Joint Qutrit-Boson Wigner Function

• Joint qutrit-boson Wigner function is a hybrid quasiprobability representation that integrates discrete three-level and continuous bosonic modes while preserving entanglement correlations.
• It utilizes both continuous-discrete Weyl-Wigner-Moyal and finite-Heisenberg-group frameworks to diagnose hybrid cat states in ℤ3-symmetric qutrit-boson models.
• The method enables clear phase-space analysis to distinguish genuine qutrit-boson cat states from classical mixtures, advancing diagnostics in protected qutrit platforms.

A joint qutrit-boson Wigner function is a hybrid quasiprobability representation for systems that combine a three-level discrete subsystem with one or more continuous bosonic degrees of freedom. Its defining purpose is to retain the phase-space structure of the bosonic mode(s) while simultaneously encoding the discrete qutrit sector and, crucially, qutrit-boson correlations that are lost under partial tracing. In the literature represented here, this object appears in two closely related forms: a general continuous-discrete Weyl-Wigner-Moyal formalism on $L^2(\mathbb R^3)\otimes \mathcal H^{(s+1)}$, whose $s+1=3$ specialization yields a qutrit construction (Przanowski et al., 2018), and a finite-Heisenberg-group-based phase-space representation tailored to $\mathbb Z_3$-symmetric qutrit-boson Rabi models, where it serves as a direct diagnostic of hybrid cat states (Lotkov et al., 10 Sep 2025).

1. Conceptual role and motivation

The central motivation for a joint qutrit-boson Wigner function is that an ordinary bosonic Wigner function is generally insufficient for hybrid discrete-continuous states. In the $\mathbb Z_3$-symmetric Rabi models studied in (Lotkov et al., 10 Sep 2025), taking the partial trace over the finite-dimensional subsystem and then plotting a bosonic Wigner function “discards all information about the qudit-boson entanglement.” As a result, a reduced bosonic phase-space plot cannot determine which bosonic coherent-state component is correlated with which qutrit state, whether the state is a genuine hybrid qutrit-boson cat state, or whether the state is coherent rather than a classical mixture.

This issue becomes particularly acute in the parameter regime where the hybrid states of interest are explicitly cat-like. For the one-mode and two-mode $\mathbb Z_3$ Rabi models, the relevant regime is deep-strong coupling, $\lambda \gtrsim \Omega$, together with a weak magnetic field, $B \ll \Omega$. In that regime, after a canonical transformation and perturbative treatment in $B$, the three lowest eigenstates become $\mathbb Z_3$ qutrit-boson cat states with coherent amplitude $\alpha=\lambda/\Omega$ (Lotkov et al., 10 Sep 2025). The joint Wigner function is introduced precisely to make those states visible in a phase-space language without erasing their hybrid structure.

The same motivation has a broader significance. Whenever a three-level protected manifold is embedded in an oscillator-like platform, phase-space diagnostics become natural tools for identifying coherence, interference, and leakage. In this sense, the joint qutrit-boson Wigner function belongs to the wider family of hybrid quasiprobability methods that attempt to represent finite-dimensional and continuous-variable quantum data within a single symbol calculus.

2. Hybrid phase space and formal constructions

One systematic framework is the continuous-discrete Weyl-Wigner-Moyal formalism developed on the Hilbert space

$s+1=3$0

with continuous canonical variables $s+1=3$1, $s+1=3$2, and a discrete $s+1=3$3-dimensional internal space with basis $s+1=3$4 (Przanowski et al., 2018). The associated hybrid phase space is

$s+1=3$5

For a qutrit one sets $s+1=3$6, so $s+1=3$7, and the discrete sector becomes the $s+1=3$8 grid

$s+1=3$9

The construction combines continuous Weyl displacement operators

$\mathbb Z_3$0

with discrete Schwinger operators

$\mathbb Z_3$1

The resulting hybrid Stratonovich-Weyl quantizer is

$\mathbb Z_3$2

with kernels $\mathbb Z_3$3 and $\mathbb Z_3$4 controlling operator ordering. The paper imposes

$\mathbb Z_3$5

and, when $\mathbb Z_3$6, the symbol map takes the simple form

$\mathbb Z_3$7

A second construction, designed specifically for $\mathbb Z_3$8 qutrit-boson systems, uses the finite Heisenberg group generated by the qutrit shift and clock operators $\mathbb Z_3$9 and $\mathbb Z_3$0, together with the usual bosonic displacement-parity kernel (Lotkov et al., 10 Sep 2025). In that approach, the qutrit phase space is again discrete but is labeled by $\mathbb Z_3$1, rather than $\mathbb Z_3$2.

Framework	Phase-space coordinates	Characteristic feature
Continuous-discrete WWM (Przanowski et al., 2018)	$\mathbb Z_3$3	Hybrid Stratonovich-Weyl quantizer $\mathbb Z_3$4
$\mathbb Z_3$5 qutrit-boson kernel (Lotkov et al., 10 Sep 2025)	$\mathbb Z_3$6 or $\mathbb Z_3$7	Finite-Heisenberg qutrit displacement plus bosonic parity kernel

These are not identical constructions, but both implement a one-to-one operator-symbol correspondence for hybrid discrete-continuous systems and both yield a qutrit specialization when the internal dimension is three.

3. Core definition, marginals, and dynamics

In the continuous-discrete formalism, the Wigner function of a density operator $\mathbb Z_3$8 is defined by

$\mathbb Z_3$9

For the qutrit case $\mathbb Z_3$0, the preferred odd-dimensional kernel choice is

$\mathbb Z_3$1

which realizes symmetric ordering in the discrete sector (Przanowski et al., 2018).

Under this specialization, the qutrit-boson Wigner function becomes

$\mathbb Z_3$2

with $\mathbb Z_3$3. This formula makes explicit that the hybrid symbol depends simultaneously on continuous phase-space data and matrix elements in the qutrit sector.

The formalism preserves the standard structural properties expected of a Wigner representation. The Wigner function is real,

$\mathbb Z_3$4

and normalized,

$\mathbb Z_3$5

Its marginals reproduce reduced continuous and discrete probabilities: $\mathbb Z_3$6

$\mathbb Z_3$7

$\mathbb Z_3$8

$\mathbb Z_3$9

Expectation values are represented by phase-space integration against the corresponding symbol, and the evolution equation takes the hybrid Liouville-von Neumann-Wigner form

$\lambda \gtrsim \Omega$0

when the orthogonality conditions on the kernels hold (Przanowski et al., 2018).

4. Finite-Heisenberg qutrit kernel and joint displacement-parity form

For $\lambda \gtrsim \Omega$1-symmetric qutrit-boson models, (Lotkov et al., 10 Sep 2025) adopts a different but closely related representation. The bosonic sector uses the standard one-mode Wigner kernel

$\lambda \gtrsim \Omega$2

and, for two bosonic modes,

$\lambda \gtrsim \Omega$3

The qutrit sector is built from the finite Heisenberg group. The displacement operators are

$\lambda \gtrsim \Omega$4

and the qutrit Wigner function is

$\lambda \gtrsim \Omega$5

with discrete coordinates $\lambda \gtrsim \Omega$6 and qutrit parity

$\lambda \gtrsim \Omega$7

Its marginals are

$\lambda \gtrsim \Omega$8

$\lambda \gtrsim \Omega$9

For a qutrit plus two bosonic modes, the joint displacement operator is

$B \ll \Omega$0

the joint parity operator is

$B \ll \Omega$1

and the joint Wigner function is

$B \ll \Omega$2

The same source states that the natural one-boson analog is

$B \ll \Omega$3

although the paper does not separately define the one-mode joint QB Wigner function in full formal detail (Lotkov et al., 10 Sep 2025).

5. Closed-form $B \ll \Omega$4 cat-state expressions

The most explicit closed-form joint qutrit-boson Wigner formulas currently available in this literature arise in the $B \ll \Omega$5-symmetric Rabi models. For the one-mode model, the approximate low-energy states are

$B \ll \Omega$6

while for the two-mode model they are

$B \ll \Omega$7

with $B \ll \Omega$8 and $B \ll \Omega$9 (Lotkov et al., 10 Sep 2025).

For the two-mode cat state, the joint qutrit-boson Wigner function is given in closed form as

$B$0

This formula is the principal analytic expression for the topic. At fixed $B$1, it contains one Gaussian “coherent-state” contribution and one Gaussian-envelope interference contribution. The qutrit coordinate $B$2 selects which bosonic coherent component appears as the direct Gaussian, $B$3 shifts the phase of the interference fringes, and $B$4 labels the $B$5 symmetry sector (Lotkov et al., 10 Sep 2025).

A striking consequence is that the six structures present in the purely bosonic two-mode $B$6 cat Wigner function are not superposed in a single bosonic panel. Instead, they are distributed across the discrete qutrit phase space. The paper states that, at fixed $B$7, the joint Wigner function contains one corner blob and one midpoint/interference blob; changing $B$8 cyclically permutes the qutrit-phase-space rows because $B$9 and $\mathbb Z_3$0 enter only as $\mathbb Z_3$1 in the phase.

6. Diagnostics, limitations, and related directions

The joint qutrit-boson Wigner function is primarily a diagnostic tool. In the numerical study of the two-mode $\mathbb Z_3$2 Rabi model, (Lotkov et al., 10 Sep 2025) computes the hybrid Wigner function with both bosonic Hilbert spaces truncated at $\mathbb Z_3$3 and shows a progression with increasing $\mathbb Z_3$4: at $\mathbb Z_3$5 the hybrid-cat structure is weak or overlapping, at $\mathbb Z_3$6 the blobs are partially separated, and at $\mathbb Z_3$7 the $\mathbb Z_3$8-cat geometry is clearly resolved. The same analysis distinguishes three cases that a reduced bosonic Wigner function would not separate as cleanly: a genuine qutrit-two-boson cat state, a classical incoherent mixture, and a two-boson cat state unentangled with a spectator qutrit.

Several limitations and points of interpretation are equally important. First, the paper gives the exact printed closed form only for the two-mode Q2B cat state; it does not present the one-mode joint QB case as a standalone numbered equation (Lotkov et al., 10 Sep 2025). Second, the literature contains more than one mathematically natural hybrid Wigner construction. The continuous-discrete Stratonovich-Weyl approach (Przanowski et al., 2018) and the finite-Heisenberg-group construction (Lotkov et al., 10 Sep 2025) are compatible in spirit but are not the same representation. Third, a broader generalized-Wigner viewpoint is suggested by the title and description of “Properties of the Wigner distribution for $\mathbb Z_3$9 arbitrary operators” (Schwonnek et al., 2018): a reconstructed framework in the supplied material treats Wigner distributions for arbitrary tuples of Hermitian operators and suggests a possible route to hybrid qutrit-boson quasi-distributions. However, the supplied text explicitly states that the original scientific text was unavailable, so any such connection remains an inference rather than a verified claim from that paper.

The subject also connects to current experimental work on protected qutrit platforms. In a three-photon Kerr parametric oscillator, direct Wigner function measurements reveal three-component cat-like states, while breathing-like dynamics in phase space arise from temporal interference between the qutrit and excited states; the frequency of this interference corresponds to the energy gap between the qutrit and excited manifolds and is presented as an experimental hallmark of qutrit space protection (Kwon et al., 25 Jan 2026). That result does not introduce a joint qutrit-boson Wigner function in the same formal sense as (Przanowski et al., 2018) or (Lotkov et al., 10 Sep 2025), but it shows that direct phase-space tomography already plays a central role in identifying three-component cat structure and qutrit protection in oscillator-based hardware.

Taken together, these developments place the joint qutrit-boson Wigner function at the intersection of hybrid Weyl quantization, finite-dimensional discrete phase-space methods, and cat-state physics in $\alpha=\lambda/\Omega$0-symmetric and protected-qutrit systems. Its main technical value is that it retains the discrete-continuous entanglement structure that reduced bosonic plots erase, while preserving the familiar Wigner-function logic of marginals, interference fringes, and phase-space localization.