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REVIEW 3 major objections 5 minor 73 references

Gaussian Open Quantum Dynamics and Isomorphism to Superconformal Symmetry

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Pith's one-line read The paper claims that every Gaussianity-preserving open quantum generator forms a Lie algebra isomorphic to $\mathbb{R}^{2n^2+3n} \oplus_{\mathrm{S}} \mathfrak{gl}(2n,\mathbb{R})$, yielding an exact finite-matrix solution of quadratic…

desk verdict Useful finite-matrix solution to quadratic Gaussian open dynamics, but the advertised superconformal duality rests on an unsupplied and likely impossible map between field and density matrix. read the letter →

arxiv 2507.04932 v1 pith:UKXXMS3Q submitted 2025-07-07 quant-ph hep-thmath-phmath.MP

Ju-Yeon Gyhm , Dario Rosa , Dominik Šafránek This is my paper · ORCID
classification quant-phhep-thmath-phmath.MP PACS 03.65.Yz03.65.-w02.20.Sv11.30.Pb
keywords GaussianopenalgebraGaussianity-preservingsuperoperatorsquadraticRedfieldequationWignerfunctionsemidirectproductLiesuperconformalsymmetryquantumsystemscontinuous-variableinformation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper identifies the Lie algebra underlying all open quantum dynamics that preserve Gaussianity, $\mathfrak{go}(n)$, and proves it is isomorphic to $\mathbb{R}^{2n^2+3n} \oplus_{\mathrm{S}} \mathfrak{gl}(2n,\mathbb{R})$. On this basis it gives an exact finite-matrix solution of the quadratic-order Redfield equation: any initial Wigner function, Gaussian or not, evolves by one linear coordinate transformation, one displacement, and one Gaussian convolution. The result turns a class of open-system evolutions into the tracking of $6n^2+3n$ real parameters, covering states such as cat states, GKP states, and Fock states. The paper further reports that the single-mode algebra contains the three-dimensional super-Poincaré and superconformal algebras, with the CPTP condition mapping to causality plus the arrow of time.

What carries the argument

The central object is the Gaussian open algebra $\mathfrak{go}(n)$, a Lie algebra of superoperators that preserve Gaussianity. Every element is parametrized by a real matrix $\Gamma_M$, a symmetric real matrix $\Gamma_D$, and a real vector $\Gamma_v$, and the isomorphism $\mathfrak{go}(n) \cong \mathbb{R}^{2n^2+3n} \oplus_{\mathrm{S}} \mathfrak{gl}(2n,\mathbb{R})$ converts superoperator evolution into the finite matrix equations $\dot M = \Gamma_M M$, $\dot D = \Gamma_D + \Gamma_M D + D\Gamma_M^\intercal$, and $\dot v = \Gamma_v + \Gamma_M v$, whose solution feeds the closed-form Wigner evolution.

What would settle it

Evolve a single-photon Fock state under the zero-temperature thermalization generator of Eq. (26), once with the Wigner formula (60) and once by direct numerical integration of the master equation in a truncated Fock basis; a mismatch in $W(x,p,t)$ falsifies the central solution. For the superconformal isomorphism, compute the graded brackets of the fourteen generators of $\mathfrak{ego}(1)$ in a finite truncation and compare with $\mathfrak{osp}(1|4)$; one failed (anti-)commutator falsifies Theorem 10.

Watch

Extended reading notes

Core claim

Theorem 7 states that $\mathfrak{go}(n)$ is a Lie algebra isomorphic to $\mathbb{R}^{2n^2+3n} \oplus_{\mathrm{S}} \mathfrak{gl}(2n,\mathbb{R})$, spanned by unitary Gaussian generators, one-photon dissipation superoperators, and mode-mixing terms. Theorem 8 gives the exact time evolution of any initial state: the Wigner function at time $t$ is the initial Wigner function evaluated at $M^{-1}((\vec{y},\vec{q})-v)$, divided by $|\det M|$, and convolved with a Gaussian kernel whose covariance is the symmetric matrix $D$, where $(M,D,v)$ are integrated through the three matrix equations (59). Theorem 9 claims this single-mode algebra defines a supersymmetric massless free field theory in three-dimensional Minkowski spacetime; Theorem 10 extends it to an algebra $\mathfrak{ego}(1)$ isomorphic to the $N=1$ superconformal algebra, in which the density matrix satisfies the massless Klein-Gordon and Dirac equations and the CPTP condition becomes causality together with the arrow of time.

Load-bearing premise

The load-bearing premise is that $\mathfrak{go}(n)$ is exactly the algebra of Gaussianity-preserving superoperators and that the Wigner action of its group is captured by the $(M,D,v)$ parametrization; the field-theory reading additionally needs a bijective map between the field and the density matrix satisfying canonical commutation relations, which the paper concedes is not yet justified.

Editorial extensions

If this is right

  • The quadratic Redfield equation and its Lindblad subclass can be integrated exactly with $6n^2+3n$ real parameters, regardless of whether the initial state is Gaussian.
  • Every infinitesimally divisible Gaussian quantum channel is a CP element of $\mathfrak{GO}_+(n)$, and every Gaussian quantum channel is a CP element of the closure $\overline{\mathfrak{GO}}(n)$, so non-Gaussian states can be evolved through these channels exactly.
  • Any pair of Gaussian states can be connected by a CPTP evolution lying in the closure of $\mathfrak{GO}_+(n)$, giving a group-theoretic statement of the reachable set of Gaussian open dynamics.
  • In the single-mode case the CPTP condition is equivalent to the spacetime conditions $\Delta\tau^2 \geq \Delta x^2 + \Delta y^2$ and $\Delta\tau \geq 0$, so complete positivity and trace preservation become causality and time orientation.
  • The density matrix of a single bosonic mode satisfies both the massless Klein-Gordon and Dirac equations in the superoperator representation, and the extended Gaussian open algebra realizes the maximal symmetry of that field theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct numerical translation the authors do not spell out: their formula (60) is a ready-made algorithm to evolve non-Gaussian states such as a single-photon Fock state under any quadratic Lindbladian, and it can be checked against truncated-Fock-space integration as a clean test of the paper's main theorem.
  • Because the CPTP condition becomes a light-cone condition, the paper suggests a dictionary between positivity of open-system generators and causal propagation; one could classify non-Markovian or non-CPTP Redfield regimes by the spacetime type of their generators, which the authors only sketch.
  • The field-theory duality is explicitly conditional: the map $f:\phi\leftrightarrow\rho$ satisfying $[\phi,\pi]=i\delta$ is not yet justified, so the superconformal interpretation should be read as a structural correspondence of algebras rather than a proven equivalence of theories.
  • The maximality theorem for $\mathfrak{ego}(1)$ gives a practical normal form: any candidate superoperator claimed to preserve Gaussianity of all Gaussian observables must be expressible within $\mathfrak{ego}(1)$ times the identity, a checkable algebraic criterion.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper constructs a Lie algebra go(n) of Gaussianity-preserving superoperators for n bosonic modes, proves its isomorphism to R^{2n^2+3n} semidirect gl(2n,R), derives explicit CPTP conditions, and uses the finite-dimensional parametrization to give an exact Wigner-function solution for quadratic Redfield and Lindblad evolution of arbitrary initial states. It then proposes a correspondence between go(1) and three-dimensional super-Poincaré and superconformal algebras, claiming that density matrices satisfy Klein-Gordon and Dirac equations. The appendices contain the main algebra proofs, the CPTP proof, the Gaussian-channel classification proofs, and the Wigner evolution proof.

Significance. The algebraic core of the paper is sound and potentially useful: the explicit isomorphism and the finite-matrix representation of Gaussian open dynamics are nontrivial, and Theorem 8 gives a direct convolution formula for evolving even non-Gaussian states, which is a practical contribution to continuous-variable open quantum systems. The CPTP condition in Theorem 2 and the Gaussian-channel results in Theorems 4 and 5 are consistent with standard Gaussian quantum information, and no fitted parameters appear in the construction. However, the advertised field-theory duality is not established: without the map f of Eq. (71), the Klein-Gordon and Dirac equations are algebraic identities rather than equations of motion. The paper's lasting value is therefore in the open-system algebra and exact solution, not in the superconformal field theory interpretation as currently presented.

major comments (3)
  1. [The central issue is in Section V, Eq. (71), and Theorem 9.] The claimed duality to a three-dimensional superconformal field theory is not supported. Immediately after Eq. (71) the authors state that establishing a one-to-one correspondence between the field operator and the density matrix is "not yet justified"; this is precisely the load-bearing step for Theorem 9 and for the title and abstract. Moreover, Eq. (71) demands one canonical pair (phi(x,y,tau), pi(x,y,tau)) for every spatial point, whereas the single-mode density matrix carries exactly one canonical pair; no bijective, CCR-preserving map f between a field on R^2 and a one-mode density matrix is supplied, and the natural degree-of-freedom count makes such a map impossible without artificially reducing the field to one mode. The algebraic isomorphism of go(1) to a super-Poincaré subalgebra does not by itself define a field theory.
  2. [The issue is in Section V, Eqs. (67) and (80).] The Klein-Gordon and Dirac "equations" are algebraic identities, not equations of motion. With the identifications in Eq. (62) and with L+_xx = ad_x^2, L+_pp = ad_p^2, L+_xp = ad_x ad_p, Eq. (67) reduces to (ad_x^2 ad_p^2 - (ad_x ad_p)^2) rho = 0, which holds for every rho because ad_x and ad_p commute. The same is true for the matrix identity in Eq. (80). These identities therefore impose no dynamical condition and cannot serve as evidence that the density matrix describes a massless bosonic or fermionic field. This strengthens the concern in the previous comment that Theorem 9 overclaims.
  3. [The issue is in Theorems 6 and 8, Eqs. (50) and (60).] The stated general solution is not well-defined for all cases covered by the theorem. The Gaussian kernel contains D^{-1} and 1/sqrt(det D), so singular diffusion matrices D, including D = 0 which occurs for unitary evolution, are excluded, yet the theorem claims a solution for any initial state and any generator in go(n). Similarly, the prefactor |det M|^{-1} and M^{-1} require det M != 0, while the closure GO+(n) includes limiting elements with det M = 0; Appendix F discusses the det M = 0 case but the theorem statements do not. The formulas should be amended with a distributional limiting prescription or with an explicit separate treatment of singular D and singular M.
minor comments (5)
  1. [The issue is in Appendix C, Eq. (C22).] The equality in Eq. (C22) is reversed; it should read rho' = E_{rho -> rho'} rho rather than rho = E_{rho -> rho'} rho'.
  2. [The issue is in Appendix A, Eq. (A17).] The last listed bi-mode generator duplicates L+_{xi xj}; it should be L+_{pi pj}.
  3. [The issue is in Section III, around Theorem 3 and Eq. (40).] The notation GO+(1) is used both for the group and for its topological closure; please introduce and use a consistent notation, for example GO+(1) for the closure, throughout the theorem statements.
  4. [The issue is in Appendix H, Eq. (H14).] In the Hermite polynomial expansion following Eq. (H13), the index in H_j(sqrt(Lambda_xx) x) should be m, matching the summation index in the decomposition, rather than j.
  5. [The issue is across the Introduction and Appendix F.] There are several typos, including "pseduomode" in the Introduction and "Wingner" in Appendix F, which should be corrected.

Circularity Check

3 steps flagged · score 4.0 of 10

The finite-matrix Redfield solution and the go(n) isomorphism are self-contained; the advertised field-theory duality rests on stipulated operator identifications and an explicitly unproven map f, so the Klein-Gordon and Dirac 'findings' are by construction rather than independent predictions.

  1. self definitional [Section V, Eqs. (62), (67)-(68)]
    "Furthermore, it can be verified that L+ˆxˆxL+ˆpˆp = (L+ˆxˆp)2 and the following identity holds for any density matrix ˆρ: ((L+ˆxˆx+L+ˆpˆp)^2/4 − (L+ˆxˆx−L+ˆpˆp)^2/4 −(L+ˆxˆp)^2) ˆρ=0. (67) Using Eq. (62), we find that this identity is equivalent to the massless Klein-Gordon equation for the field φ(x,y,τ): (∂2t −∂2x−∂2y)φ=0. (68)"

    The operators L+ are defined in Eq. (20) as bilinears in x and p, and the dictionary Pτ↔..., Px↔..., Py↔... in Eq. (62) is a stipulated isomorphism. The 'identity holds for any density matrix' is therefore an algebraic consequence of those definitions, not a dynamical equation derived from a field action or from an independently given map f. The Klein-Gordon equation is just a transcription of that operator identity, so the claimed statement that a density matrix satisfies the KG equation is true for every ρ by construction. The physical field interpretation additionally requires the canonical-commutation-relation map f of Eq. (71), which the paper itself concedes is not justified.

  2. self definitional [Section V, Eqs. (72), (78)-(81)]
    "Similarly to the discussion leading to Eq. (67), the following identity is verified for any density matrix ˆρ: ( L+ xp L+xx −L+pp −L+xp )(−adˆx adˆp ) ˆρ= 0. (80) By applying Eq. (72) and γµ elements defined above, this identity gets translated into the following conditions for the field ψ: γµ∂µψ = 0. (81) Eq. (81) is nothing but the celebrated Dirac equation for a massless spinorial field in three dimensions."

    The spinor ψ is defined in Eq. (78) as (Q1,Q2)φ with the identifications Q1⇔−adˆx and Q2⇔adˆp chosen in Eq. (72). Equation (80) is verified for any density matrix in the same superoperator representation. The massless Dirac equation is therefore a restatement of that algebraic identity under the stipulated dictionary, not a prediction obtained from independent fermionic dynamics. The Majorana condition and the gamma-matrix assignments are also built into the chosen identification, so the Dirac equation reduces by construction to the input superoperator commutation relations.

1 more flagged steps
  1. other [Section V, after Eq. (71)]
    "We require that [φ(x1,y1,τ), π(x2,y2,τ)]= iδ(x1−x2,y1−y2) ... This is however not guaranteed by all maps f. This means that there is no guarantee that the density matrix satisfies the canonical commutation relations. Therefore, establishing a one-to-one correspondence between the field operator and the density matrix is, at this stage, not yet justified."

    Theorem 9, 'The representation go(1) defines a supersymmetric massless free field theory in the three-dimensional Minkowski spacetime,' depends on a bijective, canonical-commutation-relation-preserving map f, which the paper explicitly states is not established. Without such a map, the Klein-Gordon and Dirac identities are only algebraic identities in the superoperator representation and do not define a physical three-dimensional field theory. This is a load-bearing missing assumption for the field-theoretic duality, though it does not affect the finite-matrix Redfield solution in Theorems 6-8.

full rationale

The core Lie-algebra and dynamical results are self-contained and not circular. go(1) and go(n) are defined by explicit superoperator bases, and Theorems 1 and 7 are proven by direct matrix representations; no fitted parameter is later renamed as a prediction. Theorem 6/Theorem 8 are derived from the explicit phase-space action of the generators, and the CPTP condition in Theorem 2 follows from standard Lindblad positivity. There are no load-bearing self-citations: the cited uniqueness-type facts are standard references, not prior claims by the same authors that already assume the target result. The circularity burden is confined to the advertised field-theory interpretation. The Klein-Gordon equation and the Dirac equation are algebraic identities that hold for every density matrix in the chosen representation, so presenting them as field equations is a renaming of the defining superoperator algebra rather than an independent derivation. Moreover, the paper explicitly concedes that the map f needed to turn these identities into a genuine three-dimensional field theory, with the canonical commutation relations of Eq. (71), is not yet justified. This makes the superconformal-field-theory claim structurally dependent on an unsupplied bijection, while leaving the finite-matrix solution of quadratic Redfield/Lindblad dynamics intact. Overall, the main algebraic and dynamical content is independent, so the score is moderate rather than high.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The supercharges Q, dilation D, and special conformal generators K are linear combinations of existing single-mode superoperators, so they do not count as invented entities. The ledger contains one ad hoc algebraic constant, plus several standard domain assumptions imported from the Gaussian quantum information and Lie superalgebra literature.

free parameters (1)
  • c-number shift in dilation generator = 1/4
    In Eq. (85) and Appendix G, L^-_xp is shifted by +1/2, giving D = 1/2 L^-_xp + 1/4, so that the extended algebra closes as osp(1|4). The constant is chosen ad hoc and affects the identification with the special conformal generators.
assumptions (5)
  • domain assumption Gaussian channels are fully characterized by the pair (M, v, D) with sigma' = M sigma M^T + 2D and d' = M d + v; complete positivity imposes a matrix inequality.
    Used in Appendices D and E to prove Theorems 4 and 5; imported from Refs. [1] and [48] rather than derived.
  • domain assumption Williamson decomposition: any Gaussian state is a Gaussian unitary applied to a product of thermal states, each at some inverse temperature beta_i.
    Used in Theorem 3 proof in Appendix C to reduce arbitrary Gaussian-state connectivity to thermal-state connectivity.
  • standard math The N=1 superconformal algebra in three dimensions is isomorphic to osp(1|4).
    Invoked at the start of Appendix G to identify ego(1) with the superconformal algebra via osp(1|4).
  • domain assumption On Wigner functions, the adjoint and anti-adjoint operations act as ad_x -> -d_p, ad_p -> d_x, ad+_x -> 2x, ad+_p -> 2p.
    Used to derive the closed-form Wigner solutions in Theorems 6 and 8 and in the completeness proof of Theorem 11 in Appendices F and H.
  • domain assumption Any Gaussian quantum channel that is infinitesimally divisible must have det(M) >= 0 in addition to being CPTP.
    Used in the proof of Theorem 4; relies on the classification of Gaussian channel semigroups from Ref. [48].

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Pith. "Pith review of Gaussian Open Quantum Dynamics and Isomorphism to Superconformal Symmetry." pith.science (2026). https://pith.science/paper/UKXXMS3Q

@misc{pith2026250704932,
  author       = {Pith},
  title        = {Pith review of: Gaussian Open Quantum Dynamics and Isomorphism to Superconformal Symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UKXXMS3Q}},
  note         = {Machine review of arXiv:2507.04932}
}
abstract

Understanding the Lie algebraic structure of a physical problem often makes it easier to find its solution. In this paper, we focus on the Lie algebra of Gaussian-conserving superoperators. We construct a Lie algebra of $n$-mode states, $\mathfrak{go}(n)$, composed of all superoperators conserving Gaussianity, and we find it isomorphic to $\mathbb{R}^{2n^2+3n}\oplus_{\mathrm{S}}\mathfrak{gl}(2n,\mathbb{R})$. This allows us to solve the quadratic-order Redfield equation for any, even non-Gaussian, state. We find that the algebraic structure of Gaussian operations is the same as that of super-Poincar\'e algebra in three-dimensional spacetime, where the CPTP condition corresponds to the combination of causality and directionality of time flow. Additionally, we find that a bosonic density matrix satisfies both the Klein-Gordon and the Dirac equations. Finally, we expand the algebra of Gaussian superoperators even further by relaxing the CPTP condition. We find that it is isomorphic to a superconformal algebra, which represents the maximal symmetry of the field theory. This suggests a deeper connection between two seemingly unrelated fields, with the potential to transform problems from one domain into another where they may be more easily solved.

Figures

Figures reproduced from arXiv: 2507.04932 by the authors.

Figure 1
Figure 1. FIG. 1. CPTP condition on the dual classical field. Time [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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Reference graph

Works this paper leans on

73 extracted references · 45 canonical work pages

  1. [1]

    E is an infinitesimally divisible Gaussian quantum channel [48]

    work page

  2. [2]

    E is an element of GO+(1) and satisfies the CP condition. Proof. See Appendix D. Here, infinitesimally divisible refers to a quantum channel that can be expressed as the exponential of a Lindbladian superoperator, or as the limit of such expo- nentials [48]. As a consequence, we find that GO+(1) encompasses most Gaussian quantum channels, particularly tho...

    work page

  3. [3]

    E is a Gaussian quantum channel

    work page

  4. [4]

    E is an element of GO(1) and satisfies the CP con- dition. Proof. See Appendix E. While GO+(1) captures all infinitesimally divisible Gaussian channels, GO(1) is required to describe the full space of Gaussian quantum channels, including those that are not connected to the identity component. As a consequence, we conclude that the set of Gaussian quan- tu...

    work page

  5. [5]

    (49) With this, we can write a general solution for the dy- namics of any, even non-Gaussian, initial state ˆρ(0)

    (47) Defining Γ(t)= (γ− ˆx ˆp− γ ˆX γ ˆY − γ ˆN γ ˆY + γ ˆN γ− ˆx ˆp+ γ ˆX ) , (48) 7 the time evolution of EG(t)= (D(t)⊕ v(t))⊕S M(t) is d dt M = ΓM, d dt D = ( γ+ ˆp ˆp −γ+ ˆx ˆp/2 −γ+ ˆx ˆp/2 γ+ ˆxˆx )+ ΓD+ DΓ⊺, d dt v= (−γ ˆp γˆx )+ Γv. (49) With this, we can write a general solution for the dy- namics of any, even non-Gaussian, initial state ˆρ(0). T...

    work page

  6. [6]

    Single bosonic mode : Theorem 1 go(1) is defined as the vector space over the real field, R, spanned by nine basis vectors ad ˆN , ad ˆX , ad ˆY , adˆx, ad ˆp, L+ ˆxˆx, L+ ˆx ˆp, L+ ˆp ˆp, L− ˆx ˆp. (A1) To show that go(1) is Lie algebra, we need to prove the existence of a bilinear operator, [A, B], that is closed under go(1), and satisfies the anticommu...

    work page

  7. [7]

    , ad ˆp = (−ad ˆp adˆx)(−1 0 ) . (A11) With this, we derive the commutation relation between g ∈ g and h1 ∈ h, [g, h1]= ⎡⎢⎢⎢⎢⎣ (−ad ˆp adˆx) Γ⎛ ⎝ ad+ ˆx 2 ad+ ˆp 2 ⎞ ⎠ ,(−ad ˆp adˆx) v ⎤⎥⎥⎥⎥⎦ = (ad ˆp −adˆx)(Γv)∈ h, (A12) because Γv is also a two-dimensional real vector. The quadratic ordered operators, L+ ˆxˆx, L+ ˆp ˆp, and L+ ˆx ˆp, are represented by ...

    work page

  8. [8]

    Multi bosonic mode: Theorem 7 go(n) is the vector space over the real number, R, which has 9 n basis as ad ˆNi , ad ˆXi , ad ˆYi , ad ˆxi , ad ˆpi , L+ ˆxi ˆxi , L+ ˆxi ˆpi , L+ ˆpi ˆpi , L− ˆxi ˆpi , (A16) for 1 ≤ i≤ n acting on single mode, which we dealt with before. Meanwhile, there is another basis acting on bimode, ad ˆNij+ , ad ˆNij− , ad ˆXij , ad...

    work page

Show all 73 references
  1. [9]

    We can decompose exp (go) as exp(go)= exp(h2) exp(h1) exp(g), (F1) where g ∈ g, h1 ∈ h1, and h2 ∈ h2, The definition of h1, h2, and g is equal to Eq

    Evolution of Wigner functions underEG First, we consider elements of GO +(n), which is exp(go), where go∈ go(n). We can decompose exp (go) as exp(go)= exp(h2) exp(h1) exp(g), (F1) where g ∈ g, h1 ∈ h1, and h2 ∈ h2, The definition of h1, h2, and g is equal to Eq. (A20), Eq. (A2...

  2. [10]

    EG ∈ GO+(n) can be expressed as exp (go) and, as discussed in the previous subsection, can be decomposed into exp (h2) exp(h1) exp(g)

    Evolution of EG We will discuss how the elements of GO +(n) evolve under go′. EG ∈ GO+(n) can be expressed as exp (go) and, as discussed in the previous subsection, can be decomposed into exp (h2) exp(h1) exp(g). Each of these can be expressed in terms of D, v, and M = exp(Γ),...

  3. [11]

    Weedbrook, S

    C. Weedbrook, S. Pirandola, R. Garc ´ ıa-Patr´ on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012)

    2012

  4. [12]

    Olivares, Quantum optics in the phase space, The Eu- ropean Physical Journal Special Topics 203, 3 (2012)

    S. Olivares, Quantum optics in the phase space, The Eu- ropean Physical Journal Special Topics 203, 3 (2012)

    2012

  5. [13]

    S. L. Braunstein and P. van Loock, Quantum informa- tion with continuous variables, Rev. Mod. Phys. 77, 513 (2005)

    2005

  6. [14]

    Leonhardt and H

    U. Leonhardt and H. Paul, Measuring the quantum state of light, Progress in Quantum Electronics 19, 89 (1995)

    1995

  7. [15]

    Smith, J

    G. Smith, J. A. Smolin, and J. Yard, Quantum communi- cation with gaussian channels of zero quantum capacity, Nature Photonics 5, 624 (2011)

    2011

  8. [16]

    DiGuglielmo, B

    J. DiGuglielmo, B. Hage, A. Franzen, J. Fiur´ aˇ sek, and R. Schnabel, Experimental characterization of gaussian quantum-communication channels, Phys. Rev. A 76, 012323 (2007)

    2007

  9. [17]

    C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Gaussian boson sampling, Phys. Rev. Lett. 119, 170501 (2017)

    2017

  10. [18]

    Zhong, L.-C

    H.-S. Zhong, L.-C. Peng, Y. Li, Y. Hu, W. Li, J. Qin, D. Wu, W. Zhang, H. Li, L. Zhang, Z. Wang, L. You, X. Jiang, L. Li, N.-L. Liu, J. P. Dowling, C.-Y. Lu, and J.- W. Pan, Experimental gaussian boson sampling, Science Bulletin 64, 511 (2019)

    2019

  11. [19]

    Quesada, J

    N. Quesada, J. M. Arrazola, and N. Killoran, Gaussian boson sampling using threshold detectors, Phys. Rev. A 98, 062322 (2018)

    2018

  12. [20]

    Marshall, C

    K. Marshall, C. S. Jacobsen, C. Sch¨ afermeier, T. Gehring, C. Weedbrook, and U. L. Andersen, Continuous-variable quantum computing on encrypted data, Nature Communications 7, 13795 (2016)

    2016

  13. [21]

    M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, Quantum computing with continuous- variable clusters, Phys. Rev. A 79, 062318 (2009)

    2009

  14. [22]

    Lloyd and S

    S. Lloyd and S. L. Braunstein, Quantum computation over continuous variables, Phys. Rev. Lett. 82, 1784 (1999)

    1999

  15. [23]

    Xu, Quantifying coherence of gaussian states, Phys

    J. Xu, Quantifying coherence of gaussian states, Phys. Rev. A 93, 032111 (2016). 26

    2016

  16. [24]

    Jiang, Quantum fisher information for states in expo- nential form, Phys

    Z. Jiang, Quantum fisher information for states in expo- nential form, Phys. Rev. A 89, 032128 (2014)

    2014

  17. [25]

    E. M. Graefe, B. Longstaff, T. Plastow, and R. Schubert, Lindblad dynamics of gaussian states and their superpo- sitions in the semiclassical limit, Journal of Physics A: Mathematical and Theoretical 51, 365203 (2018)

    2018

  18. [26]

    Eisert and M

    J. Eisert and M. M. Wolf, Gaussian quantum channels (2005), arXiv:quant-ph/0505151 [quant-ph]

    2005 arXiv

  19. [27]

    Brodier and A

    O. Brodier and A. Ozorio de Almeida, Markovian evolu- tion of gaussian states in the semiclassical limit, Physics Letters A 374, 2315 (2010)

    2010

  20. [28]

    K. Noh, V. V. Albert, and L. Jiang, Quantum capacity bounds of gaussian thermal loss channels and achievable rates with gottesman-kitaev-preskill codes, IEEE Trans- actions on Information Theory 65, 2563 (2019)

    2019

  21. [29]

    Prosen, Third quantization: a general method to solve master equations for quadratic open fermi systems, New Journal of Physics 10, 043026 (2008)

    T. Prosen, Third quantization: a general method to solve master equations for quadratic open fermi systems, New Journal of Physics 10, 043026 (2008)

    2008

  22. [30]

    McDonald and A

    A. McDonald and A. A. Clerk, Third quantization of open quantum systems: Dissipative symmetries and con- nections to phase-space and keldysh field-theory formu- lations, Phys. Rev. Res. 5, 033107 (2023)

    2023

  23. [31]

    Barthel and Y

    T. Barthel and Y. Zhang, Solving quasi-free and quadratic lindblad master equations for open fermionic and bosonic systems, Journal of Statistical Mechanics: Theory and Experiment 2022, 113101 (2022)

    2022

  24. [32]

    Pleasance, B

    G. Pleasance, B. M. Garraway, and F. Petruccione, Gen- eralized theory of pseudomodes for exact descriptions of non-markovian quantum processes, Phys. Rev. Res. 2, 043058 (2020)

    2020

  25. [33]

    Tamascelli, A

    D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio, Nonperturbative treatment of non-markovian dynamics of open quantum systems, Phys. Rev. Lett. 120, 030402 (2018)

    2018

  26. [34]

    Gottesman, A

    D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001)

    2001

  27. [35]

    A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Quan- tum computing with rotation-symmetric bosonic codes, Phys. Rev. X 10, 011058 (2020)

    2020

  28. [36]

    Chamberland, K

    C. Chamberland, K. Noh, P. Arrangoiz-Arriola, E. T. Campbell, C. T. Hann, J. Iverson, H. Putterman, T. C. Bohdanowicz, S. T. Flammia, A. Keller, G. Refael, J. Preskill, L. Jiang, A. H. Safavi-Naeini, O. Painter, and F. G. Brand˜ ao, Building a fault-tolerant quantum computer u...

    2022

  29. [37]

    Hastrup and U

    J. Hastrup and U. L. Andersen, All-optical cat-code quantum error correction, Phys. Rev. Res. 4, 043065 (2022)

    2022

  30. [38]

    M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, New class of quantum error-correcting codes for a bosonic mode, Phys. Rev. X 6, 031006 (2016)

    2016

  31. [39]

    S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, Efficient classical simulation of continuous variable quantum information processes, Phys. Rev. Lett. 88, 097904 (2002)

    2002

  32. [40]

    N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, Universal quantum com- putation with continuous-variable cluster states, Phys. Rev. Lett. 97, 110501 (2006)

    2006

  33. [41]

    Walschaers, Non-gaussian quantum states and where to find them, PRX Quantum 2, 030204 (2021)

    M. Walschaers, Non-gaussian quantum states and where to find them, PRX Quantum 2, 030204 (2021)

    2021

  34. [42]

    Ghose and B

    S. Ghose and B. C. S. and, Non-gaussian ancilla states for continuous variable quantum computation via gaus- sian maps, Journal of Modern Optics 54, 855 (2007), https://doi.org/10.1080/09500340601101575

    2007 doi

  35. [43]

    Huang, M

    J. Huang, M. Zhuang, and C. Lee, Non-gaussian preci- sion metrology via driving through quantum phase tran- sitions, Phys. Rev. A 97, 032116 (2018)

    2018

  36. [44]

    J. P. D. and, Quantum optical metrology – the low- down on high-n00n states, Contemporary Physics49, 125 (2008), https://doi.org/10.1080/00107510802091298

    2008 doi

  37. [45]

    G. M. Andolina, V. Stanzione, V. Giovannetti, and M. Polini, Genuine quantum advantage in non-linear bosonic quantum batteries (2024), arXiv:2409.08627 [quant-ph]

    2024 arXiv

  38. [46]

    J. E. Humphreys, Introduction to Lie Algebras and Rep- resentation Theory (Springer New York, 1972)

    1972

  39. [47]

    WEYL, The Classical Groups (Princeton University Press, 1966)

    H. WEYL, The Classical Groups (Princeton University Press, 1966)

    1966

  40. [48]

    Weyl, Quantenmechanik und gruppentheorie, Zeitschrift f¨ ur Physik46, 1 (1927)

    H. Weyl, Quantenmechanik und gruppentheorie, Zeitschrift f¨ ur Physik46, 1 (1927)

    1927

  41. [49]

    von Neumann, Mathematical Foundations of Quantum Mechanics: New Edition (Princeton University Press, 2018)

    J. von Neumann, Mathematical Foundations of Quantum Mechanics: New Edition (Princeton University Press, 2018)

    2018

  42. [50]

    Kraus, General state changes in quantum theory, An- nals of Physics 64, 311 (1971)

    K. Kraus, General state changes in quantum theory, An- nals of Physics 64, 311 (1971)

    1971

  43. [51]

    E. C. G. Sudarshan, P. M. Mathews, and J. Rau, Stochas- tic dynamics of quantum-mechanical systems, Phys. Rev. 121, 920 (1961)

    1961

  44. [52]

    M. A. Nielsen and I. L. Chuang, Quantum Computa- tion and Quantum Information: 10th Anniversary Edi- tion (Cambridge University Press, 2010)

    2010

  45. [53]

    Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

    G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

    1976

  46. [54]

    H. P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford University Press, Great Clarendon Street, 2002)

    2002

  47. [55]

    At the same time, the closed quantum dynamics is also covered, since it can also be rephrased in terms of superoperators

    We choose the name Gaussian open algebra to signify that this algebra is defined in terms of superoperators, which also describe open quantum dynamics. At the same time, the closed quantum dynamics is also covered, since it can also be rephrased in terms of superoperators

  48. [56]

    GO+(1) refers to the clo- sure of a group of superoperators acting on density matri- ces, and must therefore be understood within the topol- ogy of superoperators

    It is important to note that GO+(1) is not equal to R5⊕S GL+(2, R), While GL+(2, R) denotes the topologi- cal closure of the general linear group GL +(2, R) within the topology of real matrices. GO+(1) refers to the clo- sure of a group of superoperators acting on density matr...

  49. [57]

    The following theorem represents a form of the third ther- modynamic law: without the closure, it is not possible to create a pure state from mixed by a finite number of operations

  50. [58]

    Heinosaari, A

    T. Heinosaari, A. S. Holevo, and M. M. Wolf, The semigroup structure of gaussian channels (2009), arXiv:0909.0408 [quant-ph]

    2009 arXiv

  51. [59]

    The inverse element of exp (go) is exp (−go), but it is larger than the Lie group, because some elements on the boundary do not have an inverse element

  52. [60]

    For unitary Gaussian operators, EG(t) ≅ Sp(2, R)⊕S R2, which matches with well known result about the Gaus- sian evolution under the closed dynamics, which are de- 27 scribed by the symplectic group (for the squeezing and rotations) together with the displacement, in the phase...

  53. [61]

    Then we solve Eqs

    This is done by considering the parametrization (28) and defining time-evolved channel EG(t) = exp(go t), meaning that every coefficient is transformed γ → γt. Then we solve Eqs. (49), transform the Wigner function according to Theorem 6, and set t = 1

  54. [62]

    X. Deng, S. Li, Z.-J. Chen, Z. Ni, Y. Cai, J. Mai, L. Zhang, P. Zheng, H. Yu, C.-L. Zou, S. Liu, F. Yan, Y. Xu, and D. Yu, Quantum-enhanced metrology with large fock states, Nature Physics 20, 1874 (2024)

    2024

  55. [63]

    D. S. Schlegel, F. Minganti, and V. Savona, Quantum error correction using squeezed schr¨ odinger cat states, Physical Review A 106, 10.1103/physreva.106.022431 (2022)

    2022 doi

  56. [64]

    Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, American Journal of Mathematics 58, 141 (1936), full publication date: Jan., 1936

    J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, American Journal of Mathematics 58, 141 (1936), full publication date: Jan., 1936

    1936

  57. [65]

    Friedlander and M

    F. Friedlander and M. Joshi, Introduction to the Theory of Distributions (Cambridge University Press, 1999)

    1999

  58. [66]

    Eberhardt, Superconformal symmetry and representa- tions, Journal of Physics A: Mathematical and Theoret- ical 54, 063002 (2021)

    L. Eberhardt, Superconformal symmetry and representa- tions, Journal of Physics A: Mathematical and Theoret- ical 54, 063002 (2021)

    2021

  59. [67]

    Blank, M

    J. Blank, M. Havl ´ ıˇ cek, M. Bedn´ aˇ r, and W. Lassner, Canonical representations of the lie superalgebraosp(1,4), Czechoslovak Journal of Physics B 31, 1286 (1981)

    1981

  60. [68]

    K. E. Cahill and R. J. Glauber, Ordered expansions in boson amplitude operators, Phys. Rev. 177, 1857 (1969)

    1969

  61. [69]

    A. S. Holevo, Quantum Systems, Channels, Information (De Gruyter, Berlin, Boston, 2019)

    2019

  62. [70]

    M. E. Shirokov, Strong convergence of quantum channels: Continuity of the stinespring dilation and discontinuity of the unitary dilation, Journal of Mathematical Physics 61, 082204 (2020)

    2020

  63. [71]

    M. E. Shirokov, Strong*convergence of quantum chan- nels, Quantum Information Processing 20, 145 (2021)

    2021

  64. [72]

    Adesso and A

    G. Adesso and A. Datta, Quantum versus classical corre- lations in gaussian states, Phys. Rev. Lett. 105, 030501 (2010)

    2010

  65. [73]

    ˇSafr´ anek and I

    D. ˇSafr´ anek and I. Fuentes, Optimal probe states for the estimation of gaussian unitary channels, Phys. Rev. A 94, 062313 (2016)

    2016

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