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arxiv: 2605.31379 · v1 · pith:XMSYSDOEnew · submitted 2026-05-29 · 🪐 quant-ph · cs.IT· math-ph· math.IT· math.MP

R\'enyi divergences and binary state discrimination error exponents for fermionic quasi-free states

Mil\'an Mosonyi , G\'abor Mar\'oti-Zareczky This is my paper

Pith reviewed 2026-06-28 22:15 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath-phmath.ITmath.MP
keywords Rényi divergencesquasifree statesfermionic chainsstate discriminationerror exponentsregularizationquantum hypothesis testing
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The pith

Regularized Rényi divergences for translation-invariant gauge-invariant quasifree fermionic states on infinite chains admit explicit formulas for multiple variants.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives closed-form expressions for a range of regularized Rényi divergences between translation-invariant and gauge-invariant quasifree fermionic states on doubly infinite chains. These cover the (α,z), log-Euclidean, maximal, measured, and integral types. In the single-mode-per-site case the setting becomes asymptotically classical with all the regularized divergences coinciding, while multiple modes per site keep non-commutativity and yield z-dependent values in general. The results also allow explicit error exponents for binary state discrimination and extend prior constructions of states with super-exponential error decay to the multi-mode setting.

Core claim

For translation-invariant gauge-invariant quasifree states on doubly infinite fermionic chains, explicit formulas exist for a range of regularized Rényi divergences including the (α,z), log-Euclidean, maximal, measured, and integral variants. When there is a single mode at each lattice site the setting becomes asymptotically classical with all regularized divergences equal, but with multiple modes per site non-commutativity persists and different z yield different values in general. The formulas also allow generalization of constructions for states with super-exponential decay of discrimination error probabilities to the multi-mode case.

What carries the argument

Explicit computation of regularized Rényi divergences using the translation and gauge invariance of quasifree fermionic states on infinite chains.

If this is right

  • Error exponents for binary i.i.d. discrimination between these states can be written in closed form using the explicit regularized divergences.
  • Single-mode-per-site discrimination reduces to a classical problem where all considered regularized Rényi divergences coincide.
  • Multi-mode-per-site cases retain quantum features with regularized (α,z) values depending on z for fixed α.
  • Constructions of states exhibiting super-exponential decay of discrimination errors extend from single-mode to multiple modes per site.
  • Different families of Rényi divergences (Petz-type, sandwiched, measured, integral) can be compared directly after regularization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit formulas could be used to study the thermodynamic limit of finite-chain discrimination tasks.
  • Similar invariance-based computations might apply to other translation-invariant lattice models beyond fermions.
  • The single-mode versus multi-mode distinction points to a sharp change in scaling of error probabilities with system size.
  • Numerical checks for small lattice segments could test whether the infinite-chain formulas approximate finite cases well.

Load-bearing premise

The states under consideration are translation-invariant and gauge-invariant quasifree states on doubly infinite fermionic chains.

What would settle it

Numerical evaluation of the n-copy Rényi divergence limit as n approaches infinity for a concrete multi-mode quasifree state and direct comparison to the claimed closed-form expression.

read the original abstract

The trade-off relations between the two types of error probabilities in binary i.i.d. quantum state discrimination can be expressed by single-copy formulas in terms of the Petz-type and the sandwiched R\'enyi divergences of the two states representing the two hypotheses. In the non-i.i.d. setting, the error exponents can usually be expressed in terms of regularized R\'enyi divergences, which do not admit explicit formulas in general. Here, we consider a class of states, translation-invariant and gauge-invariant quasifree states on doubly infinite fermionic chains, and give explicit formulas for a wide range of regularized R\'enyi divergences between such states, including $(\alpha,z)$, log-Euclidean, maximal, measured, and the recently introduced integral R\'enyi divergences. We show that the case where there is a single mode at each lattice site becomes asymptotically classical, with all the different types of regularized R\'enyi divergences being equal, while in the case of multiple modes per site, non-commutativity persists under regularization, and for any fixed $\alpha$, the regularized R\'enyi $(\alpha,z)$-divergences give different regularized values for different $z$ parameters in general. We also generalize a previous construction from [Bunth, Mar\'oti, Mosonyi, Zimbor\'as, Lett.~Math.~Phys.~113:(7), 2023] to the case of multiple modes per lattice site to obtain a large class of states exhibiting super-exponential decay of the discrimination error probabilities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper derives explicit closed-form expressions for a range of regularized Rényi divergences—including (α,z), log-Euclidean, maximal, measured, and integral variants—between translation-invariant gauge-invariant quasifree states on doubly infinite fermionic chains. These states are determined by their two-point covariance operators (multiplication operators in Fourier space), allowing the regularization limit (1/n)D(ρ_n || σ_n) to reduce to an integral over the Brillouin zone. The single-mode-per-site case is shown to become asymptotically classical (all divergences coincide), while the multi-mode case retains z-dependence. The work also generalizes a prior construction to multi-mode sites, producing states with super-exponential decay of binary discrimination error probabilities.

Significance. If the derivations hold, the explicit integral formulas constitute a rare instance of closed-form regularized quantum divergences in a non-commutative infinite-dimensional setting. This directly enables computation of error exponents for fermionic state discrimination and clarifies when regularization preserves or erases non-commutativity. The multi-mode generalization of the super-exponential construction broadens the class of examples. Credit is due for the parameter-free integral reductions and the explicit distinction between single- and multi-mode regimes.

minor comments (2)
  1. §3 (or wherever the symbol matrices are introduced): the notation for the covariance operator symbol could be clarified by explicitly stating the matrix dimension in the multi-mode case before the integral expressions.
  2. The statement that 'the single-mode case collapses to the classical case' would benefit from a one-line reference to the corresponding classical Rényi divergence formula to make the equality immediate for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. We are pleased that the contributions regarding explicit formulas for regularized Rényi divergences in the fermionic setting were found to be of interest.

Circularity Check

0 steps flagged

No significant circularity; derivation is direct computation from state structure

full rationale

The paper derives explicit formulas for the regularized Rényi divergences by exploiting that the states are determined by their two-point covariance operators, which become multiplication operators in Fourier space; the n-site regularization then reduces to an explicit integral over the Brillouin zone of a function of the symbol matrices. This is a direct, parameter-free mathematical reduction from the given state class definitions, with no fitted inputs renamed as predictions, no self-definitional loops, and no load-bearing self-citations for the core formulas (the cited prior work is used only for an ancillary generalization to super-exponential decay). The single-mode vs. multi-mode distinction and z-dependence claims follow immediately from the same integral expressions. The derivation is therefore self-contained against the external benchmark of the quasifree state structure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivations rest on the domain assumption that the states belong to the translation-invariant gauge-invariant quasifree class; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption States are translation-invariant and gauge-invariant quasifree fermionic states on doubly infinite chains.
    This class is required for the explicit regularization formulas to hold.

pith-pipeline@v0.9.1-grok · 5833 in / 1196 out tokens · 26402 ms · 2026-06-28T22:15:01.286838+00:00 · methodology

discussion (0)

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Reference graph

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