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REVIEW 3 major objections 4 minor 103 references

Disparity between multipartite entangling and disentangling powers of unitaries: Even vs Odd

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Certain non-diagonal unitaries have unequal multipartite entangling and disentangling powers, with the construction depending on whether the number of qubits is even or odd.

desk verdict New multipartite entangling/disentangling asymmetry with a parity effect, numerically supported but not yet analytically certified; worth refereeing with requests for stronger evidence. read the letter →

arxiv 2505.18539 v1 pith:FCZIUAOQ submitted 2025-05-24 quant-ph cond-mat.str-el

Mrinmoy Samanta , Sudipta Mondal , Aditi Sen De This is my paper · ORCID
classification quant-phcond-mat.str-el MSC 81P4081P4581P68 PACS 03.67.Mn03.65.Ud
keywords multipartiteentanglemententanglingpowerdisentanglinggenuinegeneralizedgeometricmeasureDzyaloshinskii-Moriyainteractionrandomunitarycircuits
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 paper asks whether a unitary operation on $N$ qubits has the same capacity to generate genuine multipartite entanglement from a fully separable state as its adjoint does, a question that matters because entanglement creation and erasure are both resources in quantum information processing. The answer it argues is no in general: a class of diagonal unitaries preserves the equality, but for two constructed families of non-diagonal unitaries the entangling power $E_N(U)$ and the disentangling power $D_N(U)=E_N(U^\dagger)$ become unequal when the optimization is over fully separable inputs. The new observation is a parity effect: one family realizes the imbalance for even $N$, a different family is needed for odd $N$, and the same behavior is found for Hamiltonian evolutions with nearest-neighbor Dzyaloshinskii-Moriya (even $N$) or Heisenberg plus DM interactions (odd $N$), as well as for two-layer random brickwork circuits.

What carries the argument

The load-bearing object is the pair of quantities $E_N(U)$ and $D_N(U)=E_N(U^\dagger)$, computed from the generalized geometric measure, which for a pure state is $G(|\psi\rangle)=1-\max|\langle\phi|\psi\rangle|^2$ over all states that are not genuinely multipartite entangled and can be evaluated from the largest Schmidt coefficients of the reduced density matrices. The constructions that break the symmetry are brickwork-style global unitaries built from two-qubit gates: $U_{ND}(\lambda)$ of Eq. (6) alternates the same two-qubit gate $U_{i,i+1}(\lambda)$ on odd and even bonds, while the odd-$N$ family of Eq. (7) supplements this with a specially designed two-qubit gate $U_w$ built from non-orthogonal single-qubit states. The paper points to the noncommutativity of the odd and even layers as the source of the directional asymmetry, since for diagonal unitaries no such noncommuting structure exists and the powers match.

What would settle it

Evaluate $E_4(U_{ND}(\lambda))$ and $D_4(U_{ND}(\lambda))$ at $\lambda=\pi/3$ using a dense grid over all four single-qubit Bloch angles followed by local refinement; if the two maxima agree within numerical precision, or if any fully separable state beats the reported value for either quantity, the parity-dependent disparity reported for even $N$ is an artifact of the optimization rather than a property of the unitary.

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Extended reading notes

Core claim

On its own terms, the paper establishes that multipartite entangling and disentangling powers, defined via the generalized geometric measure (GGM) as $E_N(U)=\max_{|\psi\rangle\in S_N} G(U|\psi\rangle)$ and $D_N(U)=E_N(U^\dagger)$, are equal for diagonal unitaries: Proposition 1 and Proposition 2 prove this for the single-phase family $U_{d,\phi}=\mathrm{diag}(1,\dots,1,e^{i\phi})$, and numerical sampling supports it for arbitrary diagonal unitaries on three, four, and five qubits. The central discovery is that this equality is not generic. For the non-diagonal family $U_{ND}(\lambda)$ of Eq. (6), built from alternating layers of the two-qubit gate $U_{i,i+1}(\lambda)$, the paper finds $E_N^{\mathrm{even}}\neq D_N^{\mathrm{even}}$ for even $N$ (shown for $N=4,6$ and checked up to $N=10$), while the same family gives equal powers for odd $N$. For odd $N$, replacing one layer with the two-qubit gate $U_w$ of Eq. (7) restores the disparity (shown for $N=3,5$). These inequalities are reported as numerical results, obtained by maximizing GGM over fully separable states, with no analytical proof presented.

Load-bearing premise

The claim rests on the assumption that the numerical maximization of the generalized geometric measure over all fully separable input states returns the true global maximum for both $U$ and $U^\dagger$; the paper gives no analytical proof of the inequalities for the non-diagonal families, so a missed better input state for either quantity could erase the reported disparity.

Editorial extensions

If this is right

  • Diagonal unitary operators do not distinguish entangling from disentangling direction, so resource generation and resource erasure are equally easy for them.
  • For even numbers of qubits, two identical noncommuting layers of the same two-qubit gate are enough to create the disparity; for odd numbers, the two layers must differ.
  • The disparity can be produced by physical Hamiltonians: nearest-neighbor Dzyaloshinskii-Moriya interactions for even $N$, and a combination of Heisenberg and DM interactions for odd $N$, at evolution times away from multiples of $\pi/2$ (or $\pi/4$ in the mixed case).
  • Random two-qubit Haar gates arranged in two alternating layers reproduce the effect, meaning the inequality is not an artifact of a specially fine-tuned gate alone.
  • In the random-circuit setting, even-$N$ systems show the asymmetry with identical gate sets in both layers, while odd-$N$ systems require distinct gate sets, sharpening the parity contrast.

Reading between the lines

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

  • If the numerical inequalities survive more thorough optimization, a practical consequence follows: circuit designers who want to use a gate to erase genuine multipartite entanglement cannot assume that running the inverse evolution is equivalent to reversing the resource dynamics.
  • The parity dependence may be tied to the bipartition structure of the brickwork geometry; a testable extension is to ask whether the odd-$N$ effect disappears if $U_w$ is replaced by any local-unitary equivalent, which would indicate that only the global layer structure matters.
  • A natural next check is whether the same even-odd asymmetry appears when the generalized geometric measure is replaced by another genuine-multipartite-entanglement measure, such as one based on von Neumann entropy; if it does not, the disparity is measure-dependent rather than an operational feature of the unitaries.
  • The Hamiltonian simulations suggest a concrete experimental signature: in a chain with DM interactions, the amount of genuine multipartite entanglement generated by $e^{-iHt}$ and by $e^{+iHt}$ from the best product states should differ for times in specific windows, which could be tested in cold-atom or trapped-ion platforms.
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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

3 major / 4 minor

Summary. The paper defines multipartite entangling and disentangling powers of a unitary operator as the maximum GGM generated from fully separable states by U and by U†, respectively. It claims that diagonal unitaries have equal entangling and disentangling powers, while certain non-diagonal unitary families show a disparity that depends on the parity of the number of qubits: the family in Eq. (6) for even N and the family in Eq. (7) for odd N. The paper further claims that this asymmetry can be realized through nearest-neighbor Dzyaloshinskii–Moriya Hamiltonians for even N and a combination of Heisenberg and DM interactions for odd N, and that two-layer random circuits exhibit analogous behavior. The evidence for the non-diagonal and Hamiltonian claims is numerical, based on ISRES maximization of the GGM over fully separable states.

Significance. If the numerical results are correct, the paper identifies a novel and genuinely multipartite phenomenon: unitary operations can have different capacities for generating versus destroying genuine multipartite entanglement, with a qualitative dependence on the parity of the number of parties. This would extend the known bipartite entangling/disentangling disparity to the multipartite setting and connect it to physically realizable spin Hamiltonians. The paper is clearly written and introduces a natural multipartite definition based on the GGM. However, the central claims are not yet rigorously established: the proof for diagonal unitaries is incomplete, and the non-diagonal and Hamiltonian claims rest on unverified numerical optima and unproven symmetry reductions.

major comments (3)
  1. [Sec. III.1, Construction 1 and Fig. 2] The central claim that E_N(U_ND(λ)) ≠ D_N(U_ND(λ)) for even N (and the analogous claim for odd N in Construction 2) rests entirely on ISRES numerical maximization without any global optimality certificate. The asserted symmetry reductions for the optimal inputs (|ψ_1⟩ = |ψ_{2m+2}⟩ for U_ND, and the two-parameter reduction for U_ND†) are not proved; since restricting the search space can only lower the computed maximum, an asymmetric restriction could create a spurious gap. For instance, if the true global maximum of D_N is higher than the reported two-parameter value, the apparent E_N < D_N could disappear under full optimization. Please provide an analytical proof of the inequality (e.g., separate lower and upper bounds on the two quantities) or certified global optimization results such as multi-start statistics, a second independent optimizer, or interval-based bounds.
  2. [Appendix B 2, Eq. (B9) and Proposition 2] The claimed proof that E_N(U_d,ϕ) = D_N(U_d,ϕ) for the single-parameter diagonal unitaries is incomplete. The coefficients α_k(N), β(N), γ_k(N) in Eq. (B9) are never specified, and the reduction to equal angles θ_1 = ... = θ_N is justified only by a numerical observation ('We numerically find out'). Moreover, for N ≥ 4 the GGM depends on the largest eigenvalues of reduced density matrices over all subsystem sizes up to N/2, whereas the proof analyzes only single-party reduced density matrices ρ_i. Consequently Proposition 2 is not established by the given argument; the authors should either supply a complete derivation or explicitly present this as a numerical conjecture with adequate supporting evidence.
  3. [Sec. III.2, Hamiltonian simulations] The parity-dependent asymmetry reported for the Hamiltonian-generated unitaries U_DM and U_DM,H is supported only by plots of Δ_N, without quantitative specification of the optimized values, and the statement in footnote 1 that the optimal fully separable states coincide with those of Construction 1 is unproven. Because the same global-maximization issue applies here, the physical claim that these Hamiltonians simulate the disparity is not yet established. Please report the numerical values, the number of independent optimization runs, and the sensitivity of the optima to starting points, or provide an analytical argument that the reported Δ_N is a rigorous lower bound on |E_N − D_N|.
minor comments (4)
  1. [Appendix A 1] The definition of the generalized geometric measure as 1 − max{...} over sets of maximum eigenvalues of l-site reduced density matrices is ambiguous; the maximum should be taken over all eigenvalues of all reduced density matrices for subsystem sizes l = 1, ..., ⌊N/2⌋. Please restate the formula more precisely.
  2. [Construction 2, Eq. (7)] The definition of U_w is difficult to parse (e.g., 'ω2' appears to be a typesetting error for ω^2, and the orthogonality of |β_t⟩ and |γ_t⟩ is stated tersely). Please provide an explicit matrix representation for U_w or a clearer basis notation to ensure reproducibility.
  3. [Sec. III.1] The sentence about the qubit-qutrit entangling power reaching its maximum while the disentangling power remains lower should cite Ref. [45] at that point, rather than only in the introduction.
  4. [Fig. 2] The curves for E and D are visually close in some λ-ranges; adding markers or an inset would make the disparity clearer. The absence of any indication of optimization uncertainty (e.g., error bars or number of runs) is also a concern, though this is already noted in the major comments.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the entangling and disentangling powers are independently maximized quantities, and the central inequality is not an input to the computation.

full rationale

The paper defines E_N(U) and D_N(U) independently as maxima of the GGM over fully separable inputs for U and U^† respectively (Eqs. (1)-(2)), and computes each quantity separately from the Schmidt-coefficient formula for GGM. The central claim E_N(U) ≠ D_N(U) for the non-diagonal families in Eqs. (6)-(7) is an output of these independent numerical maximizations, not an input: no fitted parameter is used to enforce the inequality, and the diagonal-unitary equality in Propositions 1-2 is derived analytically, with the expression depending only on cos φ so that U and U^† give identical optimization problems. The paper's self-citations, such as [62] and [72]-[74], supply the GGM measure and the entangling-power definition; these are background definitions and standard tools, and the disputed numerical results do not rely on an unverified result imported from those citations. The main weakness is numerical robustness: the ISRES optimizer is stochastic, and the unproven symmetry reductions in Construction 1 could in principle make the reported gap an artifact. However, that is a correctness and rigor concern, not a circularity concern. No step in the derivation reduces by construction, by definition, or by self-citation to its own inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claims rest on numerical optimization; the only tunable parameters are the gate angle λ and evolution time t, which are scanned rather than fitted. The key unproven premises are that the optimizer achieves global maxima and that local phases are irrelevant to GGM.

free parameters (2)
  • λ in U_ND(λ) = scanned over [0, π]
    Parameter of the two-qubit gate family in Eq. (5); varied to show a range where entangling and disentangling powers differ, not fitted to external data.
  • evolution time t in Hamiltonian simulations = scanned over [0, π]
    Time in U_DM and U_DM,H of Sec. III.2; varied to show nonvanishing Δ_DM or Δ_DM,H, not fitted to data.
assumptions (4)
  • standard math GGM for pure states is computed from the largest eigenvalues of reduced density matrices for all bipartitions
    Standard result used throughout, cited from Sen(De) and Sen [62] and Wei-Goldbart [84].
  • domain assumption Local phases ξ_i of input product states do not affect the GGM of the output state
    Stated in the proof of Proposition 1 and used in Appendix B2; justified only numerically up to N=8, not proven.
  • ad hoc to paper The ISRES optimizer finds the global maximum of GGM over the fully separable set S_N
    Used in all numerical maximizations for E_N and D_N; no multi-start or convergence guarantee is reported.
  • domain assumption The symmetric optimal-input structure assumed in the constructions is valid
    For example, equal angles θ_i for the adjoint in Construction 1 reduce the optimization dimension; this is an observed pattern, not a proven symmetry.

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Pith. "Pith review of Disparity between multipartite entangling and disentangling powers of unitaries: Even vs Odd." pith.science (2026). https://pith.science/paper/FCZIUAOQ

@misc{pith2026250518539,
  author       = {Pith},
  title        = {Pith review of: Disparity between multipartite entangling and disentangling powers of unitaries: Even vs Odd},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FCZIUAOQ}},
  note         = {Machine review of arXiv:2505.18539}
}
read the original abstract

We compare the multipartite entangling and disentangling powers of unitary operators by assessing their ability to generate or eliminate genuine multipartite entanglement. Our findings reveal that while diagonal unitary operators can exhibit equal entangling and disentangling powers, certain non-diagonal unitaries demonstrate an imbalance when acting on fully separable states, thereby extending the known disparity from bipartite systems to those with any number of parties. Counterintuitively, we construct classes of unitaries and their adjoints that display unequal entanglement generation capacities, behaving differently when applied to systems with an even number of qubits compared to those with an odd number. Further, we illustrate that this asymmetry can be simulated using physically realizable Hamiltonians: systems with an even number of qubits employ nearest-neighbor Dzyaloshinskii-Moriya (DM) interactions, while those with an odd number utilize a combination of Heisenberg and DM interactions. Additionally, we present a circuit composed of random noncommuting unitaries, constructed from alternating layers of two-qubit Haar-random gates, to illustrate the discrepancy in the entangling and disentangling capabilities of unitaries.

Figures

Figures reproduced from arXiv: 2505.18539 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. ). Governing Hamiltonian involving odd sites. As it was shown that the unitaries which act on odd number of qubits showing dissimilarities are much more involved than the ones for even number of qubits. This can also be visible in the in￾teracting Hamiltonian. The (i, i + 1)- pair of the system (with 1 To compute E even N (UDM) and Deven N (UDM), we again optimize over the set of fully separable initial states, and … view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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    As the number of parties increases, determining whether the entangling and disentangling powers can differ becomes increasingly nontrivial due to the increase of dimension

    Construction of non-diagonal unitaries: Even vs odd It is known that in the case of unitary operators acting on a qubit-qutrit input product state, the entangling power can reach its maximum, while the disentangling power remains strictly lower [45] than the maximum, showing their disparity from the perspective of entanglement generation. As the number of...

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    Simulating unitaries through interacting Hamiltonians Let us illustrate how the unitary operator, described in Eqs. (6) and (7) for even and odd sites, respectively, can be simulated on the currently available experimental platforms, like cold atoms in optical lattices and trapped ions [86–95]. Dynamics with even number of sites.We first present our re- s...

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