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arxiv: 2606.23102 · v1 · pith:3JMJNYACnew · submitted 2026-06-22 · 🪐 quant-ph · math-ph· math.MP

Understanding Squeezed States of Light Through Wigner's Phase-Space

Sibel Baskal , Marilyn E. Noz This is my paper

Pith reviewed 2026-06-26 08:12 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords Wigner functionsqueezed statesphase spacequantum opticssymmetry groupscoherent statesentanglementdecoherence
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The pith

The Wigner phase-space distribution explains squeezed states of light using symmetry groups including Lorentz and symplectic transformations.

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

This review establishes that the Wigner distribution function, derived from the density matrix, supplies a phase-space framework for quantum optics by linking classical canonical transformations to quantum states. It shows how coherent states, one- and two-mode squeezed states, and the squeezed vacuum of the harmonic oscillator are represented and transformed using symmetry groups such as the Lorentz group, the symplectic group in two and four dimensions, and the Euclidean group. The paper demonstrates that squeezed states generate entanglement between modes and applies the same picture to coupled oscillators and optical decoherence via a reformulated Poincaré sphere. A reader would care because the approach offers a unified language for quantum optics based on the simplest form of the Wigner function.

Core claim

The phase-space picture of quantum mechanics, using the Wigner function from the density matrix together with symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group, provides a sufficient framework for understanding coherent and squeezed states of light, the natural generation of entanglement in two-mode cases, coupled harmonic oscillators, and the decoherence of optical fields.

What carries the argument

The Wigner phase-space distribution function, arising from the density matrix and applied with symmetry groups to the harmonic oscillator.

If this is right

  • Squeezed states naturally generate entanglement between the two modes.
  • Coupled harmonic oscillators can be analyzed directly in the Wigner phase-space picture.
  • Decoherence of an optical field can be examined through a reformulation of the Poincaré sphere using the density matrix.

Where Pith is reading between the lines

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

  • The symmetry-based approach may extend to geometric descriptions of state evolution in other quantum optical systems.
  • It could offer simplified calculations for quantum information tasks that rely on squeezed light.

Load-bearing premise

That the simplest form of the Wigner function combined with the listed symmetry groups provides a sufficient and insightful framework for understanding squeezed states without requiring additional quantum-optical assumptions.

What would settle it

A calculation or measurement showing that entanglement generation or decoherence in squeezed states cannot be reproduced using only the basic Wigner function and the symmetry groups, instead requiring further quantum-optical details.

Figures

Figures reproduced from arXiv: 2606.23102 by Marilyn E. Noz, Sibel Baskal.

Figure 1
Figure 1. Figure 1: The fact that uncertainty in one quadrature can be reduced at the expense of increased uncertainty in the other quadrature is one of the most prominent features of squeezed light. We shall discuss this further in terms of the squeezed states of light in Section 6. x p ∆x ∆p ∆x ∆p [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The transformations of Sp(2) produce rotations and squeezes. The circle in this figure corresponds to a Gaussian distribution. As shown on the left, the Lorentz boost of SO(2 , 1) along the z-direction corresponds to the squeeze of Sp(2) along the x-axis of the phase space. On the right is the action of the rotated squeeze of Sp(2), which transforms the circle into a tilted ellipse whose major axis is alig… view at source ↗
Figure 3
Figure 3. Figure 3: The spread of the Gaussian wave packet. The circle represents the Gaussian distribution, while the tilted ellipse represents the time development [68]. the time evolution of the Wigner function becomes W(x, p, t) = 1 π exp{−[(x − pt/m) 2 , p2 ]} . (48) This distribution is concentrated around the Gaussian distribution. A transformation of this type is also known as ”shear”. This is illustrated in [PITH_FU… view at source ↗
Figure 4
Figure 4. Figure 4: Wigner functions (with right-angle cross sections) of the harmonic oscillator for the ground state and for the first three excited states. The symmetry of the Wigner function with respect to x and p is apparent. It can also be observed that higher the excited state the function has more ripples around the origin. The second and third excited states are for n = 2 and n = 3 resulting from Equation (75), have… view at source ↗
Figure 5
Figure 5. Figure 5: The Wigner function for the harmonic oscillator in the first excited state. It is negative at the origin, becomes positive as (x 2 + p 2 ) increases, and becomes vanishingly small as (x 2 + p 2 ) becomes very large. If this Wigner function is integrated over p, then it becomes the quantum probability distribution in x, which is ρ1(x) = 2x 2 e −x 2 / √ π, and is positive for all values of x. This is then a … view at source ↗
Figure 6
Figure 6. Figure 6: For a Euclidean transformation in phase space, a rotation around the origin results in two successive translations [68]. 6.1 Single-Mode Squeezed States of light The squeeze and rotation generators used here are those given in Equation (31). These generators are provided in [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The Wigner function for the squeezed vacuum and the first excited squeeze. Here, W (sq) 0 (x, p) = 1 π exp  −(1/2) e −λ (p + x) 2 + e λ (p − x) 2 and W (sq) 1 (x, p) = 1 π exp  −(1/2) e −λ (p + x) 2 + e λ (p − x) 2 e −λ (p + x) 2 + e λ (p − x) 2 − 1 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Pictured is the squeezed vacuum and vacuum states. Here, the circle corresponds to the vacuum state, which is a coherent state, and the tilted ellipse is for the squeezed vacuum. The squeezed vacuum is not a zero-photon state, but is a minimum uncertainty state in the sense that the area of the ellipse is the same as that of the circle. This interpretation is similar to the case of wave-packet spreads disc… view at source ↗
Figure 9
Figure 9. Figure 9: Overlapping Wigner functions. On the left, the blue ellipse represents a squeezed state, while the red ellipse represents a rotated squeeze in the phase space. On the right, the red ellipse represents a rotated squeeze in the phase space, while the ellipse on the top is for the translated squeeze. The probability of the transition of one squeezed state into another is encoded in the overlap. It is seen tha… view at source ↗
Figure 10
Figure 10. Figure 10: Variable radius of the Poincaré sphere. From Equation (212) and Equation (222), the variable radius R takes its maximum value S0, when χ = 0◦ . It becomes minimum when the decoherency angle reaches 90◦ . Its minimum value is S3 as is illustrated in the left figure. The degree of polarization is maximum when R = S0, and is minimum when R = S3. According to Equation (213), S3 becomes 0 when A = B, thus the … view at source ↗
read the original abstract

This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincar\'e sphere as a further illustration of the density matrix.

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 / 3 minor

Summary. The manuscript is a pedagogical review that introduces the Wigner phase-space distribution starting from the classical-to-quantum transition and canonical transformations, defines the Wigner function via the density matrix, treats the harmonic oscillator, and then applies the formalism to coherent states, one- and two-mode squeezed states, the squeezed vacuum, and the generation of entanglement. It further discusses coupled oscillators and employs symmetry groups (Lorentz, symplectic in 2D/4D, Euclidean) together with a Poincaré-sphere reformulation of the density matrix to address decoherence.

Significance. If the derivations and group-theoretic illustrations are accurate and clearly presented, the paper offers a compact, symmetry-based route to visualizing squeezed states and entanglement in phase space. Such expositions can strengthen intuition in quantum optics, especially for readers already familiar with the Wigner function but seeking explicit connections to the listed groups; however, the work contains no new derivations, parameter-free results, or falsifiable predictions.

minor comments (3)
  1. [Abstract] Abstract and closing paragraph both state that 'the phase-space picture of quantum mechanics has become an important scientific language'; the repetition is unnecessary and could be consolidated into a single, more precise sentence.
  2. The text repeatedly emphasizes use of 'the simplest form of the Wigner function'; a short paragraph or footnote clarifying when this form is insufficient (e.g., for non-Gaussian states or strong decoherence) would prevent readers from over-generalizing the framework.
  3. Figure captions and axis labels for the phase-space plots of squeezed states should explicitly state the quadrature scaling and the value of the squeezing parameter r used in each panel.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript as a pedagogical review and for recommending minor revision. The work is explicitly framed as an expository treatment using the Wigner function and symmetry groups to build intuition for squeezed states, entanglement, and decoherence; it makes no claim to new derivations or predictions. No specific major comments appear in the report, so we provide no point-by-point responses below.

Circularity Check

0 steps flagged

No significant circularity

full rationale

This is a pedagogical review paper that presents the Wigner function, coherent and squeezed states, and symmetry groups (Lorentz, symplectic, Euclidean) as established tools for quantum optics. No novel derivations, predictions, or parameter fits are claimed; the text relies on prior literature for definitions and applications without any steps that reduce by construction to self-citations, fitted inputs renamed as outputs, or self-definitional loops. The central observation about phase space as a language for quantum optics is expository rather than deductive.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an expository review; the abstract introduces no new free parameters, axioms, or invented entities beyond standard quantum mechanics and group theory.

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Works this paper leans on

122 extracted references · 92 canonical work pages

  1. [1]

    Goldstein, Classical Mechanics, 2nd edn

    H. Goldstein, Classical Mechanics, 2nd edn. Addison-Wesley series in physics (Addison-Wesley Pub. Co, Reading, MA, USA, 1980). ISBN 978-0-201-02918-5. (Originally published 1952.)

    1980

  2. [2]

    Arnold, Mathematical methods of classical mechanics , 2nd edn

    V.I. Arnold, Mathematical methods of classical mechanics , 2nd edn. No. 60 in Graduate texts in mathematics (Springer, New York, NY, USA, 1997). ISBN 978-0-387-96890-2. (This is an English 37 translation by K. Vogtmann and A. Weinstein of the Russian original edition: Maremaricheskie Merody Klassicheskoi Mekhaniki, Nauka, Moscow, 1974.)

    1997

  3. [3]

    Abraham, E

    R. Abraham, E. Marsden, J, Foundations of mechanics , 2nd edn. (AMS Chelsea Pub./American Mathematical Society, Providence, RI, USA, 2008). ISBN 978-0-8218-4438-0. (Originally published 1978; OCLC: ocn191847156.)

    2008

  4. [4]

    F. Xu, C.C. Martens, Y. Zheng, Entanglement dynamics with a trajectory-based formulation, Physical Review A 96(2), 022,138 (2017). DOI 10.1103/PhysRevA.96.022138. URL https://link.aps.org/ doi/10.1103/PhysRevA.96.022138

    work page doi:10.1103/physreva.96.022138 2017

  5. [5]

    Kim, M.E

    Y.S. Kim, M.E. Noz, Phase space picture of quantum mechanics: group theoretical approach . No. 40 in Lecture notes in physics series (World Scientific Publishing Co., Singapore; Hackensack, NJ, USA, 1991). ISBN 978-981-02-0360-3,978-981-02-0361-0. URL https://doi.org/10.1142/1197

    work page doi:10.1142/1197 1991

  6. [6]

    Zachos, D.B

    C.K. Zachos, D.B. Fairlie, T.L. Curtright, Quantum Mechanics in Phase Space: An Overview with Se- lected Papers, World Scientific Series in 20th Century Physics , vol. 34 (WORLD SCIENTIFIC, 2005). DOI 10.1142/5287. ISBN 97898123838469789812703507. URL https://www.worldscientific.com/ worldscibooks/10.1142/5287

    work page doi:10.1142/5287 2005

  7. [7]

    Weyl, The classical groups: their invariants and representations , 2nd edn

    H. Weyl, The classical groups: their invariants and representations , 2nd edn. Princeton landmarks in mathematics and physics Mathematics (Princeton University Press, Princeton, N.J. USA, 1997). ISBN 978-0-691-07923-3;978-0-691-05756-9. (Originally published 1947.)

    1997

  8. [8]

    Gilmore, Lie groups, Lie algebras, and some of their applications (Dover Publications, Mineola, NY, USA, 2005)

    R. Gilmore, Lie groups, Lie algebras, and some of their applications (Dover Publications, Mineola, NY, USA, 2005). ISBN 978-0-486-44529-8. (Originally published: 1974, John Wiley and Sons, New York, NY, USA.)

    2005

  9. [9]

    Guillemin, S

    V. Guillemin, S. Sternberg, Symplectic techniques in physics , reprinted edn. (Cambridge Univ. Press, Cambridge, UK, 2001). ISBN 978-0-521-38990-7. (Originally published 1984; OCLC: 248721606.)

    2001

  10. [10]

    Wigner, On the Quantum Correction For Thermodynamic Equilibrium, Physical Review 40(5), 749–759 (1932)

    E.P. Wigner, On the Quantum Correction For Thermodynamic Equilibrium, Physical Review 40(5), 749–759 (1932). DOI 10.1103/PhysRev.40.749. URL https://link.aps.org/doi/10.1103/ PhysRev.40.749

    work page doi:10.1103/physrev.40.749 1932

  11. [11]

    J.v. Neumann, Wahrscheinlichkeitstheoretischer aufbau der quantenmechanik, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1927, 245–272 (1927). URL http://eudml.org/doc/59230

    1927

  12. [12]

    Neumann, Thermodynamik quantenmechanischer gesamtheiten, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1927, 273–291 (1927)

    J.v. Neumann, Thermodynamik quantenmechanischer gesamtheiten, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1927, 273–291 (1927). URL http://eudml.org/doc/59231

    1927

  13. [13]

    Landau, Das Dämpfungsproblem in der Wellenmechanik, Zeitschrift für Physik 45(5-6), 430–441 (1927)

    L.D. Landau, Das Dämpfungsproblem in der Wellenmechanik, Zeitschrift für Physik 45(5-6), 430–441 (1927). DOI 10.1007/BF01343064. URL http://link.springer.com/10.1007/BF01343064

    work page doi:10.1007/bf01343064 1927

  14. [14]

    Dirac, Note on Exchange Phenomena in the Thomas Atom, Mathematical Proceedings of the Cambridge Philosophical Society 26(3), 376–385 (1930)

    P.A.M. Dirac, Note on Exchange Phenomena in the Thomas Atom, Mathematical Proceedings of the Cambridge Philosophical Society 26(3), 376–385 (1930). DOI 10.1017/S0305004100016108. URL https://www.cambridge.org/core/product/identifier/S0305004100016108/type/journal_ article

    work page doi:10.1017/s0305004100016108 1930

  15. [15]

    Blum, Density Matrix Theory and Applications , 3rd edn

    K. Blum, Density Matrix Theory and Applications , 3rd edn. No. 64 in Springer Series on Atomic, Optical, and Plasma Physics (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012). ISBN 97836422056069783642205613. (Originally published 1981.)

    2012

  16. [16]

    Fano, A Stokes-Parameter Technique for the Treatment of Polarization in Quantum Mechanics, Physical Review 93(1), 121–123 (1954)

    U. Fano, A Stokes-Parameter Technique for the Treatment of Polarization in Quantum Mechanics, Physical Review 93(1), 121–123 (1954). DOI 10.1103/PhysRev.93.121. URL https://link.aps. org/doi/10.1103/PhysRev.93.121 38

    work page doi:10.1103/physrev.93.121 1954

  17. [17]

    Leonhardt, Measuring the quantum state of light

    U. Leonhardt, Measuring the quantum state of light . Cambridge studies in modern optics (Cambridge University Press, Cambridge, UK ; New York, 1997). ISBN 9780521497305

    1997

  18. [18]

    Schleich, Quantum Optics in Phase Space , 1st edn

    W.P. Schleich, Quantum Optics in Phase Space , 1st edn. (Wiley, 2001). DOI 10.1002/3527602976. ISBN 97835272943509783527602971. URL https://onlinelibrary.wiley.com/doi/book/10.1002/ 3527602976

    work page doi:10.1002/3527602976 2001

  19. [19]

    Simon, N

    R. Simon, N. Mukunda, Optical phase space, wigner representation, and invariant quality parameters, J. Opt. Soc. Am. A 17(12), 2440–2463 (2000). DOI 10.1364/JOSAA.17.002440. URL https://opg. optica.org/josaa/abstract.cfm?URI=josaa-17-12-2440

    work page doi:10.1364/josaa.17.002440 2000

  20. [20]

    Quasi-local photon surfaces in general spherically symmetric spacetimes.Eur

    T. Colas, J. Grain, V. Vennin, Four-mode squeezed states: two-field quantum systems and the symplectic group Sp(4,R), The European Physical Journal C 82(1), 6 (2022). DOI 10.1140/epjc/ s10052-021-09922-y. URL https://link.springer.com/10.1140/epjc/s10052-021-09922-y

    work page doi:10.1140/epjc/ 2022

  21. [21]

    DOI 10.1140/epjc/s10052-023-12021-9

    ATLAS Collaboration, Search for heavy Majorana or Dirac neutrinos and right-handed W gauge bosons in final states with charged leptons and jets in pp collisions at √s=13 TeV with the ATLAS detector, The European Physical Journal C 83(12), 1164 (2023). DOI 10.1140/epjc/s10052-023-12021-9. URL https://doi.org/10.1140/epjc/s10052-023-12021-9

    work page doi:10.1140/epjc/s10052-023-12021-9 2023

  22. [22]

    A. Mari, J. Eisert, Positive Wigner Functions Render Classical Simulation of Quantum Computation Efficient, Physical Review Letters 109(23), 230,503 (2012). DOI 10.1103/PhysRevLett.109.230503. URL https://link.aps.org/doi/10.1103/PhysRevLett.109.230503

    work page doi:10.1103/physrevlett.109.230503 2012

  23. [23]

    Bianucci, C

    P. Bianucci, C. Miquel, J. Paz, M. Saraceno, Discrete Wigner functions and the phase space representa- tion of quantum computers, Physics Letters A 297(5-6), 353–358 (2002). DOI 10.1016/S0375-9601(02) 00391-2. URL https://linkinghub.elsevier.com/retrieve/pii/S0375960102003912

    work page doi:10.1016/s0375-9601(02 2002

  24. [24]

    Miquel, J.P

    C. Miquel, J.P. Paz, M. Saraceno, Quantum computers in phase space, Physical Review A 65(6), 062,309 (2002). DOI 10.1103/PhysRevA.65.062309. URL https://link.aps.org/doi/10.1103/ PhysRevA.65.062309

    work page doi:10.1103/physreva.65.062309 2002

  25. [25]

    Sánchez-Soto, A

    L.L. Sánchez-Soto, A. Muñoz, P. De La Hoz, A.B. Klimov, G. Leuchs, Phase Space Insights: Wigner Functions for Qubits and Beyond, Applied Sciences 15(9), 5155 (2025). DOI 10.3390/app15095155. URL https://www.mdpi.com/2076-3417/15/9/5155

    work page doi:10.3390/app15095155 2025

  26. [26]

    K.L. Ng, R. Polkinghorne, B. Opanchuk, P.D. Drummond, Phase-space representations of thermal Bose-–Einstein condensates, Journal of Physics A: Mathematical and Theoretical 52(3), 035,302 (2019). DOI 10.1088/1751-8121/aaeeb1. URL https://iopscience.iop.org/article/10.1088/ 1751-8121/aaeeb1

    work page doi:10.1088/1751-8121/aaeeb1 2019

  27. [27]

    Liang, M

    P. Liang, M. Marthaler, L. Guo, Floquet many-body engineering: topology and many-body physics in phase space lattices, New Journal of Physics 20(2), 023,043 (2018). DOI 10.1088/1367-2630/aaa7c3. URL https://iopscience.iop.org/article/10.1088/1367-2630/aaa7c3

    work page doi:10.1088/1367-2630/aaa7c3 2018

  28. [28]

    Y. Yuan, K. Li, J.H. Wang, K. Ma, The Wigner Functions for a Spin-1/2 Relativistic Particle in the Presence of Magnetic Field, International Journal of Theoretical Physics 49(9), 1993–2001 (2010). DOI 10.1007/s10773-010-0262-0. URL http://link.springer.com/10.1007/s10773-010-0262-0

    work page doi:10.1007/s10773-010-0262-0 1993

  29. [29]

    Mahlein, L

    M. Mahlein, L. Barioglio, F. Bellini, L. Fabbietti, C. Pinto, B. Singh, S. Tripathy, A real- istic coalescence model for deuteron production, The European Physical Journal C 83(9), 804 (2023). DOI 10.1140/epjc/s10052-023-11972-3. URL https://link.springer.com/10.1140/epjc/ s10052-023-11972-3

    work page doi:10.1140/epjc/s10052-023-11972-3 2023

  30. [30]

    Baker, I.E

    G.A. Baker, I.E. McCarthy, C.E. Porter, Application of the Phase Space Quasi-Probability Distribu- tion to the Nuclear Shell Model, Physical Review 120(1), 254–264 (1960). DOI 10.1103/PhysRev.120

    work page doi:10.1103/physrev.120 1960

  31. [31]

    URL https://link.aps.org/doi/10.1103/PhysRev.120.254 39

    work page doi:10.1103/physrev.120.254

  32. [32]

    Lee, Theory and application of the quantum phase-space distribution functions, Physics Reports 259(3), 147–211 (1995)

    H.W. Lee, Theory and application of the quantum phase-space distribution functions, Physics Reports 259(3), 147–211 (1995). DOI 10.1016/0370-1573(95)00007-4. URL https://linkinghub.elsevier. com/retrieve/pii/0370157395000074

    work page doi:10.1016/0370-1573(95)00007-4 1995

  33. [33]

    Kurtsiefer, T

    C. Kurtsiefer, T. Pfau, J. Mlynek, Measurement of the Wigner function of an ensemble of helium atoms, Nature 386(6621), 150–153 (1997). DOI 10.1038/386150a0. URL https://www.nature.com/ articles/386150a0

    work page doi:10.1038/386150a0 1997

  34. [34]

    Brody, E.M

    D.C. Brody, E.M. Graefe, R. Melanathuru, Phase-Space Measurements, Decoherence, and Classicality, Physical Review Letters 134(12), 120,201 (2025). DOI 10.1103/PhysRevLett.134.120201. URL https: //link.aps.org/doi/10.1103/PhysRevLett.134.120201

    work page doi:10.1103/physrevlett.134.120201 2025

  35. [35]

    R.F. O Connell, Wigner distribution function approach to dissipative problems in quantum me- chanics with emphasis on decoherence and measurement theory, Journal of Optics B: Quantum and Semiclassical Optics 5(3), S349–S359 (2003). DOI 10.1088/1464-4266/5/3/369. URL https: //iopscience.iop.org/article/10.1088/1464-4266/5/3/369

    work page doi:10.1088/1464-4266/5/3/369 2003

  36. [36]

    Chun, H.W

    Y.J. Chun, H.W. Lee, Measurement-induced decoherence and Gaussian smoothing of the Wigner distribution function, Annals of Physics 307(2), 438–451 (2003). DOI 10.1016/S0003-4916(03)00116-7. URL https://linkinghub.elsevier.com/retrieve/pii/S0003491603001167

    work page doi:10.1016/s0003-4916(03)00116-7 2003

  37. [37]

    Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of Modern Physics 75(3), 715–775 (2003)

    W.H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of Modern Physics 75(3), 715–775 (2003). DOI 10.1103/RevModPhys.75.715. URL https://link.aps.org/ doi/10.1103/RevModPhys.75.715

    work page doi:10.1103/revmodphys.75.715 2003

  38. [38]

    Zurek, Sub-Planck structure in phase space and its relevance for quantum decoherence, Na- ture 412(6848), 712–717 (2001)

    W.H. Zurek, Sub-Planck structure in phase space and its relevance for quantum decoherence, Na- ture 412(6848), 712–717 (2001). DOI 10.1038/35089017. URL https://www.nature.com/articles/ 35089017

    work page doi:10.1038/35089017 2001

  39. [39]

    Rundle, M.J

    R.P. Rundle, M.J. Everitt, Overview of the Phase Space Formulation of Quantum Mechanics with Application to Quantum Technologies, Advanced Quantum Technologies 4(6), 2100,016 (2021). DOI 10.1002/qute.202100016. URL https://onlinelibrary.wiley.com/doi/10.1002/qute.202100016

    work page doi:10.1002/qute.202100016 2021

  40. [40]

    Seyfarth, A.B

    U. Seyfarth, A.B. Klimov, H.D. Guise, G. Leuchs, L.L. Sanchez-Soto, Wigner function for SU(1,1), Quantum 4, 317 (2020). DOI 10.22331/q-2020-09-07-317. URL https://quantum-journal.org/ papers/q-2020-09-07-317/

    work page doi:10.22331/q-2020-09-07-317 2020

  41. [41]

    Cahill, R.J

    K.E. Cahill, R.J. Glauber, Density Operators and Quasiprobability Distributions, Physical Review 177(5), 1882–1902 (1969). DOI 10.1103/PhysRev.177.1882. URL https://link.aps.org/doi/10. 1103/PhysRev.177.1882

    work page doi:10.1103/physrev.177.1882 1902

  42. [42]

    Gerry, P

    C.C. Gerry, P. Knight, Introductory quantum optics , second edition edn. (Cam- bridge University Press, Cambridge, United Kingdom New York, NY, 2024). ISBN 978100941529397811391512079781107653948

    2024

  43. [43]

    Linowski, L

    T. Linowski, L. Rudnicki, Relating the Glauber-Sudarshan, Wigner, and Husimi quasiprobability distributions operationally through the quantum-limited amplifier and attenuator channels, Physical Review A 109(2), 023,715 (2024). DOI 10.1103/PhysRevA.109.023715. URL https://link.aps. org/doi/10.1103/PhysRevA.109.023715

    work page doi:10.1103/physreva.109.023715 2024

  44. [44]

    Andreev, D.M

    V.A. Andreev, D.M. Davidović, L.D. Davidović, M.D. Davidović, V.I. Man’ko, M.A. Man’ko, A transformational property of the Husimi function and its relation to the wigner function and symplectic tomograms, Theoretical and Mathematical Physics 166(3), 356–368 (2011). DOI 10.1007/s11232-011-0028-8. URL http://link.springer.com/10.1007/s11232-011-0028-8

    work page doi:10.1007/s11232-011-0028-8 2011

  45. [45]

    Weedbrook, S

    C. Weedbrook, S. Pirandola, R. García-Patrón, N.J. Cerf, T.C. Ralph, J.H. Shapiro, S. Lloyd, Gaussian quantum information, Reviews of Modern Physics 84(2), 621–669 (2012). DOI 10.1103/RevModPhys. 84.621. URL https://link.aps.org/doi/10.1103/RevModPhys.84.621 40

    work page doi:10.1103/revmodphys 2012

  46. [46]

    Braunstein, P

    S.L. Braunstein, P. Van Loock, Quantum information with continuous variables, Reviews of Modern Physics 77(2), 513–577 (2005). DOI 10.1103/RevModPhys.77.513. URL https://link.aps.org/ doi/10.1103/RevModPhys.77.513

    work page doi:10.1103/revmodphys.77.513 2005

  47. [47]

    Braunstein, A.K

    S.L. Braunstein, A.K. Pati (eds.), Quantum information with continuous variables (Kluwer Acad. Publ, Dordrecht, 2003). ISBN 9781402011955

    2003

  48. [48]

    Rahman, A.x

    A.u. Rahman, A.x. Liu, M.Y. Abd-Rabbou, C.f. Qiao. From Local Nonclassicality to Entanglement: A Convexity Law for Single-Excitation Dynamics (2025). DOI 10.48550/arXiv.2511.10470. URL http://arxiv.org/abs/2511.10470. ArXiv:2511.10470 [quant-ph]

    work page doi:10.48550/arxiv.2511.10470 2025

  49. [49]

    Flũhmann, T.L

    C. Flũhmann, T.L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, J.P. Home, Encoding a qubit in a trapped-ion mechanical oscillator, Nature 566(7745), 513–517 (2019). DOI 10.1038/ s41586-019-0960-6. URL https://www.nature.com/articles/s41586-019-0960-6

    2019

  50. [50]

    Campagne-Ibarcq, A

    P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N.E. Frattini, V.V. Sivak, P. Rein- hold, S. Puri, S. Shankar, R.J. Schoelkopf, L. Frunzio, M. Mirrahimi, M.H. Devoret, Quantum error correction of a qubit encoded in grid states of an oscillator, Nature 584(7821), 368–372 (2020). DOI 10.1038/s41586-020-2603-3. URL https://www.nature.com/artic...

    work page doi:10.1038/s41586-020-2603-3 2020

  51. [51]

    Vlastakis, G

    B. Vlastakis, G. Kirchmair, Z. Leghtas, S.E. Nigg, L. Frunzio, S.M. Girvin, M. Mirrahimi, M.H. Devoret, R.J. Schoelkopf, Deterministically Encoding Quantum Information Using 100-Photon Schrödinger Cat States, Science 342(6158), 607–610 (2013). DOI 10.1126/science.1243289. URL https://www.science.org/doi/10.1126/science.1243289

    work page doi:10.1126/science.1243289 2013

  52. [52]

    Hofheinz, H

    M. Hofheinz, H. Wang, M. Ansmann, R.C. Bialczak, E. Lucero, M. Neeley, A.D. O’Connell, D. Sank, J. Wenner, J.M. Martinis, A.N. Cleland, Synthesizing arbitrary quantum states in a superconducting resonator, Nature 459(7246), 546–549 (2009). DOI 10.1038/nature08005. URL https://www.nature. com/articles/nature08005

    work page doi:10.1038/nature08005 2009

  53. [53]

    C.F. Lo, R. Sollie, Generalized multimode squeezed states, Phys. Rev. A 47, 733–735 (1993). DOI 10.1103/PhysRevA.47.733. URL https://link.aps.org/doi/10.1103/PhysRevA.47.733

    work page doi:10.1103/physreva.47.733 1993

  54. [54]

    Simon, N

    R. Simon, N. Mukunda, B. Dutta, Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms, Phys. Rev. A 49, 1567–1583 (1994). DOI 10.1103/PhysRevA.49.1567. URL https://link.aps.org/doi/10.1103/PhysRevA.49.1567

    work page doi:10.1103/physreva.49.1567 1994

  55. [55]

    L. Ma, Y.D. Zhang, J.W. Pan, X.B. Wang, THE GENERAL EXPONENTIAL QUADRATIC MULTIMODE SQUEEZED STATES, Modern Physics Letters A 10(10), 837–840 (1995). DOI 10.1142/S0217732395000892. URL https://www.worldscientific.com/doi/abs/10.1142/ S0217732395000892

    work page doi:10.1142/s0217732395000892 1995

  56. [56]

    S. Dai, Q. Kang, L. Hu, C. Liu, T. Zhao, Entanglement Improvement of Three-mode Squeezed Vacuum State Via Number-conserving Operation, International Journal of Theoretical Physics 64(6), 158 (2025). DOI 10.1007/s10773-025-06025-2. URL https://link.springer.com/10.1007/ s10773-025-06025-2

    work page doi:10.1007/s10773-025-06025-2 2025

  57. [57]

    Feynman, Statistical Mechanics: a set of lectures

    R.P. Feynman, Statistical Mechanics: a set of lectures . Advanced book classics (Westview Press, Boul- der, CO, USA, 1998). ISBN 978-0-201-36076-9. (Originally published 1972 by Benjamin Cummings, Reading, MA, USA; OCLC: ocm60679997.)

    1998

  58. [58]

    Kim, E.P

    Y.S. Kim, E.P. Wigner, Canonical transformation in quantum mechanics, American Journal of Physics 58(5), 439–448 (1990). DOI 10.1119/1.16475. URL http://aapt.scitation.org/doi/10.1119/1. 16475

    work page doi:10.1119/1.16475 1990

  59. [59]

    Wigner, On Unitary Representations of the Inhomogeneous Lorentz Group, The Annals of Mathematics 40(1), 149–204 (1939)

    E.P. Wigner, On Unitary Representations of the Inhomogeneous Lorentz Group, The Annals of Mathematics 40(1), 149–204 (1939). DOI 10.2307/1968551. URL http://www.jstor.org/stable/ 1968551?origin=crossref 41

    work page doi:10.2307/1968551 1939

  60. [60]

    Başkal, Y.S

    S. Başkal, Y.S. Kim, M.E. Noz, Theory and Applications of the Poincaré Group , 2nd edn. (Springer Nature Switzerland, Cham, 2024). DOI 10.1007/978-3-031-64376-7. ISBN 978-3-031-63488-8, 978-3-031-64376-7

    work page doi:10.1007/978-3-031-64376-7 2024

  61. [61]

    Perelomov, Generalized Coherent States and Their Applications (Springer Berlin Heidelberg, Berlin, Heidelberg, 1986)

    A. Perelomov, Generalized Coherent States and Their Applications (Springer Berlin Heidelberg, Berlin, Heidelberg, 1986). ISBN 978-3-642-61629-7. URL http://public.eblib.com/choice/ publicfullrecord.aspx?p=3094196. OCLC: 851731914

    1986

  62. [62]

    Han, Y.S

    D. Han, Y.S. Kim, D. Son, E(2)–like little group for massless particles and neutrino polarization as a consequence of gauge invariance, Physical Review D 26(12), 3717–3725 (1982). DOI 10.1103/ PhysRevD.26.3717. URL https://link.aps.org/doi/10.1103/PhysRevD.26.3717

    work page doi:10.1103/physrevd.26.3717 1982

  63. [63]

    Gerry, Remarks on the use of group theory in quantum optics, Opt

    C.C. Gerry, Remarks on the use of group theory in quantum optics, Opt. Express 8(2), 76–85 (2001). DOI 10.1364/OE.8.000076. URL https://opg.optica.org/oe/abstract.cfm?URI=oe-8-2-76

    work page doi:10.1364/oe.8.000076 2001

  64. [64]

    Wünsche, Symplectic groups in quantum optics, Journal of Optics B: Quantum and Semiclassi- cal Optics 2(2), 73 (2000)

    A. Wünsche, Symplectic groups in quantum optics, Journal of Optics B: Quantum and Semiclassi- cal Optics 2(2), 73 (2000). DOI 10.1088/1464-4266/2/2/302. URL https://dx.doi.org/10.1088/ 1464-4266/2/2/302

    work page doi:10.1088/1464-4266/2/2/302 2000

  65. [65]

    Kramer, M

    P. Kramer, M. Saraceno, Semicoherent states and the group ISp(2, R), Physica A: Statistical Mechanics and its Applications 114(1-3), 448–453 (1982). DOI 10.1016/0378-4371(82)90330-2. URL https: //linkinghub.elsevier.com/retrieve/pii/0378437182903302

    work page doi:10.1016/0378-4371(82)90330-2 1982

  66. [66]

    Ellinas, A.J

    D. Ellinas, A.J. Bracken, Phase space regions operators and ISp(2) maps, Journal of Physics: Conference Series 1194, 012,033 (2019). DOI 10.1088/1742-6596/1194/1/012033. URL https: //iopscience.iop.org/article/10.1088/1742-6596/1194/1/012033

    work page doi:10.1088/1742-6596/1194/1/012033 2019

  67. [67]

    Von Neumann, R.T

    J. Von Neumann, R.T. Beyer, N.A. Wheeler, Mathematical foundations of quantum me- chanics, new edn. (Princeton University Press, Princeton, NJ, USA, 2018). ISBN 978-0-691-17856-1~978-0-691-17857-8. (Originally published: Princeton University Press, 1996, 1955, and in German by Springer, Berlin, Germany 1932; OCLC: on1004934776.)

    2018

  68. [68]

    Wigner, The general properties of the distribution function and remarks on its weakness, in First International Conference on the Physics of Phase Space , ed

    E.P. Wigner, The general properties of the distribution function and remarks on its weakness, in First International Conference on the Physics of Phase Space , ed. by Y.S. Kim, W.W. Zachary (Springer-Verlag, Heidelberg, Univeristy of Maryland, College Park, MD USA, 1987), 161–170. ISBN 978-3-540-47901-7

    1987

  69. [69]

    Başkal, Y

    S. Başkal, Y. Kim, M. Noz, Mathematical Devices for Optical Sciences. (IOP Publishing, Bristol, UK, 2019). ISBN 978-0-7503-1612-5. URL https://dx.doi.org/10.1088/2053-2563/aafe78. (OCLC: 1034620988.)

    work page doi:10.1088/2053-2563/aafe78 2019

  70. [70]

    Hillery, R.F

    M. Hillery, R.F. O’Connell, M.O. Scully, E.P. Wigner, Distribution functions in physics: Fun- damentals, Physics Reports 106(3), 121–167 (1984). DOI 10.1016/0370-1573(84)90160-1. URL http://linkinghub.elsevier.com/retrieve/pii/0370157384901601

    work page doi:10.1016/0370-1573(84)90160-1 1984

  71. [71]

    Schleich, D.F

    W. Schleich, D.F. Walls, J.A. Wheeler, Area of overlap and interference in phase space versus Wigner pseudoprobabilities, Physical Review A 38(3), 1177–1186 (1988). DOI 10.1103/PhysRevA.38.1177. URL https://link.aps.org/doi/10.1103/PhysRevA.38.1177

    work page doi:10.1103/physreva.38.1177 1988

  72. [72]

    Arfken, H.J

    G.B. Arfken, H.J. Weber, F.E. Harris, Mathematical methods for physicists: a comprehensive guide , 7th edn. (Elsevier, Amsterdam NL; Boston, MA, USA, 2013). ISBN 978-0-12-384654-9. (Originally published in 1966.)

    2013

  73. [73]

    Bell, Epr correlations and epw distributions, Annals of the New York Academy of Sciences 480(1), 263–266 (1986)

    J.S. Bell, Epr correlations and epw distributions, Annals of the New York Academy of Sciences 480(1), 263–266 (1986). DOI https://doi.org/10.1111/j.1749-6632.1986.tb12429.x. URL https://nyaspubs. onlinelibrary.wiley.com/doi/abs/10.1111/j.1749-6632.1986.tb12429.x 42

    work page doi:10.1111/j.1749-6632.1986.tb12429.x 1986

  74. [74]

    Mallick, S

    B. Mallick, S. Chakrabarty, S. Mukherjee, A.G. Maity, A.S. Majumdar, Efficient detection of non- classicality using moments of the Wigner function, Physical Review A 111(3), 032,406 (2025). DOI 10.1103/PhysRevA.111.032406. URL https://link.aps.org/doi/10.1103/PhysRevA.111.032406

    work page doi:10.1103/physreva.111.032406 2025

  75. [75]

    Fock, Quanten Elecktrodynamik, Physikalische Zeitschrift der Sovietunion 6, 425–469 (1934)

    V. Fock, Quanten Elecktrodynamik, Physikalische Zeitschrift der Sovietunion 6, 425–469 (1934)

    1934

  76. [76]

    Stoler, Equivalence Classes of Minimum Uncertainty Packets, Physical Review D 1(12), 3217–3219 (1970)

    D. Stoler, Equivalence Classes of Minimum Uncertainty Packets, Physical Review D 1(12), 3217–3219 (1970). DOI 10.1103/PhysRevD.1.3217. URL https://link.aps.org/doi/10.1103/PhysRevD.1. 3217

    work page doi:10.1103/physrevd.1.3217 1970

  77. [77]

    Glauber, Coherent and Incoherent States of the Radiation Field, Physical Review 131(6), 2766– 2788 (1963)

    R.J. Glauber, Coherent and Incoherent States of the Radiation Field, Physical Review 131(6), 2766– 2788 (1963). DOI 10.1103/PhysRev.131.2766. URL https://link.aps.org/doi/10.1103/PhysRev. 131.2766

    work page doi:10.1103/physrev.131.2766 1963

  78. [78]

    Sudarshan, Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams, Physical Review Letters 10(7), 277–279 (1963)

    E.C.G. Sudarshan, Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams, Physical Review Letters 10(7), 277–279 (1963). DOI 10.1103/PhysRevLett.10.277. URL https://link.aps.org/doi/10.1103/PhysRevLett.10.277

    work page doi:10.1103/physrevlett.10.277 1963

  79. [79]

    Klauder, B.S

    J.R. Klauder, B.S. Skagerstam, Coherent States (World Scientific Publishing Co., Singapore; Hacken- sack, NJ, USA, 1985)

    1985

  80. [80]

    Arecchi, Measurement of the Statistical Distribution of Gaussian and Laser Sources, Physical Review Letters 15(24), 912–916 (1965)

    F.T. Arecchi, Measurement of the Statistical Distribution of Gaussian and Laser Sources, Physical Review Letters 15(24), 912–916 (1965). DOI 10.1103/PhysRevLett.15.912. URL https://link.aps. org/doi/10.1103/PhysRevLett.15.912

    work page doi:10.1103/physrevlett.15.912 1965

Showing first 80 references.