Sandwiched Quantum Rényi Divergence for Density-Matrix Diagnostics¶
A Colab session read Müller-Lennert et al. ("On quantum Rényi entropies: a new generalization and some applications", arXiv:1306.3142) and proposed adding the Sandwiched Quantum Rényi Divergence to dense_evolution as a noise/state-distance diagnostic. Two problems were already found with the original proposal in an earlier evaluation pass: a real implementation bug, and a disproven original use case (replacing the JSD-based truncation criterion in mps.py's bond-dimension search -- the Colab's own benchmarking showed Rényi and JSD induce the exact same truncation ordering on that diagonal singular-value spectrum, since rho and sigma commute there and there's nothing for a non-commuting-aware divergence to add).
This page fixes and validates the divergence itself for the case where it does have something to add: full density-matrix diagnostics, where states genuinely don't commute.
Bug 1: the clamp that zeroed the signal¶
The formula (Definition 13 of the paper): D_α(ρ‖σ) := 1/(α−1) · log Tr[(σ^e ρ σ^e)^α], e = (1-α)/(2α).
The original code computed the inner trace correctly, then clamped it: tr_inner = jnp.maximum(tr_inner, 1.0). Since a sub-1 trace is the normal case for non-commuting states, this silently forced log2(1) = 0 for the majority of real inputs -- confirmed directly in the Colab's own printed output: alpha=1.5 gave exactly 0.000000 for every rotation angle in a sweep, including a large rotation where a real divergence should have been substantial.
Fix: floor at a small numerical epsilon (to avoid log2(0)), not at 1.0.
Bug 2: infinite divergence mishandled as a wrong-signed finite number¶
Fixing bug 1 alone was not enough. For α > 1, the sandwiched Rényi divergence is only finite when supp(ρ) ⊆ supp(σ) -- exactly analogous to the classical relative entropy diverging to +∞ outside full support overlap. Two different pure states (both rank-1) generically violate this: verified by hand that Tr[Q^1.5] = 0.6759, matching the closed-form prediction (|⟨σ|ρ⟩|²)^α exactly, and plugging that into the naive formula gives a finite negative divergence (-1.13) -- worse than the original bug's silent zero, since it's a wrong-signed answer that looks plausible instead of visibly broken.
Fix: detect tr_inner < 1 at α > 1 and return +∞, matching the standard convention for quantum divergences under support mismatch. Verified this branch does not fire spuriously for α < 1, where sub-1 traces are the ordinary, always-finite case (confirmed against the classical Rényi formula below).
Validation against three independent references¶
- α→1 limit matches an independent numpy computation of
Tr[ρ(log ρ − log σ)]to 4 decimal places (0.234028 both ways), on states specifically chosen to be full-rank (avoiding a separate, legitimate edge case: exactly-singular density matrices makescipy.linalg.logmitself raiseLogmExactlySingularWarning, an inherent ill-conditioning of relative entropy near degenerate support, not an implementation bug). - Commuting/diagonal case reproduces the classical Rényi divergence formula
D_α(p‖q) = 1/(α−1) log Σ pᵢ^α qᵢ^(1−α)exactly, at α = 0.7, 1.5, 2.0, 3.0. - Two different pure states at α > 1 correctly give
+∞(support mismatch), not a finite number of either sign.
Comparison against the existing uhlmann_fidelity metric¶
On a Bell state degraded by amplitude damping, the fixed divergence tracks uhlmann_fidelity's noise sensitivity in the well-behaved (mid-noise) regime, and correctly diverges to +∞ at high noise (p=0.9, α=1.5) as the damped state approaches a support boundary relative to the ideal state -- the mathematically expected behavior, not a bug. One open observation, reported honestly rather than smoothed over: the α=1.5 curve dips between p=0.5 and p=0.7 before jumping to infinity. No theorem in Müller-Lennert et al. requires monotonicity along an arbitrary one-parameter noise sweep (the data-processing inequality governs a fixed channel applied to both states, not a channel-parameter scan on one of them), so this isn't necessarily wrong -- just not yet fully explained.
Status¶
The clamp bug and the infinity-handling bug are both fixed and validated in scripts/sandwiched_renyi_density_matrix.py. Not yet promoted to dense_evolution -- pending a decision on the right integration point (likely alongside zne_density_matrix/uhlmann_fidelity in dense_evolution.mitigation as an additional, α-tunable distance metric for density-matrix ZNE diagnostics).
Reproduce¶
Produces data/sandwiched_renyi_bugfix_confirmation.csv, data/sandwiched_renyi_noise_scaling.csv, data/sandwiched_renyi_vs_uhlmann_fidelity.csv.