Eternal Non-Markovian Weyl Dephasing Maps

• The paper demonstrates that higher-dimensional Weyl dephasing maps can independently exhibit eternal non-Markovianity without requiring convex mixtures, unlike qubit Pauli channels.
• It uses Weyl operator algebra and subgroup geometry to analyze the time-local generators and CP-divisibility conditions in finite-dimensional quantum channels.
• Eternal non-Markovianity is established by showing persistent negativity in canonical decay rates, with cyclic subgroup orders dictating the emergence of memory effects.

Eternally non-Markovian Weyl dephasing maps are finite-dimensional random-unitary quantum channels built from Weyl operators whose time-local generator violates CP-divisibility for every positive time, typically through at least one canonical decay rate that is negative for all $t>0$. In the Weyl setting, the decisive structure is the discrete phase space $\mathbb{Z}_d\times\mathbb{Z}_d$, its subgroup geometry, and the diagonal action of the channel on the Weyl operator basis. Recent work shows that, unlike the qubit Pauli case, higher-dimensional Weyl dephasing already supports irreducible eternal non-Markovianity at the level of a single dephasing map, while convex combinations can either suppress or generate eternal memory effects depending on how subgroup supports combine in phase space (Xu et al., 22 May 2026).

1. Algebraic framework of Weyl dynamical maps

For a $d$-dimensional system, the Weyl operators are

$$U_{kl}=\sum_{m=0}^{d-1}\omega^{km}\,|m\rangle\langle m+l|,\qquad \omega=e^{2\pi i/d},$$

with labels in $\mathbb{Z}_d\times\mathbb{Z}_d$. They form a unitary operator basis and satisfy

$$U_{kl}U_{rs}=\omega^{lr-ks}U_{rs}U_{kl},\qquad U_{kl}^\dagger=U_{-k,-l},\qquad \mathrm{Tr}(U_{kl}^\dagger U_{rs})=d\,\delta_{kr}\delta_{ls}.$$

Using $u=(i,j)$, $v=(k,l)$, the symplectic product is

$u\wedge v:=jk-il \pmod d,$

and the commutation relation becomes $U_uU_v=\omega^{u\wedge v}U_vU_u$ (Xu et al., 22 May 2026).

A Weyl dynamical map is a random-unitary channel

$\mathbb{Z}_d\times\mathbb{Z}_d$0

with $\mathbb{Z}_d\times\mathbb{Z}_d$1 and $\mathbb{Z}_d\times\mathbb{Z}_d$2. The Weyl operators are eigenoperators: $\mathbb{Z}_d\times\mathbb{Z}_d$3 This diagonalization is the basis of all divisibility and non-Markovianity analyses in the Weyl setting (Xu et al., 22 May 2026).

The subgroup structure of $\mathbb{Z}_d\times\mathbb{Z}_d$4 is not auxiliary; it organizes the channel family itself. Every subgroup $\mathbb{Z}_d\times\mathbb{Z}_d$5 admits a unique Hermite normal form generated by

$\mathbb{Z}_d\times\mathbb{Z}_d$6

with $\mathbb{Z}_d\times\mathbb{Z}_d$7, $\mathbb{Z}_d\times\mathbb{Z}_d$8, $\mathbb{Z}_d\times\mathbb{Z}_d$9, and $d$0. The associated symplectic dual

$d$1

controls the spectral degeneracies of isotropic Weyl maps and, through them, the time-local generator (Xu et al., 22 May 2026).

2. Weyl dephasing as a distinguished Weyl-diagonal subclass

The basic Weyl dephasing map is the single-generator random-unitary channel

$d$2

where $d$3. It is diagonal in the Weyl basis,

$d$4

Its generator eigenvalues are

$d$5

and the canonical rates follow by finite Fourier inversion over phase space (Xu et al., 22 May 2026).

The order of the cyclic subgroup generated by $d$6,

$d$7

is the key arithmetic invariant. It determines how many distinct symplectic phases appear in $d$8, and hence whether one of the canonical rates can remain negative for all $d$9 (Xu et al., 22 May 2026).

A closely related, but structurally narrower, dephasing sector arises when only diagonal Weyl operators are present in the Kraus decomposition. In the special generalized Weyl channel

$$U_{kl}=\sum_{m=0}^{d-1}\omega^{km}\,|m\rangle\langle m+l|,\qquad \omega=e^{2\pi i/d},$$0

all other Kraus operators vanish, so only $$U_{kl}=\sum_{m=0}^{d-1}\omega^{km}\,|m\rangle\langle m+l|,\qquad \omega=e^{2\pi i/d},$$1 survive. In that sense the dynamics is genuinely dephasing-type: it preserves diagonal matrix elements and damps off-diagonal ones (Xu et al., 13 Mar 2025).

3. CP-divisibility and the exact criterion for eternal non-Markovianity

For Weyl maps with time-local generator

$$U_{kl}=\sum_{m=0}^{d-1}\omega^{km}\,|m\rangle\langle m+l|,\qquad \omega=e^{2\pi i/d},$$2

the rates are reconstructed from the spectral logarithmic derivatives

$$U_{kl}=\sum_{m=0}^{d-1}\omega^{km}\,|m\rangle\langle m+l|,\qquad \omega=e^{2\pi i/d},$$3

Markovianity is identified with CP-divisibility, equivalently $$U_{kl}=\sum_{m=0}^{d-1}\omega^{km}\,|m\rangle\langle m+l|,\qquad \omega=e^{2\pi i/d},$$4 for all $$U_{kl}=\sum_{m=0}^{d-1}\omega^{km}\,|m\rangle\langle m+l|,\qquad \omega=e^{2\pi i/d},$$5 and all $$U_{kl}=\sum_{m=0}^{d-1}\omega^{km}\,|m\rangle\langle m+l|,\qquad \omega=e^{2\pi i/d},$$6. Eternal non-Markovianity is the stronger condition that there exists at least one channel $$U_{kl}=\sum_{m=0}^{d-1}\omega^{km}\,|m\rangle\langle m+l|,\qquad \omega=e^{2\pi i/d},$$7 such that

$$U_{kl}=\sum_{m=0}^{d-1}\omega^{km}\,|m\rangle\langle m+l|,\qquad \omega=e^{2\pi i/d},$$8

The dynamics then fails CP-divisibility immediately after initialization and continues to fail it for all positive times (Xu et al., 22 May 2026).

For the Weyl dephasing family with

$$U_{kl}=\sum_{m=0}^{d-1}\omega^{km}\,|m\rangle\langle m+l|,\qquad \omega=e^{2\pi i/d},$$9

the criterion is exact. The map $\mathbb{Z}_d\times\mathbb{Z}_d$0 is eternally non-Markovian if either

1. $\mathbb{Z}_d\times\mathbb{Z}_d$1 is odd, or
2. $\mathbb{Z}_d\times\mathbb{Z}_d$2 is even and $\mathbb{Z}_d\times\mathbb{Z}_d$3,

where $\mathbb{Z}_d\times\mathbb{Z}_d$4 (Xu et al., 22 May 2026).

This result isolates the genuinely higher-dimensional phenomenon. In the qubit Pauli setting, a single dephasing semigroup cannot realize eternal non-Markovianity; for Pauli maps, ENM is obtained only by a convex combination of two Pauli dephasing semigroups, while mixing all three yields only quasi-ENMity (Jagadish et al., 12 Jan 2025). By contrast, higher-dimensional Weyl dephasing admits irreducible eternal non-Markovianity: the single map

$\mathbb{Z}_d\times\mathbb{Z}_d$5

may already be ENM without any mixing mechanism (Xu et al., 22 May 2026).

4. Isotropic Weyl maps, semigroups, and convexity effects

A broader family is obtained by distributing the nontrivial weight uniformly over a subgroup $\mathbb{Z}_d\times\mathbb{Z}_d$6: $\mathbb{Z}_d\times\mathbb{Z}_d$7 Its spectrum simplifies to

$\mathbb{Z}_d\times\mathbb{Z}_d$8

From this, the map is a Markovian semigroup iff

$\mathbb{Z}_d\times\mathbb{Z}_d$9

in which case the nonzero canonical rates are constant and nonnegative (Xu et al., 22 May 2026).

The same isotropic semigroup can be rewritten as the equal convex combination

$$U_{kl}U_{rs}=\omega^{lr-ks}U_{rs}U_{kl},\qquad U_{kl}^\dagger=U_{-k,-l},\qquad \mathrm{Tr}(U_{kl}^\dagger U_{rs})=d\,\delta_{kr}\delta_{ls}.$$0

so convexity becomes a structural operation on dephasing directions rather than a mere probabilistic reinterpretation. One of the paper’s central theorems is that a Markovian semigroup can arise as a convex combination of eternally non-Markovian Weyl dephasing maps when the relevant cyclic subgroup orders are odd. The converse phenomenon also occurs: convex mixtures of $$U_{kl}U_{rs}=\omega^{lr-ks}U_{rs}U_{kl},\qquad U_{kl}^\dagger=U_{-k,-l},\qquad \mathrm{Tr}(U_{kl}^\dagger U_{rs})=d\,\delta_{kr}\delta_{ls}.$$1 distinct isotropic Weyl semigroups are ENM whenever

$$U_{kl}U_{rs}=\omega^{lr-ks}U_{rs}U_{kl},\qquad U_{kl}^\dagger=U_{-k,-l},\qquad \mathrm{Tr}(U_{kl}^\dagger U_{rs})=d\,\delta_{kr}\delta_{ls}.$$2

where $$U_{kl}U_{rs}=\omega^{lr-ks}U_{rs}U_{kl},\qquad U_{kl}^\dagger=U_{-k,-l},\qquad \mathrm{Tr}(U_{kl}^\dagger U_{rs})=d\,\delta_{kr}\delta_{ls}.$$3 is the common subgroup order and $$U_{kl}U_{rs}=\omega^{lr-ks}U_{rs}U_{kl},\qquad U_{kl}^\dagger=U_{-k,-l},\qquad \mathrm{Tr}(U_{kl}^\dagger U_{rs})=d\,\delta_{kr}\delta_{ls}.$$4 is the number of subgroups of order $$U_{kl}U_{rs}=\omega^{lr-ks}U_{rs}U_{kl},\qquad U_{kl}^\dagger=U_{-k,-l},\qquad \mathrm{Tr}(U_{kl}^\dagger U_{rs})=d\,\delta_{kr}\delta_{ls}.$$5 (Xu et al., 22 May 2026).

The qutrit case exhibits all three regimes explicitly. For $$U_{kl}U_{rs}=\omega^{lr-ks}U_{rs}U_{kl},\qquad U_{kl}^\dagger=U_{-k,-l},\qquad \mathrm{Tr}(U_{kl}^\dagger U_{rs})=d\,\delta_{kr}\delta_{ls}.$$6, mixtures with $$U_{kl}U_{rs}=\omega^{lr-ks}U_{rs}U_{kl},\qquad U_{kl}^\dagger=U_{-k,-l},\qquad \mathrm{Tr}(U_{kl}^\dagger U_{rs})=d\,\delta_{kr}\delta_{ls}.$$7 or $$U_{kl}U_{rs}=\omega^{lr-ks}U_{rs}U_{kl},\qquad U_{kl}^\dagger=U_{-k,-l},\qquad \mathrm{Tr}(U_{kl}^\dagger U_{rs})=d\,\delta_{kr}\delta_{ls}.$$8 distinct order-3 semigroups are eternally non-Markovian, while the uniform $$U_{kl}U_{rs}=\omega^{lr-ks}U_{rs}U_{kl},\qquad U_{kl}^\dagger=U_{-k,-l},\qquad \mathrm{Tr}(U_{kl}^\dagger U_{rs})=d\,\delta_{kr}\delta_{ls}.$$9 mixture is Markovian. This establishes that non-Markovianity is not additive under mixing: convexity can either suppress or generate eternal memory effects (Xu et al., 22 May 2026).

5. Relation to Pauli channels and generalized Pauli/MUB constructions

For $u=(i,j)$0, Weyl maps reduce to Pauli maps. The canonical qubit ENM generator is

$u=(i,j)$1

One rate is negative for all $u=(i,j)$2, yet the dynamical map is CPTP and can be written as the random-unitary mixture

$u=(i,j)$3

This is the prototype of eternal non-CP-divisibility without pathological loss of complete positivity (Megier et al., 2016).

The Pauli prototype also clarifies what does not generalize. A single qubit dephasing map

$u=(i,j)$4

cannot itself be eternally non-Markovian, because a rate that is negative for all $u=(i,j)$5 would force an unphysical $u=(i,j)$6. In qubits, ENM is therefore a property of structured mixtures, not of a lone dephasing branch (Jagadish et al., 12 Jan 2025).

Higher-dimensional generalized Pauli channels based on mutually unbiased bases provide an intermediate setting between Pauli and full Weyl theory. There the dynamics is generated by dephasing maps

$u=(i,j)$7

with $u=(i,j)$8. Convex mixtures of these Markovian dephasing semigroups remain CPTP while producing time-local generators with eternally negative rates, and up to $u=(i,j)$9 always-negative mode rates out of $v=(k,l)$0 total can occur when multiplicities are counted (Siudzińska et al., 2020). This generalized-Pauli result strongly parallels the convexity mechanisms later established directly for Weyl maps.

6. Singular thresholds, quasi-eternal tails, and common misconceptions

Not every Weyl-dephasing model with a long non-Markovian regime is eternally non-Markovian in the strict sense. A solvable counterpoint is the special class of generalized Weyl channels with only diagonal Weyl Kraus operators,

$v=(k,l)$1

where

$v=(k,l)$2

Its time-local master equation has equal rates

$v=(k,l)$3

and the sign structure is explicit: the map is Markovian for $v=(k,l)$4, singular at $v=(k,l)$5, and non-Markovian for $v=(k,l)$6. The singularity is the zero of

$v=(k,l)$7

hence the point where the intermediate map becomes noninvertible and the Choi eigenvalues of the propagator cross over. Physically, it corresponds to complete dephasing into the computational basis rather than to a breakdown of the density-operator description (Xu et al., 13 Mar 2025).

This threshold structure is best described as a persistent non-Markovian tail, not as strict eternal non-Markovianity. The same distinction appears in earlier qubit Pauli literature under the label quasi-eternal non-Markovianity: a decay rate may become negative after a singular finite time and remain negative thereafter, yet the dynamics is initially CP-divisible (Utagi et al., 2020). The term “eternal” is therefore reserved for the stronger condition of negativity for all $v=(k,l)$8, not merely for all late times.

Diagnostic methods reflect this difference. In the Weyl-diagonal solvable model, CP-divisibility is decided by the Choi matrix of the intermediate map, while quantitative diagnostics include the Hall-Cresser-Li-Andersson measure and the Breuer-Laine-Piilo trace-distance measure. The former is sensitive to the generator singularity and requires normalization; the latter detects the post-threshold revival of distinguishability but is insensitive to the singular point itself (Xu et al., 13 Mar 2025). This suggests that “eternal non-Markovianity” is fundamentally a statement about canonical rates and intermediate-map complete positivity, not merely about long-lived recoherence.

Taken together, the modern picture is sharply stratified. In qubits, strict ENM is a Pauli random-unitary phenomenon generated by mixing. In generalized-Pauli/MUB dynamics, convex mixtures already permit macroscopically many permanently negative modes. In full finite-dimensional Weyl theory, a single dephasing map can itself be irreducibly ENM, and convexity can either erase or create eternal memory according to the subgroup geometry of $v=(k,l)$9 (Xu et al., 22 May 2026).