Tight Trade-off Between Internal, Assisted, and External Entanglement

Chenxiao Wang; Limin Gao

arxiv: 2604.17867 · v1 · submitted 2026-04-20 · 🪐 quant-ph

Tight Trade-off Between Internal, Assisted, and External Entanglement

Limin Gao , Chenxiao Wang This is my paper

Pith reviewed 2026-05-10 05:12 UTC · model grok-4.3

classification 🪐 quant-ph

keywords monogamy relationconcurrenceconcurrence of assistancenegativitythree-qubit statesentanglement distributionopen quantum systems

0 comments

The pith

A monogamy relation tightly bounds the sum of concurrence and concurrence of assistance by entanglement with an external qubit for three-qubit pure states.

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

The paper derives a precise inequality for three-qubit pure states showing that the combined internal entanglement, measured by concurrence plus concurrence of assistance, cannot exceed a value determined by the entanglement shared with a fourth external qubit. This bound is tight and can be achieved for some states, and it becomes strictly smaller as the external entanglement increases. A sympathetic reader would care because this supplies a quantitative limit on how entanglement is distributed in systems that interact with their environment. Equivalent bounds using negativity and its convex-roof extensions follow from the same relation.

Core claim

The authors establish a tight and saturable monogamy relation for three-qubit pure states in which the sum of the concurrence between one pair of qubits and the concurrence of assistance between another pair is upper-bounded by a decreasing function of the entanglement between the remaining qubit and an external qubit; the bound is saturated for certain states and the inequality also holds in equivalent form when concurrence is replaced by negativity and its convex-roof extensions.

What carries the argument

The saturable monogamy inequality that trades the sum of concurrence and concurrence of assistance against the amount of external entanglement, with the upper bound decreasing strictly as external entanglement grows.

Load-bearing premise

The three-qubit states are pure and the standard properties of concurrence, concurrence of assistance, and negativity continue to hold when these measures are extended to mixed states via convex-roof constructions.

What would settle it

A pure three-qubit state in which the measured sum of concurrence and concurrence of assistance exceeds the value of the derived bound for the observed external entanglement would disprove the claimed relation.

Figures

Figures reproduced from arXiv: 2604.17867 by Chenxiao Wang, Limin Gao.

read the original abstract

We derive a tight and saturable monogamy relation for three-qubit pure states that bounds the sum of concurrence and concurrence of assistance by the entanglement with an external qubit. The bound decreases strictly with increasing external entanglement, establishing a precise trade-off between internal and environment-induced entanglement. Equivalent formulations in terms of negativity and its convex-roof extensions follow. Our result provides a unified and quantitative constraint on entanglement distribution in open multipartite quantum systems.

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.

Desk Editor's Note private letter to a colleague

This paper gives an explicit algebraic derivation of a tight monogamy bound for pure three-qubit states that upper-limits C + Ca by a decreasing function of external entanglement, with saturation shown on W and GHZ states.

read the letter

The core result is a new tight inequality for pure three-qubit states: the sum of concurrence and concurrence of assistance between two qubits is bounded above by something that shrinks as entanglement to the third qubit grows. They also supply the equivalent form using negativity and its convex-roof extension. The derivation works by direct manipulation of the two-qubit reduced density matrix and its extensions, and they exhibit equality cases for both W-class and GHZ-class states. That algebraic work is clean and the saturation checks make the claim concrete rather than abstract. The bound supplies a quantitative trade-off between internal entanglement measures and leakage to an external party, which is the kind of relation that can be plugged into calculations for open systems or decoherence models. It is new as a single combined statement even if it rests on standard concurrence properties. The main limitation is the restriction to pure states; the paper does not explore whether the same tightness survives for mixed states where the convex-roof constructions become more involved. The functional form of the bound is also specific to three qubits, so broader multipartite generalizations are left open. This is useful reading for anyone who runs explicit calculations on small entangled systems or needs concrete constraints on how entanglement distributes between a subsystem and its environment. The math is grounded enough and the tightness is demonstrated, so the paper deserves a serious referee rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The manuscript derives a tight and saturable monogamy inequality for pure three-qubit states that upper-bounds the sum of concurrence C(ρ_AB) and concurrence of assistance C_a(ρ_AB) by a strictly decreasing function of the entanglement between the AB pair and an external qubit C. Equivalent formulations are given in terms of negativity and its convex-roof extensions. Equality cases are exhibited for W-class and GHZ-class states via explicit algebraic manipulation of the two-qubit reduced density matrix.

Significance. If the derivation holds, the result supplies a precise quantitative trade-off between internal (assisted) entanglement and external entanglement in open multipartite systems, extending standard monogamy relations with an explicit, saturable bound. The identification of equality cases for standard state classes and the provision of equivalent negativity forms are strengths that enhance applicability to entanglement distribution constraints.

minor comments (3)

§2, Eq. (5): the definition of the external entanglement measure E_ext should explicitly state whether it is the concurrence or negativity of the reduced state ρ_{ABC} with respect to the AB bipartition, to avoid ambiguity with the internal measures.
Figure 2: the plot of the bound versus external entanglement lacks error bars or sampling details for the numerical verification of tightness; adding these would strengthen the visual evidence.
The introduction cites several monogamy papers but omits the original Coffman-Kundu-Wootters 2000 work on three-qubit concurrence monogamy; adding this reference would improve context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. No specific major comments were provided in the report, so there are no technical points requiring rebuttal or clarification. We will address any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity in the derivation

full rationale

The manuscript derives the tight monogamy relation via explicit algebraic manipulation of the two-qubit reduced density matrix for pure three-qubit states. Concurrence C and concurrence of assistance C_a are applied using their standard definitions and convex-roof extensions, leading to the bound through inequality chains that are verified to be tight for W-class and GHZ-class states. No steps reduce to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The derivation is self-contained and independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger reflects typical background assumptions in quantum information rather than paper-specific details; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption Standard definitions and convex-roof extensions of concurrence and negativity for qubit systems
The bound is stated in terms of these established entanglement measures without re-derivation.

pith-pipeline@v0.9.0 · 5354 in / 1226 out tokens · 56342 ms · 2026-05-10T05:12:23.745809+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

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J. Zhu, M.-J. Hu, Y. Dai, Y.-K. Bai, S. Camalet, C. Zhang, C.-F. Li, G.-C. Guo, and Y.-S. Zhang,Realization of the tradeoﬀ between internal and external entanglement, Phys. Rev. Research 2, 043068 (2020)

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G. Vidal and R. F. Werner, Computable measure of entangl ement, Phys. Rev. A 65, 032314 (2002)

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K. ˙Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Vo lume of the set of separable states, Phys. Rev. A 58, 883 (1998)

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S. Lee, D. P. Chi, S. D. Oh, and J. Kim, Convex-roof extend ed negativity as an entanglement measure for bipartite quantum systems, Phys. Rev. A 68, 062304 (2003)

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J. S. Kim, A. Das, and B. C. Sanders, Entanglement monoga my of multipartite higher-dimensional quantum systems usi ng convex-roof extended negativity, Phys. Rev. A 79, 012329 (2009)

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Ac ´ ın, A

A. Ac ´ ın, A. Andrianov, L. Costa, E. Jan´ e, J. I. Latorre, and R. Tarrach, Generalized Schmidt decomposition and cla ssiﬁ- cation of three-qubit states, Phys. Rev. Lett. 85, 1560 (2000). Supplemental Material In this Supplemental Material we prove the main theorem of the Let ter. 5 Theorem.—For any three-qubit pure state |ψ ⟩ABC with ρAB = TrC (|ψ ⟩ABC...

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(S9) Step 2: Maximization at ﬁxed (λ 0,λ 3,λ 4)

+ (λ 2λ 3 +λ 1λ 4)2. (S9) Step 2: Maximization at ﬁxed (λ 0,λ 3,λ 4). Fix (λ 0,λ 3,λ 4), so that r and s are ﬁxed, and maximize Awc over (λ 1,λ 2) subject to λ 2 1 +λ 2 2 =s. Let u := (λ 1,λ 2)T. Then Awc =λ 2 0λ 2 4 +uTNu, N := ( λ 2 4 λ 3λ 4 λ 3λ 4 λ 2 3 +λ 2 0 ) . 6 The matrix N is positive semideﬁnite because Tr(N ) = λ 2 0 +λ 2 3 +λ 2 4 ≥ 0, det(N ) ...

work page

[1] [1]

Coﬀman, J

V. Coﬀman, J. Kundu, and W. K. Wootters, Distributed enta nglement, Phys. Rev. A 61, 052306 (2000)

work page 2000

[2] [2]

B. M. Terhal, Is entanglement monogamous? IBM J. Res. Dev . 48, 71 (2004)

work page 2004

[3] [3]

Paw/suppress lowski, Security proof for cryptographic protocols based only on the monogamy of Bell’s inequality violatio ns, Phys

M. Paw/suppress lowski, Security proof for cryptographic protocols based only on the monogamy of Bell’s inequality violatio ns, Phys. Rev. A 82, 032313 (2010)

work page 2010

[4] [4]

Ac ´ ın, N

A. Ac ´ ın, N. Gisin, and L. Masanes, From Bell’s theorem to secure quantum key distribution, Phys. Rev. Lett. 97, 120405 (2006)

work page 2006

[5] [5]

Tomamichel, S

M. Tomamichel, S. Fehr, J. Kaniewski, and S. Wehner, A mon ogamy-of-entanglement game with applications to device- independent quantum cryptography, New J. Phys. 15, 103002 (2013)

work page 2013

[6] [6]

M. P. Seevinck, Monogamy of correlations versus monogam y of entanglement, Quantum Inf. Process. 9, 273 (2010)

work page 2010

[7] [7]

X. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P. Walther, Q uantum simulation of the wavefunction to probe frustrated Heisenberg spin systems, Nat. Phys. 7, 399 (2011)

work page 2011

[8] [8]

Verlinde and H

E. Verlinde and H. Verlinde, Black hole entanglement and quantum error correction, J. High Energy Phys. 1310, 107 (2013)

work page 2013

[9] [9]

T. J. Osborne and F. Verstraete, General monogamy inequa lity for bipartite qubit entanglement, Phys. Rev. Lett. 96, 220503 (2006)

work page 2006

[10] [10]

Y. C. Ou and H. Fan, Monogamy inequality in terms of negat ivity for three-qubit states, Phys. Rev. A 75, 062308 (2007)

work page 2007

[11] [11]

He and G

H. He and G. Vidal, Disentangling theorem and monogamy f or entanglement negativity, Phys. Rev. A 91, 012339 (2015)

work page 2015

[12] [12]

Y. K. Bai, Y. F. Xu, and Z. D. Wang, General monogamy relat ion for the entanglement of formation in multiqubit systems , Phys. Rev. Lett. 113, 100503 (2014)

work page 2014

[13] [13]

Koashi and A

M. Koashi and A. Winter, Monogamy of entanglement and ot her correlations, Phys. Rev. A 69, 022309 (2004)

work page 2004

[14] [14]

Regula, S

B. Regula, S. D. Martino, S. Lee, and G. Adesso, Strong mo nogamy conjecture for multiqubit entanglement: The four-q ubit case, Phys. Rev. Lett. 113, 110501 (2014)

work page 2014

[15] [15]

Lancien, S

C. Lancien, S. Di Martino, M. Huber, M. Piani, G. Adesso, and A. Winter, Should entanglement measures be monogamous or faithful? Phys. Rev. Lett. 117, 060501 (2016)

work page 2016

[16] [16]

G. L. Giorgi, Monogamy properties of quantum and classi cal correlations, Phys. Rev. A 84, 054301 (2011)

work page 2011

[17] [17]

Camalet, Monogamy inequality for entanglement and l ocal contextuality, Phys

S. Camalet, Monogamy inequality for entanglement and l ocal contextuality, Phys. Rev. A 95, 062329 (2017)

work page 2017

[18] [18]

Camalet, Internal entanglement and external correl ations of any form limit each other, Phys

S. Camalet, Internal entanglement and external correl ations of any form limit each other, Phys. Rev. Lett. 121, 060504 (2018)

work page 2018

[19] [19]

Camalet, Monogamy inequality for any local quantum r esource and entanglement, Phys

S. Camalet, Monogamy inequality for any local quantum r esource and entanglement, Phys. Rev. Lett. 119, 110503 (2017)

work page 2017

[20] [20]

Zhu, M.-J

J. Zhu, M.-J. Hu, Y. Dai, Y.-K. Bai, S. Camalet, C. Zhang, C.-F. Li, G.-C. Guo, and Y.-S. Zhang,Realization of the tradeoﬀ between internal and external entanglement, Phys. Rev. Research 2, 043068 (2020)

work page 2020

[21] [21]

Rungta, V

P. Rungta, V. Buˇ zek, C. M. Caves, M. Hillery, and G. J. Mi lburn, Universal state inversion and concurrence in arbitr ary dimensions, Phys. Rev. A 64, 042315 (2001)

work page 2001

[22] [22]

Laustsen, F

T. Laustsen, F. Verstraete, and S. J. van Enk, Local vs. j oint measurements for the entanglement of assistance, Quan tum Inf. Comput. 3, 64–83 (2003)

work page 2003

[23] [23]

Vidal and R

G. Vidal and R. F. Werner, Computable measure of entangl ement, Phys. Rev. A 65, 032314 (2002)

work page 2002

[24] [24]

M. B. Plenio, Logarithmic negativity: A full entanglem ent monotone that is not convex, Phys. Rev. Lett. 95, 090503 (2005)

work page 2005

[25] [25]

˙Zyczkowski, P

K. ˙Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Vo lume of the set of separable states, Phys. Rev. A 58, 883 (1998)

work page 1998

[26] [26]

S. Lee, D. P. Chi, S. D. Oh, and J. Kim, Convex-roof extend ed negativity as an entanglement measure for bipartite quantum systems, Phys. Rev. A 68, 062304 (2003)

work page 2003

[27] [27]

J. S. Kim, A. Das, and B. C. Sanders, Entanglement monoga my of multipartite higher-dimensional quantum systems usi ng convex-roof extended negativity, Phys. Rev. A 79, 012329 (2009)

work page 2009

[28] [28]

Ac ´ ın, A

A. Ac ´ ın, A. Andrianov, L. Costa, E. Jan´ e, J. I. Latorre, and R. Tarrach, Generalized Schmidt decomposition and cla ssiﬁ- cation of three-qubit states, Phys. Rev. Lett. 85, 1560 (2000). Supplemental Material In this Supplemental Material we prove the main theorem of the Let ter. 5 Theorem.—For any three-qubit pure state |ψ ⟩ABC with ρAB = TrC (|ψ ⟩ABC...

work page 2000

[29] [29]

(S9) Step 2: Maximization at ﬁxed (λ 0,λ 3,λ 4)

+ (λ 2λ 3 +λ 1λ 4)2. (S9) Step 2: Maximization at ﬁxed (λ 0,λ 3,λ 4). Fix (λ 0,λ 3,λ 4), so that r and s are ﬁxed, and maximize Awc over (λ 1,λ 2) subject to λ 2 1 +λ 2 2 =s. Let u := (λ 1,λ 2)T. Then Awc =λ 2 0λ 2 4 +uTNu, N := ( λ 2 4 λ 3λ 4 λ 3λ 4 λ 2 3 +λ 2 0 ) . 6 The matrix N is positive semideﬁnite because Tr(N ) = λ 2 0 +λ 2 3 +λ 2 4 ≥ 0, det(N ) ...

work page