Wigner Phase-Space Formulation of 2-RDM

• The paper demonstrates a rigorous Wigner phase-space technique for representing and evolving two-body reduced density matrices, enhancing computational tractability.
• It applies spectral discretization and Chebyshev methods to efficiently handle high-dimensional phase-space integrals and capture dynamic correlations beyond mean-field approximations.
• The study reveals significant implications such as quantifiable entropy growth in two-body scattering, preservation of entanglement, and potential extensions to time-dependent density functional theory.

The Wigner phase-space formulation of the two-body reduced density matrix (2-RDM) provides a rigorous and computationally tractable framework for representing and evolving two-particle quantum correlations in continuous and discrete many-body systems. By encoding the 2-RDM as a quasiprobability function in phase space, both analytic and numerically efficient routes are opened for the paper of two-body phenomena that go beyond one-body mean-field descriptions, including entanglement, dynamic correlation, collision integral effects, and irreversibility.

1. Foundations: The Wigner Transform and the 2-RDM

Let $\rho$ be a density operator on the $N$-particle Hilbert space $\mathcal H=(L^2(\mathbb R^d))^{\otimes N}$, with canonical coordinates $\mathbf x=(x_1,...,x_N)$ and momenta $\mathbf p=(p_1,...,p_N)$. The $N$-body Wigner function $W_{\rm tot}(\mathbf x;\mathbf p)$ is defined by the Weyl–Wigner integral,

$$W_{\rm tot}(x_1,...,x_N; p_1,...,p_N) = \frac{1}{(2\pi\hbar)^{dN}}\int_{\mathbb R^{dN}} e^{-\frac{i}{\hbar}\mathbf p \cdot \mathbf y}\left\langle \mathbf x + \frac{\mathbf y}{2} \Big| \rho \Big| \mathbf x - \frac{\mathbf y}{2}\right\rangle\, d^N y,$$

which recasts the quantum state as a function on $\mathbb R^{2dN}$.

The two-body reduced density matrix (2-RDM) is obtained by tracing out all but two particles, labeled $i,j$: $$\rho^{(2)} = \operatorname{Tr}_{\rm rest}(\rho), \qquad \rho^{(2)}(x_i, x_j; x'_i, x'_j) = \int \rho(x_i, x_j, \mathbf{x}_{\rm rest}; x'_i, x'_j, \mathbf{x}_{\rm rest})\, d^{d(N-2)}x_{\rm rest}.$$ Its Wigner function, $W_2(z_i,z_j)$ with $z_k = (q_k,p_k)$, becomes

$$W_2(q_i,p_i; q_j,p_j) = \int \prod_{k\neq i,j} dq_k\,dp_k\, W_{\rm tot}(q_1,...,q_N; p_1,...,p_N),$$

i.e., the marginalization of $W_{\rm tot}$ over all degrees of freedom except the specified pair (Gosson et al., 2021).

2. Rigorous Phase-Space Marginalization and Weyl–Wigner–Moyal Structure

The operator-level partial trace is, under suitable regularity conditions, rigorously equivalent to phase-space marginalization of the Wigner function. This equivalence follows from the Weyl correspondence:

• For any bounded operator $Q$ on the two-particle subspace,

$$\operatorname{Tr}(\rho\, Q \otimes I_{\rm rest}) = \operatorname{Tr}(\rho^{(2)} Q),$$

• With Weyl symbol $q_{ij}(z_i,z_j)$ independent of traced-out variables, the trace relation is

$$\operatorname{Tr}(\rho\,\operatorname{Op}_W(q)) = (2\pi\hbar)^{dN} \int W_{\rm tot}(\mathbf z)\, q(\mathbf z)\, d^{2dN} \mathbf z.$$

Comparing with the analogous formula for $\rho^{(2)}$ yields the marginalization result.

Wigner functions are central to the Weyl–Wigner–Moyal formalism:

• They obey star-product composition,

$$a \star b = a(z)\exp\left[\frac{i\hbar}{2} \left(\overleftarrow{\partial_x}\cdot\overrightarrow{\partial_p} - \overleftarrow{\partial_p}\cdot\overrightarrow{\partial_x}\right)\right] b(z).$$

• Quantum commutators become Moyal brackets,

$$\frac{1}{i\hbar}[A,B] \longleftrightarrow \{a,b\}_{\rm M} = \frac{1}{i\hbar}(a\star b - b\star a),$$

so that quantum evolution in this representation closely parallels classical phase-space dynamics (Gosson et al., 2021).

3. Gaussian States and Explicit 2-Body Wigner Functions

For Gaussian states, the Wigner function is itself Gaussian: $$W_{\rm tot}(\mathbf z) =\frac{1}{(2\pi)^{dN}\sqrt{\det\Sigma}}\exp\left[-\frac{1}{2}\mathbf z^T\Sigma^{-1}\mathbf z\right],$$ where $\Sigma$ is the $2dN\times 2dN$ covariance matrix. Marginalization yields another Gaussian: $$W_2(z_i,z_j) = \frac{1}{(2\pi)^{2d}\sqrt{\det\Sigma_{[ij]}}} \exp\left[-\frac12 (z_i,z_j)^T\Sigma_{[ij]}^{-1} (z_i,z_j)\right],$$ with $\Sigma_{[ij]}$ the $4d\times4d$ block corresponding to the distinguished pair. For such states, phase-space marginalization gives an explicit, closed-form description of all two-body correlations.

This highlights a particular strength of the Wigner formalism: for all Gaussian states (including thermal and squeezed states), the 2-RDM Wigner function is efficiently computed and completely characterized by appropriate covariance blocks (Gosson et al., 2021).

4. Discrete Phase-Space Formulation for Finite Systems

For finite-dimensional (e.g., spin or qubit) systems, the Wigner 2-RDM formalism is constructed using discrete phase-space points and Schwinger operator bases. The mapping kernel $A(\mu,\nu)$ for an $N$-level system,

$$A(\mu,\nu) = \frac{1}{\sqrt{N}} \sum_{\eta,\xi=0}^{N-1} \omega^{\Phi(\eta,\xi;N) - \mu \eta - \nu \xi}\,\hat U^\eta \hat V^\xi,$$

generates an orthonormal operator basis. For two qubits, either an $\mathrm{SU(2)}\otimes\mathrm{SU(2)}$ “product” phase space or a single $\mathrm{SU(4)}$ “ququart” phase space may be employed (Marchiolli et al., 2019).

The discrete Wigner function is given by

$$W(\mu,\nu) = \operatorname{Tr}[\rho A(\mu,\nu)] = \frac{1}{N} + \frac{1}{2} \sum_{i=1}^{N^2-1} g_i\,g_i(\mu,\nu),$$

where $g_i$ are Bloch vector components and $g_i(\mu,\nu)$ are fixed functions of the phase-space coordinates.

Marginals over a subset of phase-space coordinates yield the Wigner function of the reduced density matrix, enabling reconstruction of partial states and diagnostics of nonseparability or entanglement (Marchiolli et al., 2019).

5. Wigner Phase-Space Dynamics of the 2-RDM

Phase-space evolution of the 2-RDM is governed by equations obtained via Wigner transformation of the BBGKY hierarchy, with the highest-body correlations truncated. The time-dependent Wigner 2-RDM $f_{12}(r_1,r_2,p_1,p_2,t)$ satisfies

$$\frac{\partial f_{12}}{\partial t} + \frac{p_1}{m} \cdot \nabla_{r_1}f_{12} + \frac{p_2}{m} \cdot \nabla_{r_2}f_{12} = \Theta_{V_{HXC}[n_1]}[f_{12}] + \Theta_{V_{HXC}[n_2]}[f_{12}] + \Theta_{V_{12}}[f_{12}] + \Theta_{W_{12}}[f_{12}],$$

where:

• $\Theta_{V_{HXC}[n_j]}[f_{12}]$ encodes Hartree–exchange–correlation on particle $j$, as a convolution in momentum,
• $\Theta_{V_{12}}$ is the two-body collision integral,
• $\Theta_{W_{12}}$ is the effect of external potentials (Liang et al., 14 Nov 2025).

The explicit kernels incorporate Fourier transforms of the relevant potentials and action of shifted arguments in the momenta, mimicking both collision and mean-field effects in phase space.

6. Numerical Methods for High-Dimensional 2-RDM Wigner Equations

To address the high-dimensionality inherent in the 2-RDM Wigner equation (in $4d$ phase-space dimensions for $d$ spatial degrees), efficient spectral discretization and parallelism are essential.

Two complementary families of spectral approximations are employed:

• Periodic case: Pseudo-difference operators, with momentum discretization and FFT acceleration, yield difference formulas such as

$$\Theta_{V_{HXC}[n_1]}[f_{12}] = \frac{1}{i\hbar} \sum_{m \in \mathbb Z^d} \widehat V_m[n_1] e^{-\frac{2\pi i}{L_x}m \cdot r_1} [f_{12}(p_1-m\Delta p,...) - f_{12}(p_1+m\Delta p,...)],$$

and similar representations for the two-body collision term. Truncation to $|m|\le N_p$ and selection $N_x=2N_p$ enables $\mathcal O(N_x^{2d}N_p^{2d}\log N_p)$ scaling (Liang et al., 14 Nov 2025).

• Non-periodic case: Chebyshev spectral-element methods are adopted, expanding $f_{12}$ in tensorized Chebyshev polynomials within subdomains of momentum space, reducing the action of the complex nonlocal kernels to manageable sparse tensor contractions.

A fully distributed characteristic–spectral integration scheme (termed CHASM in this context) is used for time-stepping, combining semi-Lagrangian advection with localized spline interpolation, domain decomposition, and minimal cross-domain communication via perfectly-matched boundary conditions.

7. Applications, Numerical Experiments, and Physical Implications

Direct simulation in 1D-2RDM Wigner phase space elucidates two-body corrections beyond mean-field theory:

• In nonlinear Landau damping, local exchange–correlation has negligible effect on the linear damping rate, whereas the inclusion of the two-body collision operator modifies the rate and suppresses fine-scale filamentation in phase space.
• In the quantum two-stream instability, two-body repulsion regularizes saturation patterns and suppresses field energy growth.
• For strongly correlated initial conditions, mean-field or Hartree dynamics cause rapid dispersal and do not affect the natural occupation spectrum $\{n_k\}$. Inclusion of the two-body operator impedes wavepacket coalescence, preserves two-particle correlations, and leads to quantitatively irreversible entropy growth.

Correlation entropy, defined as $s(t)=-\sum_k n_k(t)\ln n_k(t)$ where $\{n_k\}$ are the eigenvalues of the one-body density matrix, remains constant under mean-field, but increases monotonically under two-body scattering. This provides deterministic evidence that 2-RDM Wigner dynamics can generate entropy and correct kinetic rates beyond mean-field approximations (Liang et al., 14 Nov 2025).

A plausible implication is that these Wigner phase-space approaches may be used to extend time-dependent density functional theory (TDDFT) to include fully dynamical two-body observables in both quantum and semiclassical systems, potentially enabling accurate simulation of processes involving dynamic correlation, double excitations, and quantum thermalization.

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Table: Key Wigner 2-RDM Procedures

Procedure	Formula Type	Physical/Computational Effect
Continuous marginalization	$\int W_{\rm tot}(\cdot)d^{2d(N-2)}z_{\rm rest}$	Extracts 2-RDM Wigner function from $N$-body state
Discrete marginal (qubits)	$\sum_{\rm rest} W(\cdot)$	Extracts reduced discrete Wigner function, finite $N$
2-RDM Wigner dynamics	PDEs involving $\Theta_{V_{12}}$	Simulates time evolution including explicit collisions
FFT pseudo-difference	Discrete convolutions (periodic)	Efficient high-dimensional computation
Chebyshev spectral method	Polynomial element expansion	Non-periodic domains, sparse contraction computation

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The Wigner phase-space formulation of the 2-RDM unifies operator-theoretic manipulations, rigorous phase-space representations, and efficient high-dimensional computation, providing a powerful framework for the investigation of many-body quantum dynamics, irreversibility, and multipartite entanglement.