Finite-horizon resource accounting: what survives beyond the memory box

Finite-horizon resource accounting: what survives beyond the memory box

An exact streaming advantage survives the expanded accounting, but it is an advantage in reusable causal memory, not a claim that an entire closed apparatus has four dimensions. For the noiseless trine process and any declared horizon4, four quantum dimensions and nine classical states are the exact causal-memory minima when every history-correlated controller, reference and environment that can influence the future is included at the causal cut. In contrast, the retained archive and an isometric implementation of its classicalization require resources growing with the horizon. These resources can be calculated exactly for the supplied quantum construction. This corrects a possible overreading of the earlier four-versus-nine result. The calculation concerns ordinary finite-dimensional instruments and exact channels. It assumes Born probabilities, does not select actual events, and does not assign hardware size, energy, temperature or a physical reset cost to a Hilbert dimension. The preparation, interface and accuracy must be the same for every comparison; the finite-accuracy comparator elsewhere in this companion changes the appropriate conclusion when an exact law is unnecessary.

Declared task and phase encoding

For the emitted-symbol construction, use phases

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|ψj⟩= |0⟩+ eiθj|1⟩
√
2
,
|ψ⊥
j ⟩= |0⟩−eiθj|1⟩
√
2
,
Ej = 2
m|ψ⊥
j ⟩⟨ψ⊥
j |,

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with ρj = |ψj⟩⟨ψj| and
Tij = Tr(Ejρi) = 1 −cos(θi −θj)
m
.

Starting from4, the iterated instrument

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Ψj(X) = Tr1[(Ej ⊗I2)X] ⊗ρj
(37)

has the full stationary word law

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n−2
Y
p(w1 · · · wn) = 1
t=1
Twt,wt+2,
n ≥2.
(38)
m2

The circular phase representation is informed by [6, abstract and Sections 2–3] and [7, pp. 64–68]; the stochastic process and instrument are defined above, and their resource costs are proved below. The task here retains all outputs in their original order, their settings and timing metadata, and every declared null/loss outcome. The ideal alphabet contains no loss; a failed physical implementation must add it and changes the target law. The archive is append-only: it is not consulted to decide future source transitions. Future fresh stock is initially uncorrelated with the history and memory. Any violation must be explicitly added to the causal memory or to the task’s intervention interface. There are two distinct output tasks. A coherent archive retains the amplitudes between different output words. A classical archive is diagonal in the word basis, including when the input memory is entangled with an arbitrary retained reference. The passive word probabilities agree, but the channels and their resource requirements differ.

An exact single-step dilation

Let swap the two memory qubits and set

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Lj =
q
2/m|ψj⟩⟨ψ⊥
j |,
Kj = S(Lj ⊗I2).

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Then Ψj(X) = KjXK†
j and P
j K†
jKj = I4. Thus

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m−1
X
j=0
|j⟩A ⊗Kj : C4 −→Cm
A ⊗C4
M
(39)
V =

is an isometry. A fresh blank-level output register and a fixed unitary extension implement the coherent-output step with no additional discarded system. This does not say that the output register is free. For comparison the dephased complete-record channel is

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C1(X) =
X
j
|j⟩⟨j|A ⊗KjXK†
j.

Its Choi rank is exactly: each nonzero output block has rank one. Consequently a globally isometric implementation with a pure initially independent environment requires an environment of dimension at least, and dimension suffices by copying the label,

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F : |j⟩A 7→|j⟩A|j⟩E.

The environment must be retained elsewhere or traced; tracing implements classicalization. Conditioning on one actual outcome is a different operation from replacing this complete channel by a pure branch.

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The memory channel with its record discarded, Λ(X) = P
j KjXK†
j, has Choi rank exactly three,
independent of m. Indeed the three functions 1, eiθj, e−iθj are linearly independent for m ≥3, and

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1
−e−iθj
eiθj
−1
!
Lj =
1
√
2m

span a three-dimensional operator space. Multiplication by and tensoring with preserve this span dimension. Dropping the record can therefore reduce a dilation cost from to three, but solves a weaker task. These minimal-environment statements are standard Choi–Kraus–Stinespring theory, applied here explicitly. If a pure-environment dilation has dimension, its environment-basis Kraus representation has at most terms, so rank; a spectral decomposition of the positive Choi matrix attains equality. A primary textbook statement is Watrous, The Theory of Quantum Information, Corollary 2.27, printed pp.89–90, author’s chapter PDF pp.16–17: https://cs.uwaterloo.ca/~watrous/TQI /TQI.double.2.pdf.

Exact horizon cost and the obstruction to indefinite reuse

For a word, let. Define the complete classical-record channel

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Cn(X) =
X
w
|w⟩⟨w|A ⊗KwXK†
w.
(40)

This is a channel on arbitrary input states and hence includes all initially retained input references. It is stronger than requesting a distribution only for

Horizon dilation theorem. For the noiseless phase-circle instrument,2, put

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Nn = m2(m −1)n−2.

Then has Choi rank exactly. Its minimum pure initial environment dimension is. For the fixed stationary input4, the final memory–classical-archive state also has rank. Conversely the coherent-output isometry requires no additional bath beyond its output registers, while producing a different retained-output channel. Proof. After the first two steps,

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K(a,b) = 2
m|ψaψb⟩⟨ψ⊥
a ψ⊥
b |.

Every further step shifts the two states and contributes a scalar whose squared modulus is Therefore has operator rank one and is nonzero precisely when at every later step. There are first pairs and1 later choices, giving positive words. Each word block of the Choi matrix is the nonzero rank-one matrix; distinct archive blocks are orthogonal. Their ranks add, proving the Choi claim and the pure-environment minimum. On4 every word block is a positive scalar times, so the fixed-output state has the same rank. Finally the ordered iteration of is an isometry before dephasing and introduces no bath. The lower bound is sharp for the declared final complete channel:

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X 7→
X
w
|w⟩A|w⟩E ⊗KwX

is its Stinespring isometry. It is not a claim that is the minimum physical bath for every source having only the same passive output law, or that independently optimizing each intermediate cut gives

a minimum protocol. It is a concrete finite-horizon implementation and lower bound for (40). The distinction between a channel’s dilation rank and a whole strategy’s memory is essential; see Bisio et al., Phys. Rev. A 85, 032333 (2012), DOI 10.1103/PhysRevA.85.032333, Definitions 3–4 and Theorem 3, https://arxiv.org/abs/1112.3853. A fixed reusable environment of dimension, initially pure and with no reset, export or new environment, cannot realize these dephased channels at arbitrarily large: it must satisfy For

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$
log(D/m2)
log(m −1)
%
n ≤2 +

whenever a horizon2 is feasible. This is a finite-dimensional isometric obstruction, not a claim about the speed or thermodynamics of erasure. If an initially mixed environment has rank and output dimension, a Kraus expansion gives only; its purifying stock cannot be silently declared free. If the apparatus exports the purifier or outcome copies, those registers belong in the resource ledger. If a fundamental nonunitary law is introduced, this dilation accounting no longer describes that law and must not be used to rule it out. The raw-symbol archive has dimension for either realization. Its positive-word subspace has dimension, allowing an exactly reversible lossless word code of that capacity. A controller implementing such global compression is additional circuitry; it is not produced by the rank bound. With stationary input, the required archive entropy is

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H(W1 · · · Wn) = 2 log2 m + (n −2)hm,
hm = −
X
j
T0j log2 T0j.

This follows by the chain rule from the order-two law and its uniform pair distribution. For the noiseless trine, = 1 and all positive words are equiprobable. The purifying environment for the stationary output can have exactly this entropy; entropy in bits is not an energy estimate.

An explicitly budgeted classical comparator

For = 3, the nine-state orthogonal-pair source achieves the exact classical minimum for this law. A declared quantum-channel embedding of that classical update on is

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Dj = diag(T0j, . . . , Tm−1,j),
Qj(X) = Tr1[(Dj ⊗Im)X] ⊗|j⟩⟨j|.

It preserves the same passive word law from. It retains second-register coherence in this extension rather than gratuitously dephasing the entire memory. Its independent Kraus operators are

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Fj,a =
q
Taj Sm(|j⟩⟨a| ⊗Im),
a ̸= j.

Thus its one-step complete-record Choi rank is1), and for2 its horizon Choi rank is . To see the latter, a word map has pure final memoryand effect proportional to , of rank (; archive ranks again add. These environment counts are exact for these specified CP updates, not a proved optimum over every nine-state classical HMM and every physical extension of its diagonal-state behavior. For the trine, the following ledger separates resources rather than hiding them in a single memory number. Its dilation rows concern the specified final channel; they must not be multiplied with a per-step memory optimum and called a simultaneously optimized closed protocol. The simple streaming implementation uses independent fresh per-step ports; global compression needs its own controller.

Resource, exact horizon Quantum shift Canonical classical pair Reusable causal memory 4 dimensions 9 states Raw ordered symbol archive dimensions dimensions Positive-word archive capacity same Pure bath for the specified final com-plete channel

Pure bath for one specified recorded step 3 6

Purifier of stationary mixed memory 4 9 Clock, switch routing, calibration separately supplied separately supplied

The purifier row refers to each stated stationary mixed-memory preparation; it is not evidence that the two process models have the same response to arbitrary interventions on those different purifiers. Bath ranks compare specific channels, not an optimization of all closed realizations. For example = 16 needs = 147456 positive-word labels (18 binary storage positions suffice for their lossless fixed-length code), despite a reusable quantum memory of dimension four. Repeated independent per-step baths of dimension three give a simple, larger upper bound 3; finite-horizon Kraus compression reduces the final-channel bath requirement to. Neither count is the same as a device’s number of elementary components.

A causal-cut separation that includes correlated controllers

There is a precise sense in which the memory advantage is not an artifact of drawing a small box around the source. At the cut between outputs two and three, define to include all jointly history-correlated systems that can influence future source operations: memory, feedback controller, reused environment, and any history-bearing phase or clock register. A history-bearing part of a future tape belongs here, even if physically outside the nominal source. Independent fresh stock is a declared preparation resource, while an append-only archive and environments never accessed again are retained-history resources, not reusable causal memory.

Expanded causal-memory proposition. For the exact noiseless trine four-output law, a quantum needs dimension at least four, and a classical needs at least nine states. Both bounds are attainable while using the explicitly budgeted fresh resources and archives above. They therefore hold for every exact horizon4 under this interface. Proof. Future operations, including independent fresh ancillas, define positive effects on The unnormalized state after outputs is0. Hence

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Hab,cd = p(a, b, c, d) = Tr(AabBcd) = TacTbd
9
.

For trine rank rank = 9. A classical-state cut factors through nonnegative coordinates, giving9. A quantum cut has3. To exclude = 3, use the zeros whenever or. Positive matrices with zero trace pairing have orthogonal ranges. Put span ran and span ran. The nonzero range of is orthogonal to, hence dim2. If one has dimension two, all lie in it, so every effect is supported on a common plane; compressing all factors would give rank at most four, contradicting nine. The same holds for a two-dimensional. Otherwise all these nonzero spaces are lines. Each nonzero lies in both and, forcing all lines equal and rank at most one, again a contradiction. Thus4. The two-qubit shift and nine-state pair source attain the respective bounds.

Certified noisy-interval corollary. The same expanded interface gives exact causal-cut minima

noise interval quantum dimensions classical states 4 9 3

For the first row use the quantitative PSD-rank exclusion proved in the main resource manuscript and the four-dimensional shift. This companion retains the narrower rational-encoding interval in the second row so its stated construction remains self-contained; the main manuscript extends qutrit attainment to the larger interval beginning at. For the second use the exact qutrit instrument in Section 1; ordinary rank nine of the same four-time cut excludes a quantum dimension below three and a classical size below nine. Fresh independent ancillas and fixed clocks still define effects on so including correlated controllers does not change these factorization arguments. These are exact-law statements for4; the middle interval and minimum approximate-memory frontier are not asserted. This is the earlier proved rank/support argument with the causal interface made explicit, not a new lower-bound method. It closes an accounting loophole: moving history into a controller or returning bath does not evade the bound when that system is counted in. Allowing a pre-correlated future tape and then excluding it from would invalidate the factorization, not disprove the proposition. Allowing coherent archive feedback changes the interface and must include the interacting archive in the cut. This restricted four-versus-nine separation remains useful even when common archive storage is large. In logarithmic memory capacity it is log170 bits. Adding an identical separately budgeted storage term does not remove that difference, but the reusable-memory fraction of the total storage can become small as grows. It is not an optimization of total closed hardware, and no energy or speed ordering follows.

Phase, calibration and clock accounting

The instrument is fixed at every tick. It does not call a hidden classical controller that remembers the previous output and prepares a time-dependent phase state; the preparation and update are already included in. Nevertheless the fixed-outcome coupler, its calibration and output ports are resources. Increasing does not make those resources disappear merely because the reusable memory remains four-dimensional. A modest symmetry result limits what phase reference is actually needed in this abstract construction. Let diag, and let mod. The chosen phases of give , hence (41)

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The stationary state is invariant. The isometry therefore respects simultaneous discrete rotation of the
memory and relabelling of the ports. In particular, the common azimuth is not a separate time-varying
source variable.
There is a symmetry-preserving unitary extension with a fresh output blank |s⟩= m−1/2 P
j |j⟩,
invariant under P. To prove existence, the invariant subspace |s⟩⊗C4 and the image of V carry
equivalent representations of the finite group Cm by (41). Their orthogonal complements have the same
representation multiplicities, so choose a unitary intertwiner on each complement and combine it with
V on the input subspace. This is a finite-group constructive existence argument, not an assertion of
reference-free implementation under continuous U(1) or of a particular energy-conserving Hamiltonian.
Physical phase stability and gate precision still require calibration at the declared accuracy.
A clock supplies ticks separated by a chosen ∆t in seconds and routes the next fresh output
register. Its precision, reset and control law are independent implementation inputs. A finite routing

register with + 1 orthogonal positions is one possible explicitly budgeted schedule; no minimum clock dimension or thermodynamic cost is proved. If that clock acquires history correlations used to modify the source, it joins. An unused archival timestamp does not become hidden controller memory.

Noise, accuracy and what remains established

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The exact zero pattern is load bearing. For the earlier noisy construction Eλ
j = [I −(1 −λ)nj · σ]/m
with 0 < λ ≤1, each effect has rank two. The same four-dimensional shift channel, which need not be
the minimum realization of the noisy law, then has one-step recorded Choi rank 2m and unrecorded
rank four. For n ≥2, every word is positive and the word map has pure final memory but an input
effect proportional to Eλ
w1 ⊗Eλ
w2, of rank four. Consequently its complete-channel Choi rank is 4mn,
although its stationary fixed-output state without an initial retained reference has rank only mn. This
is a concrete reason not to replace a retained-reference channel task by one fixed-input density matrix
when counting environment.
These ranks are exact channel identities, not robust physical costs at finite tolerance. At λ = 1 the
output law is IID uniform and a one-dimensional source suffices; the four-dimensional shift’s dilation
ranks are plainly not process-wide minima. The exact and finite-accuracy realization results elsewhere
in this companion must be used for the corresponding task and tolerance. No discontinuous exact rank
jump is offered as an experimentally discontinuous hardware demand.

The rational encoded-pair qutrit has a larger specified bath. The exact rational certificate in Section 1 establishes that all three 9-by-9 Choi matrices of its qutrit branch maps are strictly positive throughout 7657 1; the Bernstein control matrices are positive definite. This independently determines that construction’s dilation ranks, rather than borrowing the four-dimensional shift’s formula. If 0 and 0, choose 0 with. Complete positivity and strict positivity of Θ() give

Here Θ( 0 follows, for example, by partially tracing a strictly positive Choi matrix. Induction proves that every word branch of the specified qutrit source has Choi rank nine. There are 3 positive words; orthogonal classical archive blocks therefore give exact recorded-channel Choi rank 9 3 stationary input3, every conditional word state is strictly positive, giving fixed-output rank 3 3

instead. For these specified implementations on the rational encoded-pair interval 7657 1, the comparable ledger is

Same passive trine word law Qutrit Shift Classical pair Reusable memory dimension 3 4 9 One-step complete-record Choi rank 27 6 9 Horizon complete-record Choi rank,

Stationary fixed-output rank, reference omit-ted

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Raw classical symbol archive
3n
3n
3n

The classical column uses the explicit orthogonal-pair CP extension above; for positive noise its effects have rank three and its word effects have rank nine. It is again a specified implementation, not a universal optimum over classical physical extensions. The source with one-step rank nine and the mixed-support history source in the main manuscript improve on this rational construction. This table preserves the distinct, exactly specified channels above; it is not the current process-wide frontier.

The lower memory dimension and higher minimum pure bath for these particular channels form a real resource tradeoff. The two complete memory/bath channels are different; only the declared passive symbol task is the same. This is not a lower bound on the bath of every possible qutrit realization, nor a proof that the four-dimensional source minimizes total hardware. It is precisely why one should report a resource vector before choosing a cost function. An implementation allowed only finite output accuracy can use a different comparator, as quantified in Section 2. The completed benefit is an explicit audit of where the earlier compression does and does not apply: a sharp exact reusable-causal-memory separation, exact finite-horizon bath requirements for its complete channel, a proof that finite unreset environments cannot produce indefinitely many exact classical records in the isometric model, and a symmetry-respecting fixed implementation. No statement here closes the measurement programme’s outstanding physical obligations or identifies a conscious perspective.

Verification. The historical diagnostic research3/accounting/verify_accounting.py, preserved in the separate evidence archive, checks Kraus completeness, Choi ranks, word probabilities, conditional states and discrete covariance. It tests noiseless grids 5 through horizons 7,5,5 respectively, and the positive trine shifts at 1 through horizon four. The canonical classical channels are tested separately. Numerical compositions of the qutrit certificate at35 through horizon four independently check strict Choi ranks and the new tradeoff. All 7,198 checks pass; the maximum completeness or state/probability error is 1. Exact proofs provide the universal claims; floating-point tests diagnose implementation mistakes. These archived diagnostic counts are not additional checks executed by the compact release’s nine-job memory reproduction command.

A conditional normal form for a mixed-support equality profile

The following reduces a candidate equality branch to a finite spectral form. It is not an existence proof for the mixed support profiles.

Lemma. Let Φ : be CP, trace preserving, of linear rank three, with faithful Φ(). If two distinct pure input rays have singular nonzero outputs, then after an invertible output congruence its image is the full diagonal algebra. There are a density matrix0 and three scalars, at least two distinct and of unit modulus, with1, such that in a suitable input basis

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x
y
z
w
!
= √ρ U diag(x + w + λiy +  ̄λiz)3
i=1U†√ρ.
Φ

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Its Choi rank is three or four, according as the remaining scalar has unit modulus or strict modulus
below one.
Proof. The Hermitian kernel is one-dimensional and contains a nonzero traceless matrix because
the map is trace preserving. Diagonalize it. The two diagonal input projectors then have the same
output ρ, which is faithful. Writing the off-diagonal response as C gives Φ(X) = (x + w)ρ + yC + zC†.
Choi positivity is equivalent to
 
I
K
K†
I
!

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⪰0,
K = ρ−1/2Cρ−1/2,
∥K∥≤1.

For a normalized pure input, its output is congruent to sin Re). A zero eigenvalue requires sin = 1 and a unit vector with Re1. Equality in the contraction and Cauchy–Schwarz inequalities gives. A unit-circle eigenvector of a contraction is reducing: since its saturated norm implies ( = 0. Distinct pure rays give distinct phases and therefore

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orthogonal reducing eigenvectors. The remaining orthogonal space is one-dimensional, so K is normal.
Diagonalizing it gives the displayed form.
Linear rank three makes the three diagonal response
functionals independent. The Choi matrix then splits into three positive 2-by-2 blocks
 1 λi
 ̄λi 1

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under
congruence. Two have rank one and the third has rank one or two, proving the last statement.
□
For a non-trace-preserving branch with faithful input effect on its two-dimensional input support,
an invertible input whitening makes it trace preserving without changing ranks or distinct pure rays.
In a putative equality profile (a, b) = (1, 2), two other output planes project to distinct rays in this
branch’s input plane, and equality requires their posterior outputs to have rank two. Thus the lemma
applies to that branch. It is only a necessary normal form: its output-congruenced diagonal structure
is not literal commutativity of the physical output states, and no global full-law realization follows
from it alone.