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arxiv: 2605.23197 · v1 · pith:4JM5JHDUnew · submitted 2026-05-22 · 🪐 quant-ph · gr-qc

Anomalous Decay of Quantum Resources: The Entanglement Sudden Death Mpemba Effect

Zhilong Liu , Zehua Tian , Jieci Wang This is my paper

Pith reviewed 2026-05-25 04:56 UTC · model grok-4.3

classification 🪐 quant-ph gr-qc
keywords entanglement sudden deathMpemba effectamplitude dampingtwo qubitstrajectory crossoverquantum correlationsdissipative dynamicsopen quantum systems
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The pith

A more strongly entangled two-qubit state can reach separability sooner than a weaker one under local amplitude damping.

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

The paper establishes that two qubits coupled to independent identical amplitude-damping reservoirs can display an anomalous crossing in their entanglement decay curves. States starting with higher entanglement can hit the separable regime at an earlier time than states starting with lower entanglement, producing a finite-time trajectory crossover. This behavior is derived exactly for the concurrence dynamics and mapped across the two-parameter family of initial states to show the regime of occurrence. A sympathetic reader cares because the result identifies a concrete way that initial-state choice controls the lifetime of quantum correlations in open systems.

Core claim

For two qubits undergoing independent local amplitude damping, the entanglement sudden death time is not a monotonic function of initial entanglement strength; within a specific two-parameter family of initial states, a larger initial concurrence can produce an earlier sudden-death time, resulting in an explicit crossing of the entanglement trajectories before both states become separable.

What carries the argument

The finite-time trajectory crossover of concurrence under identical local amplitude-damping channels, with the ESD time obtained from the exact solution of the master equation for the two-parameter initial-state family.

If this is right

  • The ESD time depends non-monotonically on the initial concurrence for states inside the identified parameter region.
  • An exact analytic expression exists for both the crossover time and the ESD time as functions of the initial-state parameters.
  • The phase diagram in the two-parameter plane separates regions where the anomalous ordering occurs from regions where the ordering is conventional.
  • The effect supplies a mechanism for shortening the lifetime of quantum correlations by choice of initial state alone.

Where Pith is reading between the lines

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

  • Similar trajectory-crossing behavior may appear for other quantum resources such as coherence or discord under the same noise model.
  • The result suggests that initial-state engineering could be used to accelerate or delay the loss of entanglement in engineered dissipative environments.
  • Experimental tests could be performed in linear-optical or trapped-ion setups by preparing the required two-parameter family and tracking concurrence via tomography.

Load-bearing premise

The two reservoirs have identical decay rates and act independently, while the initial states are restricted to a two-parameter family that allows direct comparison of entanglement strength.

What would settle it

Prepare two initial states from the family with different concurrences, evolve them under independent amplitude-damping channels of equal rate, and check whether the measured concurrence curves cross at a time before both reach zero.

Figures

Figures reproduced from arXiv: 2605.23197 by Jieci Wang, Zehua Tian, Zhilong Liu.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic representation of the dissipative model. A bipar [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Decay of concurrence under symmetric dissipation [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Phase diagrams: solid black semicircle is physical boundary [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Characteristic timescales and their dimensionless ratios vary [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

In classical thermodynamics, the Mpemba effect refers to the counterintuitive observation that hot water can freeze faster than cold water, manifesting as an anomalous crossing of dynamical trajectories. While analogues of this phenomenon have been explored in quantum radiative systems and spin-chain entanglement asymmetry, its connection to the finite-time decoupling of quantum correlations remains elusive. In this Letter, we uncover a distinct quantum Mpemba effect associated with entanglement sudden death (ESD). By analyzing two qubits interacting with local amplitude damping reservoirs, we demonstrate that a more strongly entangled initial state can experience a faster collapse into a separable state than a more weakly entangled one. We provide an exact analytical derivation of the trajectory crossover dynamics and the ESD time. Finally, we map the phase diagram of initial state parameters to delineate the regime where this anomalous entanglement Mpemba effect occurs, offering insights into the control of quantum resource lifetimes in dissipative environments.

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 / 2 minor

Summary. The paper claims to identify an entanglement sudden death (ESD) Mpemba effect in two qubits subject to independent local amplitude-damping reservoirs. It asserts that, within a two-parameter family of initial states, a more strongly entangled state can reach zero concurrence in shorter time than a less entangled state, supported by an exact analytical expression for the concurrence trajectory C(t) and a phase diagram in the initial-state parameter plane that delineates the regime of anomalous ordering.

Significance. If the central claim holds under the stated model, the work supplies a concrete, analytically tractable example of anomalous relaxation ordering for a quantum resource. The exact derivation of crossover times and the explicit phase diagram constitute verifiable, falsifiable content that could guide further studies of resource lifetimes in open systems.

major comments (1)
  1. [Analytical derivation and phase diagram] The exact solution and the reported trajectory crossing are derived under the assumption of identical reservoir decay rates (both equal to a single γ). The functional form reduces to C(t) = max(0, f(α,β) e^{-γ t} - g(α,β) e^{-2γ t}); when the rates differ the exponents become asymmetric and the algebraic condition for ordering of ESD times changes. The manuscript contains no analytic continuation or numerical check for γ1 ≠ γ2, leaving open whether the anomalous regime survives this physically relevant perturbation.
minor comments (2)
  1. [Abstract] The abstract and main text should explicitly state the assumption of equal decay rates and note its necessity for the reported effect.
  2. [Phase diagram figure] Figure captions for the phase diagram should indicate whether boundaries are obtained solely from the analytic root or include numerical sampling.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The single major comment is addressed point-by-point below. We agree that extending the analysis to unequal decay rates strengthens the work and will incorporate this in the revision.

read point-by-point responses

  1. Referee: The exact solution and the reported trajectory crossing are derived under the assumption of identical reservoir decay rates (both equal to a single γ). The functional form reduces to C(t) = max(0, f(α,β) e^{-γ t} - g(α,β) e^{-2γ t}); when the rates differ the exponents become asymmetric and the algebraic condition for ordering of ESD times changes. The manuscript contains no analytic continuation or numerical check for γ1 ≠ γ2, leaving open whether the anomalous regime survives this physically relevant perturbation.

    Authors: We agree that the closed-form derivation and phase diagram are obtained for γ1 = γ2 = γ. This symmetric case yields the exact crossover condition and is the natural starting point for an analytically tractable demonstration of the ESD Mpemba effect. For γ1 ≠ γ2 the concurrence takes the more general form involving independent exponentials, and the ordering of ESD times is no longer governed by the same algebraic relation. Nevertheless, the underlying mechanism—stronger initial entanglement leading to faster decay of the relevant off-diagonal terms—remains operative. In the revised manuscript we will add a numerical section that scans a range of γ1/γ2 ratios (including values up to 2) and shows that a substantial fraction of the original anomalous region in the (α,β) plane survives, thereby confirming robustness under this perturbation. revision: yes

Circularity Check

0 steps flagged

No circularity; exact analytical derivation from standard model

full rationale

The paper derives the concurrence C(t) = max(0, f(α,β) e^{-γ t} - g(α,β) e^{-2γ t}) directly from the amplitude-damping master equation for two independent reservoirs, then obtains ESD times as algebraic roots and maps the (α,β) phase diagram by direct inspection. No fitted parameters are relabeled as predictions, no self-citations are load-bearing for the crossover condition, and the anomalous ordering is an explicit consequence of the closed-form expression under the stated identical-γ assumption rather than a definitional identity or imported uniqueness theorem. The derivation chain is therefore self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard amplitude-damping master equation and a two-parameter family of initial two-qubit states; no free parameters are introduced in the abstract and no new entities are postulated.

axioms (1)
  • domain assumption Two qubits interact via independent local amplitude-damping channels with identical decay rates
    This is the standard Markovian noise model invoked to produce entanglement sudden death.

pith-pipeline@v0.9.0 · 5680 in / 1246 out tokens · 31547 ms · 2026-05-25T04:56:34.501086+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · 9 internal anchors

  1. [1]

    E. B. Mpemba and D. G. Osborne, Cool?, Phys. Educ.4, 172 (1969)

    work page 1969

  2. [2]

    Lu and O

    Z. Lu and O. Raz, Nonequilibrium thermodynamics of the markovian mpemba effect and its inverse, Proc. Natl. Acad. Sci. U.S.A.114, 5083 (2017)

    work page 2017

  3. [3]

    Beato and G

    N. Beato and G. Teza, Relaxation Control of Open Quantum Systems, Phys. Rev. Lett.136, 070401 (2026), arXiv:2507.15948 [quant-ph]

    work page arXiv 2026

  4. [4]

    Bao, Initial-State Typicality in Quantum Relaxation, Phys

    R. Bao, Initial-State Typicality in Quantum Relaxation, Phys. Rev. Lett.136, 070402 (2026), arXiv:2511.01709 [quant-ph]

    work page arXiv 2026

  5. [5]

    Nava and R

    A. Nava and R. Egger, Pontus-Mpemba Effects, Phys. Rev. Lett. 135, 140404 (2025), arXiv:2505.14622 [quant-ph]

    work page arXiv 2025

  6. [6]

    Carollo, A

    F. Carollo, A. Lasanta, and I. Lesanovsky, Exponentially accel- erated approach to stationarity in Markovian open quantum sys- tems through the Mpemba effect, Phys. Rev. Lett.127, 060401 (2021), arXiv:2103.05020 [quant-ph]

    work page arXiv 2021

  7. [7]

    Longhi, Photonic Mpemba effect, Opt

    S. Longhi, Photonic Mpemba effect, Opt. Lett.49, 5188 (2024), arXiv:2408.03296 [physics.optics]

    work page arXiv 2024

  8. [8]

    Van Vu and H

    T. Van Vu and H. Hayakawa, Thermomajorization Mpemba Effect, Phys. Rev. Lett.134, 107101 (2025), arXiv:2410.06686 [cond-mat.stat-mech]

    work page arXiv 2025

  9. [9]

    Bao and Z

    R. Bao and Z. Hou, Accelerating Quantum Relaxation via Tem- porary Reset: A Mpemba-Inspired Approach, Phys. Rev. Lett. 135, 150403 (2025), arXiv:2212.11170 [quant-ph]

    work page arXiv 2025

  10. [10]

    Wang and J

    X. Wang and J. Wang, Mpemba effects in nonequilibrium open quantum systems, Phys. Rev. Res.6, 033330 (2024), arXiv:2401.14259 [quant-ph]

    work page arXiv 2024

  11. [11]

    D. J. Strachan, A. Purkayastha, and S. R. Clark, Non-Markovian Quantum Mpemba Effect, Phys. Rev. Lett.134, 220403 (2025), arXiv:2402.05756 [quant-ph]

    work page arXiv 2025

  12. [12]

    Moroder, O

    M. Moroder, O. Culhane, K. Zawadzki, and J. Goold, Thermo- dynamics of the Quantum Mpemba Effect, Phys. Rev. Lett.133, 140404 (2024), arXiv:2403.16959 [quant-ph]

    work page arXiv 2024

  13. [13]

    A. K. Chatterjee, S. Takada, and H. Hayakawa, Quantum Mpemba Effect in a Quantum Dot with Reservoirs, Phys. Rev. Lett.131, 080402 (2023), arXiv:2304.02411 [cond-mat.stat- mech]

    work page arXiv 2023

  14. [14]

    Torrente, M

    A. Torrente, M. A. L ´opez-Casta˜no, A. Lasanta, F. V . Reyes, A. Prados, and A. Santos, Large mpemba-like effect in a gas of inelastic rough hard spheres, Phys. Rev. E99, 060901(R) (2019)

    work page 2019

  15. [15]

    Kumar and J

    A. Kumar and J. Bechhoefer, Exponentially faster cooling in a colloidal system, Nature584, 64 (2020)

    work page 2020

  16. [16]

    Aharony Shapira, Y

    S. Aharony Shapira, Y . Shapira, J. Markov, G. Teza, N. Akerman, O. Raz, and R. Ozeri, Inverse Mpemba Effect Demonstrated on a Single Trapped Ion Qubit, Phys. Rev. Lett.133, 010403 (2024), arXiv:2401.05830 [quant-ph]

    work page arXiv 2024

  17. [17]

    Zhanget al., Observation of quantum strong Mpemba effect, Nature Commun.16, 301 (2025), arXiv:2401.15951 [quant-ph]

    J. Zhanget al., Observation of quantum strong Mpemba effect, Nature Commun.16, 301 (2025), arXiv:2401.15951 [quant-ph]

    work page arXiv 2025

  18. [18]

    F. Ares, P. Calabrese, and S. Murciano, The quantum Mpemba ef- fects, Nature Rev. Phys.7, 451 (2025), arXiv:2502.08087 [cond- mat.stat-mech]

    work page arXiv 2025

  19. [19]

    G. Teza, J. Bechhoefer, A. Lasanta, O. Raz, and M. Vucelja, Speedups in nonequilibrium thermal relaxation: Mpemba and related effects, Phys. Rept.1164, 1 (2026), arXiv:2502.01758 [cond-mat.stat-mech]

    work page arXiv 2026

  20. [20]

    Bechhoefer, A

    J. Bechhoefer, A. Kumar, and R. Ch´etrite, A fresh understanding of the mpemba effect, Nat. Rev. Phys.3, 534 (2021)

    work page 2021

  21. [21]

    H. Yu, S. Liu, and S.-X. Zhang, Quantum Mpemba effects from symmetry perspectives, AAPPS Bull.35, 17 (2025), arXiv:2507.02301 [quant-ph]

    work page arXiv 2025

  22. [22]

    Chang, S

    W.-X. Chang, S. Yin, S.-X. Zhang, and Z.-X. Li, Imaginary-Time Mpemba Effect in Quantum Many-Body Systems, Phys. Rev. Lett.136, 100403 (2026), arXiv:2409.06547 [cond-mat.str-el]

    work page arXiv 2026

  23. [23]

    H. Yu, J. Hu, and S.-X. Zhang, Quantum pontus-Mpemba effects in real- and imaginary-time dynamics, Phys. Rev. B113, 134304 (2026), arXiv:2509.01960 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  24. [24]

    F. Ares, S. Murciano, and P. Calabrese, Entanglement asymmetry as a probe of symmetry breaking, Nature Commun.14, 2036 (2023), arXiv:2207.14693 [cond-mat.stat-mech]

    work page arXiv 2036

  25. [25]

    Yu, Z.-X

    H. Yu, Z.-X. Li, and S.-X. Zhang, Symmetry Breaking Dynamics in Quantum Many-Body Systems, Chin. Phys. Lett.42, 110602 (2025), arXiv:2501.13459 [quant-ph]

    work page arXiv 2025

  26. [26]

    Klobas, C

    K. Klobas, C. Rylands, and B. Bertini, Translation symmetry restoration under random unitary dynamics, Phys. Rev. B111, L140304 (2025), arXiv:2406.04296 [cond-mat.stat-mech]

    work page arXiv 2025

  27. [27]

    Summer, M

    A. Summer, M. Moroder, L. P. Bettmann, X. Turkeshi, I. Mar- vian, and J. Goold, Resource-Theoretical Unification of Mpemba Effects: Classical and Quantum, Phys. Rev. X16, 011065 (2026), arXiv:2507.16976 [quant-ph]

    work page arXiv 2026

  28. [28]

    Liu, H.-K

    S. Liu, H.-K. Zhang, S. Yin, S.-X. Zhang, and H. Yao, Symmetry restoration and quantum Mpemba effect in many-body local- ization systems, Sci. Bull.70, 3991 (2025), arXiv:2408.07750 [cond-mat.dis-nn]

    work page arXiv 2025

  29. [29]

    Foligno, P

    A. Foligno, P. Calabrese, and B. Bertini, Nonequilibrium Dynam- ics of Charged Dual-Unitary Circuits, PRX Quantum6, 010324 (2025), arXiv:2407.21786 [cond-mat.stat-mech]

    work page arXiv 2025

  30. [30]

    Bertini, P

    B. Bertini, P. Calabrese, M. Collura, K. Klobas, and C. Ry- lands, Nonequilibrium Full Counting Statistics and Symmetry- Resolved Entanglement from Space-Time Duality, Phys. Rev. Lett.131, 140401 (2023), arXiv:2212.06188 [cond-mat.stat- mech]

    work page arXiv 2023

  31. [31]

    Yamashika, F

    S. Yamashika, F. Ares, and P. Calabrese, Entanglement asymme- 6 try and quantum Mpemba effect in two-dimensional free-fermion systems, Phys. Rev. B110, 085126 (2024), arXiv:2403.04486 [cond-mat.stat-mech]

    work page arXiv 2024

  32. [32]

    Murciano, F

    S. Murciano, F. Ares, I. Klich, and P. Calabrese, Entanglement asymmetry and quantum Mpemba effect in the XY spin chain, J. Stat. Mech.2401, 013103 (2024), arXiv:2310.07513 [cond- mat.stat-mech]

    work page arXiv 2024

  33. [33]

    Y . Yu, T. Jin, L. Zhang, K. Xu, and H. Fan, Tuning the quantum Mpemba effect in an isolated system by initial-state engineering, Phys. Rev. B112, 094315 (2025), arXiv:2505.02040 [quant-ph]

    work page arXiv 2025

  34. [34]

    Rylands, K

    C. Rylands, K. Klobas, F. Ares, P. Calabrese, S. Murciano, and B. Bertini, Microscopic Origin of the Quantum Mpemba Effect in Integrable Systems, Phys. Rev. Lett.133, 010401 (2024), arXiv:2310.04419 [cond-mat.stat-mech]

    work page arXiv 2024

  35. [35]

    Xuet al., Observation and Modulation of the Quantum Mpemba Effect on a Superconducting Quantum Processor, (2025), arXiv:2508.07707 [quant-ph]

    Y . Xuet al., Observation and Modulation of the Quantum Mpemba Effect on a Superconducting Quantum Processor, (2025), arXiv:2508.07707 [quant-ph]

    work page arXiv 2025

  36. [36]

    A. Rath, V . Vitale, S. Murciano, M. V otto, J. Dubail, R. Kueng, C. Branciard, P. Calabrese, and B. Vermersch, Entanglement Bar- rier and its Symmetry Resolution: Theory and Experimental Ob- servation, PRX Quantum4, 010318 (2023), arXiv:2209.04393 [quant-ph]

    work page arXiv 2023

  37. [37]

    L. K. Joshiet al., Observing the Quantum Mpemba Effect in Quantum Simulations, Phys. Rev. Lett.133, 010402 (2024), arXiv:2401.04270 [quant-ph]

    work page arXiv 2024

  38. [38]

    Turkeshi, P

    X. Turkeshi, P. Calabrese, and A. De Luca, Quantum Mpemba Effect in Random Circuits, Phys. Rev. Lett.135, 040403 (2025), arXiv:2405.14514 [quant-ph]

    work page arXiv 2025

  39. [39]

    Liu, H.-K

    S. Liu, H.-K. Zhang, S. Yin, and S.-X. Zhang, Symmetry Restora- tion and Quantum Mpemba Effect in Symmetric Random Cir- cuits, Phys. Rev. Lett.133, 140405 (2024), arXiv:2403.08459 [quant-ph]

    work page arXiv 2024

  40. [40]

    Yamashika and F

    S. Yamashika and F. Ares, Quantum Mpemba Effect in Long- Range Spin Systems, Phys. Rev. Lett.136, 090402 (2026), arXiv:2507.06636 [cond-mat.stat-mech]

    work page arXiv 2026

  41. [41]

    Medina, O

    I. Medina, O. Culhane, F. C. Binder, G. T. Landi, and J. Goold, Anomalous Discharging of Quantum Batteries: The Ergotropic Mpemba Effect, Phys. Rev. Lett.134, 220402 (2025), arXiv:2412.13259 [quant-ph]

    work page arXiv 2025

  42. [42]

    Sapui, T

    T. Sapui, T. K. Konar, and A. S. De, Ergotropic Mpemba crossings in finite-dimensional quantum batteries, (2026), arXiv:2602.11056 [quant-ph]

    work page arXiv 2026

  43. [43]

    Y . Li, W. Li, and X. Li, Ergotropic mpemba effect in non- markovian quantum systems, Phys. Rev. A112, 032209 (2025)

    work page 2025

  44. [44]

    Yu and J

    T. Yu and J. H. Eberly, Sudden death of entanglement: Classi- cal noise effects, Opt. Commun.264, 393 (2006), arXiv:quant- ph/0602196

    work page arXiv 2006

  45. [45]

    Finite-Time Disentanglement via Spontaneous Emission

    T. Yu and J. H. Eberly, Finite-Time Disentanglement Via Spontaneous Emission, Phys. Rev. Lett.93, 140404 (2004), arXiv:quant-ph/0404161

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  46. [46]

    M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Wal- born, P. H. S. Ribeiro, and L. Davidovich, Environment-Induced Sudden Death of Entanglement, Science316, 1139892 (2007), arXiv:quant-ph/0701184

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  47. [47]

    Li, S.-H

    S.-H. Li, S.-H. Shang, and S.-M. Wu, Does acceleration always degrade quantum entanglement for tetrapartite Unruh-DeWitt detectors?, JHEP05, 214, arXiv:2502.05881 [gr-qc]

    work page arXiv

  48. [48]

    X. Liu, W. Liu, and S.-M. Wu, Entanglement degradation of static black holes in effective quantum gravity, Phys. Lett. B875, 140334 (2026), arXiv:2511.12245 [gr-qc]

    work page arXiv 2026

  49. [49]

    Gallock-Yoshimura and R

    K. Gallock-Yoshimura and R. B. Mann, Entangled detectors nonperturbatively harvest mutual information, Phys. Rev. D104, 125017 (2021), arXiv:2109.07495 [quant-ph]

    work page arXiv 2021

  50. [50]

    Entanglement dynamics of two independent qubits in environments with and without memory

    B. Bellomo, R. L. Franco, and G. Compagno, Entanglement dy- namics of two independent qubits in environments with and with- out memory, Phys. Rev. A77, 032342 (2008), arXiv:0711.4799 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  51. [51]

    Non-Markovian effects on the dynamics of entanglement

    B. Bellomo, R. L. Franco, and G. Compagno, Non-Markovian Effects on the Dynamics of Entanglement, Phys. Rev. Lett.99, 160502 (2007), arXiv:0804.2377 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  52. [52]

    Y . Pan, B. Zhang, and Q. Cai, Entanglement is protected by acceleration-induced transparency in thermal field, Phys. Rev. D 113, 025004 (2026), arXiv:2512.06043 [quant-ph]

    work page arXiv 2026

  53. [53]

    M. F. Cavalcante, M. V . S. Bonanc ¸a, E. Miranda, and S. Deffner, Emergence of X states in a quantum impurity model, Phys. Rev. Res.7, L022027 (2025), arXiv:2501.13914 [cond-mat.str-el]

    work page arXiv 2025

  54. [54]

    Entanglement dynamics and performance of two-qubit gates for superconducting qubits under non-Markovian effects

    K. Nakamura and J. Ankerhold, Entanglement dynamics and performance of two-qubit gates for superconducting qubits un- der non-Markovian effects, Phys. Rev. Res.8, 013337 (2026), arXiv:2510.05872 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  55. [55]

    A. R. P. Rau, Algebraic characterization of X-states in quantum information, J. Phys. A42, 412002 (2009), arXiv:0906.4716 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  56. [56]

    Evolution from Entanglement to Decoherence of Bipartite Mixed "X" States

    T. Yu and J. H. Eberly, Evolution from entanglement to decoher- ence of bipartite mixed ”X” states, Quant. Inf. Comput.7, 459 (2007), arXiv:quant-ph/0503089

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  57. [57]

    Entanglement of a Pair of Quantum Bits

    S. Hill and W. K. Wootters, Entanglement of a pair of quantum bits, Phys. Rev. Lett.78, 5022 (1997), arXiv:quant-ph/9703041

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  58. [58]

    W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett.80, 2245 (1998), arXiv:quant- ph/9709029

    work page arXiv 1998

  59. [59]

    Klich, O

    I. Klich, O. Raz, O. Hirschberg, and M. Vucelja, Mpemba index and anomalous relaxation, Phys. Rev. X9, 021060 (2019)

    work page 2019

  60. [60]

    Kumar, R

    A. Kumar, R. Ch ´etrite, and J. Bechhoefer, Anomalous heat- ing in a colloidal system, Proc. Natl. Acad. Sci. U.S.A.119, e2118484119 (2022). Appendix In this Appendix, we provide the explicit derivation steps for the asymmetric dissipation framework and present supple- mentary analysis regarding the geometric boundaries of the phase diagrams. WhenΓ A , ΓB...

    work page 2022