Distributed Entanglement

Joydip Kundu; Valerie Coffman; William K. Wootters

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Distributed Entanglement

Valerie Coffman, Joydip Kundu, William K. Wootters

Quantum Physics · Submitted 13 Jul 1999 · 20 references

Pith's verdict: The CKW paper: the derivation is correct, the tightness family really does cover the whole boundary, and the flaws are typographic rather than mathematical.

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arXiv IDquant-ph/9907047 (version 2)

Pith IDV32LWRWU, Pith's permanent record of this paper and its review

CitationJoydip Kundu, Valerie Coffman, William K. Wootters. "Distributed Entanglement." arXiv:quant-ph/9907047 (1999). https://pith.science/paper/V32LWRWU

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Editor's note

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This is the paper that gives you monogamy of entanglement for three qubits, and it holds up. Eq. (12), τ_AB + τ_AC ≤ τ_A(BC) = 4 det ρ_A, comes straight out of the tangle definition: bound each pairwise tangle by Tr(ρ ρ̃) using the two-eigenvalue structure of ρ_AB ρ̃_AB, sum, and the sum collapses to 4 det ρ_A. No fitting, no circular step, nothing imported that the inequality secretly depends on. The pure-state result stands on its own algebra, and I checked the pieces that matter: Eq. (7) on product and GHZ states, and the saturating family α|100⟩ + β|010⟩ + γ|001⟩, which does cover every (x, y) with x + y ≤ 1. The abstract's tightness claim [abstract:s4] is earned, not asserted. The three-way tangle in Eq. (24) is the real payoff — permutation-invariant, zero on the saturating family, one on GHZ, and the cube picture is a genuinely useful way to see why the d_i are symmetric. The optimality argument is scoped honestly to functions of τ, and the n-qubit extension at the end is labelled a conjecture rather than smuggled in as a result. That is the right way to end a paper like this. The mixed-state extension (Eq. 14) leans on convexity of the tangle from Ref. [7], a published and independently checkable source; footnote 2's one-liner is valid, since the square of a nonnegative convex function is convex. Soft spots are small. The algebra from Eq. (19) to Eq. (22) is summarized rather than shown — I verified the endpoints (GHZ gives 1, the saturating family gives 0) and both are consistent, so I believe it, but a reader reproducing it will have to work. There is a typo where |v0⟩, |v1⟩ become |v1⟩, |v2⟩ in the sentence defining the 2×2 action. One thing to disregard: the stress-test's worry about a spurious factor of 4 in the ǫ identity between Eqs. (6) and (7) is a page number from the preprint bleeding into the text, not an error. The identity as used is standard and Eq. (7) is correct. Send it to referees; it deserves the time. Anyone working on multipartite entanglement, resource theories, or security arguments built on monogamy should read it and cite it.

Referee report

Minor objections (5)

[Between Eqs. (6) and (7)] The printed identity ǫ_nn′ ǫ_j′j = 4δ_nj′ δ_n′j − δ_nj δ_n′j′ carries a spurious factor of 4; the standard 2×2 identity is ǫ_ab ǫ_cd = δ_ac δ_bd − δ_ad δ_bc. The resulting Eq. (7) is independently correct, so this is a typographical slip, but it should be corrected so that a reader reproducing the step is not misled.
[Eq. (18)] The text writes the general vector in the range as x|v1⟩ + y|v2⟩, but the range is spanned by |v0⟩ and |v1⟩ as defined two sentences earlier; the labels should be |v0⟩ and |v1⟩.
[Eqs. (19)–(22)] The passage from Eq. (19) to Eq. (22) is summarized as “somewhat tedious but straightforward algebra.” Because Eq. (22) is a central result of the paper, a short appendix giving the intermediate determinant calculation would materially improve reproducibility.
[Optimality argument following Eq. (15)] The argument that τ is the unique function of τ compatible with the tight family is correct, but it applies only to functions of τ, as the text itself notes. The abstract's phrase “as strong as it could be” could be qualified to make this scope explicit, since a reader of the abstract alone might take it to exclude non-τ-based measures.
[Eq. (25)] The permutation invariance of τ_ABC is not manifest in the form of Eq. (25), and the text acknowledges this. A one-line remark pointing back to the cube picture, or an explicit statement of how the invariance is recovered from this form, would help the reader.

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The paper

The paper claims

For three qubits, the amount of entanglement one qubit shares with a second plus what it shares with a third can never exceed what it shares with the pair — and the leftover is a genuinely three-way quantity.

The reading

Entanglement is a resource that cannot be handed out freely: if qubit A is perfectly entangled with qubit B, it has nothing left over for qubit C. This paper turns that folklore into an exact accounting rule for three qubits. Using a measure called the tangle — the square of the concurrence, which is a monotone function of the entanglement of formation — the authors prove that $τ_{AB} + τ_{AC} ≤ τ_{A(BC)}$, where $τ_{A(BC)} = 4$ det $ρ_A$ is the tangle between A and the pair BC treated as a single two-level object. They then show the bound cannot be improved: a one-excitation family of states saturates it for every allowed triple of values, and any alternative measure that also saturates it must coincide with the tangle. The difference between the two sides is the part of A's entanglement with BC that is not owed to B or C separately, and it turns out to be symmetric in all three qubits, so it deserves the name three-way tangle.

What carries the argument

The tangle, $τ_{AB} = [max{λ_1 - λ_2 - λ_3 - λ_4, 0}]^2$, built from the square roots of the eigenvalues of $ρ_{AB} ρ̃_{AB}$ with $ρ̃_{AB} = (σ_y ⊗ σ_y) ρ^*_{AB} (σ_y ⊗ σ_y)$ the spin-flipped density matrix. Two identities carry the argument. First, for a pair inside a pure three-qubit state, $τ_{AB} ≤ Tr(ρ_{AB} ρ̃_{AB})$, since the trace equals $λ_1^2 + λ_2^2$ while the tangle is $(λ_1 - λ_2)^2$. Second, the two traces sum to $4$ det $ρ_A$, obtained by rewriting the amplitude sum using $ε_{nn'} ε_{j'j} = δ_{nj'} δ_{n'j} - δ_{nj} δ_{n'j'}$ and expressing the result as $2( $det $ρ_A + $det $ρ_B - $det $ρ_C)$ plus its symmetric partner. The residual, computed as the determinant of the restriction of $ρ_{AB} ρ̃_{AB}$ to its two-dimensional range, is the three-way tangle.

Extended reading notes

Core claim

The paper establishes a monogamy relation for three qubits in a pure state: $τ_{AB} + τ_{AC} ≤ τ_{A(BC)}$, with equality achievable for any consistent set of values. The proof runs by bounding each pairwise tangle from above by $Tr(ρ_{AB} ρ̃_{AB})$, where $ρ̃$ is the spin-flipped density matrix, and then evaluating the sum of the two traces in closed form as $4$ det $ρ_A$ — the tangle of A with the pair BC. Because the state of ABC is pure, the pair BC lives effectively in two dimensions, so the tangle between A and BC is well defined even though BC has a four-dimensional state space. The authors then identify the gap $τ_{A(BC)} - τ_{AB} - τ_{AC}$ as a residual that can be written explicitly in terms of the eight amplitudes $a_{ijk}$ as $4|d_1 - 2d_2 + 4d_3|$, where the three terms collect products of four amplitudes whose indices sit at the corners of a cube in three characteristic configurations. That residual is unchanged under permutations of the qubits, giving the symmetric decomposition $τ_{A(BC)} = τ_{AB} + τ_{AC} + τ_{ABC}$; for the GHZ state this reads $1 = 0 + 0 + 1$. A mixed-state version, $τ_{AB} + τ_{AC} ≤ τ^{min}_{A(BC)}$, follows by applying the pure-state inequality to each member of an optimal decomposition and using convexity of the tangle. The authors close with a conjecture that the analogous inequality holds for any number of qubits, backed by an exact equality for the $n$-qubit one-excitation states.

Load-bearing premise

The inequality itself is proved by algebra on the tangle, so its force as a statement about entanglement rests on the tangle — the squared concurrence — being a faithful stand-in for the entanglement of formation, an identification imported from earlier work rather than re-derived here; the mixed-state version additionally leans on the tangle being convex.

Editorial extensions

If this is right

Any protocol that requires qubit A to be entangled with two separate partners must budget against a hard ceiling set by A's entanglement with the pair, so a fully entangled pair leaves zero for a third party.
The residual $τ_{ABC}$ gives a permutation-invariant number that separates genuine three-way entanglement from pairwise entanglement: GHZ states score 1 with all pairwise tangles zero, while the one-excitation states of the form $α|100⟩ + β|010⟩ + γ|001⟩ score zero.
The entanglement of formation does not obey the same inequality — the paper's example gives $E_{AB} = E_{AC} = 0.601$ against $E_{A(BC)} = 1$ — so accounting in ebits and accounting in tangle give different pictures of what a single qubit can afford to share.
For mixed states the bound survives in convex-roof form, $τ_{AB} + τ_{AC} ≤ τ^{min}_{A(BC)}$, so the trade-off is not an artifact of purity.
The authors conjecture, with exact equality verified for $n$-qubit one-excitation states, that $τ_{12} + τ_{13} + ⋯ + τ_{1n} ≤ τ_{1(23…n)}$ holds for all pure states of $n$ qubits.

Reading between the lines

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

Read as a resource-accounting statement, the inequality says that entanglement with a group is the total budget and each individual partner draws from it — a picture that suggests treating the three-way tangle as the part of the budget that can only be spent on genuinely tripartite tasks, such as GHZ-correlated protocols, and never on bipartite ones.
The failure of the entanglement of formation to satisfy the same inequality while remaining a monotone function of the tangle is a concrete case where two measures that agree on ordering disagree on how a resource decomposes; it would be worth checking whether other convex-roof measures show the same split.
The permutation invariance of $τ_{ABC}$ means the quantity can be computed from any qubit's point of view, which makes it a candidate invariant for classifying three-qubit states under local unitary transformations, alongside the quantities catalogued in the invariant-theory literature the paper cites.
The natural test of the paper's conjecture is a numerical search over pure four-qubit states for a violation of $τ_{12} + τ_{13} + τ_{14} ≤ τ_{1(234)}$; the one-excitation family only confirms the equality case, not the inequality's general validity.

Original abstract

Consider three qubits A, B, and C which may be entangled with each other. We show that there is a trade-off between A's entanglement with B and its entanglement with C. This relation is expressed in terms of a measure of entanglement called the "tangle," which is related to the entanglement of formation. Specifically, we show that the tangle between A and B, plus the tangle between A and C, cannot be greater than the tangle between A and the pair BC. This inequality is as strong as it could be, in the sense that for any values of the tangles satisfying the corresponding equality, one can find a quantum state consistent with those values. Further exploration of this result leads to a definition of the "three-way tangle" of the system, which is invariant under permutations of the qubits.

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, 4,914 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.85 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Consider three qubits A, B, and C which may be entangled with each other. We show that there is a trade-off between A's entanglement with B and its entanglement with C. This relation is expressed in terms of a measure of entanglement called the "tangle," which is related to the entanglement of formation. Specifically, we show that the tangle between A and B, plus the tangle between A and C, cannot be greater than the tangle between A and the pair BC. This inequality is as strong as it could be, in the sense that for any values of the tangles satisfying the corresponding equality, one can find a quantum state consistent with those values. Further exploration of this result leads to a definition of the "three-way tangle" of the system, which is invariant under permutations of the qubits.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Consider three qubits A, B, and C which may be entangled with each other.	background 0.99	0.06	0.02	hedged possibility 0.98
s1	We show that there is a trade-off between A's entanglement with B and its entanglement with C.	result 0.87	0.94	0.81	asserted 1.00
s2	This relation is expressed in terms of a measure of entanglement called the "tangle," which is related to the entanglement of formation.	method 0.61	0.41	0.06	asserted 1.00
s3	Specifically, we show that the tangle between A and B, plus the tangle between A and C, cannot be greater than the tangle between A and the pair BC.	result 1.00	0.94	0.85	asserted 1.00
s4	This inequality is as strong as it could be, in the sense that for any values of the tangles satisfying the corresponding equality, one can find a quantum state consistent with those values.	result 0.67	0.90	0.13	asserted 0.98
s5	Further exploration of this result leads to a definition of the "three-way tangle" of the system, which is invariant under permutations of the qubits.	contribution 0.62	0.91	0.07	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
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levels run

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S sections
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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

References

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How later papers used it

319 later papers cite it

Evidence: substantial. 319 of the citing papers share no author with it, and 53 of those carry a confident label from Pith's citation classifier.

Pith read 39 of the 502 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

10 later papers by other authors are listed below: 7 used its method or tool, 1 compared its own results with it and 2 cited it as support.

Uses its method or tool 7 of 33 papers with this label

The tripartite entanglement is quantified by the minimum residual contangle $ℛ_τ^min$ that is defined as [ref] ,

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We measure the entanglement of individual spins using the tangle $τ(x,t)$ [ref] : [equation] which is proportional to the purity $τ_x = 2 - 2ρ_x^2$.

Entanglement dynamics and ergodicity breaking in a quantum cellular automaton (2022) · Chiral quasiparticles · 1.00
Used the genuine multipartite entanglement measure originally introduced in this paper to quantify the entanglement in its system.

Strain induced coupling and quantum information processing with hexagonal boron nitride quantum emitters (2021) · 0.98 · paraphrased: the licence does not allow quoting
Computed a quantum correlation measure based on the monogamy inequality of this paper to classify the generated states.

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Following this paper, introduced the measure.

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Chose the 3-tangle of this paper as the entanglement measure.

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We start from the definition [ref] , write down all the terms and combine them to obtain the $f_i$s that are finally expressed as a function of the entangling phases.] The explicit formula is given by [equation] where [equation] where $p,q,r∈{1,2,3}$.

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Our results reveal the polygamous nature of distributed quantum incompatibility, in stark contrast to the monogamy of entanglement [ref] and coherence [ref] across subsystems of multipartite quantum states.

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How the literature uses it

70 arXiv papers cite this in 103 sentences

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Papers citing it · 63 sentences without a confident label

Contests it 1 paper

Let us also mention here that there is a well-known difficulty with the approach of Page and of the work reviewed in [ref] which is that, by monogamy [ref] , the late emitted particles cannot simultaneously purify the early emitted particles and also be entangled (as one might expect them to be on a semiclassical understanding) with their partner-quanta which fell through the horizon.

The black hole information loss puzzle, matter-gravity entanglement entropy and the second law · Introduction · contest 0.71

Uses its method or tool 4 papers

For the final state in Eq. [ref], the three single-qubit reduced density matrices are defined by [equation] For a pure three-qubit state, the one-versus-rest concurrence between subsystem $X$ and the remaining two-qubit system $Y$ is [equation] We then define the three squared concurrences as [equation] and regard $a,b,c$ as the three sides of the concurrence triangle [ref] .

Genuine Tripartite Entanglement Selects Gauge-Invariant Theories · Concurrence fill · 0.92
The properties and peculiarities of the three-body system can be analyzed by means of the monogamy inequality [ref] C_i(kj)^2 ≤C_ij^2+C_ik^2 and the genuine multiparticle entanglement quantified by the concurrence triangle given by [ref] F_3= 4/√(3) [ Q (Q-C_1(23)) (Q-C_2(13)) (Q-C_3(12)) ]^1/2 with $F_3$ takes values from 0 and 1.

Quantum entanglement and Bell inequality violation at colliders · Three-body decays · 0.86
The W-type entanglement can be detected with the help of 3-tangle $τ_ABC$ introduced in [ref] .

Entanglement Classification from a Topological Perspective · Tripartite entanglement · 0.90

Compares with it 4 papers

Its persistence at large $r$—$h(A:B)→0.75$ for Bell and $0.16$ for GHZ—signals genuine three-party structure among $A,B,$, paralleling the fermionic finding that the Markov gap detects correlations invisible to the residual tangle [ref] .

Surviving correlations across a horizon: reflected entropy for bosonic fields in non-inertial frames and black hole spacetimes · Markov gap Results · 0.91
In figure [ref] we show the correlation of the ladder $τ_3$ ([ref]) with the measure of triparite entanglement known as 3-tangle [ref] .

Connectomes and Properties of Quantum Entanglement · Measures of entanglement · 0.98
We point out that <ref> (i) is reminiscent of the Coffman-Kundu-Wootters inequality [ref] .

Monogamy of correlations and entropy inequalities in the Bloch picture · Monogamy of Correlations from the Bloch Picture · 1.00

Cites it as support 3 papers

The two-body entanglement is not enough to describe the tripartite entanglement [ref] .

Tripartite Entanglement and Quantum Correlation · Introduction · 0.97
The appearance of 3-body entanglement quantity implies that the 2-body entanglement quantities are not enough [ref] .

Non-Locality$\neq$Quantum Entanglement · Entanglement Measures · 0.98
Given this distinguishing ability, it is perhaps unsurprising that D-CTC circuits are also capable of cloning arbitrary quantum states [ref] (though, of course, entanglement is not cloned as this is forbidden by monogamy [ref] ).

Reality, Causality, and Quantum Theory · Overview of D-CTCs · 0.96

Cites it as background 23 papers

In quantum theory, quantum systems are currently modelled by density matrices $(ρ)$ and entanglement is recognized to be monogamous [ref] .

A paradox regarding monogamy of entanglement · Introduction · 0.95
In this scenario, monogamy of entanglement [ref] typically has been discussed in terms of the paradox in [ref] , as commented on in [ref] .

Replica wormhole and island incompatibility with monogamy of entanglement · Introduction · 0.99
The three-tangle [ref] is a genuine tripartite entanglement measure for three-qubit pure states [ref] .

Separability from Multipartite Measures · Tripartite Entanglement Measures · 0.94

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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