Continuous majorization in quantum phase space for Wigner-positive states and proposals for Wigner-negative states

TL;DR

The paper develops continuous majorization in quantum phase space (Wigner majorization) for bosonic CV systems, extending prior single‑mode results to the general $N$‑mode, Wigner‑positive case and proposing three approaches to extend the ordering to Wigner‑negative states. It proves a simple determinant criterion for Gaussian‑state majorization, shows the pure $N$‑mode Gaussian state with minimal symplectic spectrum majorizes the convex hull of Gaussian states, and establishes a phase‑space analogue of Uhlmann’s majorization theorem for a broad class of channels, including random Gaussian unitaries. It then introduces three preorders (Proposal 1, Proposal 2, and a tautological preorder) to handle Wigner negativity, investigates their relation to the Wigner logarithmic negativity, and analyzes how Gaussian channels and non‑Gaussian channels affect majorization under these preorders. The results provide a structured way to compare CV states in resource theories of non‑Gaussianity and Wigner negativity, with potential applications to quantum information processing and CV quantum thermodynamics.

Abstract

In quantum resource theory, one is often interested in identifying which states serve as the best resources for particular quantum tasks. If a relative comparison between quantum states can be made, this gives rise to a partial order, where states are ordered according to their suitability to act as a resource. In the literature, various different partial orders for a variety of quantum resources have been proposed. In discrete variable systems, vector majorization of Wigner functions in discrete phase space provides a natural partial order between quantum states. In the continuous variable case, a natural counterpart would be continuous majorization of Wigner functions in quantum phase space. Indeed, this concept was recently proposed and explored (mostly restricting to the single-mode case) in Van Herstraeten, Jabbour, Cerf, Quantum 7, 1021 (2023). In this work, we develop the theory of continuous majorization in the general $N$-mode case. In addition, we propose extensions to include states with finite Wigner negativity. For the special case of the convex hull of $N$-mode Gaussian states, we prove a conjecture made by Van Herstraeten, Jabbour and Cerf. We also prove a phase space counterpart of Uhlmann's theorem of majorization.

Figures (5)

• Figure 1: Differences of the partial sums (\ref{['eq:partialsums']}) of eigenvalues of density matrices representing two-mode Gaussian states as functions of the number $m$ of terms involved in partial sums. The data are reported for five pairs of Gaussian states, whose symplectic eigenvalues are displayed in the parametrization (\ref{['eq:twomode_par']}). When one of the curves crosses the horizontal axis, the density matrix majorization between the two corresponding Gaussian states is ruled out.
• Figure 2: Left: The functional in (\ref{['eq:firstcond']}) evaluated on the absolute value of the Wigner functions (\ref{['eq:wignerHarmonic']}) of the harmonic oscillator eigenstates and plotted as a function of $t$. Five different eigenstates are considered. The fact that each curve intersects all the others indicates the absence of Wigner majorization among these eigenstates. The inset shows $\ln I_{0}$ evaluated on the same class of Wigner functions plotted as a function of $\ln n$. The dashed line corresponds to $0.44 \ln n + 0.12$ and is obtained through a fit procedure. Right: The functional in (\ref{['eq:firstcond']}) is evaluated on the mixture (\ref{['eq:Wigner_HCmix_ex']}) for four choices of $u$. The non-intersection of the curves signals the relation $|\widetilde{W}_{1}|\ggcurly|\widetilde{W}_{9/10}|\ggcurly|\widetilde{W}_{3/4}|\ggcurly |\widetilde{W}_{3/5}|$ and the consequent Wigner majorization among the considered states.
• Figure 3: Test of Proposal 2 for Wigner majorization through the difference (\ref{['eq:difference_WignermajoProp2']}) evaluated for a pair of Wigner functions $W_1$ and $W_2$. Left: $W_1=W_{|0\rangle}$ and $W_2=W_{|1\rangle}$, where $W_{|n\rangle}$ is defined in (\ref{['eq:wignerHarmonic']}). Right: $W_1=\widetilde{W}_{3/5}$ and $W_2=\widetilde{W}_{9/10}$, where both the Wigner functions are defined in (\ref{['eq:Wigner_HCmix_ex']}). In both panels, the curves are shown as functions of $t$ for various values of $\Lambda$, exhibiting collapse for large $\Lambda$. In the left panel the asymptotic curve has no definite sign, implying the absence of majorization order between $W_{|0\rangle}$ and $W_{|1\rangle}$, while in the right panel the asymptotic curve takes only negative values, meaning that $W_2=\widetilde{W}_{9/10}\succ W_1=\widetilde{W}_{3/5}$.
• Figure 4: Left: The difference between $S^{(\alpha)}_{W}$ evaluated on the output state obtained by applying a thermal noise channel (\ref{['eq:noisechannel']}) and the corresponding input state given by the mixture (\ref{['eq:Wigner_HCmix_ex']}). The curves are plotted as functions of the parameter $s$ of the thermal noise channel, with $c=0.75$. Two different input states corresponding to two different values of $u$ and three distinct values of the Rényi index $\alpha=2p/(2q-1)$, with $p$ and $q$ integers, are considered. The fact that the curves have no definite sign shows that $S^{(\alpha)}_{W}$ (and the Wigner Renyi entropy with even indices) are not in general monotonic under Gaussian channels. Right: The logarithm of the determinant of the matrix $X$ characterizing the thermal noise channel is subtracted from the quantity plotted in the left panel. The curves confirm the inequality (\ref{['eq:ineq_Renyinew']}).
• Figure 5: Comparison of the functional defined in (\ref{['eq:def_functionalI']}) evaluated on the absolute values of input single-mode Wigner functions (\ref{['eq:Wigner_HCmix_ex']}) (orange) and of the corresponding output Wigner functions (blue) after the application of a thermal noise channel (\ref{['eq:noisechannel']}). The values of the parameters of the input state and the applied channel are reported in each panel. The panels show that input and output states do not have any fixed Wigner majorization relation according to Proposal 1 in Result 8. Wigner majorization can be observed depending on the choice of the parameters: $\hat{\rho}_{\textrm{\tiny out}}\succ_{\textrm{\tiny W}}\hat{\rho}_{\textrm{\tiny in}}$ in the left panel, $\hat{\rho}_{\textrm{\tiny in}}\succ_{\textrm{\tiny W}}\hat{\rho}_{\textrm{\tiny out}}$ in the middle panel, no Wigner majorization relation in the right panel.