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arxiv: 2206.07731 · v2 · submitted 2022-06-15 · 🪐 quant-ph · cond-mat.str-el

Controlling gain with loss: Bounds on localizable entanglement in multi-qubit systems

Jithin G. Krishnan , Harikrishnan K. J. , Amit Kumar Pal This is my paper

Pith reviewed 2026-05-24 12:08 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el
keywords localizable entanglementmulti-qubit systemsGHZ statesW statesDicke statesquantum spin modelsentanglement boundsphase-flip noise
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The pith

Localizable entanglement on a qubit subsystem is bounded by the bipartite entanglement lost when measuring the rest of the system.

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

The paper derives relations between the entanglement that can be localized onto a chosen group of qubits through local measurements on the remaining qubits and the amount of bipartite entanglement that vanishes in the process. For generalized GHZ and W states it supplies explicit analytical bounds connecting the two quantities. For Dicke states the localized value approaches the pre-measurement bipartite entanglement as the total number of qubits grows. Numerical work on the transverse-field XY and XXZ spin chains reveals a cubic dependence of localized on lost entanglement that survives both phase-flip noise and disorder in the external field.

Core claim

For generalized GHZ and W states, localizable entanglement is analytically bounded by functions of the bipartite entanglement present before measurement; for Dicke states the two quantities become equal in the large-system limit; in one-dimensional spin models the localized entanglement scales cubically with the lost entanglement.

What carries the argument

Localizable entanglement obtained by local projective measurements performed on all qubits outside a chosen subsystem, quantified relative to the bipartite entanglement across the cut in the pre-measurement pure state.

If this is right

  • For any generalized GHZ or W state the localized entanglement cannot exceed the derived functions of the lost bipartite entanglement.
  • In the thermodynamic limit of Dicke states the localizable entanglement equals the pre-measurement bipartite entanglement across the chosen cut.
  • In the transverse XY and XXZ chains the localized entanglement grows as the cube of the lost entanglement.
  • The cubic relation remains unchanged when single-qubit phase-flip noise acts on every qubit or when the external field strength is disordered.

Where Pith is reading between the lines

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

  • The same loss-localization trade-off may appear in other one-dimensional critical systems whose ground states are well approximated by matrix-product states.
  • The bounds could be used to design measurement protocols that concentrate entanglement onto a small register while minimizing the information sacrificed.
  • Numerical checks on random pure states suggest the cubic scaling is not limited to translation-invariant models.

Load-bearing premise

The multi-qubit states remain pure before measurement and local projective measurements on the complement alone are sufficient to realize the localization without extra optimization or post-selection.

What would settle it

An explicit computation for a three-qubit generalized GHZ state in which the localizable entanglement after measuring one qubit exceeds the stated analytical upper bound in terms of the initial bipartite entanglement.

Figures

Figures reproduced from arXiv: 2206.07731 by Amit Kumar Pal, Harikrishnan K. J., Jithin G. Krishnan.

Figure 1
Figure 1. Figure 1: FIG. 1. Consider a multi-qubit system divided into three subsystems, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

We investigate the relation between the amount of entanglement localized on a chosen subsystem of a multi-qubit system via local measurements on the rest of the system, and the bipartite entanglement that is lost during this measurement process. We study a number of paradigmatic pure states, including the generalized GHZ, the generalized W, Dicke, and the generalized Dicke states. For the generalized GHZ and W states, we analytically derive bounds on localizable entanglement in terms of the entanglement present in the system prior to the measurement. Also, for the Dicke and the generalized Dicke states, we demonstrate that with increasing system size, localizable entanglement tends to be equal to the bipartite entanglement present in the system over a specific partition before measurement. We extend the investigation numerically in the case of arbitrary multi-qubit pure states. We also analytically determine the modification of these results, including the proposed bounds, in situations where these pure states are subjected to single-qubit phase-flip noise on all qubits. Additionally, we study one-dimensional paradigmatic quantum spin models, namely the transverse-field XY model and the XXZ model in an external field, and numerically demonstrate a cubic dependence of the localized entanglement on the lost entanglement. We show that this relation is robust even in the presence of disorder in the strength of the external field.

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 paper investigates the relation between localizable entanglement (LE) localized on a subsystem via measurements on the complement and the bipartite entanglement lost in the process. It analytically derives bounds relating LE to pre-measurement bipartite entanglement for generalized GHZ and W states, shows that LE approaches the pre-measurement bipartite entanglement for large Dicke and generalized Dicke states, extends the analysis to phase-flip noise, and numerically demonstrates a cubic dependence of localized on lost entanglement in the transverse-field XY and XXZ spin models (robust to disorder).

Significance. If the central relations hold for the optimized definition of LE, the analytical bounds for GHZ/W states and the cubic scaling in spin models would provide concrete, falsifiable connections between entanglement gain and loss, useful for designing measurement-based entanglement distribution protocols. The noise robustness and size-scaling results add practical value. The work ships explicit analytical expressions for paradigmatic states, which is a strength.

major comments (2)
  1. [§§3–4] §§3–4 (GHZ and W derivations): the reported bounds are derived using specific projective measurements (computational basis or fixed angles) on the complement; the manuscript does not demonstrate that these measurements achieve the maximization required by the standard definition of localizable entanglement. If the bounds are not for the optimized quantity, they do not constrain true LE and the central claim is affected.
  2. [§6] §6 (XY/XXZ models): the cubic dependence is obtained numerically for the chosen partitions and measurement strategies; it is unclear whether the relation survives maximization over all local measurements on the complement or is an artifact of the specific protocol used. A concrete test (e.g., comparison with optimized LE for small N) is needed to support the scaling claim.
minor comments (2)
  1. [§2] The definition of LE (maximization step) should be stated explicitly in §2 with a reference to the original literature (e.g., Verstraete et al.).
  2. [Figure 5] Figure captions for the spin-model plots should specify the exact measurement basis and partition used to generate the data points.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point-by-point below and commit to revisions that strengthen the connection to the optimized definition of localizable entanglement.

read point-by-point responses

  1. Referee: [§§3–4] §§3–4 (GHZ and W derivations): the reported bounds are derived using specific projective measurements (computational basis or fixed angles) on the complement; the manuscript does not demonstrate that these measurements achieve the maximization required by the standard definition of localizable entanglement. If the bounds are not for the optimized quantity, they do not constrain true LE and the central claim is affected.

    Authors: We acknowledge that the standard definition of localizable entanglement (LE) requires maximization over all possible local measurements. In §§3–4 we employed specific projective measurements that are optimal for the generalized GHZ and W states (computational-basis measurements for GHZ by symmetry, and fixed-angle measurements for W that saturate the concurrence), but the manuscript does not contain an explicit optimality proof. In the revised version we will add a short argument (or appendix) showing that these choices achieve the maximum, thereby ensuring the derived bounds apply directly to the optimized LE. revision: yes

  2. Referee: [§6] §6 (XY/XXZ models): the cubic dependence is obtained numerically for the chosen partitions and measurement strategies; it is unclear whether the relation survives maximization over all local measurements on the complement or is an artifact of the specific protocol used. A concrete test (e.g., comparison with optimized LE for small N) is needed to support the scaling claim.

    Authors: The cubic scaling was observed for the specific measurement protocols and partitions used in the numerical simulations of the XY and XXZ models. To address the concern, we will add a concrete test in the revision: for small system sizes (N ≤ 6) we will numerically optimize the measurement angles on the complement and compare the resulting LE versus lost entanglement with the non-optimized protocol. The outcome of this comparison will be reported to confirm whether the cubic relation persists under maximization. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives bounds analytically from the algebraic structure of generalized GHZ and W states and demonstrates numerical relations for Dicke states and spin models without any reduction of predictions to fitted inputs or self-citations by construction. All load-bearing steps are direct calculations or simulations on the states themselves, with no evidence of self-definitional loops, ansatzes smuggled via citation, or uniqueness theorems imported from prior author work. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are indicated in the abstract; the work rests on standard definitions of localizable entanglement, bipartite entanglement measures, and the listed paradigmatic states.

axioms (1)
  • standard math Standard definitions and properties of localizable entanglement via local projective measurements and bipartite entanglement quantifiers hold for the studied pure states.
    Implicit background of quantum information theory invoked throughout.

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Forward citations

Cited by 1 Pith paper

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