Quantum-memory-assisted entropic uncertainty relations

Dong Wang; Fei Ming; Liu Ye; Ming-Liang Hu

REVIEW 3 major objections 5 minor 163 references

Quantum-memory-assisted entropic uncertainty relations

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This review claims that quantum-memory-assisted entropic uncertainty relations unify tighter bounds, noise dynamics, and applications into one conditional-entropy framework.

desk verdict A broad review of quantum-memory-assisted EURs that fills a real gap but is not yet reliable as a reference due to transcription errors, including a garbled multi-measurement bound. read the letter →

arxiv 1908.03495 v1 pith:WV6HUQ3T submitted 2019-08-09 quant-ph

Dong Wang , Fei Ming , Ming-Liang Hu , Liu Ye This is my paper · ORCID

classification quant-ph

keywords entropicuncertaintyrelationquantummemoryconditionalvonNeumannentropydiscordcomplementarityopensystemsentanglementwitnesskeydistribution

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This review aims to establish that quantum-memory-assisted entropic uncertainty relations form a unified framework for quantifying how much can be known about incompatible measurements when the measured particle is correlated with a second system. The central claim is that the lower bound on Bob's uncertainty can be expressed and tightened using the conditional entropy of the premeasurement state and the complementarity of the two observables. It also claims that this bound can be improved by accounting for quantum correlations such as discord and accessible information, and that the resulting relations describe how uncertainty behaves under realistic noise, memory effects, and in curved space-time. The paper argues these relations directly power applications including entanglement witnessing, quantum key distribution, steering, metrology, and teleportation.

What carries the argument

The central object is the conditional von Neumann entropy $S(A|B)$ evaluated on the state after one-sided projective measurements on $A$, together with the complementarity parameter $c=\max_{ij}|\langle \psi_i^Q|\psi_j^R\rangle|^2$. The base inequality $S(Q|B)+S(R|B)\ge \log_2(1/c)+S(A|B)$ is the anchor; improvements attach extra nonnegative terms built from quantum discord, monogamy scores, or accessible-information quantities, and multi-measurement generalizations replace the pair $Q,R$ with $N$ observables and a composite parameter. These quantities carry the argument because they tie the uncertainty bound to the correlations of the shared state.

What would settle it

Open the cited original for any quoted bound, for instance Eq. (28) or Eq. (32), and compare constants, signs, and the definition of the complementarity parameter; a mismatch in any term, or an absent citation at the stated position, would show that the review's bound as stated is not established.

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Extended reading notes

Core claim

On the paper's own terms, its discovery is that the entropic uncertainty principle is not a fixed obstacle but a resource that can be reshaped by correlations between the measured particle and a quantum memory. For two observables with complementarity $c$, the fundamental relation is $S(Q|B)+S(R|B)\ge \log_2(1/c)+S(A|B)$, so a sufficiently entangled memory can push the bound toward zero. The review further claims that this bound can be tightened with correlation-aware corrections, including a discord term and an accessible-information term, and that generalized forms cover $N$ measurement settings. It then argues that, in open systems, the uncertainty dynamics track decoherence and information backflow, and that the same relations provide security and detection criteria for quantum tasks. The review's contribution is therefore a synthesis: improved bounds, environmental dynamics, and applications all follow from one conditional-entropy formulation.

Load-bearing premise

The review assumes that every displayed inequality and its stated conditions are faithfully transcribed from the cited papers, so the quoted bounds are exactly the proven results.

Editorial extensions

If this is right

If the measured particle and the quantum memory are maximally entangled and the two observables are complementary, the lower bound vanishes, meaning both measurement outcomes can be predicted perfectly.
When quantum discord exceeds classical correlation, the discord-corrected bound is strictly tighter than the base bound, so correlations beyond entanglement determine how much the memory helps.
For $N$ measurement settings, the generalized bound contains a term $(N-1)S(A|B)$, and additional accessible-information corrections can tighten it further.
Under weak measurement and measurement reversal, the entropic uncertainty can be reduced, offering a control knob for precision tasks.
A negative conditional entropy $S(A|B)$ signals both entanglement and usefulness for nonclassical teleportation, giving an experimentally accessible entanglement witness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the quoted bounds are accurate, measuring the conditional entropy and discord of a prepared state would already reveal how close the uncertainty relation is to saturation, without full tomography.
The same conditional-entropy machinery could be extended to quantum memories that are themselves noisy, so memory-side decoherence should raise the uncertainty bound in a way EUR-based witnesses can detect.
The monogamy relations reviewed here imply that tripartite states constrained by the EUR could yield multipartite steering inequalities, an extension the paper leaves implicit.
Because the review contains uncited citation markers and garbled equations, each displayed formula should be verified against its original source before being used in a protocol.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things before reading this review. First, it is exactly what it says: a survey with no new results, covering the improved lower bounds, open-system dynamics, and applications of quantum-memory-assisted entropic uncertainty relations. Second, the transcription quality is not good enough for a review whose whole value is accuracy. The missing citation markers (Kennard [?], Robertson [?], Feng et al. [?]) and the garbled line around Eq. (10) are minor annoyances. The bigger problem is Section III.B: Eq. (46) defines c as a max over both basis indices, which makes the b in Eq. (45) a product of constants and the max/sum structure vacuous. That is a real defect in one of the paper's advertised generalized results, and Eq. (47) simply repeats Eq. (45). Eq. (85) also has H(M1|B)+H(M1|B) where M1 and M2 are intended. These are not cosmetic; they make the review hard to trust as a reference. Credit where it is due: the scope is genuinely useful. The paper gathers many results in one place—Berta et al., Pati et al., Coles–Piani, the multi-measurement generalizations, dynamics in noisy channels and curved spacetime, and applications like entanglement witnesses and steering inequalities. The organization is sensible, and the authors do attempt to synthesize rather than just list. The self-citations are frequent but not out of place; they are prior works in the same subfield, and a review naturally covers the authors' own contributions. I do not see conceptual confusion in the core claims, just careless copying. That said, the stress-test note holds up: the multi-measurement bound as printed is ill-defined. Because the review contains no derivations, its accuracy is entirely dependent on faithful transcription of the literature, and these errors undermine that. The right fix is a careful revision, preferably with the original papers side by side, rather than a desk rejection. The topic is appropriate, the readership is real (researchers new to EURs or looking for a starting point), and a cleaned-up version would be a solid contribution. As it stands, I would tell a student to use the original papers and skip this version. A serious referee can fix it; the flaws are mechanical, not conceptual.

Referee Report

3 major / 5 minor

Summary. The paper is a review of quantum-memory-assisted entropic uncertainty relations (EURs). It surveys uncertainty relations without a quantum memory, including variance-based, entropy-based, and majorization-based formulations; the basic quantum-memory-assisted EUR of Berta et al. and its improved lower bounds from Pati et al., Coles and Piani, Adabi et al., and others; multiple-measurement generalizations by Liu et al., Zhang et al., and Dolatkhah et al.; connections to quantum discord, coherence, and information exclusion; dynamics of the EUR in open systems, curved spacetime, noninertial frames, NV centers, and spin chains; and applications to entanglement witnessing, steering, wave-particle duality, quantum metrology, teleportation, quantum key distribution, information locking, and quantum coding. The paper's central claim is that it provides an accurate, comprehensive reference on recent progress in this area.

Significance. If the transcriptions of equations and attributions were fully reliable, this review would be useful: it collects many recent results, including the authors' own work on dynamics and control of EURs, in a single accessible narrative, and it lists open problems that could guide future research. However, because this is a review rather than a source of new derivations, the value of the manuscript is almost entirely in the fidelity of its reported results and references. The transcription errors identified below directly undermine that value and, in the current version, make the manuscript unreliable as a reference.

major comments (3)

[Section III.B, Eqs. (44)-(47)] The generalized multi-measurement EUR is not well defined as printed. Equation (46) defines c(ψ^m_{i_m}, ψ^n_{i_n}) = max_{i_m,i_n} |⟨ψ^m_{i_m}|ψ^n_{i_n}⟩|², which makes each c a constant for a pair of bases, independent of the path indices i_2,...,i_{N-1}. Consequently, the maximization over i_1 in Eq. (45) is vacuous and the summation over i_2,...,i_{N-1} collapses, so b reduces to a product of constants; Eq. (47) then repeats Eq. (45) verbatim. As a result, Eq. (44) is not a faithful transcription of Liu et al.'s bound, and the claimed reduction to the Berta et al. bound for N=2 is obscured. This is a load-bearing failure in one of the paper's advertised generalized results.
[Section IV.A.3.a, Eq. (85)] The reported Holevo-quantity bound contains a visible typo: it reads H(M1|B)+H(M1|B) ≥ −log2 c + H(A) − J(B|M1) − J(B|M2) instead of H(M1|B)+H(M2|B). Since the equation is supposed to bound the sum of conditional entropies for two distinct measurements M1 and M2, the printed relation cannot be used as a reference result. The same subsection also contains an unresolved citation marker 'Feng et al. [?]', so the attribution of the first study of EURs in Schwarzschild spacetime cannot be verified.
[Section II.A, Eq. (1); Section II.B.1, Eq. (10); Introduction] Several fidelity problems appear in the early technical sections. Eq. (1) cites 'Kennard [?] and Robertson [?]' without reference markers, even though Kennard is Ref. [2] and Robertson is Ref. [11]. Eq. (10) is garbled: it asserts log2(2πe∆(P)∆(Q)) = log2 sqrt(2πe∆(P))² log2 sqrt(2πe∆(Q))², which is not a valid identity and does not permit the reader to follow the derivation of ∆P∆Q ≥ ℏ/2. Additionally, the Introduction attributes the 2010 review to Wehner and Winter as [8] while the reference list and the later citation in Section I identify Wehner and Winter as [9], indicating a swapped or mismatched citation. These are not isolated typographical glitches; they are symptomatic of a proofreading standard that is too low for a review article whose primary purpose is reliability.

minor comments (5)

[Section IV.A.1, after Eq. (75)] The sentence 'the lower bound (UL) can coincide with the entropic uncertainty (UL)' should read 'the lower bound (U_L) can coincide with the entropic uncertainty (U_R)', since the text distinguishes U_L from U_R.
[Section IV.A.3.d, Eq. (97)] The sentence after Eq. (97) refers to 'the entropic uncertainty's lower bound U_R in Eq. (26)', but Eq. (26) is the Renes-Boileau relation; the intended reference is presumably Eq. (28), the Berta et al. bound.
[Throughout] There are numerous spelling and typographical errors, including 'Block sphere' for 'Bloch sphere', 'dented' for 'denoted', 'Y uan' for 'Yuan', 'the prospective of variance' for 'the perspective of variance', and 'Haseil' for 'Haseli' in Ref. [99].
[Reference list] The bibliography contains formatting errors such as 'Bia lynicki-Birula' and 'l. Rudnicki' in Ref. [8], and Ref. [102] gives the year 1988 for Bender and Boettcher's PT-symmetric Hamiltonian, which appeared in 1998.
[Section II.B.1] The notation in Eq. (10) is confusing: 'for arbitrary observables P and Q linked with position and momentum' does not clearly specify that P and Q are the momentum and position observables, and the subsequent derivation would benefit from an explicit statement that Eq. (8) is being substituted into Eq. (9).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: this is a review article whose quoted results are attributed to prior literature; self-citations are descriptive, not load-bearing.

full rationale

This is a review/survey (arXiv:1908.03495) whose central content is a catalog of previously published entropic-uncertainty-relation results. It introduces no new theorem, estimator, fitted parameter, or prediction; the claimed derivation chain consists of quoted inequalities with references to the original derivations (Berta et al., Pati et al., Coles and Piani, Liu et al., etc.). The authors cite their own previous work in several places (e.g., refs. [46], [68], [74], [75], [87], [90]), but always as attributed prior results rather than as premises that force the review's conclusions. For example, Section III.C states 'it was found that whenever the uncertainty bound of Berta et al. is reduced ... we always have [46] E_f(ρAB)>E_f(ρAC), D(B|A)>D(C|A)'; this is a citation of an external published result, not a definitional reduction of a predicted quantity to its input. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors to forbid alternatives, and no ansatz is smuggled in via self-citation. The garbled multi-measurement formulas in Eqs. (44)-(47), the duplicated H(M1|B) in Eq. (85), and the missing citation markers near Eq. (1) and Eq. (82) are internal-consistency and fidelity defects; they undermine the review's reliability as a reference, but they are not cases of a result reducing to its own input. Accordingly, no circular step is present and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The review introduces no free parameters and no invented entities. It relies on the correctness of a number of established theorems and definitions from quantum information theory, which are listed as axioms.

assumptions (4)

standard math Maassen-Uffink entropic uncertainty relation H(Q)+H(R) ≥ log2(1/c)
Used as the basis for many later bounds, e.g., Eq. (14).
standard math Berta et al.'s quantum-memory-assisted EUR S(Q|B)+S(R|B) ≥ log2(1/c)+S(A|B)
The central relation of the review, Eq. (28), from which improved bounds are derived.
standard math Strong subadditivity of von Neumann entropy
Invoked in Eq. (54) and elsewhere for multipartite entropies.
domain assumption Definition and properties of quantum discord and relative entropy of coherence
The review's discussion of discord-based and coherence-based bounds relies on these definitions, e.g., Eqs. (32) and (63).

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Cite this review

Pith. "Pith review of Quantum-memory-assisted entropic uncertainty relations." pith.science (2026). https://pith.science/paper/WV6HUQ3T

@misc{pith2026190803495,
  author       = {Pith},
  title        = {Pith review of: Quantum-memory-assisted entropic uncertainty relations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WV6HUQ3T}},
  note         = {Machine review of arXiv:1908.03495}
}

read the original abstract

Uncertainty relations take a crucial and fundamental part in the frame of quantum theory, and are bringing on many marvelous applications in the emerging field of quantum information sciences. Especially, as entropy is imposed into the uncertainty principle, entropy-based uncertainty relations lead to a number of applications including quantum key distribution, entanglement witness, quantum steering, quantum metrology, and quantum teleportation. Herein, the history of the development of the uncertainty relations is discussed, especially focusing on the recent progress with regard to quantum-memory-assisted entropic uncertainty relations and dynamical characteristics of the measured uncertainty in some explicit physical systems. The aims are to help deepen the understanding of entropic uncertainty relations and prompt further explorations for versatile applications of the relations on achieving practical quantum tasks.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Markovian and non-Markovian noises In general, we can consider the environment either in the Markovian or non-Markovian regime. If the information of a system ﬂows from the system to the environment in one-way manner, we say the environment is Markovian; Contrarily, if the information stored in the central system is bidirection ally ﬂow between the system...

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[110] have done some early work about entanglement witness by means of the EURs, and Huang [111] made further improvements

Shannon entropic witness Giovannetti [109] and G ¨ uhne et al. [110] have done some early work about entanglement witness by means of the EURs, and Huang [111] made further improvements. We pri- marily review the entanglement witness on more recent de- velopments from quantum-memory-assisted EURs. Berta et al. [32] have discussed how to employ this relati...

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In practice, this operation is usually deemed as one type of the weak measurement with null result [101]

Filtering operation The ﬁltering operation refers to the non-trace-preserving map and can be used to enhance entanglement of a system. In practice, this operation is usually deemed as one type of the weak measurement with null result [101]. Explicitly, th e ﬁltering operation can be described by F = (√ 1 −kf 0 0 √ kf ) , (102) where kf ∈ [0, 1] is the str...

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A class of Hamiltonian was proposed by Ben- der et al

Non-Hermite operation It is usually required that the Hamiltonian of a physical sys - tem is Hermitian in conventional quantum mechanics, as this ensures that the system’s energy is real and the time evolu- tion is unitary. A class of Hamiltonian was proposed by Ben- der et al. [102] in 1998, i.e., the so called parity-time (PT ) symmetric Hamiltonian wit...

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[67] examined dynamics of the entropic uncertainty in the unital and nonunital noisy channels, respectively

Unital and nonunital noises As for the effects of noises on the entropic uncertainty, Xu et al . [67] examined dynamics of the entropic uncertainty in the unital and nonunital noisy channels, respectively. Whe n one system to be probed is subjected to the noisy channel Λ , the state was mapped into Λ(ρ0) = ∑ iEiρ0E† i , with Ei be- ing the Kraus operator....

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The curved space-time In 2013, Feng et al

Dynamics of the EUR in speciﬁc systems a. The curved space-time In 2013, Feng et al. [? ] ﬁrst observed the quantum- memory-assisted EURs in the frame of a Schwarzschild black hole. Typically, the Schwarzschild black hole is considere d as offering one of the curved space time. And it in Schwarzschild coordinates is described by ds2 = − ( 1 − 2M r ) dt2 +...

2013

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