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Uncertainty relation for the position of an electron in a uniform magnetic field from quantum estimation theory

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Pith's one-line read The paper shows that estimating both position coordinates of an electron in a uniform magnetic field obeys a nontrivial uncertainty trade-off even though the position operators commute.

desk verdict A clean and honest model calculation showing that commuting position observables can have a non-trivial estimation-theoretic trade-off; the thermal-state transition is real but the gauge-independence claim stays unproved. read the letter →

arxiv 1908.04868 v2 pith:N4DK5BDM submitted 2019-08-13 quant-ph

Shin Funada , Jun Suzuki This is my paper · ORCID
classification quant-ph
keywords quantumestimationtheoryCramér-RaoboundLandaulevelsuncertaintyrelationangularmomentumGaussianshiftmodelsymmetriclogarithmicderivativeright
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 claims that the usual Heisenberg–Robertson uncertainty relation says nothing about the joint position of an electron in a uniform magnetic field, because the position operators $X$ and $Y$ commute. It then shows that a two-parameter quantum estimation formulation produces a meaningful trade-off for the mean square error matrix of unbiased position estimates. For a pure lowest-Landau-level state, the canonical-momentum shift model gives a floor $V_{11},V_{22}\geq \lambda^2/2$ with no trade-off, while the mechanical-momentum model gives a hyperbolic trade-off $(V_{11}-\lambda^2/4)(V_{22}-\lambda^2/4)\geq \lambda^4/16$. For a thermal state with fixed angular momentum, the trade-off is governed by both the SLD and RLD Cramér–Rao bounds and changes shape at $|\langle L\rangle_0|=1/2$. This establishes that uncertainty relations can exist for commuting observables and that the choice of shift generator changes the achievable estimation accuracy.

What carries the argument

The load-bearing object is the mean square error matrix $V$ for the two parameters $\theta_1,\theta_2$ together with the two quantum Cramér–Rao inequalities. For any two-parameter model the inequality $V\ge G^{-1}$ yields component lower bounds and a product inequality involving $|\operatorname{Im} g_{12}|$, where $G^{-1}=[g_{ij}]$. The authors compute the SLD and RLD Fisher information matrices for the two unitary shift models; the imaginary off-diagonal part of the inverse Fisher information is what creates a trade-off between $V_{11}$ and $V_{22}$. In Model 2 the noncommuting mechanical momenta introduce a phase factor in the shifted wavefunction, which produces that imaginary off-diagonal term and hence the hyperbolic bound; in the pure-state Model 1 the term vanishes, so only independent component floors remain.

What would settle it

Prepare a thermal Landau electron with a known $\langle L\rangle_0$, estimate the two position shifts with unbiased measurements, and record the achievable mean square error pairs. The paper predicts that for $|\langle L\rangle_0|\le 1/2$ the SLD floor $V_{11},V_{22}\ge g_{11}^S$ is the relevant bound, while for $|\langle L\rangle_0|>1/2$ no estimator should enter the region below both the SLD lines and the RLD hyperbola. Finding a measurement with a mean square error pair below the predicted union of bounds would falsify the claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the mean square error matrix $V$ for estimating the two position-shift parameters is bounded by explicit quantum Cramér–Rao inequalities that depend on which momenta generate the shift. In the pure-state case the reference state is the $|0,0\rangle$ lowest Landau level with zero angular momentum. Model 1, generated by the commuting canonical momenta $p_x,p_y$, is quasi-classical: its SLD bound is $V_{11},V_{22}\ge \lambda^2/2$, and because its inverse Fisher information has no imaginary off-diagonal entry there is no trade-off between the two variances. Model 2, generated by the noncommuting mechanical momenta $\pi_x,\pi_y$, is a Gaussian shift model whose generalized RLD bound is $(V_{11}-\lambda^2/4)(V_{22}-\lambda^2/4)\ge \lambda^4/16$ and is achievable. With a thermal reference state constrained to fixed $\langle L\rangle_0$, Model 1's bound is determined by both the SLD and RLD inequalities: for $|\langle L\rangle_0|\le 1/2$ the SLD bound dominates, while for $|\langle L\rangle_0|>1/2$ both contribute and the bounds intersect twice; the bound is achievable only at $\langle L\rangle_0=0$. Model 2 remains a Gaussian shift model, so its RLD bound is achievable. The paper also shows that Model 2's bound is lower than Model 1's in both pure and thermal settings.

Load-bearing premise

The thermal-state analysis assumes that a degenerate thermal equilibrium is uniquely determined by fixing the expectation value of angular momentum with a chemical potential and that one root of the resulting equation is unphysical; if a preparation does not enforce this constraint, the derived bounds and the crossover at $|\langle L\rangle_0|=1/2$ need not apply.

Editorial extensions

If this is right

  • Joint position estimation in a Landau system is limited by a genuine trade-off even though $X$ and $Y$ commute, so the Heisenberg–Robertson relation is not the right diagnostic for this sensing problem.
  • The mechanical-momentum encoding (Model 2) gives a smaller, achievable Cramér–Rao bound than the canonical-momentum encoding in both pure and thermal regimes, so position measurements that exploit kinetic momentum can in principle be more accurate.
  • In the thermal canonical-momentum model the uncertainty relation changes shape at $|\langle L\rangle_0|=1/2$, giving a signature of the angular momentum in the achievable estimation accuracy.
  • For $\langle L\rangle_0\neq0$ the thermal Model 1 bound is not achievable, so optimal joint position estimation requires a strategy beyond separate position measurements; at $\langle L\rangle_0=0$ the SLD bound is achievable.

Reading between the lines

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

  • Beyond the paper, the same estimation-theoretic construction should yield trade-offs for any pair of commuting observables that are linear combinations of noncommuting canonical variables; the Landau system is a concrete instance of a more general recipe.
  • The crossover at $|\langle L\rangle_0|=1/2$ may reflect a qualitative change in the set of physical thermal preparations; a natural check is whether the same threshold appears in other degenerate harmonic-oscillator systems with a conserved angular momentum.
  • Beyond the paper, the Model 2 advantage suggests an experimental direction: encode position information through kinetic-momentum displacements in a two-dimensional electron gas or a two-mode linear optical system, keeping in mind that the optimal measurement may require noncanonical variables.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 6 minor

Summary. This paper studies two-parameter quantum estimation of the position shift of a single electron in a uniform magnetic field, using either the canonical momenta (Model 1) or the mechanical momenta (Model 2) as displacement generators. For a pure lowest-Landau-level reference state, the authors derive quantum Cramér-Rao bounds for the mean-square-error matrix: Model 1 gives V11,V22 ≥ λ²/2 with no trade-off, while Model 2 gives (V11−λ²/4)(V22−λ²/4) ≥ λ⁴/16. For thermal reference states with a fixed expectation value of the angular momentum, Model 1 exhibits a bound determined by both the symmetric and right logarithmic derivative Cramér-Rao bounds, with a structural transition at |⟨L⟩0| = 1/2, and Model 2 is identified with a Gaussian shift model whose RLD bound is achievable. The paper concludes that quantum estimation theory can yield non-trivial uncertainty relations for two commuting observables.

Significance. If the technical issues below are resolved, the paper makes a useful conceptual contribution: it demonstrates explicitly that a trade-off relation for two commuting observables can arise from multi-parameter quantum estimation theory, in contrast to the Heisenberg-Robertson relation. The appendices provide detailed and largely checkable computations of the relevant Fisher information matrices, and the pure-state results are physically transparent. The authors are also careful to identify which bounds are achievable and which are not (e.g., Model 1 thermal bounds are not tight except at ⟨L⟩0 = 0). The thermal-state results are conditional on the explicit fixed-⟨L⟩0 grand-canonical ensemble, which is a stated modeling choice. The significance is moderate: the work is a clear application of known quantum estimation tools to a specific physical model, rather than a new general theorem.

major comments (1)
  1. [III A 2 and B 2 b] The matrix \tilde G^π_R reported in Appendix B 2 b is singular: det[[1,i],[-i,1]] = 0. Consequently ( \tilde G^π_R )^{-1} does not exist, and Eq. (29) cannot be the inverse of \tilde G^π_R as written. This affects the derivation of the central pure-state bound Eq. (30). The authors should clarify whether Eq. (29) is intended as a Moore-Penrose pseudoinverse and derive the bound accordingly, or present Eq. (30) as the zero-temperature limit of the thermal bound Eq. (51), which is non-singular for κ_a² > 0. The final inequality appears correct, but the derivation in the pure-state section is not.
minor comments (6)
  1. [II A] The assertion that the choice of gauge gives no change in the quantum Fisher information for a uniform magnetic field is stated without proof or reference; please provide a derivation or a citation.
  2. [IV A, Eq. (37)] The statement that one solution of Eq. (37) is unphysical because it gives a negative temperature state is not demonstrated; a short proof that the discarded branch violates μ>0 and βω>μ would strengthen the exposition.
  3. [IV B 1, Eq. (47)] The off-diagonal sign in the displayed matrix for (G_S)^{-1} − (G_R)^{-1} appears to be the opposite of what one obtains from the preceding explicit formulas; the condition |⟨L⟩0| ≤ 1/2 is unaffected, but the sign should be corrected.
  4. [Figure 1 caption] The caption's last sentence appears to confuse Model 1 and Model 2; please clarify which allowed region corresponds to which model.
  5. [IV C 1] The relation ξ2 = ξ*1 is inconsistent with Eq. (A10), where the second factor is e^{ξ* b† − ξ b}; either ξ2 = ξ or the expression should be adjusted.
  6. [Eq. (28)] The Gaussian exponent in Eq. (28) is missing parentheses; it should read exp[ −((x−θ1)² + (y−θ2)²)/λ² ].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the uncertainty relations are derived by direct application of standard quantum Cramér-Rao inequalities to explicit state families, with no fitted parameters renamed as predictions.

full rationale

The derivation chain is self-contained. The pure-state bounds (Eqs. (18)-(19), (29)-(30)) follow from computing SLD and generalized RLD Fisher information matrices for the explicit unitary shift models (3) and (6); the matrices are obtained by elementary operator algebra in Appendix B and are not fitted to any data. The thermal-state bounds (Eqs. (44)-(51)) likewise come from direct evaluation of RLD and SLD operators for the grand-canonical state (31) with fixed ⟨L⟩0; the chemical potential is solved from Eq. (37), and the transition at |⟨L⟩0|=1/2 is an algebraic consequence of the matrix inequality (G_S^{p thermal})^{-1} - (G_R^{p thermal})^{-1} ≥ 0. No parameter is estimated from a subset of data and then used to predict a closely related quantity. The self-citations ([27], [32], by one of the authors) are used only as external, independently published mathematical criteria for D-invariance and SLD achievability; the main lower bounds do not depend on them, and the paper explicitly discloses where its bounds are not tight (Model 1 thermal case, except ⟨L⟩0=0). Identification of Model 2 with the known Gaussian shift model is a stated reduction to an existing result, not a circular renaming. Accordingly, no load-bearing circular step was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard quantum estimation theory and on specific modeling choices for the Landau system. No new entities are introduced; the only model-specifying ingredient beyond the Hamiltonian is the fixed-angular-momentum chemical potential used to select a thermal reference state.

assumptions (5)
  • domain assumption The symmetric gauge A = B(-y/2, x/2, 0) is adopted, and gauge choice is claimed not to affect quantum Fisher information (Sec. II A).
    This justifies using one vector potential for all calculations; the proof is not shown, only asserted.
  • domain assumption Ladder operators a,b and vacuum state |0,0> describe the lowest Landau level with zero angular momentum (Secs. II A-II B).
    The pure-state results depend on the reference state being the LLL Gaussian ground state.
  • standard math The Baker-Campbell-Hausdorff formula and Gaussian phase-space integrals are used to evaluate shifts and thermal states (Eqs. (26), (35), (B5)).
    Standard mathematical tools; no new mathematics is introduced.
  • standard math Existence and properties of SLD, RLD, and generalized RLD Fisher information and the D-invariance achievability criterion are taken from Fujiwara-Nagaoka, Suzuki, Yuen-Lax.
    The paper applies known results rather than re-deriving them.
  • ad hoc to paper The thermal state is fixed by the constraint <L>0 = constant through a chemical potential μ, and the physically relevant solution of Eq. (37) is selected (Sec. IV A).
    This constraint is introduced to resolve degeneracy of Landau levels; the paper calls it a chemical potential but it is a modeling choice.

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Pith. "Pith review of Uncertainty relation for the position of an electron in a uniform magnetic field from quantum estimation theory." pith.science (2026). https://pith.science/paper/N4DK5BDM

@misc{pith2026190804868,
  author       = {Pith},
  title        = {Pith review of: Uncertainty relation for the position of an electron in a uniform magnetic field from quantum estimation theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N4DK5BDM}},
  note         = {Machine review of arXiv:1908.04868}
}
read the original abstract

We investigate the uncertainty relation for estimating the position of one electron in a uniform magnetic field in the framework of the quantum estimation theory. Two kinds of momenta, canonical one and mechanical one, are used to generate a shift in the position of the electron. We first consider pure state models whose wave function is in the ground state with zero angular momentum. The model generated by the two-commuting canonical momenta becomes the quasi-classical model, in which the symmetric logarithmic derivative quantum Cram\'er-Rao bound is achievable. The model generated by the two non-commuting mechanical momenta, on the other hand, turns out to be a Gaussian model, where the generalized right logarithmic derivative quantum Cram\'er-Rao bound is achievable. We next consider mixed-state models by taking into account the effects of thermal noise. The model with the canonical momenta now becomes genuine quantum mechanical, although its generators commute with each other. The derived uncertainty relationship is in general determined by two different quantum Cram\'er-Rao bounds in a non-trivial manner. The model with the mechanical momenta is identified with the well-known Gaussian shift model, and the uncertainty relation is governed by the right logarithmic derivative Cram\'er-Rao bound.

Figures

Figures reproduced from arXiv: 1908.04868 by the authors.

Figure 1
Figure 1. FIG. 1: The uncertainty relation of Model 1 and Model 2 given by the inequalities Eqs. (18, 19). The allowed region of Model [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. shows µ as a function of hLi0 at βω = 0.1, 1, and 5 from top to bottom. The chemical potential µ as a function of hLi0 diverges for hLi0 ≥ 0 as βω goes to infinity, i.e., the zero temperature limit. At a special case, hLi0 = 0, we see µ = βω/2 from Eq. (38). Explicitly, the zero temperature limit is lim β→∞ µ =    ∞ (hLi0 ≥ 0) log h hLi0−1 hLi0 i (hLi0 < 0) . (39) For Model 2, the two-parameter family of the stat… view at source ↗
Figure 3
Figure 3. FIG. 3: Uncertainty relation of Model 1 and Model 2 given by the quantum Cram´er-Rao inequalites. The temperature [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: shows ∆V R−S as a function of hLi0 at three different βω’s which are the same as [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

62 extracted references · 50 canonical work pages

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    RLD LR,i (θ) and RLD Fisher information matrix: GR(θ) RLD LR,i (θ) is given as a solution of the equation below if one exists. ∂ρθ ∂θi =ρθLR,i (θ). The RLD Fisher information matrix GR(θ) = [gR,ij (θ)] is defined by gR,ij (θ) = tr [ρθLR,j (θ)L† R,i (θ)]. (20)

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    Generalized RLD In general, the RLD does not exist when a pure state is the reference state [21]. We can show that this holds for Model 1 and Model 2. Instead of the RLD Fisher information matrix, we are able to obtain the generalized RLD Fisher information matrix by the method introduced by [21]. Let the generalized RLD Fisher information matrix ˜GR be ˜...

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    Z matrix LS i(θ) is defined by LS i(θ) = ∑ j gS ji(θ)LS,j (θ). where G−1 S (θ) = [gSij(θ)]. Then, Z matrix, Z(θ) = [zij(θ)] is defined by zij(θ) = tr [ρθLj S (θ)Li† S (θ)]. It is worth noting the relationship between Z matrix and the expectation value of the commutator of SLD’s, tr(ρ0[LS,i (θ), LS,j (θ)]) [27]. By using the ( i,j ) component of the Z matrix...

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    Reference state Since the energy eigenstate of Hamiltonian (8) is infinitely degenerated, we choose the tensor product of the vacuum states as the reference state ρ0 which is denoted by ρ0 =|0⟩aa⟨0|⊗| 0⟩bb⟨0| =|0, 0⟩⟨ 0, 0|. (24)

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    Unitary transformations We introduce two kinds of unitary transformations, e −iθ1pxe−iθ2py and e−iθ1πxe−iθ2πy. We consider that we have them act on the LLL, ψ00(x,y ). Since we have e−iθ1pxe−iθ2pyψ00(x,y ) =ψ00(x−θ1,y−θ2), (25) this unitary transformation e−iθ1pxe−iθ2py makes a shift in x−y coordinate of ψ00(x,y ) from (x, y) to (x−θ1,y−θ2). We also have ...

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