arxiv: 2605.01832 · v1 · submitted 2026-05-03 · 🪐 quant-ph

Recognition: unknown

Tight Entropic Uncertainty Relations

Alberto Riccardi , Lorenzo Maccone

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:55 UTC · model grok-4.3

classification 🪐 quant-ph

keywords entropic uncertainty relationsShannon entropyRényi entropyMaassen-Uffink boundquantum observablesstate-independent boundsincompatible measurements

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The pith

A new parameter s allows a tighter state-independent lower bound on the sum of Shannon entropies for any incompatible quantum observables A and B.

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

Entropic uncertainty relations set a positive lower limit on the combined Shannon entropies of measurement outcomes from incompatible observables, depending only on outcome probabilities rather than the values themselves. The paper derives a family of such bounds γ_s that exceeds the standard Maassen-Uffink expression for every pair of incompatible A and B. In the limit where the free parameter s approaches 2, γ_s approaches the actual minimum value of the entropy sum that any state can achieve. The same construction extends directly to Rényi entropies of varying order.

Core claim

Entropic uncertainty relations H(A)+H(B)≥γ give a nonzero lower bound γ to the sum of the Shannon entropies H of the outcome probabilities of incompatible observables A and B. They are better than the variance-based uncertainty relations because they only depend on the Born statistics of the outcomes and not on the outcomes themselves, and because bounds γ typically are state independent. Here we provide a state-independent lower bound γ_s that is better than the textbook Maassen-Uffink bound and, in the limit of the parameter s→2, becomes asymptotically tight for all A,B. The bound can be extended to Renyi entropies.

What carries the argument

The parameterized state-independent lower bound γ_s on the entropy sum, constructed to exceed the Maassen-Uffink value while approaching the true minimum as s tends to 2.

If this is right

For every pair of incompatible observables the sum of Shannon entropies is bounded below by a number strictly larger than the Maassen-Uffink value.
As the parameter s approaches 2 the bound γ_s converges to the smallest entropy sum that can ever be realized by any state.
The identical construction supplies improved lower bounds when Rényi entropies replace Shannon entropies.
The bound remains valid without reference to any particular quantum state or to the explicit form of the observables beyond their incompatibility.

Where Pith is reading between the lines

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

Because the bound is asymptotically tight, the value at s=2 may serve as a benchmark against which future exact uncertainty relations can be tested.
The construction may be applied to any pair of observables once their outcome probability distributions are known, without needing additional state-dependent information.

Load-bearing premise

The tighter bound γ_s follows from the standard definitions of Shannon and Rényi entropies together with the mutual incompatibility of the observables A and B.

What would settle it

Compute the greatest lower bound on H(A) + H(B) over all quantum states for any chosen incompatible pair A and B; if that value lies below γ_s for any such pair, the claimed improvement is false.

Figures

Figures reproduced from arXiv: 2605.01832 by Alberto Riccardi, Lorenzo Maccone.

**Figure 2.** Figure 2: FIG. 2. Upper: comparison between Montecarlo and NPIM [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Left: Depiction of the qubit lower bound (B13) as [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

Entropic uncertainty relations $H(A)+H(B)\geqslant \gamma$ give a nonzero lower bound $\gamma$ to the sum of the Shannon entropies $H$ of the outcome probabilities of incompatible observables $A$ and $B$. They are better than the variance-based uncertainty relations because they only depend on the Born statistics of the outcomes and not on the outcomes themselves, and because bounds $\gamma$ typically are state independent. Here we provide a state-independent lower bound $\gamma_s$ that is better than the textbook Maassen-Uffink bound and, in the limit of the parameter $s\to 2$, becomes asymptotically tight for all $A,B$. The bound can be extended to Renyi entropies.

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.

Desk Editor's Note private letter to a colleague

This paper gives a parameterized family of state-independent entropic bounds that improve on Maassen-Uffink and approach tightness as the parameter goes to 2.

read the letter

The main takeaway is a new family of lower bounds γ_s on H(A) + H(B) for incompatible observables A and B. These beat the textbook Maassen-Uffink value for finite s and recover asymptotic tightness for every pair as s approaches 2, with a parallel version for Rényi entropies. The construction stays within standard entropy definitions and incompatibility, without extra state dependence. That is the concrete advance over prior work. It is useful in the subfield because tighter state-independent bounds directly help security analyses in quantum cryptography and some foundational estimates. The paper earns credit for identifying a limit where the bound saturates for all observables rather than just special cases. The math appears to rest on ordinary properties of the entropies, so the central claim looks internally consistent on its face. The soft spot is that the abstract supplies no derivation, no proof outline, and no numerical check on even a simple pair of observables. Without those, it is impossible to gauge how large the improvement actually is or whether the s to 2 limit holds uniformly. If the full text contains only a short calculation that works, the contribution is modest but still worth recording; if the steps are more involved, the paper would benefit from an explicit example. This is for people already working on entropic uncertainty relations or using them in protocols. A reader who needs the best available constant for a specific application would check the details and possibly adopt γ_s. I would send it to peer review. The claim is narrow and falsifiable, so referees can verify the construction quickly and ask for the missing checks if needed.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to derive a state-independent lower bound γ_s on the sum of Shannon entropies H(A) + H(B) for incompatible observables A and B. This bound is asserted to be strictly better than the Maassen-Uffink bound for finite values of the parameter s, and to become asymptotically tight for all pairs of observables as s approaches 2. The result is extended to Rényi entropies as well.

Significance. Should the derivation prove correct, this work offers a meaningful advancement in entropic uncertainty relations by providing a family of improved, state-independent bounds with a desirable asymptotic tightness property. Such bounds are valuable in quantum information science for tasks involving measurement incompatibility, and the extension to Rényi entropies broadens the applicability. The approach relies on standard entropy definitions and observable incompatibility, which is a strength in terms of generality.

minor comments (2)

The abstract would benefit from a concise statement of the key technical step used to construct γ_s, to help readers immediately grasp how the improvement over Maassen-Uffink is achieved.
Notation for the family of bounds and the limit s → 2 should be introduced with a short reminder of the relevant entropy definitions in the main text to improve readability for non-specialists.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and the recommendation for minor revision. We appreciate the recognition that the parameterized bound γ_s provides a strict improvement over the Maassen-Uffink relation for finite s while becoming asymptotically tight for all observable pairs as s approaches 2, together with the extension to Rényi entropies. This is acknowledged as a meaningful contribution to state-independent entropic uncertainty relations in quantum information.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives a family of state-independent lower bounds γ_s on H(A) + H(B) that strictly improve the Maassen-Uffink bound for finite s while recovering asymptotic tightness for every pair of observables as s → 2 (with a parallel statement for Rényi entropies). The abstract and available text invoke only the standard definitions of Shannon/Rényi entropy together with the incompatibility of A and B; no fitted parameters are renamed as predictions, no self-citation chain is load-bearing for the central claim, and no ansatz or uniqueness theorem is smuggled in. The explicit construction of γ_s is presented as an independent mathematical improvement, not a re-expression of its own inputs. This is the normal, non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete and based on standard background assumptions of quantum information theory.

axioms (2)

standard math Standard definitions of Shannon and Rényi entropies for discrete probability distributions
Invoked implicitly when stating H(A) + H(B) ≥ γ_s
domain assumption Observables A and B are incompatible (non-commuting) quantum measurements
Required for any nontrivial entropic uncertainty relation

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discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 1 canonical work pages · 1 internal anchor

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Generate a random complex vector (seed)u (k) ∈C d withk= 0
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Apply the operatorU:w (k) j =P i Ujiv(k) i
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Rescale it asx (k) j =w (k) j |w(k) j |s′−2, wheres ′ −2>0, sinces ′ >2
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Apply the adjoint to go back:y i =P j U ∗ jixj
[28]

Rescale it again asz (k) j =y (k) j |y(k) j |s′−2
[29]

Iterate from step 2, choosingv (k+1) =z (k), and stop when the iteration converges: ∥w(k+1)∥s′ − ∥w(k)∥s′ < ϵ, withϵsome (small) accuracy parameter
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Repeat the whole procedureN seeds times from step 1 and select the largestLof the obtained norms∥w (k)∥s′. This is needed because the algorithm converges to alocal maximum and one needs to repeat it multiple times with Nseeds random seeds to find the global maximum. The final quantityLis the estimated norm∥U∥ s→s′. A good check to see whetherN seeds is su...