A noisy delayed trine process: exact qutrit realizations

A noisy delayed trine process: exact qutrit realizations

The formerly unresolved positive-noise case admits an exact three-dimensional quantum source in a nonempty, explicitly certified regime. This improves the two-qubit upper bound. The result preserves every finite output word, and its cone constraints are certified on an entire parameter interval by exact rational arithmetic. A separate analytic construction gives a narrower interval without computer-assisted positivity checks. Both constructions use ordinary quantum instruments; neither derives the Born rule or an actual-event selection law.

Process, source class and baseline bounds

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For 0 ≤λ ≤1, write r = 1 −λ and

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T λ
aa = 1 −r
3
,
T λ
ab = 1 + r/2
3
(a ̸= b).
(1)

All parameters and probabilities are dimensionless. Interleave two independent stationary chains with this symmetric doubly stochastic transition matrix. The stationary word law is

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n−2
Y
Pλ(y1 · · · yn) = 1
t=1
T λ
yt,yt+2,
n ≥2,
Pλ(y1) = 1/3.
(2)
9

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The source class is a time-homogeneous quantum instrument on Cd: CP maps Φj with P
j Φj trace
preserving, a stationary density matrix ω, and word probabilities Tr[Φyn · · · Φy1(ω)]. Outputs are
emitted into a retained classical archive. The current symbol need not occupy an orthogonal sector of
the internal memory. Fresh Markov environments are admitted explicitly; the cost d counts internal
memory, not archives, purifying environments, or a hidden history-dependent controller.

The eigenvalues of are 12. For 1 the past-two/future-two cut matrix

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Hab,cd = 1
9T λ
acT λ
bd
(3)

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has ordinary rank nine. Classical finite HMMs factor this matrix through their hidden states, so require
at least nine. Explicitly, the pair source has states (a, b), transition (a, b) 7→(b, j) with probability
T λ
aj, uniform stationary weights 1/9, and emitted symbol j at its update; it attains the full law with
nine states. A d-dimensional instrument gives factors Aab = ΦbΦa(ω) and Bcd = Φ∗
cΦ∗
d(I), with
Hab,cd = Tr(AabBcd), in the d2-dimensional real space of Hermitian matrices, so d ≥3.
A four-dimensional construction exists for every λ. With equatorial trine Bloch vectors na, let
ρa = (I + na · σ)/2 and Ej = (I −rnj · σ)/3. Then

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Φ(4)
j (X) = Tr1[(Ej ⊗I)X] ⊗ρj
(4)

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is CP, has effects Ej ⊗I summing to identity, and fixes I4/4 after summing j. It sends ρa ⊗ρb to
T λ
ajρb ⊗ρj, establishing every word of (2). Thus

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3 ≤dmin(λ) ≤4
(0 ≤λ < 1),
(5)

whereas at = 1 the law is iid uniform and = 1. The rank argument and quantum-process framework have established antecedents [5, 8]; no new rank method is claimed.

A general exact encoded-pair construction

Let have zero sum, and let a diagonal metric = diag() obey

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2
X
(
1
a = j,
−1/2
a ̸= j.
μ=1
gμxaμxjμ =

Let be any basis of traceless Hermitian 3 3 matrices. Use the contractions

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Aa =
X
μ
xaμAμ,
Bb =
X
ν
xbνBν,
Cab =
X
μ,ν
xaμxbνCμν,
Cν,j =
X
η
xjηCνη.

For nonzero positive coefficients, define candidate pair states

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Sab = I3/3 + uAa + vBb + wCab,
Rj = I3/3 + vBj.
(6)

Specify a Hermiticity-preserving complex-linear map by its values on the Hermitian basis:

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Φj(I3/3) = Rj/3,
(7)
Φj(Aμ) = −r
3ugμxjμRj,
(8)
Φj(Bν) = u
3vAν + w
3vCν,j,
(9)
Φj(Cμν) = −ru
3wgμxjμAν −r
3gμxjμCν,j.
(10)

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Lemma 1.1 (Exact update and full-law realization). If all Sab are positive and all maps in (10) are
CP, then they form a quantum instrument with stationary state I3/3 and generate (2) exactly. In
particular,
Φj(Sab) = T λ
ajSbj.
(11)

For this attains the minimum dimension three.

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Proof. All basis matrices except I are traceless, so the candidates have trace one. Expand (10) on (6),
writing qaj = P
μ gμxaμxjμ. The constant and a terms give (1 −rqaj)Rj/3; the b and ab terms give
(1 −rqaj)(uAb + wCbj)/3. This is T λ
ajSbj, proving the update.
Summing the trace of the maps on the basis yields trace preservation: the I/3 traces sum to one,
the A traces vanish because P
j xj = 0, and the B, C images are traceless. Summing Φj(I/3) = Rj/3
gives I/3, proving stationarity. The first output j has probability 1/3 and leaves state Rj. A direct
application gives Φk(Rj) = Sjk/3, so the second output is independent uniform and leaves Sjk. Every
later update is (11); induction proves all finite words. This does not infer temporal independence from
a reset: it uses the explicitly specified CP instrument at every step. Finally (3) gives the matching
dimension-three lower bound.

For positive, the nine candidate matrices span the full Hermitian qutrit space. The functions form a basis of functions on three labels; their tensor products provide the nine independent coordinate functions used above. Thus the construction is not silently collapsing an unobserved classical coordinate or omitting an internal controller.

An analytic high-noise interval

Choose the ordinary unit trine coordinatescos sin = 1, and an orthonormal traceless Hermitian basis with Tr(, split as the basis above. Put

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r = ε3,
u = ε2,
v = ε,
w = ε3,
0 < ε ≤1/7.
(12)

The state perturbation has exact squared Frobenius norm 9, so every is positive. For the unnormalized Choi matrix of (10), the depolarizing branchTr9 has Choi matrix 9. Orthonormality of the input basis makes squared Frobenius norms add, giving

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9 (1/3 + ε2) + 2(ε2 + ε4)
9
+ 2(ε4 + ε6)
∥Jj −I9/9∥2
F = ε2
3 + ε2
9
= 16ε2
27 + 5ε4
9 + 2ε6
9 .
(13)

At7, eighty-one times this expression is 117471 1; the expression increases with The operator norm is at most the Frobenius norm, so is strictly positive and the map is CP by the Choi criterion [1]. Lemma 1.1 therefore proves

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dmin(λ) = 3
for 342/343 ≤λ < 1.
(14)

This is a complete analytic interval proof, independent of the sharper finite certificate below.

A larger interval certified entirely by rational arithmetic

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Take rational trine coordinates

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x0 = (1, 0),
x1 = (−1/2, 1/2),
x2 = (−1/2, −1/2),
(g1, g2) = (1, 3).

Let denote matrix units and set

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A1 = E01 + E10,
A2 = −iE01 + iE10,
B1 = E02 + E20,
B2 = −iE02 + iE20,
C11 = E12 + E21,
C12 = −iE12 + iE21,
C21 = diag(1, −1, 0),
C22 = diag(1, 1, −2).

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These are orthogonal, with squared Frobenius norms two except ∥C22∥2
F = 6. This is a different,
explicitly specified frame from the preceding analytic construction. Put

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r = ε3,
u = αε2,
v = βε,
w = γε3,
(α, β, γ) = (3/8, 3/4, 3/4),
0 < ε ≤h := 7/20.
(15)

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The maps and states are now degree-three matrix polynomials in ε, with Gaussian rational coefficients.
Here are explicit coefficients, making the positivity certificate reconstructible without fitting
numerical samples. Superscript ∨denotes the Hilbert–Schmidt dual basis (F ∨= F/2, except C∨
22 =
C22/6). Define
Ag,∨
j
=
X
μ
gμxjμA∨
μ,
Cg,∨
j,ν =
X
μ
gμxjμC∨
μν.

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In the convention Jj = P
pq Epq ⊗Φj(Epq), write Jj(ε) = P3
l=0 εlJjl. Then

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Jj0 = I9/9,
(16)
Jj1 = β
3 I ⊗Bj −1
9α(Ag,∨
j
)T ⊗I + α
ν
(B∨
ν )T ⊗Aν,
(17)
X
3β
Jj2 = −β
3α(Ag,∨
j
)T ⊗Bj + γ
ν
(B∨
ν )T ⊗Cν,j
X
3β
−α
ν
(Cg,∨
j,ν )T ⊗Aν,
(18)
X
3γ
Jj3 = −1
ν
(Cg,∨
j,ν )T ⊗Cν,j.
(19)
X
3

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For any degree-three matrix polynomial P(ε) = P
l εlPl, its Bernstein representation on [0, h] is

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3
X
k
X
k
l


3
k
!
tk(1 −t)3−kMk,
Mk =
3
l

hlPl.
(20)
P(ht) =
k=0
l=0

The coefficients outside are nonnegative and sum to one. Thus positive control matrices certify positivity on the whole interval, not at sampled values.

Theorem 1.2 (Exact high-noise qutrit interval). For the noisy delayed trine law,

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dmin(λ) = 3
7657/8000 ≤λ < 1.
(21)

The concrete states and maps are (6), (10), and (15).

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Proof. For the twelve Choi Bernstein control matrices from (19)–(20), and the thirty-six state control
matrices from (6), exact rational LDLT certificates establish strict positivity. The certificate uses the
realification
!

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Re Z
−Im Z
Im Z
Re Z
R(Z) =
;

a Hermitian matrix is positive definite precisely when its realification is. In the fixed computational-basis ordering, the certified bounds on every LDL pivot are:

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Bernstein index k
0
1
2
3

Choi control matrices (all State control matrices (all

These are pivot bounds, not eigenvalue bounds. They are verified with exact Fraction arithmetic in supplement/memory/three/memory/exact_qutrit_certificate.py; the explicit rational control matrices and all pivots are recorded in supplement/memory/expected_outputs/three/memory/qutr it_bernstein_matrices.json and supplement/memory/expected_outputs/three/memory/qutrit _certificate.json. For clarity the finite algebraic verification is the standard recurrence

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l<k
L2
klDl,
Lik = Mik −P
l<k LilLklDl
Dk
,
Lkk = 1.
Dk = Mkk −
X

It checks 0 and reconstructs every entry of exactly. This computer-assisted part involves only 48 explicitly defined rational matrices; no numerical eigensolver or tolerance certifies their signs. Bernstein convexity proves positivity throughout 0 The same code independently verifies 197 exact polynomial identities by applying the Choi polyno-mial directly: all 27 reset identities through degree six, and trace-preservation and stationarity through degree three. The universal symbolic proof of those identities is already Lemma 1.1. It follows that this is a stationary CP instrument reproducing every word. The ordinary-rank lower bound gives minimality. Since, the interval is 8000, excluding the iid endpoint from the dimension-three assertion.

Finite observations versus exact memory requirements

The exact dimension remains at least three for every 1, even arbitrarily close to the iid endpoint. This does not create a finite-data discontinuity. Let be the uniform iid law on trits and Define

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d(r) = 1 −r
3
log(1 −r) + 2(1 + r/2)
3
log(1 + r/2).

With 0 log 0 = 0, direct conditional-likelihood calculations give

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D(Pλ,n∥Un) = (n −2)d(r),
χ2(Pλ,n∥Un) = (1 + r2/2)n−2 −1.
(22)

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Indeed the likelihood ratio is Qn−2
t=1 3T λ
yt,yt+2. The KL chain rule gives one equal row contribution d(r) per
link. Under Un, summing the newest symbol in the squared likelihood contributes 3 P
j(T λ
aj)2 = 1+r2/2,
independently of a; recursion proves the chi-square identity. Therefore

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TV(Pλ,n, Un) ≤min

1,
q
(1 + r2/2)n−2 −1

.
(23)
q
(n −2)d(r)/2, 1
2

The second bound is Pinsker and the third is Cauchy–Schwarz. Since), fixed finite records approach the iid law continuously. These are established divergence inequalities evaluated exactly for this process, not new information measures. The comparison concerns the complete declared symbol sequence, including its ordering; extra retained apparatus channels would require their joint law as well.

What the construction establishes

The original trine phase geometry supplied a four-dimensional realization; its zero-noise optimality proof did not settle noisy qutrit models. The present construction supplies such models explicitly and proves exact minimality on a full interval. It introduces a hierarchical encoding of the two individual labels and their joint Fourier components, with amplitudes that shrink at different rates. This permits the update maps to remain inside the CP cone while retaining the complete order-two law. The sharper

interval uses a rational qutrit frame and a convex polynomial certificate; it does not claim the endpoint is optimal. The finite circular phase representation draws on the geometric phase and angular-register descrip-tions in [6, abstract and Sections 2–3] and [7, pp. 64–68]. The noisy qutrit encoding, update maps and interval certificates are the constructions established here. The CP-realization problem, nonorthogonal quantum process memories, quantum dimension advantages and lower bounds are established research areas [3–5, 8, 11]. The contribution is the explicit full-law construction, interval certificate and matching dimension bound for this declared delayed process. Targeted primary-literature checks do not establish publication priority. No claim is made about reduced complete environment memory, a Hamiltonian mechanism, CSL dynamics, Born-rule selection, actual-event affinity, or conscious perspective.