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arxiv: 2603.20786 · v2 · submitted 2026-03-21 · 🪐 quant-ph · cond-mat.other

The typicality of symmetry-induced entanglement

Christian Boudreault , Nicolas Levasseur This is my paper

Pith reviewed 2026-05-15 06:53 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.other
keywords symmetric separabilitynumber entanglementconserved chargesuperselection rulesrandom quantum statestypicalityquantum entanglementreference frames
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The pith

A conserved charge makes almost all separable states fail to decompose into charge-conserving parts.

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

The paper defines the Symmetric Separability Problem as whether a separable state can be written as a mixture of components that each preserve a globally conserved charge N. For random states drawn from the natural measure, this decomposition exists with probability zero for almost all values of N. The authors apply the number entanglement witness to show that the typical distance from symmetric separability is strictly positive and follows a Gaussian distribution around its mean. These facts constrain the effective size of the set of states that remain separable once the charge is fixed and affect any quantum task performed without a shared reference frame or under a superselection rule.

Core claim

On random states, the Symmetric Separability Problem is answered negatively with probability one for almost all N. Most symmetric and separable states are actually far from being symmetrically separable, with the number entanglement featuring Gaussian concentration around a strictly positive mean value.

What carries the argument

The number entanglement witness, which quantifies the failure of a symmetric state to admit a decomposition into charge-conserving product states.

If this is right

  • Any quantum task that assumes separability under a superselection rule will typically encounter hidden entanglement induced by the charge constraint.
  • The volume of the set of states that are both separable and symmetrically separable is negligible compared with the full separable set once N is fixed.
  • The complexity of deciding symmetric separability is expected to be hard in the same regimes where ordinary separability is hard.
  • Multiparty states without a common reference frame will generically require charge-entanglement resources even when they appear separable in the usual sense.

Where Pith is reading between the lines

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

  • Symmetry constraints of this kind may serve as a generic source of entanglement that is easier to prepare than ordinary entangled states.
  • Protocols that certify entanglement or separability must incorporate an explicit check for the symmetric decomposition when a conserved charge is present.
  • The same concentration phenomenon could appear in other symmetry-protected settings, such as particle-number or angular-momentum superselection.

Load-bearing premise

The results rest on a particular probability measure for random states and on the number entanglement witness correctly detecting every failure of symmetric separability.

What would settle it

Explicitly construct, for some fixed N greater than one, a symmetric separable state whose number entanglement evaluates to zero, or compute the distribution of number entanglement on small-dimensional systems and find that it does not concentrate away from zero.

Figures

Figures reproduced from arXiv: 2603.20786 by Christian Boudreault, Nicolas Levasseur.

Figure 1
Figure 1. Figure 1: Pictorial representation of some classes of states considered in this work. Class D contains all states on HA ⊗ HB. A lower index Nˆ indicates symmetries, and an upper index ‘sep’ or ‘symsep’ indicates that states are separable or symmetrically separable, respectively. We algebraically show that D sep Nˆ local = D symsep Nˆ local = D symsep Nˆ . For a degenerate Nˆ, D sep Nˆ local is of measure zero in D s… view at source ↗
Figure 2
Figure 2. Figure 2: Sets and maps for the construction of the purifying manifold. i is a C ∞-diffeomorphism, j is C ∞ injective, while π, τ , and TrHA⊗HB are C ∞ surjective, but not injective. (·) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of NE values for randomly generated 2- [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Convex hull of two compact convex sets belonging to mutually orthogonal spaces. C1 ⊂ W and C2 ⊂ W⊥ are compact convex sets in W and W⊥, respectively, two mutually orthogonal subspaces of R n . (In the figure, W = span{z} and W⊥ = span{x, y}.) S1 and S2 are convex subsets of C1 and C2, respectively. If both vol S1/vol C1 and vol S2/vol C2 tend to zero, so does vol conv(S1 ∪ S2)/vol conv(C1 ∪ C2). In general… view at source ↗
Figure 5
Figure 5. Figure 5: Distribution of NE values for 2-qudit states. Each subsystem [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Distribution of NE values for multi-qubit states. Each subsystem [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
read the original abstract

In the presence of a globally conserved charge $N$, a natural question is whether a given separable state can be separated into charge-conserving components. We dub this problem the Symmetric Separability Problem (SSP). On random states, the SSP is answered negatively with probability one for almost all $N$. Using a witness to the failure of symmetric separability, namely the number entanglement (NE) introduced in arXiv:2110.09388, we show that most symmetric and separable states are actually far from being symmetrically separable, with the NE featuring Gaussian concentration around a strictly positive mean value. We discuss some consequences of our results for quantum tasks in the presence of a superselection rule or in the absence of a common reference frame. Progress is made on the question of the size of the separable space constrained by $N$. We also touch upon the question of the complexity of SSP, and multiparty entanglement.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the Symmetric Separability Problem (SSP) for bipartite states subject to a globally conserved charge N. It claims that, under a natural measure on random states, the SSP is answered negatively with probability one for almost all N. Employing the number entanglement (NE) witness imported from arXiv:2110.09388, the authors show that NE on symmetric separable states exhibits Gaussian concentration around a strictly positive mean, implying that most such states lie far from the symmetrically separable set. The work also discusses consequences for quantum tasks under superselection rules or without a common reference frame, and reports progress on the dimension of the N-constrained separable set together with remarks on the complexity of SSP.

Significance. If the central claims are substantiated, the result supplies a probabilistic characterization of symmetry-induced entanglement and sharply constrains the effective size of the separable set under charge conservation. The Gaussian concentration statement is a strong, quantitative typicality result with direct implications for reference-frame-free quantum information processing. The manuscript makes explicit use of concentration-of-measure techniques and builds on an existing witness, which are positive features when the witness's detection properties are fully characterized.

major comments (2)
  1. [§3] §3 (main typicality theorem): The probability-one claim that SSP fails for almost all random states rests on NE serving as a complete detector of symmetric-separability failure. Because NE is introduced only as a witness (NE > 0 implies failure), the argument does not rule out a positive-measure set of states with NE = 0 that nevertheless violate SSP; this would undermine the measure-zero statement for the symmetrically separable set. An independent argument that the kernel of NE coincides with the symmetrically separable states, or an explicit bound on the undetected set, is required.
  2. [§2.2] §2.2 (definition of the typicality measure): The measure on random symmetric states is taken from prior work without re-derivation of its invariance properties or concentration constants under the N-conservation constraint. The Gaussian concentration result for NE therefore inherits any hidden assumptions of that measure; the dependence on N and the precise scaling of the variance with system size should be stated explicitly.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'almost all N' is used without specifying the range or probability measure on N; a brief clarification would help readers.
  2. [§2] Notation: the symbol for the symmetric-separable set is introduced without an explicit equation number; adding Eq. (X) would improve traceability when the set is later compared with the NE kernel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address the major comments point by point below, providing the strongest honest defense of the manuscript while making revisions where they improve clarity.

read point-by-point responses

  1. Referee: [§3] §3 (main typicality theorem): The probability-one claim that SSP fails for almost all random states rests on NE serving as a complete detector of symmetric-separability failure. Because NE is introduced only as a witness (NE > 0 implies failure), the argument does not rule out a positive-measure set of states with NE = 0 that nevertheless violate SSP; this would undermine the measure-zero statement for the symmetrically separable set. An independent argument that the kernel of NE coincides with the symmetrically separable states, or an explicit bound on the undetected set, is required.

    Authors: We thank the referee for highlighting this subtlety. Because NE is a valid witness, every symmetrically separable state necessarily satisfies NE = 0, so the symmetrically separable set is contained in the kernel of NE. The Gaussian concentration result establishes that the set where NE > 0 has measure one, hence the kernel {NE = 0} has measure zero. It follows immediately that the symmetrically separable set, being a subset of a measure-zero set, itself has measure zero. Any additional states that may lie in the kernel but outside the symmetrically separable set (the undetected set) are likewise contained in the measure-zero kernel and do not affect the conclusion. The probability-one claim therefore holds on the basis of the one-sided implication alone. We will insert a short clarifying paragraph in §3 emphasizing the subset relation. revision: partial

  2. Referee: [§2.2] §2.2 (definition of the typicality measure): The measure on random symmetric states is taken from prior work without re-derivation of its invariance properties or concentration constants under the N-conservation constraint. The Gaussian concentration result for NE therefore inherits any hidden assumptions of that measure; the dependence on N and the precise scaling of the variance with system size should be stated explicitly.

    Authors: We agree that the presentation would benefit from greater explicitness. In the revised manuscript we will add a concise re-statement of the measure’s invariance properties under the global N-conservation constraint (drawing directly from the cited prior work) and will record the explicit N-dependence of the concentration parameters, including the scaling of the variance with total system size. These additions will appear in §2.2 and will not alter the technical arguments. revision: yes

Circularity Check

0 steps flagged

No circularity: typicality analysis is independent of imported witness

full rationale

The paper imports the number entanglement witness as an established tool from prior work and applies it to compute concentration properties under a chosen measure on random states. No step redefines the witness in terms of the new typicality result, fits parameters to the target distribution and renames the output a prediction, or relies on a self-citation chain whose validity is presupposed by the present derivation. The probability-one claim is presented as a direct consequence of the witness properties plus the measure, without internal reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard quantum mechanics with a globally conserved charge operator, the definition of number entanglement from the cited reference, and an implicit uniform or Haar measure over random states; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Quantum states are described by density operators on a finite-dimensional Hilbert space with a globally conserved charge operator N.
    Invoked throughout the abstract when defining symmetric states and the SSP.
  • domain assumption Number entanglement serves as a witness for failure of symmetric separability.
    Taken from arXiv:2110.09388 and used to quantify distance from symmetric separability.

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Reference graph

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