A continuous family of minimum-Kraus qutrit sources

A continuous family of minimum-Kraus qutrit sources

A coordinate-plane Weyl encoding gives exact stationary qutrit sources with recorded one-step Choi rank nine throughout an explicit noise interval. Its two real seed weights and complex phase can be classified exactly within the minimum-Kraus subfamily. This class is not the class of all qutrit sources. In particular, its fixed full-history cost does not preclude the mixed-support improvement constructed below. The source laws and CP maps are defined explicitly below.

Process, resources and the exact update interface

For 0 1 define the trit transition matrix

Searchable transcription

{︄
(1 −r)/3
a = j,
(1 + r/2)/3
a ̸= j,
λ = 1 −r.
(14)
Taj(r) =

The stationary delayed-trine law is

Searchable transcription

n−2
∏︂
Pr(y1 · · · yn) = 1
v=1
Tyv,yv+2(r),
n ≥2,
Pr(y1) = 1/3.
(15)
9

It is the interleaving of two independent stationary chains. A source is a stationary density matrix on a-dimensional memory with time-homogeneous completely positive maps Φ, whose sum is trace preserving. The complete output at each step is the observed classical trit. The emitted-record channel is

Searchable transcription

2
∑︂
j=0
|j⟩⟨j| ⊗Φj(X),
k = rank J(A) =
∑︂
A(X) =
j
rank J(Φj).

Here is the minimum pure-environment dimension for a Stinespring dilation of this complete cq channel. A coherent emitted record is a different channel. The retained archive is external to the reusable memory, independent fresh ports are admitted, and all reusable history-bearing controllers belong in the source cut. This is an information-theoretic resource comparison, not a claim about hardware dimensions, energy, preparation cost or mixed baths. The four-symbol cut9 has rank nine, because has eigenvalues 1 -dimensional quantum instrument factors this cut through a space of dimension at most; a classical -state HMM factors it through dimension. Thus3 and9. The classical pair source attains nine. These are standard representation-rank methods [7, 11]; our contribution is explicit completely positive attainment with controlled environmental resources, not the rank inequality itself.

A variable seed and its Choi blocks

We use the established finite Weyl framework [5] and the standard Choi criterion for complete positivity [2]. Let diag) and mod. Choose 0

Searchable transcription

and initially set ζ = −tω2 = teiπ/3. Define

Searchable transcription

⎛
⎞
⎜
⎝
0
0
0
0
a
ζ
0
 ̄ζ
1 −a
⎟
⎠,
Suv = ZuXvχX−vZ−u.
(16)
χ =

Here are trit indices and is the real seed weight. Write

Searchable transcription

q = a(1 −a),
D = 1 −3q > 0,
Db = Xb diag(0, a, 1 −a)X−b,
Fb = |b + 1⟩⟨b + 2|.

Searchable transcription

Then
Suv = Dv + ζω−uFv +  ̄ζωuF †
v .

Searchable transcription

The three Dv span the diagonal matrices: their coordinate determinant is a3 + (1 −a)3 = D. The
nonconstant Fourier components in u span the six Fv, F †
v . Therefore the nine states form a Hermitian
basis. Each has rank two, with support on the plane missing |v⟩, and ∑︁
uv Suv = 3I.
Define the unique linear maps on this basis by

Searchable transcription

Φj(Suv) = Tuj(r)Svj.
(17)

Searchable transcription

Equivalently,

Searchable transcription

Φj(Dv) = Svj/3,
(18)
6ζ Svj,
Φj(F †
v ) = −rω−j
Φj(Fv) = −rωj
6 ̄ζ Svj.
(19)

Searchable transcription

These identities specify the map on every qutrit operator, including coherences. They are not merely
an update on a finite list of states. Covariance gives Φj = AdXj ◦Φ0 ◦AdZ−j.
Use J(Φ) = ∑︁
pq |p⟩⟨q| ⊗Φ(|p⟩⟨q|). In the input Fourier basis fk = 3−1/2 ∑︁
p ωpk|p⟩, the nonzero
part of J(Φ0) consists of three blocks on span{fk ⊗1, fk+1 ⊗2}:

Searchable transcription

(︄
ae−
c0
 ̄c0
(1 −a)e−
)︄
(︄
ae−
c1
 ̄c1
(1 −a)e+
)︄
(︄
ae+
c1
 ̄c1
(1 −a)e−
)︄
B0 =
,
B1 =
,
B2 =
,
(20)
e−= 1
3 −r
6t,
e+ = 1
3 + r
3t,
L = (a −1) + (2a −1)ω,
(21)
c0 = −ω2(Dr + tL)
3D
,
c1 = −ω2(−Dr + 2tL)
6D
.
(22)

All other output-directions vanish. Since and Re

Searchable transcription

|c0|2 = Dr2 −rt + t2
9D
,
|c1|2 = Dr2 + 2rt + 4t2
36D
.

Consequently the two distinct determinants are

Searchable transcription

36t2
−Dr2 −rt + t2
d0 = q(2t −r)2
9D
,
(23)
18t2
−Dr2 + 2rt + 4t2
d1 = d2 = q(2t −r)(t + r)
36D
.
(24)

These are derived by exact change of basis from (19); the verification code independently reconstructs the entire Choi matrix.

All three blocks have rank one on a continuous curve

Theorem 4.1 (A continuous minimum-Kraus construction). For, set

Searchable transcription

⎛
√︄
⎞
q =
h
3h + 2,
a = 1
2 −h
3h + 2
⎠,
t2 =
h(2 −h)
2(h + 1)(3h + 2),
r = ht.
(25)
⎝1 +
2

The states and maps above are a stationary qutrit instrument generating every finite word of (15). Each branch has Choi rank three and the complete emitted-record channel has rank nine. The attained processes include every

Searchable transcription

71 −17
√
√︂
17
4
= 0.2381181907818797 . . . ,
0 < r ≤rmax :=
(26)
λmin := 1 −rmax = 0.7618818092181203 . . . ≤λ < 1.

The minimum internal memory dimension is exactly three throughout this interval. The iid endpoint has minimum dimension one and is not included in that assertion.

Proof. For 04 and 1. Thus all seed states are positive of rank two and their basis is nonsingular. Also Multiplication of (24 and substitution reduce the determinant equations to

Searchable transcription

qD(2 −h)2 = 4t2(Dh2 −h + 1),
2qD(2 −h)(1 + h) = t2(Dh2 + 2h + 4).

Searchable transcription

Both are identities under (25). Every block therefore has positive diagonal and zero determinant: it is
positive of rank one. Covariance transfers this to every branch, proving complete positivity and the
stated Choi ranks.
Trace preservation follows on the basis Suv from ∑︁
j Tuj = 1.
With initial state I/3, direct
application gives
Φj(I/3) = Dj/3,
Φk(Dj) = Sjk/3,
∑︂
j
Dj = I.

The initial state is stationary, and the first two symbols have probability 19 and normalized posterior state. Thereafter (17) supplies exactly the prescribed conditional probability and posterior state at every step. Induction proves the full word law, not only its first few marginals. Finally

Searchable transcription

r(h)2 =
h3(2 −h)
2(h + 1)(3h + 2),
d
dhr(h)2 = −3h2(h + 2)(2h2 −h −2)
2(h + 1)2(3h + 2)2
.

Searchable transcription

17)/16. On (0, h∗]
the function is strictly increasing from zero, so every 0 < r ≤rmax has a unique parameter in this
subinterval. A bracketed root finder constructs it. The observed rank-nine cut proves that the admitted
dimension-three realization is minimal.
The unique maximum in (0, 2) is at h∗= (1 +
√
17)/4, where r2 = (71 −17
√

Searchable transcription

There are exactly three Kraus operators per branch. Explicitly, for the positive rank-one kth block
(︂αk ck
 ̄ck βk
)︂
, its Choi vector is

Searchable transcription

vk = √αk fk ⊗|1⟩+
 ̄ck
√αk
fk+1 ⊗|2⟩.

Unvectorize to obtain one Kraus operator; rotate by covariance for other emitted symbols. Their completeness follows from the trace-preserving proof. This is a defined admitted quantum interaction, preparation and readout law using standard Born-form instruments; it does not derive the Born rule or provide an autonomous collapse mechanism.

The complete arbitrary-phase class and a sharp scoped boundary

The family is not merely a numerical choice among unspecified phases. Replace by while retaining 0 1 and 0. The same exact-reset definition remains unique. The three blocks now have diagonals, cyclically, where

Searchable transcription

3 + r(x −
√
3 + r(x +
√
e0 = 1
3 −rx
3t2 ,
e1 = 1
3y)
6t2
,
e2 = 1
3y)
6t2
.
(27)

Their upper off-diagonal entries are

Searchable transcription

6 ̄ζ
,
M = −1 + i
√
3(2a −1),
(G0, G1, G2) = (x +
√
3y, −2x, x −
√
ck = t2M/D + rGk
3y).
(28)

These formulas follow by the same direct Fourier transform, not by imposing complete positivity after a fit.

Searchable transcription

Lemma 4.2 (No singular one-step effect in this Weyl class). At r > 0, every completely positive
exact-reset source of this class has strictly positive effects Ej = Φ∗
j(I).

Searchable transcription

Proof. The effects have eigenvalues e0, e1, e2. If e0 = 0, positivity of the blocks forces c0 = c2 = 0. Their
difference is r
√

Searchable transcription

3y/(3 ̄ζ), so y = 0. Then e0 = 0 gives r = x > 0. The numerator of c0 is t2M/D + rx.
Its imaginary part vanishes only if a = 1/2, in which case D = 1/4, t2 = x2 and the numerator is
−4x2 + x2 = −3x2 ̸= 0, a contradiction. For a zero e1 or e2, the two forced zero off-diagonals instead
give y = ∓
√

0 and again the numerator. Its imaginary part forces 2 and its real part is= 0. Positive semidefiniteness of the effects therefore makes them positive definite.

Theorem 4.3 (Exhaustion of the minimum-Kraus Weyl class). Among all nonsingular rank-two coordinate-plane Weyl exact-reset sources above, with arbitrary seed diagonal weight and phase, branch Choi rank three is attained exactly by (25), up to phase rotation by and exchange of with Thus (26) is the sharp noise boundary within this minimum-Kraus Weyl class. It is not established as the boundary for arbitrary qutrit sources, nor for members of the same Weyl class with higher Kraus rank.

Proof. Put, and

Searchable transcription

u0 = x/t,
u1 = (−x +
√
3y)/(2t),
u2 = (−x −
√
3y)/(2t).

These are three cosines spaced by 23; their sum is zero and their squares sum to 32. By Lemma 4.2, every block has positive diagonal and hence rank at least one. Choi rank three therefore means that all three block determinants vanish. Directly from (27)–(28), each determinant, indexed by its opposite cosine, has the form

Searchable transcription

9d(u) = A + Bu + Cu2,
A = q(1 −3h2/4) −v/D,
B = h(q −v/D),
C = h2(q −v).
(29)

If all three cosines are distinct, a quadratic zero at all three is zero identically. As implies, impossible for 0 and 1. Thus two cosines coincide and is an integer multiple of For an even multiple the cosine set is. Positivity of the effect requires Eliminating from2) = 0 gives2); the excluded factors are 3. For 0 1, this value is either nonpositive or exceeds 14. Thus it cannot be a seed with For an odd multiple the cosines are. Positive effects require 0 2. Elimination instead gives + 2), and either remaining determinant yields These are precisely (25). There are no omitted solutions at = 2: an effect would be singular, already excluded. The maximum of on this curve was proved in Theorem 4.1.

What the improved source does and does not settle

Searchable transcription

For every member of this class, effects are positive definite and every pair posterior has rank two. With
Fjk = Φ∗
j(Ek), Ek ⪰ckI for some ck > 0, so Fjk ⪰ckEj ≻0. Basis-complete reset implies

Searchable transcription

ΦkΦj(A) = Tr(FjkA)Sjk
on all M3.

Therefore every length-2 word branch has Choi rank rank rank 2 = 6; all intervening transition weights are positive. The complete-record cq channel has rank 6 3. For the single stationary preparation its output state instead has rank 2 3. These are different operational tasks. This remains true for arbitrary phases and higher branch Kraus ranks in this coordinate-plane class. Consequently phase and diagonal freedom can remove one-step excess without reducing the globally compressed finite-horizon channel below 6 3. In particular it cannot improve its finite-horizon cost merely by changing these seed parameters. Any reduction below 6 3 requires a different encoding. The stronger universal lower bound 437 3 for4, proved in Section 8.5, leaves a gap below this construction; in particular the simpler 4 3 bound is not attainable at those horizons. The general result in Appendix A that each branch of a minimal qutrit source for this process has Choi rank at least three makes = 9 optimal among all qutrit realizations, not just this ansatz. Combined with the pure-latitude two-qubit source of ( 3) and the three-output floor the one-step Pareto frontier is now exactly throughout (26). This invocation uses the full CP rank lemma in Appendix A; it is not derived from the more limited exhaustion theorem above. The algebraic identities and all-word induction, rather than finite numerical fits, establish the result. The singular endpoints 2 are not admitted finite encodings. No global monotonicity of the unrestricted qutrit-admissible set is asserted.