---
title: Weyl Dephasing Maps in Quantum Channels
url: https://www.emergentmind.com/topics/weyl-dephasing-maps
type: topic
---

# Weyl Dephasing Maps in Quantum Channels

Searching arXiv for recent papers on Weyl dephasing maps and related Weyl channels.
Weyl dephasing maps are finite-dimensional quantum channels built from discrete Weyl operators and characterized by a basis-selective action on Weyl components of a density operator. In the multipartite setting, they arise as a direct generalization of Pauli-diagonal channels to tensor-product systems of qudits, while in the one-parameter dynamical setting they appear as random-unitary evolutions supported on cyclic subgroups of the discrete phase space \(\mathbb Z_d\times \mathbb Z_d\). Their defining feature is that they preserve a distinguished sector of Weyl coordinates, possibly with roots-of-unity phase factors, while suppressing or averaging over complementary sectors; in this sense they form a generalized dephasing or phase-erasing family in Weyl coordinates [2310.10947][2605.23852].

## 1. Weyl-operator framework

For a single \(d\)-dimensional system, the Weyl operators are
\[
U(m,n)=\sum_{k=0}^{d-1}\omega^{mk}\,|k\rangle\langle k+n|,
\qquad
\omega=e^{2\pi i/d},
\]
with all arithmetic modulo \(d\). For an \(N\)-particle system of \(N\) qudits, the tensor-product Weyl operators are
\[
U(\vec m,\vec n)=\bigotimes_{\alpha=1}^N U(m_\alpha,n_\alpha),
\]
equivalently,
\[
U(\vec m,\vec n)=\sum_{\vec k}\omega^{\vec m\cdot \vec k}\,|\vec k\rangle\langle \vec k+\vec n|.
\]
The same construction extends to different local dimensions \(d_\alpha\), with \(\omega_\alpha=e^{2\pi i/d_\alpha}\) [2310.10947].

These operators form an orthogonal unitary basis of Hilbert–Schmidt space. Their algebra is governed by
\[
U(m,n)^\dagger U(m',n')=d\,\delta_{mm'}\delta_{nn'},
\]
\[
U(m,n)U(m',n')=\omega^{m'n}U(m+m',n+n'),
\]
\[
U(m,n)U(m',n')=\omega^{m'n-mn'}U(m',n')U(m,n),
\]
and
\[
U(m,n)^\dagger=\omega^{mn}U(-m,-n).
\]
In the single-qudit dynamical notation, one often writes \(U_{kl}\) for the Weyl basis elements indexed by \((k,l)\in\mathbb Z_d\times\mathbb Z_d\); the corresponding projective commutation law is
\[
U_{kl}U_{rs}=\omega^{lr-ks}U_{rs}U_{kl}.
\]
This indexing makes explicit that Weyl dephasing maps are organized by the discrete phase space \(\mathbb Z_d\times\mathbb Z_d\) [2605.23852].

## 2. Diagonal Weyl maps and complete positivity

Any density matrix on \((\mathbb C^d)^{\otimes N}\) admits the expansion
\[
\rho=\frac{1}{d^N}\sum_{\vec m,\vec n}\alpha(\vec m,\vec n)U(\vec m,\vec n),
\]
with hermiticity condition
\[
\alpha(\vec m,\vec n)=\omega^{\vec m\cdot\vec n}\alpha^*(-\vec m,-\vec n).
\]
A Weyl map is diagonal in this basis:
\[
\rho\rightarrow \rho'=[\rho]=\frac{1}{d^N}\sum_{\vec m,\vec n}\tau(\vec m,\vec n)\,\alpha(\vec m,\vec n)\,U(\vec m,\vec n).
\]
The coefficients \(\tau(\vec m,\vec n)\) are therefore the eigenvalues of the superoperator in the Weyl basis. Trace preservation and hermiticity preservation require
\[
\tau(-\vec m,-\vec n)=\tau(\vec m,\vec n)^*,
\qquad
\tau(\vec 0,\vec 0)=1
\]
[2310.10947].

Complete positivity is encoded by the Choi–Jamiołkowski matrix
\[
D=\frac1{d^N}\sum_{\vec m,\vec n}\tau(\vec m,\vec n)\, U(\vec m,\vec n)\otimes U(\vec m,\vec n)^*,
\]
which must satisfy \(D\succeq 0\). Because the operators \(U(\vec m,\vec n)\otimes U(\vec m,\vec n)^*\) commute, \(D\) is diagonalized by a Fourier-transform basis, yielding Choi eigenvalues
\[
\lambda(\vec r,\vec s) = d^{-N}\sum_{\vec m,\vec n}\tau(\vec m,\vec n)\,\omega^{\vec m\cdot\vec r-\vec n\cdot\vec s}\ge 0,
\]
together with the inverse relation
\[
\tau(\vec m,\vec n)=d^{-N}\sum_{\vec r,\vec s}\lambda(\vec r,\vec s)\, \omega^{-\vec m\cdot\vec r+\vec n\cdot\vec s},
\]
and normalization
\[
\sum_{\vec r,\vec s}\lambda(\vec r,\vec s)=d^N.
\]
Thus valid Weyl channels are exactly those whose Fourier-transformed coefficients are nonnegative and normalized. The same condition may be written as a linear transform involving quantum Fourier matrices,
\[
\tau(\vec m,\vec n) = \sum_{\vec r,\vec s} \bigotimes_{\alpha} \bigl[F_\alpha\otimes F_\alpha^*\bigr](\vec m,\vec n;\vec r,\vec s)\, \lambda(\vec r,\vec s).
\]
This Fourier-positivity criterion is the basic structural constraint behind Weyl dephasing maps [2310.10947].

## 3. Weyl erasing channels as generalized dephasing maps

The dephasing interpretation becomes explicit in the subclass of Weyl erasing channels, defined by
\[
|\tau(\vec m,\vec n)|=0\ \text{or}\ 1.
\]
These channels “completely erase, preserve or introduce specific phases to the projections of the density matrix onto the Weyl basis.” They are the direct analogue of component-erasing maps, but in the Weyl basis rather than the Pauli basis [2310.10947].

| Weyl coefficient | Action on the corresponding Weyl component |
|---|---|
| \(\tau(\vec m,\vec n)=1\) | preserved |
| \(\tau(\vec m,\vec n)=0\) | erased |
| \(|\tau(\vec m,\vec n)|=1\) with phase | preserved up to phase |

Operationally, these maps selectively suppress some Weyl-basis coherences while leaving a subgroup of components untouched up to phase. The paper does not use “phase-damping channel” as a formal definition, but the operational meaning is exactly a generalized dephasing or phase-erasing channel in Weyl coordinates [2310.10947].

The nonzero sector has a precise algebraic description. The set of indices with \(|\tau(\vec m,\vec n)|=1\) forms a subgroup \(H\subseteq G\), where
\[
G=\mathbb Z_d^{\oplus N}\oplus \mathbb Z_d^{\oplus N},
\]
or, for different local dimensions,
\[
G=\left(\bigoplus_{\alpha=1}^N\mathbb Z_{d_\alpha}\right)\oplus \left(\bigoplus_{\alpha=1}^N\mathbb Z_{d_\alpha}\right).
\]
On that subgroup, \(\tau\) acts as a homomorphism into roots of unity,
\[
\tau(\vec m,\vec n)=\omega^{\phi(\vec m,\vec n)}.
\]
The coefficients outside \(H\) are set to zero. In Kraus form, the erasing channel admits a canonical interpretation in which the Kraus operators are a subset of Weyl matrices \(U(\vec r,\vec s)\), specifically those satisfying
\[
(\vec r-\vec r_0,\vec s-\vec s_0)\in H^\perp,
\]
with probabilities proportional to
\[
|H^\perp|/|G|.
\]
This identifies Weyl dephasing maps with subgroup-controlled random-unitary channels whose preserved sector is phase-twisted by a homomorphism and whose erased sector is annihilated in Weyl coordinates [2310.10947].

## 4. Convex geometry and subgroup classification

Because the Choi eigenvalues \(\lambda(\vec r,\vec s)\) are nonnegative and normalized, the set of Weyl channels forms a simplex of dimension \(d^{2N}-1\). Its extreme points occur when exactly one Choi eigenvalue is nonzero,
\[
\lambda(\vec r,\vec s)=d^N\,\delta_{\vec r,\vec r_0}\delta_{\vec s,\vec s_0},
\]
which gives
\[
\tau_{\vec{r}_0,\vec{s}_0}(\vec{m},\vec{n}) = \omega^{-\vec{m} \cdot \vec{r}_0 + \vec{n} \cdot \vec{s}_0}.
\]
These extreme channels are unitary conjugations because
\[
U(\vec s_0,\vec r_0)\,U(\vec m,\vec n)\,U(\vec s_0,\vec r_0)^\dagger
=
\omega^{-\vec r_0\cdot\vec m+\vec s_0\cdot\vec n}U(\vec m,\vec n).
\]
Hence every Weyl channel is a convex combination of unitary Weyl conjugations, i.e. a random-unitary channel. The extremality criterion can be stated equivalently as
\[
|\tau(\vec m,\vec n)|=1
\quad\text{for all }\vec m,\vec n
\]
[2310.10947].

The group-theoretic classification of the dephasing sector depends on the arithmetic of the dimension. For prime \(d\), the index group \(G\) is a vector space over \(\mathbb Z_d\), so subgroup analysis reduces to linear algebra. For prime powers and composite dimensions, the construction proceeds through decomposition into cyclic \(p\)-power groups. The classification is organized by
\[
G=\bigoplus_p G_p,
\qquad
G_p=\bigoplus_{\alpha=1}^r \mathbb Z_{p^{M_\alpha}},
\]
together with partitions \(L\) satisfying
\[
L_\alpha=0\ (\alpha>r),\qquad
L_\alpha\le M_\alpha,\qquad
L_\alpha\ge L_{\alpha+1}.
\]
Automorphisms are represented by constrained matrices \(T=(t_{\alpha\beta})\), and this finite-abelian-group machinery yields an algorithmic classification of Weyl erasing channels [2310.10947].

For single-qudit dynamical maps, the subgroup structure of \(\mathbb Z_d\times\mathbb Z_d\) is classified completely via the Hermite normal form. Every subgroup \(G\) has a unique HNF representation generated by
\[
M=\begin{pmatrix} m & w\\ 0 & n \end{pmatrix},
\]
with
\[
m\mid d,\qquad n\mid d,\qquad 0\le w<n,\qquad n\mid \frac{wd}{m},
\]
so that
\[
G=\langle(m,w),(0,n)\rangle =\{(mu,wu+nv)\mid u\in\mathbb{Z}_{d/m},\,v\in\mathbb{Z}_{d/n}\},
\qquad
|G|=\frac{d^2}{mn}.
\]
The classification distinguishes cyclic, split rank-2, and non-split rank-2 subgroups. This subgroup geometry is the algebraic skeleton for the dynamical theory of Weyl dephasing maps [2605.23852].

## 5. Dynamical Weyl dephasing maps and memory effects

A Weyl dynamical map on \(\mathbb C^d\) is
\[
\mathcal{E}(t)(\rho)=\sum_{(i,j)\in \mathbb{Z}_d\times\mathbb{Z}_d} p_{ij}(t)\,U_{ij}\rho U_{ij}^\dagger,
\]
where \(\{p_{ij}(t)\}\) is a probability distribution. Each Weyl operator is an eigenoperator,
\[
\mathcal{E}(t)(U_{kl})=\lambda_{kl}(t)U_{kl},
\]
with eigenvalues
\[
\lambda_{kl}(t)=\sum_{(i,j)\in \mathbb{Z}_d\times\mathbb{Z}_d}\omega^{jk-il}\,p_{ij}(t).
\]
The dynamics is therefore completely encoded by the time-dependent weights \(p_{ij}(t)\), or equivalently by the spectral data \(\lambda_{kl}(t)\) [2605.23852].

The time-local description uses
\[
\frac{d}{dt}\rho(t)=\mathcal{L}(t)\rho(t),
\]
with GKLS form
\[
\mathcal{L}(t)(\rho)= -i[H(t),\rho] +\sum_\alpha \gamma_\alpha(t)\left(L_\alpha\rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).
\]
For Weyl maps one sets \(L_\alpha=U_{ij}\) and typically \(H(t)=0\), obtaining
\[
\mathcal{L}(t)(\rho) = \sum_{(i,j)\neq(0,0)}\gamma_{ij}(t)\big(U_{ij}\rho U_{ij}^\dagger-\rho\big).
\]
The RHP criterion yields the standard dynamical trichotomy: CP-divisible or Markovian evolution when all \(\gamma_\alpha(t)\ge 0\), semigroup evolution when the rates are constant and nonnegative, non-Markovianity when some rate becomes negative, and eternal non-Markovianity (ENM) when at least one rate satisfies
\[
\gamma_\alpha(0)=0,\qquad \gamma_\alpha(t)<0\ \forall t>0
\]
[2605.23852].

The isotropic/anisotropic distinction is central. An anisotropic Weyl map has the form
\[
\mathcal{E}_{\mathrm{aniso}}(t)(\rho) = (1-p(t))\rho+p(t)\sum_{u\in G\setminus\{0\}} w_u\,U_u\rho U_u^\dagger,
\]
with fixed nonuniform weights \(w_u\), whereas an isotropic Weyl map is
\[
\mathcal{E}_{\mathrm{iso}}(t)(\rho) = (1-p(t))\rho+\frac{p(t)}{|G|-1}\sum_{u\in G\setminus\{0\}}U_u\rho U_u^\dagger.
\]
A key structural result is that anisotropic Weyl maps with nonuniform weights and at least two distinct nontrivial eigenvalues cannot form a semigroup. By contrast, isotropic maps have only two spectral values,
\[
\lambda_v(t)= \begin{cases} 1, & v\in G^\perp,\\[4pt] 1-\dfrac{|G|}{|G|-1}p(t), & v\notin G^\perp, \end{cases}
\]
and they generate a Markovian semigroup precisely when
\[
p(t)=\frac{|G|-1}{|G|}(1-e^{-ct}),\qquad c>0.
\]
Equivalently,
\[
\Lambda(t)=1-\frac{|G|}{|G|-1}p(t)=e^{-ct},
\qquad
\gamma_\alpha(t)=\frac{c}{|G|}\quad (\alpha\in G\setminus\{0\}),
\]
with zero rates otherwise [2605.23852].

For \(|G|=2\), the isotropic map reduces to the Weyl dephasing map
\[
\mathcal{E}_u^p(t)(\rho)=(1-p(t))\rho+p(t)\,U_u\rho U_u^\dagger.
\]
Let
\[
s=\gcd(i,j,d),\qquad \ell=\frac{d}{s},
\]
for \(u=(i,j)\), so that the cyclic subgroup generated by \(u\) has order \(\ell\). If
\[
p(t)=r(1-e^{-ct}),\qquad r\in(0,1],
\]
then the map is eternally non-Markovian if either \(\ell\ge 3\) is odd, or \(\ell\ge 2\) is even and \(0<r\le \tfrac12\). In higher-dimensional Weyl systems, therefore, a single dephasing map can already be irreducibly eternally non-Markovian [2605.23852].

## 6. Convex mixtures, qutrit structure, and relation to Pauli maps

Convexity plays a nontrivial role in Weyl dephasing dynamics. The isotropic Weyl map can be written as an equal-weight convex mixture of dephasing maps,
\[
\mathcal{E}_{\mathrm{iso}}(t) = \frac{1}{|G|-1}\sum_{u\in G\setminus\{0\}}\mathcal{E}_u^p(t).
\]
A striking consequence is that a convex combination of eternally non-Markovian Weyl dephasing maps can nevertheless generate a Markovian semigroup. More precisely, when the constituent dephasing maps are ENM with odd \(\ell\), the equal-weight isotropic mixture can still yield a semigroup provided
\[
p(t)=\frac{|G|-1}{|G|}(1-e^{-ct}).
\]
Thus non-Markovianity is not additive under mixing [2605.23852].

The converse direction is also established for mixtures of \(N\) distinct isotropic Weyl semigroups,
\[
\tilde{\mathcal{E}}(t)=\sum_{k=1}^N x_k\,\mathcal{E}_{\mathrm{iso}}^{(k)}(t),
\qquad
x_k>0,\ \sum_k x_k=1.
\]
If all semigroups have the same subgroup order
\[
|G_k|=K,
\qquad
p(t)=\frac{K-1}{K}(1-e^{-ct}),
\]
then the mixture is eternally non-Markovian whenever
\[
2\le N<\min\left\{\frac{d^2-1}{K-1},\ \mathcal{N}(K)\right\},
\]
where
\[
\mathcal{N}(K)=\sum_{(m,n)\in\mathcal{S}_K}\gcd\!\left(n,\frac{d}{m}\right).
\]
The underlying mechanism is geometric: if the union of subgroup supports leaves at least one nontrivial phase-space point uncovered, at least one decay rate remains negative for all \(t>0\) [2605.23852].

The qutrit case \(d=3\) makes these statements explicit. Here \(\mathbb Z_3\times\mathbb Z_3\) has \(9\) elements, there are \(\mathcal N(3)=4\) subgroups of order \(3\), and distinct order-\(3\) subgroups intersect only at \((0,0)\). For \(N=2\) or \(N=3\), the union of chosen subgroups does not cover the entire phase space, so the mixtures are eternally non-Markovian. For \(N=4\), the union can cover the whole phase space: uniform mixing \(x_k=\tfrac14\) yields a Markovian map, whereas nonuniform mixing can produce non-Markovian or even eternally non-Markovian dynamics [2605.23852].

Weyl dephasing maps reduce to familiar Pauli structures in the binary case. For \(d=2\), Weyl operators reduce to Pauli operators, and the extreme Weyl channels correspond to the vertices of the qubit Pauli-channel tetrahedron [2310.10947]. The Weyl framework is nevertheless strictly broader: generalized Pauli maps can be embedded into it, and for \(d=3\) they may be rewritten using Weyl operators with grouped weights over commuting subsets [2605.23852]. This places Weyl dephasing maps as the higher-dimensional extension of Pauli dephasing channels, with additional subgroup geometry and convex phenomena that do not appear in the qubit setting.

Source: https://www.emergentmind.com/topics/weyl-dephasing-maps