Analytic formulae for non-local magic in bipartite systems of qutrits and ququints

Abstract

We conjecture analytic expressions for the non-local magic of bipartite pure qudit states of prime local dimension. Our construction relies on the Schmidt-aligned state attaining the minimum over local unitaries, a hypothesis that we support with numerical evidence for pairs of qutrits and ququints. For composite local dimensions, we find that the analogous expressions do not in general reproduce the global minimum, but can still provide computationally cheap approximations to the non-local magic. We also find that relations between non-local magic and entanglement diagnostics that hold for two qubits generally do not extend to qutrit and higher-dimensional systems.

Figures (4)

• Figure 1: Non-local magic for two qutrits across the canonical simplex of Schmidt probabilities $(\lambda_0^2,\lambda_1^2, \lambda^2_2)$, where $\sum_i \lambda_i^2=1$. The colour scale shows the non-local magic $\mathscr{M}_{N=3}=-\ln F_3(\lambda)$. The plot is symmetric under permutations of $(\lambda^2_0,\lambda^2_1,\lambda^2_2)$, appearing as a threefold symmetric pattern over the simplex. The minimum value $\mathscr{M}_{N=3}=0$ appears at product-state vertices and at the maximally entangled point $\lambda_i=1/\sqrt{3}$ (blue circle), while the maximum $\mathscr{M}_{N=3}=\ln 2 \approx 0.69$ occurs along rank-two equal spectra, e.g. $\lambda = (1/\sqrt{2},1/\sqrt{2},0)$ (orange box), and permutations.
• Figure 2: Non-local magic for two ququints on the Schmidt-probability slice $(\lambda_0^2,\lambda_1^2,\lambda^2_2,0,0)$, where $\sum_i \lambda^2_i=1$. The colour scale shows the non-local magic $\mathscr{M}_{N=5}$. The plot is symmetric under permutations of $(\lambda^2_0,\lambda^2_1,\lambda^2_2)$, appearing as a threefold symmetric pattern over the triangular slice. The slice is chosen to feature the maximum value of $\mathscr{M}_{N=5}=\ln(27/11)\approx 0.90$, which occurs at $\lambda^2_0=\lambda^2_1=\lambda^2_2=1/3$ (blue circle).
• Figure 3: Comparison of the non-local magic from direct numerical minimisation over $U_A \otimes U_B$ and from the Schmidt-based formula for $N=3,4,5$. For each $N$, we sample $10^4$ points in Schmidt-parameter space and perform multi-start L-BFGS optimisation with $n_{\mathrm{starts}}=50$ ($N=3$), $200$ ($N=4$), and $500$ ($N=5$), using $\texttt{maxiter}=300$ per start. Top row: formula value versus numerically minimised value, with the dashed line showing equality. Bottom row: residual histograms for $\mathscr{M}_{\mathrm{formula}}-\mathscr{M}_{\mathrm{numerical}}$. Negative residuals are consistent with incomplete convergence of the numerical minimisation; positive residuals indicate formula values above the numerical minimum. For $N=3,5$, no positive residuals are observed in the sampled points, supporting Schmidt attainment in prime dimensions. For $N=4$, positive residuals are present, showing that the analogous Schmidt-based expression does not reproduce the global minimum in general; nevertheless, it often lies close to the minimum.
• Figure 4: Residual map for the $N=4$ Schmidt-based formula across projected slices of Schmidt-probability space. Shown is the residual $\mathscr{M}_{\mathrm{formula}}-\mathscr{M}_{\mathrm{numerical}}$ for fixed $\lambda_3^2 \in \{0,0.2,0.4,0.6\}$. For each panel, points are selected in a band $| \lambda_3^2-(\lambda_3^2)_*|\le 0.02$; the remaining components $(\lambda_0^2,\lambda_1^2,\lambda_2^2)$ are then normalised by $1-\lambda_3^2$, and the data are plotted on the canonical simplex. The largest residuals occur near simplex boundaries, i.e. where one Schmidt component vanishes. At $\lambda_3^2=0.2$ it seems that values along each boundary close to $0.5$ are more trustworthy than those close to $0.25$ or $0.75$; the opposite appears true for $\lambda_3^2 \geq 0.4$. Very few points have negative residuals, and all such points are localised to the vertices, which represent $(\lambda_0^2,\lambda_1^2,\lambda_2^2)=(1,0,0)$ and permutations.