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arxiv: 2606.17587 · v1 · pith:TLXA77VTnew · submitted 2026-06-16 · 🪐 quant-ph · hep-th· math-ph· math.MP

On the entanglement induced by the deformation of phase-space

Shilpa Nandi , Shatarupa Maity , Pinaki Patra This is my paper

Pith reviewed 2026-06-27 00:59 UTC · model grok-4.3

classification 🪐 quant-ph hep-thmath-phmath.MP
keywords entanglementnoncommutative phase spaceGaussian statesBopp shiftpositive partial transposequantum gravity signature
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The pith

Deformation parameters θ and η in noncommutative phase space induce entanglement between bipartite Gaussian states.

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

The paper extends the positive partial transpose criterion to Bopp shifts that include both position-position noncommutativity (parameter θ) and momentum-momentum noncommutativity (parameter η). It applies the extended criterion to an anisotropic two-dimensional harmonic oscillator and finds that the resulting Gaussian states satisfy the entanglement condition for almost all parameter values. The authors outline a gedankenexperiment that compares measured photocurrents against the covariance matrix expected in ordinary commutative space; any detected entanglement would be attributed to the intermediate noncommutative background.

Core claim

θ and η induce the entanglement. The bipartite Gaussian state is almost always entangled in deformed space. The Peres-Horodecki separability condition leads to a constraint equation for the parameter values of the oscillator in NC space.

What carries the argument

Extension of the PPT separability criterion to the general Bopp shift that incorporates both θ (position-position) and η (momentum-momentum) noncommutativity.

If this is right

  • States that are separable under ordinary commutative dynamics become entangled once the phase-space deformation parameters are turned on.
  • The Peres-Horodecki condition supplies an explicit algebraic constraint relating the oscillator frequencies and the deformation parameters θ and η.
  • Any experimental detection of unexpected entanglement in supposedly separable Gaussian states would be interpreted as evidence for an intermediate noncommutative phase-space structure.

Where Pith is reading between the lines

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

  • The same deformation-induced entanglement mechanism could be searched for in other quadratic systems whose covariance matrices are experimentally accessible.
  • If the gedankenexperiment succeeds, it supplies an operational route to distinguish noncommutative effects from other sources of entanglement without requiring direct access to Planck-scale physics.

Load-bearing premise

The positive partial transpose criterion remains a valid separability test when extended from ordinary phase space to the general Bopp shift that includes both position-position and momentum-momentum noncommutativity.

What would settle it

Perform the proposed photocurrent measurement on input states that are separable in commutative space; if the reconstructed covariance matrix violates the PPT condition, the states are entangled and the entanglement must be attributed to the noncommutative background.

Figures

Figures reproduced from arXiv: 2606.17587 by Pinaki Patra, Shatarupa Maity, Shilpa Nandi.

Figure 1
Figure 1. Figure 1: FIG. 1. The blue curve indicates the value of minimum eigenvalue with respect to NC parameter [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The blue curve indicates the value of minimum eigenvalue with respect to NC parameter [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The blue curve indicates the value of minimum eigenvalue with respect to NC parameter [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
read the original abstract

Most quantum gravity theories propose that the fundamental concept of space-time is mostly compatible with quantum theory in noncommutative (NC) space. In the present paper, we revisit the notion of entanglement induced by NC deformations of phase space. The positive partial transpose (PPT) criterion for separability of bipartite Gaussian states is extended to a general class of Bopp's shift. In particular, we have considered both the position-position and momentum-momentum noncommutativity, with deformation parameters $\theta$ and $\eta$, respectively. It turns out that $\theta$ and $\eta$ induce the entanglement. We have directly applied the formalism for an anisotropic two-dimensional harmonic oscillator. Peres-Horodecki separability condition leads to a constraint equation for the parameter values of the oscillator in NC space. It turns out that the bipartite Gaussian state is almost always entangled in deformed space. To implement the theoretical idea, we provide an outline for a gedankenexperiment to identify the signature of phase-space noncommutativity, i.e., quantum gravity. In particular, the gedankenexperiment is devised to test the separability of supposedly separable Gaussian states in the usual commutative space, through the covariance matrix, which is constructed via measured output photocurrents after interaction of input Gaussian states and reference states. If the experiment shows that the supposedly separable states are actually entangled, then the entanglement is created through the intermediate background noncommutative space, which is a signature of the quantum nature of gravity.

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 / 1 minor

Summary. The manuscript claims that noncommutative deformations of phase space with parameters θ (position-position noncommutativity) and η (momentum-momentum noncommutativity) induce entanglement in bipartite Gaussian states. It extends the positive partial transpose (PPT) separability criterion to a general class of Bopp shifts, applies the formalism to an anisotropic two-dimensional harmonic oscillator to obtain a constraint equation on the oscillator parameters, and concludes that the resulting states are almost always entangled. An outline of a gedankenexperiment is provided to detect this effect as a signature of quantum gravity via measured photocurrents and covariance matrices.

Significance. If the PPT extension were rigorously justified, the result would link noncommutativity directly to entanglement generation and offer a potential experimental probe of quantum-gravity-induced phase-space deformation. The work does not supply machine-checked proofs, reproducible code, or parameter-free derivations, and the central claim rests on an unverified extension of a separability criterion whose validity in the deformed algebra is not established.

major comments (2)
  1. [Abstract and main text] Abstract and main text: the claim that the PPT criterion 'is extended' to Bopp shifts with both θ and η is load-bearing for the conclusion that these parameters induce entanglement and that states are 'almost always entangled.' No derivation is supplied showing that the shifted covariance matrix preserves the correct symplectic structure under the deformed commutation relations or that the extended PPT condition remains necessary and sufficient for separability once the underlying operator algebra is noncommutative.
  2. [Main text (constraint equation)] The separability constraint derived for the anisotropic oscillator is presented as evidence that θ and η induce entanglement, yet the manuscript supplies no verification that the numerical constraint is not satisfied by construction once the deformation parameters are inserted into the covariance matrix.
minor comments (1)
  1. [Abstract] The abstract states conclusions ('almost always entangled') without referencing the explicit form of the constraint equation or any error analysis on the numerical results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major point below and will revise the manuscript to incorporate additional derivations and verifications as needed.

read point-by-point responses

  1. Referee: [Abstract and main text] Abstract and main text: the claim that the PPT criterion 'is extended' to Bopp shifts with both θ and η is load-bearing for the conclusion that these parameters induce entanglement and that states are 'almost always entangled.' No derivation is supplied showing that the shifted covariance matrix preserves the correct symplectic structure under the deformed commutation relations or that the extended PPT condition remains necessary and sufficient for separability once the underlying operator algebra is noncommutative.

    Authors: We agree that an explicit derivation of the PPT extension would strengthen the presentation. The Bopp shifts are introduced precisely to encode the deformed commutation relations [θ,η] while the covariance matrix is formed from the expectation values of the shifted operators; the symplectic form is preserved by the definition of the shifts. We will add a dedicated subsection (or appendix) that derives the extended PPT condition step by step, confirming that necessity and sufficiency carry over because the partial-transpose operation commutes with the shift in the manner required by the noncommutative algebra. revision: yes

  2. Referee: [Main text (constraint equation)] The separability constraint derived for the anisotropic oscillator is presented as evidence that θ and η induce entanglement, yet the manuscript supplies no verification that the numerical constraint is not satisfied by construction once the deformation parameters are inserted into the covariance matrix.

    Authors: The constraint is not satisfied by construction. When θ = η = 0 the deformed covariance matrix reduces exactly to the standard commutative case, and the PPT inequality holds for the chosen oscillator parameters, recovering separability. Nonzero θ and η generate additional cross terms that violate the inequality for generic frequency and mass values. We will insert an explicit check of the commutative limit together with numerical examples that isolate the contribution of the deformation parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies extended criterion to deformed covariance matrix

full rationale

The paper extends the PPT separability criterion to Bopp-shifted NC phase space incorporating both θ (position-position) and η (momentum-momentum) noncommutativity, constructs the corresponding covariance matrix for the anisotropic 2D harmonic oscillator, and derives a parameter constraint from the Peres-Horodecki condition. The claim that θ and η induce entanglement is the direct output of this calculation rather than a definitional tautology or a fitted input renamed as prediction. No load-bearing self-citation, uniqueness theorem imported from prior work, or ansatz smuggled via citation is present in the abstract or described chain. The derivation remains self-contained against the stated assumptions even if the validity of the PPT extension itself is debatable on other grounds.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The claim rests on the validity of extending PPT to Bopp shifts and on standard assumptions about Gaussian states and separability in NC space; deformation parameters θ and η are introduced as free inputs without independent determination.

free parameters (2)
  • θ
    Deformation parameter for position-position noncommutativity; introduced to deform phase space and induce the claimed entanglement.
  • η
    Deformation parameter for momentum-momentum noncommutativity; introduced to deform phase space and induce the claimed entanglement.
axioms (2)
  • domain assumption Bipartite Gaussian states remain well-defined and the PPT criterion applies after Bopp shift deformation
    Invoked when extending PPT to the general class of Bopp shifts in the abstract.
  • ad hoc to paper The anisotropic 2D harmonic oscillator in NC space yields a covariance matrix whose separability is governed by the stated constraint
    Used to reach the conclusion that the state is almost always entangled.

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Works this paper leans on

45 extracted references · 1 canonical work pages

  1. [1]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement , Rev. Mod. Phys. 81 , 865 (2009)

    2009

  2. [2]

    Dorner, and V

    R. Dorner, and V. Vedral, Correlations in Quantum Physics , International Journal of Modern Physics B 27 , 1345017 (2013)

    2013

  3. [3]

    Schlosshauer, Quantum decoherence , Physics Reports 831 , 1-57 (2019)

    M. Schlosshauer, Quantum decoherence , Physics Reports 831 , 1-57 (2019)

    2019

  4. [4]

    D. F. Walls, and G. J. Milburn, Quantum Optics , Publisher: Springer Cham, eBook ISBN: 978-3-031-84177-4 (2025)

    2025

  5. [5]

    Raz, and R

    T. Raz, and R. D. Levine, The essence of phase transitions in condensed matter by an information theoretic approach , PNAS 120 , e2310281120 (2023)

    2023

  6. [6]

    Agullo, A

    I. Agullo, A. J. Brady, A. Delhom, and D. Kranas, Entanglement from rotating black holes in thermal baths , Phys. Rev. D 110 , 025021 (2024)

    2024

  7. [7]

    Neukart, Geometry-information duality: Quantum entanglement contributions to gravitational dynamics , Annals of Physics 479 , 170044 (2025)

    F. Neukart, Geometry-information duality: Quantum entanglement contributions to gravitational dynamics , Annals of Physics 479 , 170044 (2025)

    2025

  8. [8]

    Price, and K

    H. Price, and K. Wharton, Taming Entanglement , arXiv:2507.15128 [quant-ph]

    arXiv

  9. [9]

    Marletto, J

    C. Marletto, J. Oppenheim, V. Vedral, and E. Wilson, Classical gravity cannot mediate entanglement , arXiv:2511.07348v1 [quant-ph]

    arXiv

  10. [10]

    Rovelli, Quantum Gravity , Cambridge University Press, ISBN: 0-521-83733-2 (2004)

    C. Rovelli, Quantum Gravity , Cambridge University Press, ISBN: 0-521-83733-2 (2004)

    2004

  11. [11]

    Stachel, and K

    J. Stachel, and K. Bradonji\' c , Quantum gravity: Meaning and measurement , Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 46 , 209-216 (2014)

    2014

  12. [12]

    Ashtekar, and R

    A. Ashtekar, and R. Geroch, Quantum theory of gravitation , Rep. Prog. Phys. 37 , 1211 (1974)

    1974

  13. [13]

    Stelle, The unification of quantum gravity , Nuclear Physics B - Proceedings Supplements 88 , 3-9 (2000)

    K.S. Stelle, The unification of quantum gravity , Nuclear Physics B - Proceedings Supplements 88 , 3-9 (2000)

    2000

  14. [14]

    Doplicher, K

    S. Doplicher, K. Fredenhagen, and J. E. Roberts, The quantum structure of spacetime at the Planck scale and quantum fields , Commun.Math.Phys. 172 187-220 (1995)

    1995

  15. [15]

    Hanif, D

    F. Hanif, D. Das, J. Halliwell, D. Home, A. Mazumdar, H. Ulbricht, and S. Bose, Testing Whether Gravity Acts as a Quantum Entity When Measured , Phys. Rev. Lett. 133 , 180201 (2024)

    2024

  16. [16]

    Last visited

    https://home.cern/science/engineering/restarting-lhc-why-13-tev. Last visited

  17. [17]

    Kiefer, Quantum gravity - an unfinished revolution , arXiv:2302.13047 [gr-qc] (25 Feb 2023)

    C. Kiefer, Quantum gravity - an unfinished revolution , arXiv:2302.13047 [gr-qc] (25 Feb 2023)

    arXiv 2023

  18. [18]

    S. Dey, A. Bhat, D. Momeni, M. Faizal, A. F. Ali, T. K. Dey, A. Rehman, Probing noncommutative theories with quantum optical experiments , Nuclear Physics B 924 , 578-587 (2017)

    2017

  19. [19]

    Pikovski, M

    I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, \^ C . Brukner, Probing Planck-scale physics with quantum optics , Nature Phys 8 , 393-397 (2012)

    2012

  20. [20]

    Bosso, S

    P. Bosso, S. Das, I. Pikovski, M. R. Vanner, Amplified transduction of Planck-scale effects using quantum optics , Phys. Rev. A 96 , 023849 (2017)

    2017

  21. [21]

    Liberati, Tests of Lorentz invariance: a 2013 update, Class

    S. Liberati, Tests of Lorentz invariance: a 2013 update, Class. Quantum Gravity 30 , 133001 (2013)

    2013

  22. [22]

    Seiberg and E

    N. Seiberg and E. Witten, String theory and noncommutative geometry, Journal of High Energy Physics 09 (1999)

    1999

  23. [23]

    J. M. Romero, J. D. Vergara, and J. A. Santiago, Noncommutative spaces, the quantum of time, and Lorentz symmetry, Phys. Rev. D 75 , 065008 (2007)

    2007

  24. [24]

    R. J. Szabo, Quantum field theory on noncommutative spaces, Physics Reports 378 , 207-299 (2003)

    2003

  25. [25]

    M. R. Douglas, and N. A. Nekrasov, Noncommutative field theory, Rev. Mod. Phys. 73 977 (2001)

    2001

  26. [26]

    M. B. Fr\" o b, A. Much, and K. Papadopoulos, Noncommutative geometry from perturbative quantum gravity, Phys. Rev. D 107 , 064041 (2023)

    2023

  27. [27]

    Quantum mechanics as a statistical theory

    J. E. Moyal, "Quantum mechanics as a statistical theory", Mathematical Proceedings of the Cambridge Philosophical Society 45 , 99-124 (1949)

    1949

  28. [28]

    Nandi, N

    P. Nandi, N. Debnath, S. Kala, and A. S. Majumdar, Magnetically induced Schr\" o dinger cat states: The shadow of a quantum space, Phys. Rev. A 110 , 032204 (2024)

    2024

  29. [29]

    On the entanglement of co-ordinate and momentum degrees of freedom in noncommutative space

    S. Nandi, M. Rahaman, and P. Patra, "On the entanglement of co-ordinate and momentum degrees of freedom in noncommutative space", Modern Physics Letters A 39 , 2450091 (2024)

    2024

  30. [30]

    Deformation quantization of noncommutative quantum mechanics and dissipation

    C Bastos, O Bertolami, N C Dias, and J N Prata, "Deformation quantization of noncommutative quantum mechanics and dissipation", J. Phys.: Conf. Ser. 67 , 012058 (2007)

    2007

  31. [31]

    Tuning the separability in noncommutative space

    P. Patra, "Tuning the separability in noncommutative space", J. Math. Phys. 65 , 052103 (2024)

    2024

  32. [32]

    Unitary maps on Hamiltonians of an electron moving in a plane and coherent state construction

    I. Aremua, and L. Gouba, "Unitary maps on Hamiltonians of an electron moving in a plane and coherent state construction", J. Math. Phys. 64 , 063508 (2023)

    2023

  33. [33]

    Entanglement due to noncommutativity in phase space

    C. Bastos, A. E. Bernardini, O. Bertolami, N. C. Dias, and J. N. Prata, "Entanglement due to noncommutativity in phase space", Phys. Rev. D 88 , 085013 (2013)

    2013

  34. [34]

    Entanglement in phase-space distribution for an anisotropic harmonic oscillator in noncommutative space

    P. Patra, "Entanglement in phase-space distribution for an anisotropic harmonic oscillator in noncommutative space", Quantum Inf Process 22 , 20 (2023)

    2023

  35. [35]

    Entanglement induced by noncommutativity: anisotropic harmonic oscillator in noncommutative space

    A. Muhuri, D. Sinha, and S. Ghosh, "Entanglement induced by noncommutativity: anisotropic harmonic oscillator in noncommutative space", Eur. Phys. J. Plus 136 , 35 (2021)

    2021

  36. [36]

    Emergent time crystals from phase-space noncommutative quantum mechanics

    A.E. Bernardini, and O. Bertolami, "Emergent time crystals from phase-space noncommutative quantum mechanics", Phys. Lett. B 835 , 137549 (2022)

    2022

  37. [37]

    Marletto, and V

    C. Marletto, and V. Vedral, Gravitationally Induced Entanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity , Phys. Rev. Lett. 119 , 240402 (2017)

    2017

  38. [38]

    Dutta, S

    A. Dutta, S. Ghosh, J. Kim, and R. Sengupta, Robust entanglement detection in arbitrary two-mode Gaussian state: a Stokes-like operator-based approach , arXiv:2103.12987v2 [quant-ph] (2026). https://doi.org/10.48550/arXiv.2103.12987

    work page doi:10.48550/arxiv.2103.12987 2026

  39. [39]

    Ferraro, S

    A. Ferraro, S. Olivares, and Matteo G. A. Paris, Gaussian States in Quantum Information , Napoli Series on physics and Astrophysics, Bibliopolis, ISBN: 88-7088-483-X, 978-8870884838 (2005). arXiv:quant-ph/0503237v1 (31 Mar 2005 )

    Pith/arXiv arXiv 2005

  40. [40]

    Peres-Horodecki Separability Criterion for Continuous Variable Systems

    R. Simon, "Peres-Horodecki Separability Criterion for Continuous Variable Systems", Phys. Rev. Lett. 84 , 2726 (2000)

    2000

  41. [41]

    Nokkala, R

    J. Nokkala, R. Mart\' i nez-Pe\ n a, G. L. Giorgi, V. Parigi, M. C. Soriano, and R. Zambrini, Gaussian states of continuous-variable quantum systems provide universal and versatile reservoir computing , Commun Phys 4 , 53 (2021)

    2021

  42. [42]

    Gaussian states and operations - a quick reference

    J. B. Brask, "Gaussian states and operations - a quick reference", arXiv:2102.05748v2 [quant-ph] (30 Mar 2022)

    arXiv 2022

  43. [43]

    Gaussian-Wigner distributions in quantum mechanics and optics

    R. Simon, E. Sudarshan, and N. Mukunda, "Gaussian-Wigner distributions in quantum mechanics and optics", Phys. Rev. A 36 , 3868 (1987)

    1987

  44. [44]

    Williamson, On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems , Am

    J. Williamson, On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems , Am. J. of Math. 58 , 141 (1936)

    1936

  45. [45]

    Nicacio, Williamson theorem in classical, quantum, and statistical physics , American Journal of Physics 89 , 1139-1151 (2021)

    F. Nicacio, Williamson theorem in classical, quantum, and statistical physics , American Journal of Physics 89 , 1139-1151 (2021)

    2021