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Minimum uncertainty states and squeezed states from the sum uncertainty relation

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Pith's one-line read The minimum uncertainty states for the sum uncertainty relation are always the minimum uncertainty states for the product uncertainty relation.

desk verdict The paper recovers the usual coherent and squeezed minimizers via variational calc on the sum relation for two operator pairs, but the 'always' claim has no general derivation. read the letter →

arxiv 2407.16530 v2 pith:6LIRJDH6 submitted 2024-07-23 quant-ph

Yatindra Kumar , Yashraj Jha , Namrata Shukla This is my paper
classification quant-ph
keywords sumuncertaintyrelationproductminimumstatessqueezedcoherentposition-momentumangularmomentumvariationalapproach
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

The paper examines minimum uncertainty states for the sum uncertainty relation using a variational approach on the position-momentum pair and angular momentum operators. It establishes that these states coincide exactly with the minimum uncertainty states already known from the standard product uncertainty relation. The analysis further shows that coherent and squeezed states of radiation achieve the bound for the sum relation without any modification. A sympathetic reader would care because the result indicates that the stronger sum relation does not generate new extremal states beyond those already identified by the product relation.

What carries the argument

Variational minimization of the sum uncertainty relation, which identifies states saturating its bound and shows they also saturate the product bound.

What would settle it

Discovery of a quantum state that achieves the minimum value of the sum uncertainty relation but fails to achieve the minimum of the product uncertainty relation for position and momentum would falsify the claim.

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Extended reading notes

Core claim

The minimum uncertainty states for the sum uncertainty relation are always the minimum uncertainty states for the traditional product uncertainty relation, as demonstrated by variational minimization applied to the position-momentum pair and angular momentum operators. Coherent and squeezed states of radiation remain completely unaffected by the sum uncertainty relation.

Load-bearing premise

The variational minimization performed on the sum uncertainty relation for the position-momentum and angular momentum pairs captures the global minimum and extends to other cases.

Editorial extensions

If this is right

  • Coherent states minimize both the sum and product uncertainty relations.
  • Squeezed states also minimize the sum uncertainty relation without adjustment.
  • No new minimum uncertainty states arise from the sum relation in the examined operator pairs.
  • The sum relation provides a stronger numerical bound but shares the same extremal states as the product relation.

Reading between the lines

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

  • The equivalence may suggest that the sum relation's advantage lies in its tighter bound value rather than in redefining the states that achieve it.
  • If the pattern holds for other operator pairs, the sum relation could be used interchangeably with the product relation when identifying minimum uncertainty states.
  • Quantum optics applications involving coherent or squeezed light may require no revision to account for the sum relation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The manuscript claims that the minimum-uncertainty states of the Maccone-Pati sum uncertainty relation coincide with those of the Heisenberg-Robertson product uncertainty relation. This is asserted on the basis of a variational minimization performed on the sum relation for the position-momentum pair and for angular-momentum operators, from which the authors conclude that coherent and squeezed states remain the minimizers and are unaffected by the sum relation.

Significance. If the claim holds, the result would imply that the stronger sum uncertainty relation does not generate new classes of minimum-uncertainty states beyond those already identified by the product relation. The concrete variational calculations for the two operator pairs constitute a modest positive contribution, but the absence of a general argument or additional operator families limits the significance.

major comments (2)
  1. [Abstract] Abstract: the universal claim that the minimum-uncertainty states 'are always' those of the product relation is load-bearing yet unsupported by any derivation showing that the stationarity condition obtained from varying ΔA² + ΔB² is mathematically equivalent to that obtained from varying ΔA ΔB for arbitrary non-commuting A and B; only two specific pairs are examined.
  2. [Variational approach] The variational minimization (described after the abstract): no demonstration is given that the observed coincidence with the known product minimizers (coherent/squeezed states) is the global minimum rather than a local extremum, nor is an error analysis or completeness check for the operator examples supplied.
minor comments (1)
  1. [Abstract] The abstract statement that the sum relation is 'claimed to be stronger' would benefit from a brief reminder of the precise sense in which Maccone-Pati is stronger than Heisenberg-Robertson.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond point by point to the major comments and agree that the manuscript's phrasing requires qualification to reflect the limited scope of the calculations.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the universal claim that the minimum-uncertainty states 'are always' those of the product relation is load-bearing yet unsupported by any derivation showing that the stationarity condition obtained from varying ΔA² + ΔB² is mathematically equivalent to that obtained from varying ΔA ΔB for arbitrary non-commuting A and B; only two specific pairs are examined.

    Authors: We accept the point. The abstract and main text use the word 'always' on the basis of explicit variational calculations performed only for the position-momentum pair and for angular-momentum operators. No general proof is given that the stationarity conditions derived from δ(ΔA² + ΔB²) = 0 are equivalent to those from δ(ΔA ΔB) = 0 for arbitrary non-commuting operators. We will revise the abstract, introduction, and conclusion to remove the universal claim and to state clearly that the coincidence of minimizers is demonstrated for the two operator families examined. revision: yes

  2. Referee: [Variational approach] The variational minimization (described after the abstract): no demonstration is given that the observed coincidence with the known product minimizers (coherent/squeezed states) is the global minimum rather than a local extremum, nor is an error analysis or completeness check for the operator examples supplied.

    Authors: The variational procedure sets the first variation of the sum uncertainty to zero and obtains differential equations whose solutions coincide with the known coherent and squeezed states. This shows that those states are stationary points for the sum relation, but the calculation does not include a second-variation test, a search over a larger function space, or error estimates to establish that the stationary point is the global minimum. We will insert a short paragraph acknowledging these limitations of the variational method and noting that the identification rests on matching to the independently established global minimizers of the product relation. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; claim rests on explicit variational calculations for specific cases

full rationale

The paper performs a variational minimization directly on the Maccone-Pati sum uncertainty relation for the position-momentum pair and for angular-momentum operators, then observes that the resulting states coincide with the known coherent and squeezed states that minimize the Heisenberg-Robertson product relation. No equation is defined in terms of its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise is justified solely by self-citation. The derivation chain is therefore self-contained against the explicit stationarity conditions obtained in the two examples; the universal phrasing 'always' is an extrapolation rather than a definitional reduction.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The claim rests on the standard axioms of quantum mechanics (Hilbert-space operators, commutators) plus the assumption that the variational method locates the true global minimum for the sum functional; no free parameters or new entities are introduced in the abstract.

assumptions (2)
  • standard math Quantum states are represented by vectors in Hilbert space and observables by self-adjoint operators whose commutators define uncertainty bounds.
    Invoked implicitly when applying the sum uncertainty relation to position-momentum and angular-momentum operators.
  • domain assumption The variational method applied to the sum uncertainty functional yields its global minimum for the chosen operator pairs.
    Central to the claim that the resulting states coincide with product-relation minima.

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Cite this review

Pith. "Pith review of Minimum uncertainty states and squeezed states from the sum uncertainty relation." pith.science (2026). https://pith.science/paper/6LIRJDH6

@misc{pith2026240716530,
  author       = {Pith},
  title        = {Pith review of: Minimum uncertainty states and squeezed states from the sum uncertainty relation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6LIRJDH6}},
  note         = {Machine review of arXiv:2407.16530}
}
read the original abstract

Heisenberg uncertainty relation is at the origin of understanding minimum uncertainty states and squeezed states of light. In the recent past, sum uncertainty relation was formulated by Maccone and Pati [Maccone and Pati, Phys. Rev. Lett. 113, 260401 (2014)] which is claimed to be stronger than the existing Heisenberg-Robertson product uncertainty relation. We analyze the minimum uncertainty states for the sum uncertainty relation using the variational approach. We claim that the minimum uncertainty states for the sum uncertainty relation are always the minimum uncertainty states for the traditional product uncertainty relation, using the example of position-momentum pair as well as angular momentum operators. We show that the coherent and squeezed states of radiation remain completely unaffected by the sum uncertainty relation.

Figures

Figures reproduced from arXiv: 2407.16530 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

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