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arxiv: 1907.00696 · v1 · pith:MIHADYWVnew · submitted 2019-07-01 · 🪐 quant-ph

The dynamic behaviors of local quantum uncertainty for three-qubit X states under decoherence channels

Abdallah Slaoui , Mohammed Daoud , Rachid Ahl Laamara This is my paper

Pith reviewed 2026-05-25 11:59 UTC · model grok-4.3

classification 🪐 quant-ph
keywords local quantum uncertaintyquantum discordnegativitythree-qubit X statesdecoherence channelsmonogamyphase reversal
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The pith

Local quantum uncertainty equals entropic quantum discord for three-qubit X states and exceeds negativity while satisfying monogamy.

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

The paper derives closed-form expressions for local quantum uncertainty on three-qubit X states and compares it directly to quantum discord and negativity. It reports that local quantum uncertainty reproduces the values given by entropic discord yet detects correlations beyond those captured by negativity. The work then tracks the time evolution of these quantities under phase damping, depolarizing, and phase reversal channels, finding that local quantum uncertainty remains larger and displays revival and freezing under phase reversal. Finally, it verifies that local quantum uncertainty obeys the monogamy inequality for any three-qubit state of this form.

Core claim

For three-qubit X states the local quantum uncertainty supplies exactly the same numerical value as entropic quantum discord and a strictly larger value than negativity. Under independent action of the three listed decoherence channels the local quantum uncertainty decays more slowly than the other two measures and, in the phase-reversal channel, exhibits both sudden revival and intervals of frozen dynamics. The same quantity satisfies the monogamy relation for every three-qubit X state.

What carries the argument

Closed-form analytical expressions for local quantum uncertainty evaluated on the three-qubit X-state density matrix under independent single-qubit decoherence maps.

If this is right

  • Local quantum uncertainty can replace entropic discord in any calculation that already uses the X-state parametrization.
  • Negativity underestimates the total non-classical correlations present in these states.
  • Phase-reversal noise leaves intervals in which local quantum uncertainty remains constant or returns to its initial value.
  • Any three-qubit X state obeys the monogamy bound for local quantum uncertainty, so correlations cannot be freely distributed among all three parties.

Where Pith is reading between the lines

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

  • Because local quantum uncertainty is analytically tractable on X states, it may be the preferred quantifier when experimentalists need to track correlation decay in real time.
  • The observed equality with discord on X states raises the question whether the same identity holds for other families that admit closed-form expressions.
  • Revival and freezing under phase reversal suggest that local quantum uncertainty could serve as a witness for certain non-Markovian effects in three-qubit experiments.

Load-bearing premise

The special block structure of an X-state density matrix together with independent decoherence on each qubit permits exact closed-form expressions for local quantum uncertainty, discord, and negativity.

What would settle it

Prepare a mixed three-qubit GHZ or Bell-diagonal X state, compute local quantum uncertainty and entropic discord from the measured density matrix, and check whether the two numbers agree to within experimental error; any systematic discrepancy falsifies the equality claim.

Figures

Figures reproduced from arXiv: 1907.00696 by Abdallah Slaoui, Mohammed Daoud, Rachid Ahl Laamara.

Figure 1
Figure 1. Figure 1: The local quantum uncertainty and negativity in three-qubit state of Bell type versus the parameter c. with I (2) ρi|j  = S (ρi) + S (ρj ) − S (ρij ). Here ρi (i = 1, 2, 3) is the reduced density matrix for the subsystem i and S (ρ) = −tr[ρ log (ρ)] is the von Neumann entropy. The total tripartite correlation (47) rewrites also as T (3) (ρ123) = min[I (2) ρ1|23 , I(2) ρ2|13 , I(2) ρ3|12 ]. (49) Analogo… view at source ↗
Figure 2
Figure 2. Figure 2: The quantum correlations in tripartite mixed GHZ state measured by local quantum uncertainty, quantum discord and negativity. The results plotted in [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The local quantum uncertainty and the quantum discord versus the dephasing parameter q. p=0 p=0.2 p=0.4 p=0.6 p=0.8 p=0.9 p=1 0.2 0.4 0.6 0.8 1.0 q 0.2 0.4 0.6 0.8 1.0 N (3) (ρGHZ DE ) [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The negativity versus the dephasing parameter q. The entropic quantum discord in mixed GHZ states shows more robustness against the dephasing effects in comparison with local quantum uncertainty. On the other hand, the negativity is more robust than local quantum uncertainty and quantum discord (see figure 4). Indeed, for 0.8 < q < 1, the local quantum uncertainty and quantum discord vanished whereas the n… view at source ↗
Figure 5
Figure 5. Figure 5: The local quantum uncertainty and the quantum discord versus the depolarizing strength q. p=0 p=0.2 p=0.4 p=0.6 p=0.8 p=0.9 p=1 0.1 0.2 0.3 0.4  [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The negativity versus the depolarizing strength q. Under depolarizing effects, the quantum discord is more robust in comparison with local quantum uncertainty. However, like for the dephasing effects, it is remarkable that the negativity shows more robustness than the local quantum uncertainty and the quantum discord. 3.3 Phase reversal environment: Phase reversal environment leaves the state invariant |0i… view at source ↗
Figure 7
Figure 7. Figure 7: The local quantum uncertainty and the quantum discord versus the decoherence parameter q. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The negativity versus the decoherence parameter q. For a mixing parameters p such that 0 < p < 0.6, the local quantum uncertainty shows a revival phenomenon under the phase reversal environment. It increases to become maximal for higher values of decoherence parameter q. It seems that the phase reversal tends to enhance the amount of quan￾tum correlation in the system. Also for states with a mixing paramet… view at source ↗
read the original abstract

We derive the analytical expression of local quantum uncertainty for three-qubit X-states. We give also the expressions of quantum discord and the negativity. A comparison of these three quantum correlations quantifiers is discussed in the special cases of mixed GHZ states and Bell-type states. We find that local quantum uncertainty gives the same amount of non-classical correlations as are measured by entropic quantum discord and goes beyond negativity. We also discuss the dynamics of non-classical correlations under the effect of phase damping, depolarizing and phase reversal channels. We find the local quantum uncertainty shows more robustness and exhibits, under phase reversal effect, revival and frozen phenomena. The monogamy property of local quantum uncertainty is also discussed. It is shown that it is monogamous for three-qubit states.

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

1 major / 0 minor

Summary. The paper derives analytical expressions for local quantum uncertainty (LQU) of three-qubit X-states, supplies expressions for quantum discord and negativity, compares the three measures on mixed GHZ and Bell-type states (finding LQU equivalent to discord and stronger than negativity), examines their dynamics under phase damping, depolarizing and phase reversal channels (highlighting LQU robustness, revival and frozen phenomena under phase reversal), and asserts that LQU is monogamous for three-qubit states.

Significance. If the closed-form derivations are correct, the work supplies exact, parameter-free expressions that enable direct comparison of correlation quantifiers and exact dynamical studies under standard decoherence channels. The reported revival and frozen effects under phase reversal constitute concrete, falsifiable predictions that could be tested experimentally.

major comments (1)
  1. [Abstract] Abstract: the claim that LQU 'is monogamous for three-qubit states' is not supported by the calculations, which are performed exclusively on the X-state family (density matrices with 14 specific zero entries). Monogamy inequalities are known to fail for some states outside restricted subclasses, and no general proof or verification outside the X-form is supplied.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that LQU 'is monogamous for three-qubit states' is not supported by the calculations, which are performed exclusively on the X-state family (density matrices with 14 specific zero entries). Monogamy inequalities are known to fail for some states outside restricted subclasses, and no general proof or verification outside the X-form is supplied.

    Authors: We agree that the abstract statement is stated too generally. All analytical derivations and explicit verifications of the monogamy inequality for LQU in the manuscript are performed exclusively on three-qubit X-states. No general proof or numerical checks outside this family are provided. We will revise the abstract (and the final paragraph of the conclusions) to state that LQU is shown to be monogamous for three-qubit X-states. This revision will be made in the next version of the manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives closed-form analytical expressions for LQU, discord, and negativity directly from the three-qubit X-state density matrix under the listed decoherence channels, then computes dynamics and monogamy checks from those explicit formulas. No parameter is fitted to data and then relabeled as a prediction, no quantity is defined in terms of itself, and no load-bearing step reduces to a self-citation chain or imported uniqueness theorem. All reported comparisons and phenomena follow from the independent definitions of the measures applied to the X-state family, making the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that three-qubit X-states admit closed-form correlation measures and that the decoherence channels act in a manner permitting exact dynamics; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Three-qubit X-states have a density matrix structure that permits closed-form analytical expressions for local quantum uncertainty, quantum discord, and negativity.
    This premise is required for the derivation of expressions and the subsequent comparisons and dynamics stated in the abstract.

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Reference graph

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