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Robustness of Noether's principle: Maximal disconnects between conservation laws and symmetries in quantum theory

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Pith's one-line read Rotational symmetry does not force spin conservation in quantum channels.

desk verdict A substantial paper on SU(2)-covariant channels with a real, localized error in the unequal-spin inversion formula; the core results survive. read the letter →

arxiv 1908.04254 v2 pith:QRTHKC6O submitted 2019-08-12 quant-ph

Cristina Cirstoiu , Kamil Korzekwa , David Jennings This is my paper · ORCID
classification quant-ph
keywords Noether'stheoremquantumchannelsSU(2)-covariantspininversionamplificationunitarityconservationlawsangularmomentum
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 asks how much a symmetry constraint on a quantum channel forces a conservation law, and answers: far less than for unitary dynamics. It proves that every rotationally symmetric SU(2)-covariant channel between spin systems simply scales spin polarization isotropically, and it finds the best physical approximations to two otherwise impossible operations, spin inversion and spin amplification, with explicit dimension-dependent factors. It then bounds how far any symmetric channel can violate angular momentum conservation in terms of its unitarity, the amount by which the channel departs from closed unitary evolution. The upshot is a quantitative robustness version of Noether's principle: for systems with multiplicity-free symmetry representations, approximate conservation holds if and only if the symmetric dynamics is close to a symmetric unitary.

What carries the argument

The central object is the simplex of SU(2)-irreducibly covariant channels: any such channel is a convex combination of extremal split-and-discard channels $\mathcal{E}_L$ labelled by $L\in\{|j_A-j_B|,\dots,j_A+j_B\}$, each realized by splitting the input spin into a $j_B$ part and an $L$ part and discarding the latter. Each channel is described by a scaling vector $f_l(\mathcal{E})$ acting on irreducible tensor operators, with the $l=1$ component governing spin polarization. The regulator for openness is unitarity, defined as the average output purity with the identity component removed, and the paper connects it to the purity of the Choi-Jamiolkowski state and to the complementary channel. The trade-off theorems work by expressing both unitarity and the deviation $\Delta(\mathcal{E})$ in terms of the probability distribution $p_L$ over the simplex.

What would settle it

Numerically optimize the polarization scaling factor $f_1(\mathcal{E})$ over all SU(2)-covariant channels with fixed $j_A,j_B$ by semidefinite programming on the Choi matrix; if any channel beats the claimed $\kappa_+$ or $\kappa_-$ values, the simplex characterization is incomplete and the fundamental limits fail.

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Extended reading notes

Core claim

For SU(2)-irreducibly covariant channels between spin-$j_A$ and spin-$j_B$ systems, the convex set is a simplex whose extremal points are split-and-discard channels $\mathcal{E}_L$. On the polarization vector the channel acts as $\mathbf{P}\to f(\mathcal{E})\mathbf{P}$, and optimizing $f$ over the simplex gives the maximal inversion factor $\kappa_-=-j_B(2j_B+1)/((j_A+1)(2j_A+1))$ and the maximal amplification factors $\kappa_+=j_B/j_A$ when $j_A\ge j_B$ and $\kappa_+=(j_B+1)/(j_A+1)$ when $j_A<j_B$. For equal spins $j_A=j_B=j$, the optimal inversion factor is $-j/(j+1)$, approaching $-1$ for large spins, while the fidelity of the optimal channel to the passive time-reversal operation approaches only $1/2$. The paper also proves two-sided bounds on the average deviation $\sqrt{\Delta(\mathcal{E})}$ from angular momentum conservation in terms of $1-u(\mathcal{E})$, showing that small deviation is equivalent to closeness to a symmetric unitary for such systems.

Load-bearing premise

The claimed limits assume that every rotationally symmetric channel is a convex combination of the known split-and-discard extremal channels, so if any additional extremal channel exists, the fundamental limits could be beaten.

Editorial extensions

If this is right

  • Rotational symmetry alone does not conserve angular momentum in open dynamics: a spin can be inverted by a factor down to $-j/(j+1)$ for equal spins, or amplified by $j_B/j_A$ or $(j_B+1)/(j_A+1)$ depending on which spin is larger.
  • For spin-$j$ systems, approximate angular momentum conservation is equivalent to the channel being close to a symmetric unitary, with explicit two-sided bounds of order $j^{-1/2}$ and $j^{3/2}$ on $\sqrt{\Delta(\mathcal{E})}$ in terms of $1-u(\mathcal{E})$.
  • For any connected compact Lie group, a symmetric channel close to a symmetric isometry approximately conserves the relevant charges, but a converse lower bound exists only when the symmetry representation is multiplicity-free.
  • The complementary channel of the optimal spin-inversion channel is a maximal spin-amplification channel, linking spin inversion, spin amplification, and the flow of angular momentum to the environment.
  • Because unitarity is experimentally estimable through randomized benchmarking, the bounds turn symmetry tests into practical diagnostics for quantum devices.

Reading between the lines

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

  • Beyond the paper: if the simplex characterization of SU(2)-covariant channels remains complete for reducible multi-particle spin systems, the same polarization-scaling formulas would give testable limits for collective spin manipulations in atomic ensembles.
  • Beyond the paper: the fact that optimal spin inversion approaches $-1$ while its fidelity to time-reversal approaches only $1/2$ suggests that polarization measurements alone understate how far a channel is from true time reversal; one could probe this by comparing higher-rank tensor components of the output state.
  • Beyond the paper: the bounds are likely not tight for continuous-time Markovian dynamics; a concrete extension would be to recompute Theorem 13 with Lindblad generators and see whether the $j^{3/2}$ coefficient in the upper bound can be reduced.
  • Beyond the paper: the absence of a lower bound in the U(1) case means that energy-conserving dephasing channels can hide decoherence completely; this could be used to benchmark whether an allegedly energy-conserving device is actually implementing unitary dynamics.
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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 / 3 minor

Summary. This paper quantifies how much a symmetry constraint on a quantum channel constrains the expectation values of the symmetry generators, thereby quantifying the failure of Noether's theorem for open dynamics. The authors characterize the convex structure of SU(2)-irreducibly covariant channels as a simplex spanned by split-and-discard extremal channels E_L, derive the scaling of spin polarization under such channels, and use this to give fundamental bounds on spin inversion and spin amplification. They also introduce unitarity-based trade-off inequalities, including a general upper bound for connected Lie groups (Theorem 11), lower bounds for multiplicity-free systems (Theorem 12), and explicit SU(2) equal-spin bounds (Theorem 13). The paper closes with applications to benchmarking, thermodynamics, measurement theory, and Hamiltonian simulation. The main technical apparatus is self-contained: the simplex structure follows from Schur's lemma applied to the Jamiolkowski state, and the channel scaling factors are computed via Clebsch-Gordan coefficients inside the paper.

Significance. If the technical claims are corrected, the paper is a substantial contribution. It connects symmetry principles and conservation laws for channels through an experimentally accessible quantity, unitarity, and it provides fundamental limits for spin inversion and spin amplification. The equal-spin inversion factor f = -j/(j+1), the qubit U(1) bound, and the spin-amplification formulas reproduce known special cases and pass spot-checks. The simplex characterization itself is sound because the relevant tensor-product representation is multiplicity-free. However, the paper's central general inversion formula for unequal spins is incorrect as printed, and this must be fixed before the stated results can be relied upon.

major comments (1)
  1. [Sec. VI.B, Theorem 9, Eq. (124), and Fig. 2] The printed inversion factor is inconsistent with the paper's own Eqs. (77) and (122). Combining Eq. (77) with L = j_A + j_B and the norm conversion in Eq. (122) gives kappa_- = -j_B/(j_A+1), not -j_B(2j_B+1)/[(j_A+1)(2j_A+1)]. For (j_A,j_B) = (1/2,1), the printed formula gives -1, while Eq. (122) gives -2/3; direct evaluation of E_{3/2}(|1/2,1/2>) from Eq. (73) yields output probabilities (1/6,1/3,1/2) on m = 1,0,-1, mapping P_z = 1/2 to -1/3 and confirming kappa_- = -2/3. Since Theorem 9 is the basis for the general spin-inversion limit and for the claim that the largest-environment channel achieves the maximal deviation from conservation, this is a load-bearing error. The correct formula kappa_- = -j_B/(j_A+1) reduces to -j/(j+1) for equal spins, so Result 1 and the amplification theorem are not affected, but Eq. (124) and Fig. 2 must be corrected and all dependent statements updated.
minor comments (3)
  1. [Sec. II.B and Theorem 7] The overview states that the simplex has 2 max(j_A,j_B)+1 extremal points, but the range of L in Eq. (64) gives 2 min(j_A,j_B)+1 vertices, as correctly stated in Theorem 7 and Eq. (67). This inconsistency should be fixed in the overview.
  2. [Sec. VI.B, Eq. (109) and Theorem 10] The term 'spin amplification' is used for kappa_+ even when j_A >= j_B, in which case Eq. (127) gives kappa_+ = j_B/j_A <= 1 and no amplification occurs. The formula is correct, but the terminology should be adjusted to avoid overstating the result.
  3. [Sec. VII.A, Theorem 11] The condition for the d_A < d_B case is stated only parenthetically: the bound requires tr(E(I_A/d_A)^2) >= 1/d_A. This is not automatic for arbitrary channels between unequal dimensions and should be emphasized, since several applications in Sec. VIII implicitly rely on it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the SU(2) inversion and amplification limits are derived from an explicit simplex decomposition and closed-form Clebsch-Gordan factors, not fitted or self-referential.

full rationale

The central claims (optimal spin inversion and amplification in Theorems 9 and 10, and the conservation-law trade-off bounds in Theorems 11–13) are derived inside the paper. The simplex characterization of SU(2)-irreducibly covariant channels is established via Schur's lemma applied to the Jamiolkowski state (Theorems 2–3 and 7, Eqs. (61)–(67)), and the scaling factors f1(E_L) are computed explicitly from Clebsch-Gordan coefficients in Eq. (77) and Appendix B. The inversion and amplification factors are then obtained by convex optimization over this simplex (Eqs. (120)–(123)), not by fitting any parameter to the claimed output. The equal-spin value f = -j/(j+1) reproduces the known qubit universal-NOT factor -1/3 as a special case, and the amplification bound is compared with and improves the external Marvian–Spekkens bound, so the results have independent benchmark content. The only relevant self-citation is the 'coherifying' construction of U(1)-covariant channels from Ref. [42] (one of the present authors is a coauthor), but the paper itself explicitly states that this construction does not produce all extremal channels and gives a counterexample (Sec. V C, Eq. (104)); moreover, the U(1) section uses only upper bounds, not extremal completeness, so this citation is not load-bearing. The internal inconsistency in Theorem 9's printed unequal-spin inversion formula (Eq. (124) versus Eq. (77) combined with Eq. (122), which yield -j_B/(j_A+1), e.g. -2/3 for qubit-to-qutrit rather than -1) is a mathematical correctness defect, not a circularity: the claimed limit is not assumed as an input but is miscalculated in one displayed formula. No fitted input is renamed as a prediction, and no load-bearing premise is justified solely by a self-citation.

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

No free parameters are fitted: the constants in Theorems 11-13 (M, K) are derived, dimension-dependent bounds, not numbers matched to data. The paper relies on standard representation theory (Schur's lemma, Clebsch-Gordan algebra), the convex-structure characterization of covariant channels attributed to Ref [14] and re-derived for SU(2) via the Jamiolkowski block decomposition, the extension of unitarity from Ref [17], and its own definition of average deviation Delta. No new physical entities are postulated.

assumptions (5)
  • standard math The set of G-covariant CPTP maps has the Jamiolkowski block structure J(E) = direct_sum p_lambda (I_lambda/d_lambda) tensor rho_lambda (Theorem 3, Eq (82)), and for SU(2)-irreducible channels the extremal points are exactly the split-and-discard channels E_L (Eqs (65)-(67), Theorem 7).
    Follows from Schur's lemma applied to [J(E), U_B tensor U*_A] = 0; the extremality claim is credited to Ref [14] and is re-derived in-paper for the SU(2)-irreducible case.
  • domain assumption Unitarity, introduced in Ref [17] for equal input and output dimensions, extends to unequal dimensions via the Jamiolkowski and complementary formulas (Lemma 6), and u(E)=1 if and only if E is an isometry (Lemma 14).
    Extends the randomized-benchmarking measure of Wallman et al [17]; the general-dimension proof of Lemma 14 is sketched rather than fully shown.
  • domain assumption The average total deviation Delta(E), defined by Haar-averaging the squared differences of generator expectation values (Eqs (10), (52)), is the operative definition of disconnect from a conservation law; all bounds and the robustness equivalence are statements about this specific average measure.
    This measure choice is the paper's operational definition; a worst-case or variance-based measure would give different bounds, and the authors motivate but do not axiomatize the choice.
  • standard math For connected compact Lie groups, the trivial irrep is the unique one-dimensional irrep, so the lambda=0 block is singled out in the proof of the upper bound (Appendix D).
    Standard representation theory; used to lower bound 1-u(E) in terms of (1-p_0)^2 and to argue that the lambda=0 block conserves charges.
  • domain assumption The multiplicity-free decomposition of B(H_A) and equal input and output dimensions are required for the lower bound (Theorem 12) and for the SU(2) robustness result (Theorem 13); for U(1) symmetry no lower bound exists.
    The paper shows robustness fails without these conditions via the two-qubit example E = I tensor E' and the family of partial dephasing channels D_p, so the scope of the robustness claim is narrower than the abstract's general framing.

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Pith. "Pith review of Robustness of Noether's principle: Maximal disconnects between conservation laws and symmetries in quantum theory." pith.science (2026). https://pith.science/paper/QRTHKC6O

@misc{pith2026190804254,
  author       = {Pith},
  title        = {Pith review of: Robustness of Noether's principle: Maximal disconnects between conservation laws and symmetries in quantum theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QRTHKC6O}},
  note         = {Machine review of arXiv:1908.04254}
}
abstract

To what extent does Noether's principle apply to quantum channels? Here, we quantify the degree to which imposing a symmetry constraint on quantum channels implies a conservation law, and show that this relates to physically impossible transformations in quantum theory, such as time-reversal and spin-inversion. In this analysis, the convex structure and extremal points of the set of quantum channels symmetric under the action of a Lie group $G$ becomes essential. It allows us to derive bounds on the deviation from conservation laws under any symmetric quantum channel in terms of the deviation from closed dynamics as measured by the unitarity of the channel. In particular, we investigate in detail the $U(1)$ and $SU(2)$ symmetries related to energy and angular momentum conservation laws. In the latter case, we provide fundamental limits on how much a spin-$j_A$ system can be used to polarise a larger spin-$j_B$ system, and on how much one can invert spin polarisation using a rotationally-symmetric operation. Finally, we also establish novel links between unitarity, complementary channels and purity that are of independent interest.

Figures

Figures reproduced from arXiv: 1908.04254 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
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Figure 4. FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
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Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]

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    V A gives a simple way to calculate their unitarity

    Expressions for unitarity and deviations The structure of general SU(2)-irreducibly-covariant channelsE : B(HA) → B(HB) presented in Sec. V A gives a simple way to calculate their unitarity. Em- ploying Lemma 6, using the decomposition of the Jamio lkowski state given in Eq. (65) and the fact that irreducibly-covariant channels are unital (so that E(IA/dA...

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    Deriving trade-off relations We will now show how the unitarity and deviations from conservation laws are related, and obtain both lower and upper bounds on the average deviation from a con- servation law of spin angular momenta under a rotation- ally invariant irreducible channel in terms of its unitarity. Theorem 13. LetE :B(HA)→B (HA) be an SU(2)- irred...

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    Examples Consider first the simplest example of a covariant chan- nel for j = 1/2. In this case the unitarity is given by u(E) = 1 3 ( 4p0(E)2 + 4 3(1−p0(E))2− 1 ) , (165) and the deviation by ∆(E) = 4 9 (1−p0(E))2. (166) A straightforward calculation then yields a direct relation betweenu(E) and ∆(E), u(E) = 1− 4 √ ∆(E)(1− √ ∆(E)), (167) while the bounds ...

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    Expressions for unitarity and deviations Substituting the decompositions given in Eq. (100) to Eq. (44), one obtains the following expression for unitar- ity of a general U(1)-covariant channelE: u(E) = 1 d2− 1 ( d2∑ λ γ(J (λ)(E))−bE ) , (172) where bE := 1 d ∑ m (∑ n PE mn )2 , (173) describes how far PE is from a bistochastic matrix, i.e., bE = 1 when P...

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    Deriving trade-off relations First of all, we note that the deviation ∆( E) depends only onPE, while the unitarityu(E) depends both onPE (forming diagonals of J (λ)(E)) and on L(λ⁄=0)(E) (form- ing the off-diagonal terms of J (λ)(E)). Therefore, it is impossible to lower bound unitarity given the deviation. To see this more clearly, consider the following f...

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    Liouville representation for extremal SU(2)-irreducible covariant channels In Sec. V A we have seen that the set of SU(2)-irreducibly-covariant channels between spin-jA and spin-jB systems is fully characterised by its extremal points EL :B(HA)→B (HB) with L ranging from |jA−jB| to jA +jB in increments of one. Since the input and output spaces carry irred...

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    Maximal inversion and amplification of spin polarisation vector for SU(2)-covariant channels Here, we will characterise the range of values that the coefficient f1(EL) takes while varying over all extremal channels L. This factor corresponds to how much the spin polarisation can scale (up or down) under a covariant operation. As we will see, due to the parti...

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