applies.

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Proof.
For a pure reference |r⟩= P
n rn|n⟩set pn = |rn|2, with missing charges assigned pn = 0. An
invariant measurement sees independently the total-charge blocks

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N−1
X
|vbq⟩=
1
√
x=0
rq−xeN(ax2 + bx)|x, q −x⟩,
eN(t) = e2πit/N.
N

Twirling is only a device for calculating invariant record probabilities; it does not authorize discarding an actually retained coherent charge archive. Within a block containing all terms the reciprocal vectors

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eN(ax2 + bx)
|wbq⟩=
1
√
X
r∗
q−x
|x, q −x⟩
N
x

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satisfy ⟨wbq|vb′q⟩= δbb′. Averaging a measurement over the cyclic relabelings b preserves its mean success,
invariance and zero-error property. Every averaged zero-error effect therefore has form tq|wbq⟩⟨wbq|.
Its normalization requires tq
P
x p−1
q−x|x, q −x⟩⟨x, q −x| ≤I, so tq ≤minx pq−x, which is attainable. A
block with fewer than N terms cannot give positive symmetric zero-error success: otherwise all N
vectors would be linearly independent, since applying their zero-error effects to a linear relation isolates
each coefficient. There are fewer than N available dimensions. Thus the exact optimal average success
is
η(r) =
X
q
min
0≤x<N pq−x.
(25)

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An exact state-independent heralded measurement must in particular discriminate this basis with mean
success η, so this is an upper bound for the nominated task.
For a finite subset A of integer charges, the number of contained length-N consecutive windows is
P
runs(|run| −N + 1)+ ≤(|A| −N + 1)+. Apply the layer-cake identity to At = {n : pn > t}:

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q
min
x pq−x =
Z ∞
Z ∞
0
#{length-N windows in At} dt ≤(M −N + 1)+
X
0
|At| dt = ηmax.
M

For, the inequality is strict at every nonempty superlevel set of size below, and at a nonconsecutive set of size. Equality therefore forces the stated uniform consecutive profile. Uniform has + 1 complete windows and attains the bound. Mixed references cannot improve it: decompose the reference into pure states within the same charge support. Zero-error probabilities are nonnegative, so a zero-error mixture measurement is zero-error on each component, and its success is bounded by the same number on every component. An invariant ancillary state decomposes into pure fixed-charge states. On each component its addition merely shifts the reference charge labels, with degeneracy vectors orthogonal between sectors; the same bound applies. Such ancillas therefore do not improve it. The construction below attains the full state-independent effects, not just basis-state average success. For any allowed loss ceiling 0 1, exact heralding requires and permits

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M ≥
N −1

.
(26)
ε

At the hypothetical = 180 energy-mode encoding, one-percent null probability requires at least 900 populated reference charges. This is a conditional resource requirement for the declared charge encoding; angular divisions alone do not specify physical reference states. An attaining reference on charges 01 has energy span ( and mean energy (2. No numerical energy is inferred without a physical

Full retained instrument: coherent versus classical charge record

Take, and. Define normalized vectors within total-charge sectors

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|φbq⟩=
1
√
x
eN(ax2 + bx)|x, q −x⟩,
q ∈I.
X
N

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The mutually orthogonal invariant projectors Fb = P
q∈I |φbq⟩⟨φbq| and F? = I −P
b Fb form a projective
measurement on SC. Outside the occupied input sector the identity may be extended arbitrarily
without affecting this experiment.
Retain a charge archive A by the invariant coherent copying isometry CQ|ξq⟩= |ξq⟩|q⟩A. Its basis
states are neutral; this records q without changing the charge of SC. Prepare C in |uM⟩and let
Jψ = ψ ⊗uM. The full operators from the logical input S to SCA are

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Kb = CQFbJ = √η|χb⟩⟨a, b|,
|χb⟩= K−1/2 X
q∈I
|φbq⟩|q⟩A,
K? = CQF?J.
(27)

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Direct substitution gives

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K†
?K? = (1 −η)I,
X
b
K†
bKb + K†
?K? = I.
(28)

Consequently, for every joint input

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Ib(ρSR) = η|χb⟩⟨χb| ⊗⟨a, b|ρSR|a, b⟩,
I?(ρSR) = (K? ⊗IR)ρSR(K†
? ⊗IR).
(29)

This retains the active phase reference, source residual, coherent charge archive, neutral outcome pointer and arbitrary inert reference. The normalized null operator is an isometry into the joint output. It preserves the logical source information there; it does not restore the initial hardware source or reference state, provide a free symmetry-preserving inverse, or make the reference reusable.

A genuinely classical charge record is a different instrument: apply charge dephasing to and retain the value. Then the conclusive output state becomes the fixed mixed state

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σb = K−1 X
q∈I
|φbq⟩⟨φbq| ⊗|q⟩⟨q|A,

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still multiplying η⟨a, b|ρSR|a, b⟩. The null channel is now a sum of charge-resolved Kraus branches and
is not generally an isometry. Indeed their K†
?,qK?,q are different partial diagonal projectors divided by
M; the Knill–Laflamme scalar condition fails. Thus the coherent-null conclusion is not applied to a
classically measured charge archive.

Exact complete-output comparison. Declare the completed ideal basis-measurement channel to have output state in outcome and zero null branch. For the classical-charge version use instead. These are specified completions of the ideal readout, not a microscopic identification with an arbitrary detector. The actual channel has exactly

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1
2∥I −J ∥⋄= 1 −η = N −1
M
.
(30)

In fact every normalized input, including every inert-reference extension, attains this distance.

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Proof.
Each actual conclusive branch is precisely η times its ideal branch on every ρSR by Eq. (29).
The conclusive difference is negative with total trace 1 −η. The orthogonal actual null block is positive
with the same trace. Adding the block trace norms gives 2(1 −η). The same argument survives
classical charge recording, since the conclusive factorization and null trace remain exact. This is an
exact full-output identity, not a scalar probability bound substituted for an instrument error.
□
The isometry P
y Fy ⊗|y⟩with a neutral outcome pointer commutes with total charge and extends
to a blockwise charge-preserving unitary. The coherent charge-copy map has the same property.
Hence the model has a finite unitary realization; it has not hidden a noninvariant Fourier gate in the
apparatus. A common fixed readout of the outcome pointer supplies the declared instrument. Reading
additional microscopic pointer/environment degrees of freedom requires declaring their common states
and couplings as well; no universal detector equivalence follows.
The finite resource ledger of this explicit dilation is: source dimension N, active reference dimension
M, neutral charge-archive dimension N + M −1, and neutral outcome-pointer dimension N + 1, plus
the nominated fixed controller/clock and arbitrary inert R. A coherent charge archive is quantum
memory; a classical one still retains its entire record. The optimum in Eq. (24) counts populated
phase-reference charge sectors only. It is not a bound on total apparatus or complete-archive dimension,
and it does not identify a single-use Choi rank with retained history capacity.

Reference dephasing: exact error–erasure boundary

The exact-success boundary is fragile under a specified physical noise model. Let the accessible reference state be (31)

This is partial dephasing in charge, not an unspecified decoherence rate. Its environment can be included and retained as a fixed passive purification, but is unavailable to the nominated decoder for measurement, conditioning, postselection or feedback. Retaining that quantum output does not grant this access. If a later reader can use a classical noise flag or measure the purification, that is extra accessible side information or a phase resource and the optimization must be recomputed. The following frontier is not asserted optimal for that enlarged complete-archive readout.

Give the basis letters equal priors and retain every result, including nulls and charge records. For total charge let

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Xq = {x : 0 ≤x < N, 0 ≤q −x < M},
mq = |Xq|,
wq = mq
NM ,
cq = ν + (1 −ν)mq
N
.

The weights sum to one. The subnormalized input block is

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ρbq = (1 −ν)|ubq⟩⟨ubq| + νImq
x∈Xq
eN(ax2 + bx)|x, q −x⟩.
(32)
NM
,
|ubq⟩=
X

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Its prior average is Imq/(NM).

Exact fixed-token frontier. For a specified conclusive probability 1], the maximum unconditional correct-letter probability is exactly

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X
Cmax(P) =
max
0≤tq≤1
P
q
wqcqtq.
(33)
q wqtq=P

It is obtained by accepting the highest blocks first, with fractional acceptance of the final tied block if needed. The minimum conditional label error is 1 for 0. This is globally optimal over all invariant POVMs at this fixed reference state; it is not claimed optimal over all reference profiles at intermediate

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Proof with a full POVM upper certificate.
For arbitrary conclusive effects Ebq with P
b Ebq ≤I,
the largest eigenvalue of ρbq is [ν + (1 −ν)mq]/(NM). Thus correct probability in that block is at most
cq times its conclusive probability. The latter is at most wq because the prior average is Imq/(NM).
This proves the linear-program upper bound without assuming a covariant candidate.
It is attained by the invariant effects

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Ebq = tq
N |ubq⟩⟨ubq|,
X
b
Ebq = tqImq,

and the retained complementary null effect. Each effect occupies a maximal-eigenvalue direction of its matching input block. The linear objective therefore sorts the. For a dual certificate at multiplier choose max). It obeys) for all, and its trace sum equals the attained Lagrange objective. This independently certifies the boundary against arbitrary effects. For, there are +1 blocks of size, total weight; every smaller size1 occurs twice, weight 2). Therefore for 0 the sharp error floor is (34)

Beyond, the additional slopes are [ for This is a closed piecewise-linear correct-probability boundary. At the interior-block operating point, the effects on every unknown logical input, not only the basis test states, are

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Eb = η0[(1 −ν)|a, b⟩⟨a, b| + νI/N],
E? = (1 −η0)I.

Thus for the declared classical pointer channel, retaining an arbitrary inert reference, the exact half-diamond discrepancy from the pure-token heralded pointer channel is). It is a uniform

randomization of the ideal letter with probability; the maximum column total variation of that classical postprocessing is 1, attained by a basis input. This pointer certificate is not a full active-token output bound. Any 0 makes exact zero-error heralding impossible at every positive conclusive probability: all letter states have the same full support within each block. More generally, any strictly positive reference density matrix on its populated charge space has the same obstruction: every relevant principal submatrix is strictly positive, so each letter’s total-charge block has the same full support. This is an exact-zero-error obstruction, not a claim that a small noise produces a large ordinary statistical error. The nearby faithful repair is not to discard the reader but to keep the complete null/error archive and use the exact controlled error floor. For example03 permits9 with minimum conditional error 0024; no lower-confidence filtering can beat that value. At all-conclusive readout,

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1 −N2 −1
"
#
+ ν
Cmax(1) = (1 −ν)
N ,
M ≥N.
3NM

All these formulas concern declared label statistics. A full noisy retained-output comparison additionally uses the explicit Kraus maps and retained purification; it is not inferred from label error alone. The effects and their positive square roots already give such a normalized finite instrument, tensored with for an arbitrary inert reference.

Different objective: best no-erasure reference

Exact finite eigenvalue optimization. For a pure reference on charges 01 with amplitude magnitudes0, the maximum equal-prior correct-letter probability with no inconclusive output is

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!2
= rTTN,Mr
N−1
X
Pcorrect(r) =
1
N2
X
x=0
rq−x
N2
,
(TN,M)nm = (N −|n −m|)+.
(35)
q

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Consequently the best reference has P opt
correct = λmax(TN,M)/N2, with its nonnegative principal eigen-
vector as amplitude profile.

Proof. Within each total-charge block, cyclic symmetrization makes the seed effect have diagonal entries 1 and hence off-diagonal magnitudes at most 1 by positivity. This upper-bounds correct probability by the first expression, attained by a phase-aligned rank-one seed. Summing counts pairs when, proving the matrix formula. The variational principle and Perron positivity give the optimum. Mixed references cannot improve a maximization over all pure states and measurements. Compressing any nonconsecutive set of integer charges decreases every pairwise distance, hence increases this nonnegative kernel entrywise; consecutive charges are sufficient for the global optimum at that cardinality. At = 2 this gives the known sine-state result

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s
2
M + 1 sin (n + 1)π
M + 1 ,
P opt
error = 1 −cos[π/(M + 1)]
rn =
2
.

For the binary classical outcome channel, this is also the exact minimum half-diamond error to the ideal basis measurement, uniform over inert references: the achieved effects are (2 in target-basis coordinates after the known diagonal chirp mask, with cos +1)], and any channel comparison is lower-bounded by its equal-prior basis discrimination error. This assertion does not compare retained active-token outputs. The quadratic label-error scaling and the exact-heralding loss law answer different questions.