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arxiv: 1907.00975 · v1 · pith:OYWUSKL2new · submitted 2019-07-01 · 🪐 quant-ph · cond-mat.mes-hall· math-ph· math.MP

Stationary quantum correlations in a system with mean-field PT symmetry

Federico Roccati , Salvatore Lorenzo , G. Massimo Palma , Francesco Ciccarello This is my paper

Pith reviewed 2026-05-25 11:32 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallmath-phmath.MP
keywords PT symmetryquantum correlationsGaussian discordnon-Hermitian Hamiltoniansquantum harmonic oscillatorsgain-loss systemsbroken PT phasequantum noise
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The pith

Quantum correlations in a gain-loss oscillator pair stabilize above a loss threshold and persist in the broken PT phase.

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

The paper models two coupled quantum harmonic oscillators, one with gain and one with loss, and tracks their full quantum evolution including noise. Starting from any two-mode coherent state, Gaussian discord appears without entanglement. Above a critical loss rate, these correlations stop decaying and remain constant over time. This stationary regime covers a broader parameter space than the region where the quantum state itself stays stable. When gain equals loss, the correlations decay in the PT-exact phase but remain constant once the system enters the broken phase.

Core claim

In the full quantum treatment of the coupled oscillators, Gaussian quantum discord is generated from coherent initial states and, once the loss rate exceeds a threshold, ceases to decay. This occurs across a wide parameter region that is substantially larger than the domain of stable quantum dynamics. For equal gain and loss strengths the discord decays throughout the PT-exact phase, including the exceptional point, yet becomes stationary in the broken phase.

What carries the argument

Gaussian discord computed from the covariance matrix of the two-mode Gaussian state, evolving under the quantum master equation that incorporates both the PT-symmetric mean-field Hamiltonian and quantum noise.

If this is right

  • Quantum correlations generated from coherent states become time-independent once loss exceeds the threshold.
  • The region of stationary discord is larger than the region of stable quantum dynamics.
  • For equal gain and loss rates, discord decays in the exact PT phase and stabilizes in the broken PT phase.
  • The result holds for any initial two-mode coherent state, including the vacuum.

Where Pith is reading between the lines

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

  • The broken PT phase may offer a route to protect quantum correlations in open systems even when the state itself is unstable.
  • Gaussian discord could serve as a diagnostic of the PT phase transition that is observable at the full quantum level.
  • Similar stabilization might appear in other non-Hermitian open quantum systems once quantum noise is included.

Load-bearing premise

The physical system is accurately described by two coupled quantum harmonic oscillators with one subject to linear gain and the other to linear loss.

What would settle it

Measure the time evolution of Gaussian discord in a physical realization of two coupled oscillators with tunable loss rate set above the predicted threshold and check whether the discord remains constant after its initial rise.

Figures

Figures reproduced from arXiv: 1907.00975 by Federico Roccati, Francesco Ciccarello, G. Massimo Palma, Salvatore Lorenzo.

Figure 1
Figure 1. Figure 1: FIG. 1. Sketch of the gain-loss system. A pair of quantum [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Total, classical and quantum correlations against time [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) and (c): Long-time QCs, as measured by [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

A pair of coupled quantum harmonic oscillators, one subject to a gain one to a loss, is a paradigmatic setup to implement PT-symmetric, non-Hermitian Hamiltonians in that one such Hamiltonian governs the mean-field dynamics for equal gain and loss strengths. Through a full quantum description (so as to account for quantum noise) here is shown that when the system starts in any two-mode coherent state, including vacuum, there appear quantum correlations (QCs) without entanglement, as measured by the Gaussian discord. When the loss rate is above a threshold, once generated QCs no more decay. This occurs in a wide region of parameters, significantly larger than that where the full quantum dynamics is stable. For equal gain and loss rates, in particular, QCs decay in the exact phase (including the exceptional point) and are stable in the broken phase.

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

0 major / 2 minor

Summary. The manuscript analyzes a pair of coupled quantum harmonic oscillators with one subject to gain and one to loss. Using the full quantum Lindblad master equation (which incorporates noise), it shows that Gaussian discord is generated from any initial two-mode coherent state (including vacuum). When the loss rate exceeds a threshold, the generated quantum correlations become stationary and do not decay. This stable region is significantly larger than the parameter region where the full quantum dynamics remains stable. For equal gain and loss rates, the discord decays throughout the PT-exact phase (including the exceptional point) but remains stable in the PT-broken phase.

Significance. If the central numerical result from the covariance-matrix dynamics holds, the work establishes a concrete regime in which mean-field PT symmetry protects quantum correlations against decay even when the underlying quantum system is unstable. This provides a clear separation between classical non-Hermitian phases and full-quantum stability, with direct implications for open quantum systems and quantum information measures in dissipative settings. The explicit use of Gaussian discord and the Lindblad treatment are strengths.

minor comments (2)
  1. The explicit Lindblad operators and the resulting ODEs for the covariance-matrix elements (used to compute the discord threshold) should be written out in a dedicated methods subsection to facilitate independent verification.
  2. Figure captions should explicitly mark the PT-exact/broken boundaries and the numerically determined discord-stability threshold on the same parameter plots for direct visual comparison.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Derivation is self-contained with no circular reductions

full rationale

The paper derives the persistence of Gaussian discord from the covariance matrix evolution under the Lindblad master equation, which incorporates quantum noise separately from the mean-field PT-symmetric Hamiltonian. The stability regions are explicitly distinguished, with the quantum result not reducing to the mean-field input by definition or fitting. No self-citation is load-bearing for the central claim, and no ansatz or renaming is used to force the outcome. The result follows from solving the dynamical equations starting from coherent states.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard quantum harmonic oscillator model and the definition of PT symmetry from prior literature; no new entities are introduced.

free parameters (1)
  • gain and loss rates
    Model parameters whose relative strength determines the threshold and phases; the threshold value is derived from them.
axioms (1)
  • domain assumption The mean-field dynamics for equal gain and loss is governed by a PT-symmetric non-Hermitian Hamiltonian.
    Invoked in the abstract as the paradigmatic setup for the system.

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Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    C. M. Bender and S. Boettcher, Phys. Rev. Lett.80, 5243 (1998)

    work page 1998

  2. [2]

    El-Ganainy, K

    R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Nat. Phys. 14, 11 (2018)

    work page 2018

  3. [3]

    L. Feng, R. El-Ganainy, and L. Ge, Nat. Photonics 11, 752 (2017)

    work page 2017

  4. [4]

    Longhi, Euro Phys

    S. Longhi, Euro Phys. Lett. 120, 64001 (2017)

    work page 2017

  5. [5]

    C. E. R¨ uter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010)

    work page 2010

  6. [6]

    Regensburger, C

    A. Regensburger, C. Bersch, M.-A. Miri, G. On- ishchukov, D. N. Christodoulides, and U. Peschel, Na- ture 488, 167 (2012)

    work page 2012

  7. [7]

    B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. 10, 394 (2014)

    work page 2014

  8. [8]

    Haroche and J.-M

    S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University Press, 2006)

    work page 2006

  9. [9]

    Schomerus, Phys

    H. Schomerus, Phys. Rev. Lett. 104, 233601 (2010)

    work page 2010

  10. [10]

    Yoo, H.-S

    G. Yoo, H.-S. Sim, and H. Schomerus, Phys. Rev. A 84, 063833 (2011)

    work page 2011

  11. [11]

    G. S. Agarwal and K. Qu, Phys. Rev. A 85, 031802 (2012)

    work page 2012

  12. [12]

    Longhi, Opt

    S. Longhi, Opt. Lett. 43, 5371 (2018)

    work page 2018

  13. [13]

    Vashahri-Ghamsari, B

    S. Vashahri-Ghamsari, B. He, and M. Xiao, Phys. Rev. A 96, 033806 (2017)

    work page 2017

  14. [14]

    Vashahri-Ghamsari, B

    S. Vashahri-Ghamsari, B. He, and M. Xiao, Phys. Rev. A 99, 023819 (2019)

    work page 2019

  15. [15]

    K. V. Kepesidis, T. J. Milburn, J. Huber, K. G. Makris, S. Rotter, and P. Rabl, New J. Phys. 18, 095003 (2016)

    work page 2016

  16. [16]

    Lau and A

    H.-K. Lau and A. A. Clerk, Nature Communications 9, 4320 (2018)

    work page 2018

  17. [17]

    Zhang, W

    M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, arXiv:1805.12001 [quant-ph] (2018), arXiv: 1805.12001

    work page arXiv 2018

  18. [18]

    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition , anniversary edizione ed. (Cambridge University Press, Cambridge ; New York, 2010)

    work page 2010

  19. [19]

    S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005)

    work page 2005

  20. [20]

    Wang and A

    Y.-X. Wang and A. A. Clerk, Phys. Rev. A 99, 063834 (2019)

    work page 2019

  21. [21]

    Ollivier and W

    H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001)

    work page 2001

  22. [22]

    Henderson and V

    L. Henderson and V. Vedral, J. Phys. A: Math. Gen. 34, 6899 (2001)

    work page 2001

  23. [23]

    K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Ve- dral, Rev. Mod. Phys. 84, 1655 (2012)

    work page 2012

  24. [24]

    Datta, A

    A. Datta, A. Shaji, and C. M. Caves, Phys. Rev. Lett. 100, 050502 (2008)

    work page 2008

  25. [25]

    W. Cao, X. Lu, X. Meng, J. Sun, H. Shen, and Y. Xiao, arXiv:1903.12213 [quant-ph] (2019), arXiv: 1903.12213

    work page arXiv 1903

  26. [26]

    Loudon, The Quantum Theory of Light (OUP Oxford,

    R. Loudon, The Quantum Theory of Light (OUP Oxford,

    work page

  27. [27]

    google-Books-ID: AEkfajgqldoC

    work page

  28. [28]

    D. Dast, D. Haag, H. Cartarius, and G. Wunner, Phys. Rev. A 90, 052120 (2014)

    work page 2014

  29. [29]

    Gardiner and P

    C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics , 3rd ed., Springer Series in Synergetics (Springer-Verlag, Berlin Heidelberg, 2004)

    work page 2004

  30. [30]

    T. M. Cover and J. A. Thomas, Elements of Informa- tion Theory, 2nd ed. (Wiley-Interscience, Hoboken, N.J, 2006)

    work page 2006

  31. [31]

    Pirandola, G

    S. Pirandola, G. Spedalieri, S. L. Braunstein, N. J. Cerf, and S. Lloyd, Phys. Rev. Lett. 113, 140405 (2014)

    work page 2014

  32. [32]

    Giorda and M

    P. Giorda and M. G. A. Paris, Phys. Rev. Lett. 105, 020503 (2010). 6

    work page 2010

  33. [33]

    Adesso and A

    G. Adesso and A. Datta, Phys. Rev. Lett. 105, 030501 (2010)

    work page 2010

  34. [34]

    M. S. Kim, W. Son, V. Buˇ zek, and P. L. Knight, Phys. Rev. A 65, 032323 (2002)

    work page 2002

  35. [35]

    Ciccarello and V

    F. Ciccarello and V. Giovannetti, Phys. Rev. A 85, 010102 (2012)

    work page 2012

  36. [36]

    X. Hu, H. Fan, D. L. Zhou, and W.-M. Liu, Phys. Rev. A 85, 032102 (2012)

    work page 2012

  37. [37]

    Streltsov, H

    A. Streltsov, H. Kampermann, and D. Bruß, Phys. Rev. Lett. 107, 170502 (2011)

    work page 2011

  38. [38]

    Ciccarello and V

    F. Ciccarello and V. Giovannetti, Phys. Rev. A 85, 022108 (2012)

    work page 2012

  39. [39]

    L. S. Madsen, A. Berni, M. Lassen, and U. L. Andersen, Phys. Rev. Lett. 109, 030402 (2012)

    work page 2012

  40. [40]

    Scheel and A

    S. Scheel and A. Szameit, Euro Phys. Lett. 122, 34001 (2018)

    work page 2018

  41. [41]

    Korolkova and G

    N. Korolkova and G. Leuchs, Rep. Prog. Phys.82, 056001 (2019)

    work page 2019

  42. [42]

    Ferraro and M

    A. Ferraro and M. G. A. Paris, Phys. Rev. Lett. 108, 260403 (2012)

    work page 2012

  43. [43]

    Adesso, T

    G. Adesso, T. R. Bromley, and M. Cianciaruso, J. Phys. A: Math. Theor. 49, 473001 (2016)

    work page 2016

  44. [44]

    Streltsov, Quantum Correlations Beyond Entangle- ment: and Their Role in Quantum Information Theory , SpringerBriefs in Physics (Springer International Pub- lishing, 2015)

    A. Streltsov, Quantum Correlations Beyond Entangle- ment: and Their Role in Quantum Information Theory , SpringerBriefs in Physics (Springer International Pub- lishing, 2015)

    work page 2015

  45. [45]

    M. Gu, H. M. Chrzanowski, S. M. Assad, T. Symul, K. Modi, T. C. Ralph, V. Vedral, and P. K. Lam, Nat. Phys. 8, 671 (2012)

    work page 2012

  46. [46]

    Dakic, Y

    B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, Nat. Phys. 8, 666 (2012)

    work page 2012

  47. [47]

    Piani, S

    M. Piani, S. Gharibian, G. Adesso, J. Calsamiglia, P. Horodecki, and A. Winter, Phys. Rev. Lett. 106, 220403 (2011)

    work page 2011

  48. [48]

    Adesso, V

    G. Adesso, V. D’Ambrosio, E. Nagali, M. Piani, and F. Sciarrino, Phys. Rev. Lett. 112, 140501 (2014)

    work page 2014

  49. [49]

    Streltsov, H

    A. Streltsov, H. Kampermann, and D. Bruß, Phys. Rev. Lett. 106, 160401 (2011)

    work page 2011

  50. [50]

    Croal, C

    C. Croal, C. Peuntinger, V. Chille, C. Marquardt, G. Leuchs, N. Korolkova, and L. Mista, Phys. Rev. Lett. 115, 190501 (2015)

    work page 2015

  51. [51]

    T. K. Chuan, J. Maillard, K. Modi, T. Paterek, M. Pa- ternostro, and M. Piani, Phys. Rev. Lett. 109, 070501 (2012)

    work page 2012

  52. [52]

    Peuntinger, V

    C. Peuntinger, V. Chille, L. Miˇ sta, N. Korolkova, M. F¨ ortsch, J. Korger, C. Marquardt, and G. Leuchs, Phys. Rev. Lett. 111, 230506 (2013)

    work page 2013

  53. [53]

    C. E. Vollmer, D. Schulze, T. Eberle, V. H¨ andchen, J. Fiur´ aˇ sek, and R. Schnabel, Phys. Rev. Lett. 111, 230505 (2013)

    work page 2013

  54. [54]

    Fedrizzi, M

    A. Fedrizzi, M. Zuppardo, G. G. Gillett, M. A. Broome, M. P. Almeida, M. Paternostro, A. G. White, and T. Pa- terek, Phys. Rev. Lett. 111, 230504 (2013)

    work page 2013

  55. [55]

    Girolami, A

    D. Girolami, A. M. Souza, V. Giovannetti, T. Tufarelli, J. G. Filgueiras, R. S. Sarthour, D. O. Soares-Pinto, I. S. Oliveira, and G. Adesso, Phys. Rev. Lett. 112, 210401 (2014)

    work page 2014