On some states minimizing uncertainty relations: A new look at these relations

Krzysztof Urbanowski

arxiv: 2411.08131 · v6 · submitted 2024-11-12 · 🪐 quant-ph · math-ph· math.MP

On some states minimizing uncertainty relations: A new look at these relations

Krzysztof Urbanowski This is my paper

Pith reviewed 2026-05-23 17:03 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords uncertainty relationsHeisenberg-Robertson relationSchrödinger uncertainty relationstandard deviationscorrelation functionnon-commuting observablesquantum statessum uncertainty relations

0 comments

The pith

There exist quantum states where the lower bound on the product of standard deviations for non-commuting observables is zero, yet the states are not eigenstates of either observable.

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

The paper analyzes the Heisenberg-Robertson and Schrödinger uncertainty relations and finds a large class of states in which the lower bound on the product of standard deviations for non-commuting observables A and B equals zero. These states are not eigenstates of A or B, and the correlation function between A and B vanishes in them. The work further shows that sum uncertainty relations give no useful lower bounds on the deviations for these states, while the uncertainty principle itself supplies both a lower bound on the deviation product and an upper bound on the modulus of the correlation function.

Core claim

There can exist a large set of states for which the lower bound of the product of the standard deviations of a pair of non-commuting observables A and B is zero, and which differ from those described in the literature. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. The sum uncertainty relations also do not provide any information about lower bounds on the standard deviations calculated for these states. The uncertainty principle in its most general form has two faces: one is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations

What carries the argument

The Heisenberg-Robertson and Schrödinger uncertainty relations applied to general normalized states, which connect the product of standard deviations to the commutator expectation and the correlation function.

If this is right

Sum uncertainty relations supply no lower bounds on the standard deviations for these states.
The uncertainty principle supplies an upper bound on the modulus of the correlation function between the observables.
A large set of such states exists beyond the eigenstates usually considered in the literature.
The product of standard deviations can serve as an upper limit on the correlation modulus in any state.

Where Pith is reading between the lines

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

These states may allow preparation of systems with vanishing correlation while keeping both variances positive.
Similar states could be searched for in concrete models such as the harmonic oscillator or angular momentum operators.
The dual role of the uncertainty principle suggests re-examining how correlation and variance trade off in quantum metrology.

Load-bearing premise

The identified states must be valid normalized quantum states obeying the standard commutation relations, with no extra constraints that would change the correlation or variance calculations.

What would settle it

An explicit normalized state |ψ⟩ for a pair of non-commuting operators A and B where the expectation value of the commutator is zero, neither variance is zero, the correlation function is zero, and direct substitution into the uncertainty relations confirms the lower bound is zero.

read the original abstract

Analyzing Heisenberg--Robertson (HR) and Schr\"{o}dinger uncertainty relations we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of non--commuting observables, $A$ and $B$, is zero, and which differ from those described in the literature. These states are not eigenstates of either the observable $A$ or $B$. The correlation function for these observables in such states is equal to zero. We have also shown that the so--called "sum uncertainty relations" also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: one is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the non--commuting observables in the state under consideration.

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 restates standard facts about uncertainty relations but adds no new states or results.

read the letter

Colleague, the core point here is that the paper does not identify new minimizing states for the uncertainty relations. It notes that states exist where the expectation of the commutator vanishes, making the Heisenberg-Robertson lower bound zero, and these states need not be eigenstates of A or B while having zero covariance. All of that follows directly from the usual derivation and has been known for a long time. The additional observation that the general relation also upper-bounds the modulus of the covariance by the product of standard deviations is likewise immediate from the Schrödinger form and requires no extra work. The paper does get the math right on these points and correctly notes that sum uncertainty relations give no lower bound when the commutator term is zero. The soft spot is that none of this is new or advances the literature. No explicit examples are constructed, no comparison to prior results is made in detail, and the framing as a 'new look' or 'additional states not described in the literature' does not hold up against standard textbook treatments. The analysis stays within the commutator-based framework without new derivations or falsifiable claims. This is the sort of note that might suit a short pedagogical comment for readers already familiar with the basics of quantum uncertainty, but it offers little to specialists. It does not deserve peer review as a full paper.

Referee Report

0 major / 2 minor

Summary. The manuscript analyzes the Heisenberg-Robertson (HR) and Schrödinger uncertainty relations for a pair of non-commuting observables A and B. It claims that there exist states (distinct from eigenstates of A or B) in which ⟨[A,B]⟩ = 0, so that the HR lower bound on σ_A σ_B vanishes, and for which the covariance cov(A,B) is also zero. The paper further states that sum uncertainty relations yield no lower bounds on the standard deviations for these states, and that the general uncertainty relation implies that the product σ_A σ_B upper-bounds |cov(A,B)|.

Significance. If the claims hold, the work offers a pedagogical clarification of the dual role of uncertainty relations (lower bound on variances versus upper bound on covariance when ⟨[A,B]⟩=0) and notes the existence of non-eigenstates where the HR bound is trivial. These observations are consistent with standard quantum mechanics and the Schrödinger relation σ_A²σ_B² ≥ cov² + (1/4)⟨[A,B]⟩², but the manuscript introduces no new derivations, explicit constructions, machine-checked proofs, or falsifiable predictions beyond known facts. Significance is therefore limited to a re-interpretation rather than a substantive advance.

minor comments (2)

[Abstract and main text] The abstract and introduction assert the existence of 'a large set of states' without providing an explicit example or construction (e.g., in the harmonic oscillator or finite-dimensional spin system). Adding at least one concrete normalized state with finite variances, zero ⟨[A,B]⟩, and zero covariance would strengthen verifiability.
[Main text (section discussing sum UR)] The discussion of sum uncertainty relations states they 'do not provide any information about lower bounds' for the identified states; a brief derivation or reference to the specific form of the sum relation used would clarify why this holds when ⟨[A,B]⟩=0.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for minor revision. Below we respond point by point to the comments in the report.

read point-by-point responses

Referee: The manuscript analyzes the Heisenberg-Robertson (HR) and Schrödinger uncertainty relations for a pair of non-commuting observables A and B. It claims that there exist states (distinct from eigenstates of A or B) in which ⟨[A,B]⟩ = 0, so that the HR lower bound on σ_A σ_B vanishes, and for which the covariance cov(A,B) is also zero. The paper further states that sum uncertainty relations yield no lower bounds on the standard deviations for these states, and that the general uncertainty relation implies that the product σ_A σ_B upper-bounds |cov(A,B)|.

Authors: The claims accurately summarize the content of the manuscript. The central observation is the existence of a class of states, distinct from eigenstates, for which the commutator expectation vanishes and the covariance is simultaneously zero. This leads to a vanishing lower bound in the HR relation and renders sum uncertainty relations uninformative for lower bounds on the standard deviations. The dual role—that the product of standard deviations also serves as an upper bound on |cov(A,B)|—follows immediately from the Schrödinger relation when ⟨[A,B]⟩=0. Our contribution is to draw explicit attention to these features and their implications. revision: no
Referee: If the claims hold, the work offers a pedagogical clarification of the dual role of uncertainty relations (lower bound on variances versus upper bound on covariance when ⟨[A,B]⟩=0) and notes the existence of non-eigenstates where the HR bound is trivial. These observations are consistent with standard quantum mechanics and the Schrödinger relation σ_A²σ_B² ≥ cov² + (1/4)⟨[A,B]⟩², but the manuscript introduces no new derivations, explicit constructions, machine-checked proofs, or falsifiable predictions beyond known facts. Significance is therefore limited to a re-interpretation rather than a substantive advance.

Authors: We agree that the underlying mathematics is contained in the standard Schrödinger relation and requires no new derivation. Nevertheless, the explicit identification of non-eigenstates satisfying both ⟨[A,B]⟩=0 and cov(A,B)=0, together with the observation that the uncertainty relation simultaneously provides an upper bound on the absolute value of the covariance, is not commonly emphasized in textbook or review presentations of uncertainty principles. We regard the pedagogical clarification of this dual aspect as the primary value of the work. No explicit constructions are provided because the argument is general and holds for any pair of observables once the stated conditions on the state are met; if the referee considers concrete examples necessary, we can add them in a revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives its claims directly from the standard Heisenberg-Robertson and Schrödinger uncertainty relations applied to states satisfying [A,B] commutators with vanishing expectation value. No parameters are fitted and then renamed as predictions, no self-citations form load-bearing steps, and no ansatz or uniqueness theorem is smuggled in. The existence of non-eigenstates with <[A,B]>=0 and cov=0 follows immediately from the definitions of variance and covariance in Hilbert space; the upper bound |cov| ≤ σ_A σ_B is an algebraic rearrangement of the Schrödinger relation itself. All steps remain self-contained against external QM benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard quantum mechanics axioms for observables and states; no free parameters, invented entities, or ad-hoc assumptions are evident from the abstract.

axioms (2)

standard math Standard quantum mechanical commutation relations [A,B] = iC for non-commuting observables
Invoked implicitly when deriving the uncertainty relations from the commutator.
domain assumption States are normalized vectors in Hilbert space with well-defined expectation values and variances
Required for defining standard deviations and correlations in the uncertainty relations.

pith-pipeline@v0.9.0 · 5709 in / 1330 out tokens · 27212 ms · 2026-05-23T17:03:51.982492+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the uncertainty relation in its most general form (14) is in fact an upper bound on the modulus of the correlation function Cϕ(A,B) ... ΔϕA · ΔϕB ≥ |Cϕ(A,B)|
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

states ... for which the lower bound of the product of the standard deviations ... is zero ... not eigenstates ... correlation function ... equal to zero

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Notes on some inequalities, resulting uncertainty relations and correlations. 1. General mathematical formalism
quant-ph 2026-04 unverdicted novelty 3.0

Generalized Schrödinger-Robertson uncertainty relations for multiple non-commuting observables are equivalently expressed using quantum Pearson correlation coefficients, with analysis of their consequences for observa...

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · cited by 1 Pith paper

[1]

Heisenberg,Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift for Physik, 43, 172 – 98 (1927); https://doi.org/10.1007/BF01397280

W. Heisenberg,Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift for Physik, 43, 172 – 98 (1927); https://doi.org/10.1007/BF01397280

work page doi:10.1007/bf01397280 1927
[2]

Heisenberg, The physical principles of the quantum theory, Dover Publications Inc., New York, N

W. Heisenberg, The physical principles of the quantum theory, Dover Publications Inc., New York, N. Y. 10014, 1930

work page 1930
[3]

H.P.Robertson, The Uncertainty Principle, Phys.Rev., 34, 163, (1929); https://doi.org/10.1103/PhysRev.34.163

work page doi:10.1103/physrev.34.163 1929
[4]

Schrödinger, Zum Heisenberschen Unschäfprinzip , Sitzungsber

E. Schrödinger, Zum Heisenberschen Unschäfprinzip , Sitzungsber. Preuss. Akad. Wiss.,XIX, 296 – 303, (1930) [Proceedings of The Prus- sian Academy of Sciences. Physics–Mathematical Section,XIX, 296- 303, (1930)]; English translation: About Heisenberg Uncertainty Rela- tion, Bulg. J. Phys.26, 193–203, (1999); arXiv: 9903100 [quant–ph]

work page 1930
[5]

Messiah,Quantum mechanics, vol

A. Messiah,Quantum mechanics, vol. I, North Holland Publ. Co., Am- sterdam, 1962

work page 1962
[6]

99, (American Mathematical Society, Providence, Rhode Island, 2009)

G.Teschl, Mathematical Methods in Quantum Mechanics (With Applica- tions to Schrödinger Operators), Graduate Studies in Mathematics Vol. 99, (American Mathematical Society, Providence, Rhode Island, 2009)

work page 2009
[7]

Aharonov, L

Y. Aharonov, L. Vaidman,Properties of a quantum system during the time interval between two measurements, Phys. Rev.,A 41, 11, (1990); https://doi.org/10.1103/PhysRevA.41.11 13

work page doi:10.1103/physreva.41.11 1990
[8]

Victor Pozsgay, Flavien Hirsch, Cyril Branciard and Nicolas Brunner, Covariance Bell inequalities, PhysicaL Review A96, 062128 (2017): doi: 10.1103/PhysRevA.96.062128

work page doi:10.1103/physreva.96.062128 2017
[9]

Andrei Khrennikov and Irina Basieva, Entanglement of Observables: Quantum Conditional Probability Approach, Foundations of Physics, (2023) 53:84 (2023); 10.1007/s10701-023-00725-7

work page doi:10.1007/s10701-023-00725-7 2023
[10]

1: Correlation and Entanglement

Bei Zeng, Xie Chen, Duan–Lu Zhou, Xiao–Gang Wen.Quantum Infor- mation Meets Quantum Matter: From Quantum Entanglement to Topo- logical Phases of Many-Body Systems, Springer Nature, 2019: Chap. 1: Correlation and Entanglement

work page 2019
[11]

L. E. Reichl.A modern course in statictical physics2nd ed., Wiley, New York, 1998

work page 1998
[12]

Rohatgi, A

Vijay K. Rohatgi, A. K. MD. Ehsanes Saleh,An Introduction to Proba- bility and Statistics, 2nd Ed., Wiley, New York, 2001

work page 2001
[13]

Lorenzo Maccone, Dagmar Bruß and Chiara Macchiavello,Complemen- tarity and Correlations, Physical Review Letters, 114, 130401 (2015); doi: 10.1103/PhysRevLett.114.130401

work page doi:10.1103/physrevlett.114.130401 2015
[14]

C. Jebarathinam, Dipankar Home and Urbasi Sinha, Pearson cor- relation coefficient as a measure for certifying and quantifying high- dimensional entanglement, Physical Review A,101, 022112 (2020); doi: 10.1103/PhysRevA.101.022112

work page doi:10.1103/physreva.101.022112 2020
[15]

Debasis Mondal, Shrobona Bagchi, and Arun Kumar Pati, Tighter uncertainty and reverse uncertainty relations, Physical Review A 95, 052117, (2017); DOI: 10.1103/PhysRevA.95.052117

work page doi:10.1103/physreva.95.052117 2017
[16]

A. K. Pati, P. K. Sahu,Sum uncertainty relation in quantum theory, Physics Letters A367, 177, (2007); doi:10.1016/j.physleta.2007.03.005

work page doi:10.1016/j.physleta.2007.03.005 2007
[17]

Maccone, L., Pati, A

L. Maccone, L., Pati, A. K., Stronger Uncertainty Relations for All Incompatible Observables, Phys. Rev. Lett.,113, 260401, (2014); https://doi.org/10.1103/PhysRevLett.113.260401

work page doi:10.1103/physrevlett.113.260401 2014
[18]

Chi–KwongLi, MikioNakahara,Unified Derivation of Uncertainty Rela- tions and Their Saturation Conditions, arXiv:2302.06002v2 [quant-ph]. 14

work page arXiv
[19]

Leifer, Uncertainty from the Aharonov–Vaidman identity, Quantum Stud.: Math

Matthew S. Leifer, Uncertainty from the Aharonov–Vaidman identity, Quantum Stud.: Math. Found., 10:373–397 (2023); https://doi.org/10.1007/s40509-023-00301-8

work page doi:10.1007/s40509-023-00301-8 2023
[20]

Rassias, Sin–Ei Takahasi,Characterization of a generalized triangle inequality in normed spaces, Nonlinear Analysis75, 735 (2012); doi:10.1016/j.na.2011.09.004

Farzad Dadipour, Mohammad Sal Moslehian, John M. Rassias, Sin–Ei Takahasi,Characterization of a generalized triangle inequality in normed spaces, Nonlinear Analysis75, 735 (2012); doi:10.1016/j.na.2011.09.004

work page doi:10.1016/j.na.2011.09.004 2012
[21]

Cerone and S

P. Cerone and S. S. Dragomir, Mathematical Inequalities, Chap. 5, (Chapman and Hall/CRC, London, 2011)

work page 2011
[22]

Chih–Yung Hsu, Sen-Yen Shawb and Han–Jin Wong,Refinements of generalized triangle inequalities, Journalof Mathematical Analysis and Applications, 344, 17 (2008)

work page 2008
[23]

Bin Chen, Shao-Ming Fei, Sum uncertainty relations for arbitrary N incompatible observables, Scientific Reports 5, 14238, (2015); doi: 10.1038/srep14238

work page doi:10.1038/srep14238 2015
[24]

Yunlong Xiao, Naihuan Jing, Xianqing Li-Jost and Shao-Ming Fei, Weighted Uncertainty Relations, Scientific Reports6, 23201 (2016); doi: 10.1038/srep23201 (2016)

work page doi:10.1038/srep23201 2016
[25]

Qiu–Cheng Song, Jun–Li Li, Guang–Xiong Peng and Cong-Feng Qiao, A Stronger Multi-observable Uncertainty Relation, Scientific Reports7, 44764 (2017); doi: 10.1038/srep44764

work page doi:10.1038/srep44764 2017
[26]

Toru Ohira,Zero-correlation entanglement, Progres in Theoretical and Experimental Physics, 013A01 (2020); doi: 10.1093/ptep/ptz149. Author contributions statement The author declares that there are no conflicts of interest regarding the pub- lication of this article and that all results presented in this article are the author’s own results. Additional in...

work page doi:10.1093/ptep/ptz149 2020

[1] [1]

Heisenberg,Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift for Physik, 43, 172 – 98 (1927); https://doi.org/10.1007/BF01397280

W. Heisenberg,Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift for Physik, 43, 172 – 98 (1927); https://doi.org/10.1007/BF01397280

work page doi:10.1007/bf01397280 1927

[2] [2]

Heisenberg, The physical principles of the quantum theory, Dover Publications Inc., New York, N

W. Heisenberg, The physical principles of the quantum theory, Dover Publications Inc., New York, N. Y. 10014, 1930

work page 1930

[3] [3]

H.P.Robertson, The Uncertainty Principle, Phys.Rev., 34, 163, (1929); https://doi.org/10.1103/PhysRev.34.163

work page doi:10.1103/physrev.34.163 1929

[4] [4]

Schrödinger, Zum Heisenberschen Unschäfprinzip , Sitzungsber

E. Schrödinger, Zum Heisenberschen Unschäfprinzip , Sitzungsber. Preuss. Akad. Wiss.,XIX, 296 – 303, (1930) [Proceedings of The Prus- sian Academy of Sciences. Physics–Mathematical Section,XIX, 296- 303, (1930)]; English translation: About Heisenberg Uncertainty Rela- tion, Bulg. J. Phys.26, 193–203, (1999); arXiv: 9903100 [quant–ph]

work page 1930

[5] [5]

Messiah,Quantum mechanics, vol

A. Messiah,Quantum mechanics, vol. I, North Holland Publ. Co., Am- sterdam, 1962

work page 1962

[6] [6]

99, (American Mathematical Society, Providence, Rhode Island, 2009)

G.Teschl, Mathematical Methods in Quantum Mechanics (With Applica- tions to Schrödinger Operators), Graduate Studies in Mathematics Vol. 99, (American Mathematical Society, Providence, Rhode Island, 2009)

work page 2009

[7] [7]

Aharonov, L

Y. Aharonov, L. Vaidman,Properties of a quantum system during the time interval between two measurements, Phys. Rev.,A 41, 11, (1990); https://doi.org/10.1103/PhysRevA.41.11 13

work page doi:10.1103/physreva.41.11 1990

[8] [8]

Victor Pozsgay, Flavien Hirsch, Cyril Branciard and Nicolas Brunner, Covariance Bell inequalities, PhysicaL Review A96, 062128 (2017): doi: 10.1103/PhysRevA.96.062128

work page doi:10.1103/physreva.96.062128 2017

[9] [9]

Andrei Khrennikov and Irina Basieva, Entanglement of Observables: Quantum Conditional Probability Approach, Foundations of Physics, (2023) 53:84 (2023); 10.1007/s10701-023-00725-7

work page doi:10.1007/s10701-023-00725-7 2023

[10] [10]

1: Correlation and Entanglement

Bei Zeng, Xie Chen, Duan–Lu Zhou, Xiao–Gang Wen.Quantum Infor- mation Meets Quantum Matter: From Quantum Entanglement to Topo- logical Phases of Many-Body Systems, Springer Nature, 2019: Chap. 1: Correlation and Entanglement

work page 2019

[11] [11]

L. E. Reichl.A modern course in statictical physics2nd ed., Wiley, New York, 1998

work page 1998

[12] [12]

Rohatgi, A

Vijay K. Rohatgi, A. K. MD. Ehsanes Saleh,An Introduction to Proba- bility and Statistics, 2nd Ed., Wiley, New York, 2001

work page 2001

[13] [13]

Lorenzo Maccone, Dagmar Bruß and Chiara Macchiavello,Complemen- tarity and Correlations, Physical Review Letters, 114, 130401 (2015); doi: 10.1103/PhysRevLett.114.130401

work page doi:10.1103/physrevlett.114.130401 2015

[14] [14]

C. Jebarathinam, Dipankar Home and Urbasi Sinha, Pearson cor- relation coefficient as a measure for certifying and quantifying high- dimensional entanglement, Physical Review A,101, 022112 (2020); doi: 10.1103/PhysRevA.101.022112

work page doi:10.1103/physreva.101.022112 2020

[15] [15]

Debasis Mondal, Shrobona Bagchi, and Arun Kumar Pati, Tighter uncertainty and reverse uncertainty relations, Physical Review A 95, 052117, (2017); DOI: 10.1103/PhysRevA.95.052117

work page doi:10.1103/physreva.95.052117 2017

[16] [16]

A. K. Pati, P. K. Sahu,Sum uncertainty relation in quantum theory, Physics Letters A367, 177, (2007); doi:10.1016/j.physleta.2007.03.005

work page doi:10.1016/j.physleta.2007.03.005 2007

[17] [17]

Maccone, L., Pati, A

L. Maccone, L., Pati, A. K., Stronger Uncertainty Relations for All Incompatible Observables, Phys. Rev. Lett.,113, 260401, (2014); https://doi.org/10.1103/PhysRevLett.113.260401

work page doi:10.1103/physrevlett.113.260401 2014

[18] [18]

Chi–KwongLi, MikioNakahara,Unified Derivation of Uncertainty Rela- tions and Their Saturation Conditions, arXiv:2302.06002v2 [quant-ph]. 14

work page arXiv

[19] [19]

Leifer, Uncertainty from the Aharonov–Vaidman identity, Quantum Stud.: Math

Matthew S. Leifer, Uncertainty from the Aharonov–Vaidman identity, Quantum Stud.: Math. Found., 10:373–397 (2023); https://doi.org/10.1007/s40509-023-00301-8

work page doi:10.1007/s40509-023-00301-8 2023

[20] [20]

Rassias, Sin–Ei Takahasi,Characterization of a generalized triangle inequality in normed spaces, Nonlinear Analysis75, 735 (2012); doi:10.1016/j.na.2011.09.004

Farzad Dadipour, Mohammad Sal Moslehian, John M. Rassias, Sin–Ei Takahasi,Characterization of a generalized triangle inequality in normed spaces, Nonlinear Analysis75, 735 (2012); doi:10.1016/j.na.2011.09.004

work page doi:10.1016/j.na.2011.09.004 2012

[21] [21]

Cerone and S

P. Cerone and S. S. Dragomir, Mathematical Inequalities, Chap. 5, (Chapman and Hall/CRC, London, 2011)

work page 2011

[22] [22]

Chih–Yung Hsu, Sen-Yen Shawb and Han–Jin Wong,Refinements of generalized triangle inequalities, Journalof Mathematical Analysis and Applications, 344, 17 (2008)

work page 2008

[23] [23]

Bin Chen, Shao-Ming Fei, Sum uncertainty relations for arbitrary N incompatible observables, Scientific Reports 5, 14238, (2015); doi: 10.1038/srep14238

work page doi:10.1038/srep14238 2015

[24] [24]

Yunlong Xiao, Naihuan Jing, Xianqing Li-Jost and Shao-Ming Fei, Weighted Uncertainty Relations, Scientific Reports6, 23201 (2016); doi: 10.1038/srep23201 (2016)

work page doi:10.1038/srep23201 2016

[25] [25]

Qiu–Cheng Song, Jun–Li Li, Guang–Xiong Peng and Cong-Feng Qiao, A Stronger Multi-observable Uncertainty Relation, Scientific Reports7, 44764 (2017); doi: 10.1038/srep44764

work page doi:10.1038/srep44764 2017

[26] [26]

Toru Ohira,Zero-correlation entanglement, Progres in Theoretical and Experimental Physics, 013A01 (2020); doi: 10.1093/ptep/ptz149. Author contributions statement The author declares that there are no conflicts of interest regarding the pub- lication of this article and that all results presented in this article are the author’s own results. Additional in...

work page doi:10.1093/ptep/ptz149 2020