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Entanglement in the stabilizer formalism

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David Fattal, Toby S. Cubitt, Yoshihisa Yamamoto, Sergey Bravyi, Isaac L. Chuang

Quantum Physics · Submitted 23 Jun 2004 · 17 references

Pith's verdict: The multipartite measure is genuinely new and cheap to compute; the monotonicity proof is a sketch that quietly restricts itself to stabilizer-preserving LOCC.

T0 · AI review

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arXiv IDquant-ph/0406168 (version 1)
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CitationDavid Fattal, Isaac L. Chuang, Sergey Bravyi, Toby S. Cubitt, Yoshihisa Yamamoto. "Entanglement in the stabilizer formalism." arXiv:quant-ph/0406168 (2004). https://pith.science/paper/T6ZK3R5P
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The paper

The paper claims

The entanglement of a stabilizer state can be read off its generators in polynomial time.

The reading

Stabilizer states — the states used in quantum error correction, cluster-state computation, and secret-sharing protocols — have a compact classical description: a list of a few Pauli operators that leave the state unchanged. This paper asks whether the entanglement of such a state can be computed directly from that list, rather than from the exponentially many amplitudes of the state vector. The authors answer yes for the bipartite case, proving that the entropy of entanglement equals half the rank of the subgroup of generators that act non-trivially across the cut, and they propose a generalisation to $k$ parties, $e_A = n - |S_1 S_2 cdots S_k|$, built from the local subgroups that act as identity on each party. If the claims hold, entanglement becomes a cheap, exact, and constructive quantity for a large and practically important family of states, computable in $O(k n^3)$ steps from $2n^2$ bits of data. The authors also assert, without separate proof, that the same formalism carries over to qudits and to continuous-variable Gaussian states by replacing the Pauli group with the Heisenberg–Weyl group.

What carries the argument

The load-bearing device is the canonical form theorem for stabilizer generators relative to a partition: $S = langle a_i otimes I_B, I_A otimes b_j, g_k, ar{g}_k rangle$, proved via a compatibility index $c(G)$ (the largest rank of an abelian subgroup of a Pauli group $G$) and the lemma that the center of the projected group $P_A(S_{AB})$ is trivial. That triviality is what forces the generator count $|S_A| + |S_B| + 2p + z = n$ to close with $z = 0$, converting a structural statement about a group into a numerical identity for entanglement. The same machinery yields an efficient algorithm: find anticommuting pairs among the projected generators, or equivalently compute the rank of an $n imes 2n$ matrix over $mathbb{Z}_2$.

Extended reading notes

Core claim

The central claim is a dictionary between entanglement and group structure. For a bipartite stabilizer state $|psi rangle_{AB}$ with stabilizer $S$, the authors show that $S$ can always be brought into a canonical form consisting of generators local to $A$, generators local to $B$, and $p$ anticommuting pairs $(g_k, ar{g}_k)$ whose projections onto either side anticommute. Each such pair corresponds to one shared Bell pair, so the entropy of entanglement is $E(|psi rangle) = p = |S_{AB}|/2$, where $|S_{AB}|$ is the rank of the correlation subgroup. For a $k$-way partition the authors define $e_A(|psi rangle) = n - |prod_{j=1}^{k} S_j|$, with $S_j$ the subgroup of stabilizers acting as the identity on party $j$, and argue that this is an entanglement monotone that reduces to twice the entropy of entanglement when $k = 2$ and is non-increasing as the partition is refined.

Load-bearing premise

The claim that the multipartite quantity $e_A$ is a genuine entanglement monotone rests on a single asserted step in Proof 2 — that measuring a local Pauli operator cannot shrink the local subgroup, so $e_A$ cannot increase — and that argument covers only local unitaries, local Pauli measurements, and added separable ancillas, not general adaptive local protocols with classical communication.

Editorial extensions

If this is right

  • Entanglement of any bipartite stabilizer state is computable exactly in $O(n^2 min(n_A, n_B))$ time from $2n^2$ bits, replacing an exponentially large density matrix with a rank computation.
  • The canonical form gives a constructive local-unitary procedure that turns any bipartite stabilizer state into exactly $E$ independent EPR pairs.
  • For graph states the same quantity reduces to the binary rank of the adjacency submatrix $Gamma_{B,A}$ across the cut, so entanglement is carried by graph edges crossing the partition once the graph is in canonical form.
  • The multipartite measure $e_A$ gives a cheap replacement for the Schmidt measure used to sort graph states into equivalence classes, and it is monotone under refinement of the partition.
  • The construction is claimed to extend to qudits and to continuous-variable Gaussian states by swapping the Pauli group for the Heisenberg–Weyl group of displacement operators.

Reading between the lines

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

  • Because $e_A$ is built from subgroup ranks, it looks like a practical screening tool for entanglement witnesses assembled from stabilizer generators, and for tracking how entanglement moves during a Clifford circuit.
  • The bipartite identity suggests a natural sanity check for any proposed multipartite measure on stabilizer states: it must agree with $|S_{AB}|/2$ on every two-way cut, which sharply constrains candidate measures.
  • If the monotonicity argument holds only for local Pauli measurements, the measure may be monotone under Clifford-local operations while failing for general local operations; a brute-force search over 3- and 4-qubit stabilizer states and all partitions would settle this quickly.
  • The claim of a smooth passage to continuous variables deserves its own check, since Gaussian states have infinitely ranked stabilizer groups and the counting arguments used here assume finite rank.
Original abstract

We define a multi-partite entanglement measure for stabilizer states, which can be computed efficiently from a set of generators of the stabilizer group. Our measure applies to qubits, qudits and continuous variables.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,225 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.11 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.05 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We define a multi-partite entanglement measure for stabilizer states, which can be computed efficiently from a set of generators of the stabilizer group. Our measure applies to qubits, qudits and continuous variables.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We define a multi-partite entanglement measure for stabilizer states, which can be computed efficiently from a set of generators of the stabilizer group. contribution 0.99 0.94 0.58 asserted 1.00
s1 Our measure applies to qubits, qudits and continuous variables. contribution 0.86 0.66 0.10 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.95 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1: Canonical set of generators for a stabilizer group
Figure 2. FIG. 2: Application of the measure
Figure 3. FIG. 3: Structure of a subgroup G of the Pauli group.

Figures reproduced from arXiv:quant-ph/0406168 by the authors.

References
How later papers used it

108 later papers cite it

Evidence: substantial. 106 of the citing papers share no author with it, and 37 of those carry a confident label from Pith's citation classifier.

Pith read 27 of the 173 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 7 of 33 papers with this label
How the literature uses it

108 arXiv papers cite this in 175 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 113 sentences without a confident label

Builds on it 3 papers
Uses its method or tool 31 papers
  • Notice that for every sample, $L$ convex optimisations are performed, as well as $L$ steps involving the algorithm of Fattal et al. [ref] , which has runtime polynomial in the total number of qubits.

    Quantifying magic for multi-qubit operations · Dynamic Monte Carlo · 1.00

  • Following the algorithm of Ref. [ref] , we compute the entanglement entropy $S_A(l)$ of region $A$ in Fig. [ref] for the $(3,3)$-BB code with stabilizer generators [ref] 𝒮_1 = < g r a p h i c s > , 𝒮_2 = < g r a p h i c s > .

    Resolving spurious topological entanglement entropy in stabilizer codes · Introduction · 1.00

  • From [ref] , we see that this representation can be found in the following way: Let us denote by $P_B$ the projection mapping $g_A⊗g_B$ to $ℐ_A⊗g_B$, and the list of generators by ${g_i}_i$.

    Quantum error correction codes and absolutely maximally entangled states · Entanglement corrections to the Ryu-Takayanagi formula · 1.00

Compares with it 1 paper
Cites it as support 2 papers
Cites it as background 15 papers

Discussion

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