Configurable photonic simulator for quantum field dynamics

Martin G\"arttner; Mauro D'Achille; Tobias Haas

REVIEW 3 major objections 3 minor 174 references

Configurable photonic simulator for quantum field dynamics

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A single fixed optical circuit, with time isolated in one phase-shift layer, can simulate the dynamics of many free quantum field theories, from relativistic scalars to long-range and curved-spacetime models.

desk verdict Static OTA is a sound new decomposition with a nice GBS hook; the time-dependent extension and cosmology claims are built on a false commutator argument and need to be cut or fixed. read the letter →

arxiv 2506.23838 v3 pith:I2CRSKQX submitted 2025-06-30 quant-ph hep-th

Mauro D'Achille , Martin Gärttner , Tobias Haas This is my paper · ORCID

classification quant-phhep-th

keywords OpticalTimeAlgorithmquantumfieldsimulationphotoniccircuitsGaussianbosonsamplingsymplecticdecompositionentanglementdynamicslong-rangetheoriescurved-spacetimefields

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 introduces the Optical Time Algorithm (OTA), a decomposition of quadratic scalar-field time evolution into a fixed optical circuit whose only time-dependent part is a single layer of phase shifters. The central claim is that for any quadratic Hamiltonian whose position and momentum blocks commute, the symplectic evolution operator can be written as $S(t)=S_I S_{Sq}^{-1}(z) S_{PS}(\varphi(t)) S_{Sq}(z) S_I^{-1}$, with the Hamiltonian's eigenvectors and symplectic eigenvalues setting fixed interferometer and squeezer parameters, and the phase shifts encoding all time dependence. If this holds, the same hardware could be reconfigured by parameter changes alone to simulate a wide landscape of free field theories: relativistic and non-relativistic, real and complex, short- and long-range, on flat and curved spacetimes, including cosmological and black-hole-like metrics. The paper benchmarks the scheme on quench dynamics and correlation spreading, reporting that 10 to 20 modes reproduce field-theoretic predictions for entanglement entropy and mutual information, and that for coherent-vacuum inputs the OTA reduces to the Gaussian boson sampling setup.

What carries the argument

The key machinery is symplectic diagonalization: Williamson's theorem decomposes the Hamiltonian matrix as $H = S_{SD}^\top D S_{SD}$, where $D$ contains the symplectic eigenvalues $d_j$. The OTA then uses a symplectic polar decomposition of $S_{SD}$ into a squeezer $S_{Sq}(z)$ and an orthosymplectic passive part $P$, with the orthosymplectic gauge freedom chosen so that the passive part becomes the eigenvector matrix $P_N \oplus P_N$ and the squeezer parameters $z_j$ encode the eigenvalue ratios. This separates the Hamiltonian's structure (fixed $S_I$ and $z$) from time evolution (a single phase-shift layer $S_{PS}(\varphi(t))$), yielding Eq. (15). The same identity carries the time-dependent extension and the reduction to Gaussian boson sampling.

What would settle it

Take $H(t) = \tfrac12(\lambda_\phi(t)\, q^2 + \lambda_\pi(t)\, p^2)$ with $\lambda_\phi(t) = t$ and $\lambda_\pi(t) = 1$. The eigenvector matrix is constant, yet $[H(t_1), H(t_2)] \neq 0$ whenever $\lambda_\phi/\lambda_\pi$ changes. Solving the corresponding time-dependent oscillator exactly and comparing the resulting covariance matrix with the OTA expression built from time-integrated eigenvalues would directly show whether Eq. (A21) holds or fails.

Watch

Extended reading notes

Core claim

The central discovery is a decomposition of the symplectic time-evolution matrix for free scalar field theories, stated as Eq. (15). Let $H = H_\phi \oplus H_\pi$ be a quadratic Hamiltonian matrix with commuting position and momentum blocks. Then there exists a fixed passive interferometer $S_I$, built from the orthonormal eigenvectors of $H_\phi$, and a fixed single-mode squeezer layer $S_{Sq}(z)$ with $z_j = -\tfrac12 \ln(\lambda_{\phi,j}/\lambda_{\pi,j})$, such that full time evolution under $H$ equals $S_I S_{Sq}^{-1}(z) S_{PS}(\varphi(t)) S_{Sq}(z) S_I^{-1}$, where $\varphi_j(t) = d_j t$ and $d_j = \sqrt{\lambda_{\phi,j}\lambda_{\pi,j}}$. All time dependence is confined to the phase-shift layer, while the circuit geometry remains identical across theories, with only parameter values changing. The paper further shows that for vacuum or coherent input states the first interferometer drops away and the remaining squeezer-plus-phase part becomes complex squeezers, so the OTA reduces to the Gaussian boson sampling layout. For time-dependent Hamiltonians with time-independent eigenvectors, the same structural form persists with time-integrated eigenvalues entering the phases and squeezing parameters.

Load-bearing premise

The time-dependent extension of the OTA assumes that if the eigenvectors of the Hamiltonian matrix are time-independent, the time-ordering operator can be dropped from the evolution, so that $U(t) = \exp(-i\int_0^t dt'\, H(t'))$ without additional ordering corrections. This premise can fail: a Hamiltonian with constant eigenvectors can still have noncommuting values at different times when the ratio of the position and momentum eigenvalue functions changes, so the step leading to Eq. (A21) is not generally valid.

Editorial extensions

If this is right

One reconfigurable photonic circuit can simulate several classes of free scalar field theories by changing only optical parameters, without redesigning the experimental setup for each theory or time step.
For vacuum or coherent inputs, the OTA becomes a layer of complex squeezers followed by a single time-independent interferometer, exactly the Gaussian boson sampling structure, so existing boson-sampling hardware could be repurposed for quantum field simulation.
The particle-number output distribution of the reduced OTA circuit contains Hafnians and is classically hard to sample in general, suggesting a route to practical quantum advantage in field-theoretic simulation.
Simulations with 10 to 20 modes reproduce quasi-particle predictions for entanglement entropy and mutual information after quenches, including linear growth, revivals, and the onset of correlations, and remain qualitatively robust to photon loss and preparation noise.
Varying the coupling range in fractional Laplacian theories bends the effective light cones from linear to algebraic to logarithmic to constant, matching predictions for long-range interacting systems.
Curved spacetimes, including FLRW cosmology with arbitrary scale factors, Rindler horizons, and black-hole-like metrics in tortoise coordinates, can be encoded in the same OTA layout through time-dependent phases and squeezers.

Reading between the lines

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

If the OTA is correct, the practical bottleneck shifts from circuit design to state preparation and loss, because all theory-specific complexity is compressed into parameters of fixed optical layers.
The time-dependent extension rests on dropping the time-ordering operator when eigenvectors are time-independent; verifying the OTA prediction against an exact solution of a driven harmonic oscillator with constant eigenvectors would settle whether this extension is valid.
The reduction to Gaussian boson sampling suggests that hardness arguments for sampling also apply to simulating field-theoretic correlation functions, but only in parameter regimes where classical algorithms for the lossy output distribution remain inefficient.
The observed light-cone bending classification could serve as an experimental signature for identifying effective coupling range in a photonic simulator by measuring the onset times of mutual information between distant modes.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things you should know. The static OTA, Eq. (15), is a sound new decomposition: symplectic time evolution for quadratic Hamiltonians with commuting phi and pi blocks is written as S(t)=SI S^{-1}_Sq(z) S_PS(phi(t)) S_Sq(z) S^{-1}_I, with all time dependence isolated in one phase-shift layer. For vacuum inputs the circuit collapses to a single squeezing layer plus one interferometer, exactly the Gaussian boson sampling layout. That is a real observation, and the authors back it with a clean proof in App. A2 and public code. The benchmarks against quasi-particle/GGE predictions in Sec. V are internally consistent, use independent analytic formulas, and show the expected light-cone structure and revivals at N~10-20. The loss and noise analysis is honest and useful.

Now the soft spot, and it is load-bearing: the time-dependent extension in App. A3 is wrong. Time-independent eigenvectors do not imply commuting Hamiltonians. The counterexample H(t) = (1/2)(lambda_phi(t) q^2 + lambda_pi(t) p^2) has constant eigenvectors but [H(t1),H(t2)] proportional to [q^2,p^2], nonzero whenever lambda_phi/lambda_pi changes. So Eq. (A21) does not follow, and Eq. (A23) generally does not equal the true time-ordered evolution. That kills the FLRW/cosmological simulation claim in Sec. IV E1 and the associated Table I, unless the authors supply a genuine Trotterized error bound or a different argument. The static OTA and the GBS connection are unaffected, as are the quench benchmarks.

Other notes: the paper is well written and the literature coverage is good. The complexity discussion for GBS is standard. The gauging and interferometer decomposition in App. A2c is a nice practical detail. The claim about arbitrary spacetimes is overstated beyond 1+1 dimensions or conformally flat coordinate patches; that is a minor framing issue.

Verdict: this deserves a serious referee. The static result is a legitimately new circuit decomposition with an experimental hook, and the benchmarks are solid. The time-dependent claims need to be fixed or removed. I would accept for review with the clear expectation of major revision on Sec. IIIA3 and Sec. IV E1. I would cite the static decomposition in work on CV simulation; I would not cite the time-dependent part until it is repaired.

Referee Report

3 major / 3 minor

Summary. The paper introduces the Optical Time Algorithm (OTA), a decomposition of the symplectic time-evolution matrix for free quadratic scalar field theories into a fixed interferometer-squeezer network with all time dependence isolated in a single phase-shifter layer. The static version, Eq. (15), is derived in App. A2 and is used to benchmark quench dynamics of entanglement entropy and mutual information against quasi-particle and GGE predictions, including a study of light-cone shapes for fractional-Laplacian theories and a detailed error analysis. The paper also claims a time-dependent generalization, Eqs. (21) and (A23), which is then applied to FLRW cosmology and other curved-spacetime settings in Sec. IV E.

Significance. If the static OTA is correct, it provides a clean, parameter-free design principle for programmable photonic quantum field simulators: a single circuit topology with Hamiltonian-dependent but time-independent interferometer and squeezer layers. The paper's strengths include a transparent algebraic derivation of the static decomposition, public code for computing circuit parameters, and benchmarks computed from exact covariance-matrix evolution against independent analytic predictions. However, the time-dependent extension in App. A3 is invalid, and the cosmological simulation claims in Sec. IV E1 rest on that invalid step. The static results and the numerical benchmarks in Secs. V and VI use time-independent Hamiltonians and are not affected by this error.

major comments (3)

[Appendix A3, Eq. (A21)] The claim that time-independent eigenvectors P≠P(t) imply [H(t1),H(t2)]=0 and hence allow the time-ordering operator to be dropped is false for the Hamiltonian operators actually being evolved. For N=1 with H(t)=1/2(λ_φ(t)q^2+λ_π(t)p^2), the eigenvector matrix of the quadratic-form matrix H(t)=diag(λ_φ(t),λ_π(t)) is the constant identity, yet [H(t1),H(t2)] = (1/4)(λ_φ(t1)λ_π(t2)-λ_π(t1)λ_φ(t2))(q^2p^2-p^2q^2), which is nonzero unless λ_φ/λ_π is constant. The condition needed to drop T is commutation of the symplectic generators ΩH(t), not constancy of the eigenvector matrix of H(t). Therefore Eq. (A21) does not follow.
[Appendix A3, Eq. (A23)] The claimed time-dependent OTA circuit is not merely unproved; it is generally incorrect. For a single mode, the right-hand side of Eq. (A23) takes the form S_Sq^{-1}(z(t))S_PS(φ(t))S_Sq(z(t)), whose diagonal entries are both equal to cos φ(t). The exact time-ordered symplectic evolution generated by ΩH(t) has unequal diagonal entries already at second order in time: u=1-∫_0^t b(s1)∫_0^{s1} a(s2) ds2 ds1 and x=1-∫_0^t a(s1)∫_0^{s1} b(s2) ds2 ds1, with a=λ_φ, b=λ_π. These are equal only when λ_φ/λ_π is constant. Hence no schedule φ(t), z(t) of the form (21) reproduces the target dynamics for a genuinely time-dependent ratio of the eigenvalue blocks.
[Sec. IV E1 and Table I] Because the time-dependent OTA is invalid, the FLRW/cosmological simulation claims, including the scale-factor examples in Table I, are unsupported. The passage in Sec. VII acknowledges complications for general time dependencies but explicitly preserves the time-independent-diagonalization case as valid; that is precisely the case invalidated by the counterexample above. The static OTA, Eq. (15), and the time-independent benchmarks in Secs. V-VI remain unaffected by this issue.

minor comments (3)

[Eq. (53)] The notation S2(t, ℓ∪ℓ) is ambiguous; the combined region should be written as a function of both intervals and their separation d, e.g., S2(t, [0,ℓ]∪[ℓ+d,2ℓ+d]), to match the definition in the text.
[Appendix A2] The symbol P is used both for the orthosymplectic polar factor in Eq. (A10) and for the eigenvector matrix P_N in Eqs. (A13) and (A18); this collision makes the derivation harder to follow and should be fixed.
[Sec. III A3] The sentence 'When [H(t1),H(t2)]=0 ∀t1,t2, which amounts to time-independent eigenvectors P≠P(t)' is imprecise even as a statement about matrices, because commutation also requires the ϕ- and π-blocks to share the same eigenbasis; the paper should state this assumption explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the static OTA derivation is self-contained linear algebra and the benchmarks are independent; the App. A3 time-dependent step is an unsupported mathematical claim, not a circular reduction.

full rationale

The central derivation is self-contained: Eq. (14) follows from Williamson's theorem in App. A1, and Eq. (15) follows by substituting the polar decomposition S_SD = S_Sq(z) S_I^{-1} in App. A2 (Eqs. A10-A13); no parameter is fitted to the phenomena being predicted, and the circuit parameters in Eqs. (16a)-(16c) are computed directly from the Hamiltonian matrix. The benchmarks in Secs. V-VI compare Rényi-2 entropies and mutual information against independent quasi-particle/GGE predictions (Eqs. 52, 54, D1-D3), and the light-cone regimes in Fig. 5 are checked against published spin-system results [138,139], not against quantities generated by the same circuit parameters. Self-citations such as [9,112,113,126-128] are contextual remarks, not load-bearing supports of the central claim. The genuine flaw is the time-dependent step: Sec. IIIA3 and App. A3 assert, respectively, 'When [H(t1), H(t2)] = 0 ∀t1,t2, which amounts to time-independent eigenvectors P ≠ P(t), time-ordering simplifies to the identity T = 1' and 'for time-independent eigenvectors P ≠ P(t), which ensures [H(t1), H(t2)] = 0'. This implication is false, e.g., for H(t)=1/2(λ_φ(t)q^2+λ_π(t)p^2) the eigenvector matrix is constant yet [H(t1),H(t2)]≠0 unless λ_φ/λ_π is constant, so Eq. (A21), Eq. (A23), and the cosmological application in Sec. IV E1 are unsupported. That is a validity error, not a circular reduction: the claimed output is neither equivalent to an input by construction nor a fitted parameter renamed as a prediction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central static claim rests on standard results (Williamson, polar decomposition, Givens and Reck-Clements decompositions) plus the domain assumption that physically relevant scalar Hamiltonians decompose into commuting phi/pi blocks. The time-dependent curved-spacetime claim additionally rests on a false commutativity assumption in App. A3. No free parameters are fitted and no new entities are introduced.

assumptions (6)

standard math Williamson's theorem: any positive semidefinite symmetric matrix can be symplectically diagonalized.
Used in Sec. III A1 and App. A1 to write H = S_SD^T (D_N xor D_N) S_SD, the basis of Eq. (14).
standard math Symplectic polar decomposition: any symplectic matrix can be written as a symmetric positive part times an orthosymplectic part.
Used in App. A2b to factor S_SD into a squeezer and an interferometer after a gauge choice.
domain assumption The Hamiltonian matrix of scalar field theories of interest decomposes into commuting phi and pi blocks, H = H^phi xor H^pi with [H^phi,H^pi]=0.
Stated at the start of Sec. III A2 as applying to the vast majority of physically relevant scalar theories; restricts the OTA to this class.
standard math All two-dimensional Lorentzian manifolds are conformally flat, so the curved spacetime metric can be written as g = f(x) eta.
Used in Sec. IV E to reduce the curved spacetime action to a flat kinetic term with position-dependent mass.
ad hoc to paper For the time-dependent OTA, time-independent eigenvectors of the Hamiltonian matrix imply the time-ordering operator is the identity.
Invoked in Sec. III A3 and App. A3; false for quadratic Hamiltonians with time-varying eigenvalue ratios, since the corresponding operators do not commute. This invalidates the exact form of the time-dependent OTA.
domain assumption The quasi-particle picture and generalized Gibbs ensemble prediction for entanglement growth after a quench in free integrable field theories.
Used as the analytic benchmark in Sec. V B and App. D; standard in the quantum quench literature [64-66,132].

how reviews work

0 comments

Cite this review

Pith. "Pith review of Configurable photonic simulator for quantum field dynamics." pith.science (2026). https://pith.science/paper/I2CRSKQX

@misc{pith2026250623838,
  author       = {Pith},
  title        = {Pith review of: Configurable photonic simulator for quantum field dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I2CRSKQX}},
  note         = {Machine review of arXiv:2506.23838}
}

abstract

Quantum field simulators provide unique opportunities for investigating the dynamics of quantum fields through tabletop experiments. A primary drawback of standard encoding schemes is their rigidity: altering the theory, its coupling geometry, metric structure, or simulation time typically requires redesigning the experimental setup, which imposes strong constraints on the types of dynamics and theories that can be simulated. Here, we introduce the Optical Time Algorithm (OTA) as a unifying framework, enabling the efficient simulation of large classes of free quantum field dynamics using a single optical circuit design that separates the time from the Hamiltonian's structure. By modifying the parameters of the optical elements, our method allows us to engineer timescales, coupling graphs, spacetime metrics, and boundary conditions, thereby facilitating the implementation of relativistic and non-relativistic, real- and complex-valued, short- and long-range quantum field theories on both flat and curved spacetimes. We exploit the OTA's configurability to investigate the spreading of quantum correlations in space and time for theories with continuously varying coupling ranges. Relevant features predicted by quantum field theory can be observed in systems with $10$ to $20$ modes under realistic conditions, paving the way for experimental implementations.

Figures

Figures reproduced from arXiv: 2506.23838 by the authors.

**Figure 2.** Pictorial representation of the OTA for N = 5. The first and last layers can be further decomposed via standard techniques, such as Reck’s [56] and Clements’ decompositions [57]. We provide an explicit example for the former in App. A 2 c. splitters only. There, we also show how gauge redundancies can be used to minimize the number of gates. Given a Hamiltonian matrix H, all circuit parameters can be calculated us… view at source ↗

**Figure 3.** For vacuum inputs, the OTA reduces to a complex [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: (a) Time evolution of the Rényi-2 entanglement entropy (51) (scaled by 102 ) of the subregion [0, ℓ] following a quench from the optical vacuum (49) to the relativistic theory (31) with ℓ = L/5, m = 1, ϵ = 2 for varying L = N ϵ against the field-theoretic prediction (5…

**Figure 5.** Figure 5: Spatiotemporal evolution of the Rényi-2 mutual information [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: Effects of state preparation noise for the relativistic [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: Effects of a lossy interferometer for the relativistic [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: Pictorial representation of the OTA for N = 5. The interferometer layers are decomposed here according to a Reck-inspired method. When applying the Givens rotation implicitly defined via the latter equations to PN from the right PN → PN Ojk(θ), the corresponding elemen…

**Figure 9.** Figure 9: Boundary effects relevant for identifying the finite-size Rényi-2 mutual information. We trace all trajectories of two [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]

**Figure 10.** Figure 10: Same analysis as in Fig [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

174 extracted references · 64 canonical work pages

[1]

Entanglement generation We consider a spatial subregion[0, ℓ] with ℓ ≤ L, see inset of Fig. 4(a). As the modes are coupled, the degrees of freedom in [0, ℓ] become entangled with the ones in the complementary region(ℓ, L]. Thus, while the global state ρ(t) remains pure during the unitary evolution, the local state ρ(t, l) = T rL−ℓ{ρ(t)} follows the evolut...
[2]

Correlation spreading We now study how two equally sized spatial regions of length ℓ separated by a distanced >0 become correlated after the quench, as shown in the inset of Fig. 4(b). To this end, we consider the Gaussian Rényi-2mutual information I2(t, ℓ, d) = 2S2(t, ℓ) − S2(t, ℓ∪ ℓ) = 1 2 ln det σ2(t, ℓ) det σ(t, ℓ∪ ℓ) , (53) where S2(t, ℓ∪ ℓ) and σ(t,...
[3]

(14) The isolation of the time parameter into single-mode phase shifters is achieved by generalizing Williamson’s theorem for the Hamiltonian matrixH = S⊺ SDDSSD [see Eq

Proof of Eq. (14) The isolation of the time parameter into single-mode phase shifters is achieved by generalizing Williamson’s theorem for the Hamiltonian matrixH = S⊺ SDDSSD [see Eq. (8)] to the symplectic matrixS(t) describing time evolution. After introducing the corresponding unitary transformation USD defined via SSDr = U † SDrUSD, the Hamiltonian be...
[4]

Eigensystems of H ϕ and H π For the OTA, we consider the class of block diagonal Hamiltonians with commuting blocksH ϕ and H π

Proof of the OTA a. Eigensystems of H ϕ and H π For the OTA, we consider the class of block diagonal Hamiltonians with commuting blocksH ϕ and H π. Given Eq. (A3), our goal is to further decompose the SD matrix by means of optical transformations. We make the ansatz SSD = ON AN BN ON , (A4) with AN and BN being further constrained asSSD must diagonalize t...
[5]

Starting once again from Williamson’s theorem(8), i.e., H(t) = S⊺ SD(t)D(t)SSD(t), and using the polar decomposition SSD = SSq(t)P [see Eqs

Time-dependent case We generalize the OTA to time-dependent Hamiltonian matrices H = H(t) for time-independent eigenvectors P ̸= P (t), which ensures [H(t1), H(t2)] = 0. Starting once again from Williamson’s theorem(8), i.e., H(t) = S⊺ SD(t)D(t)SSD(t), and using the polar decomposition SSD = SSq(t)P [see Eqs. (A10) and (A12)] results in H(t) = 1 2 UIr⊺S⊺ ...
[6]

IIIB2) in the more general setting of coherent inputs by proving(25) and a generalization of (26)

Coherent inputs Here, we show that the OTA dramatically simplifies (as discussed in Sec. IIIB2) in the more general setting of coherent inputs by proving(25) and a generalization of (26). We define coherent states as displaced vacuum states |α⟩ = D(α) |0⟩, where D(α) = eαa†−α∗a with a = ( a1, . . . ,aN )⊺ (analogously for the vector of com- plex phases α ...
[7]

We discuss the two most prominent choices: periodic and open boundaries

Finite size effects In a finite volumeL <∞, quasi-particles have discrete momenta and their dynamics are influenced by boundary conditions. We discuss the two most prominent choices: periodic and open boundaries. a. Periodic boundary conditions (PBCs) For periodic boundaries, quasi-particles reenter from the left (right) when traveling to the right (left)...
[8]

To computenk, we first determine the eigenmodes of the relativistic theory(31)

Entropy density of the GGE and mode populations The entropy densitys2,k follows from computing the Rényi-2 entanglement entropy(51) of the GGEρ(∞, l) ∝ exp(− P k yknk) in the eigenbasis of the quasi-particle number operator, leading to [65, 132] S2(∞, ℓ) = S2,GGE = X k∈K s2,k, (D7) with coefficients s2,k = ln n2 k − (1 − nk)2 , (D8) where nk = ⟨0|nk|0⟩ sp...

Show all 174 references

[9]

Formally, the corresponding pre-quench Hamiltonian(49) attains the form H0 = 1 2 Z dx h π2 + (lim ϵ→0 ϵ−2)ϕ2 i (D14) in the continuum

Comment on divergences in the continuum limit The input state chosen in the main text, which is experi- mentally motivated, leads to divergences in the continuum. Formally, the corresponding pre-quench Hamiltonian(49) attains the form H0 = 1 2 Z dx h π2 + (lim ϵ→0 ϵ−2)ϕ2 i (D1...
[10]

W. G. Unruh, Experimental Black-Hole Evaporation?, Phys. Rev. Lett.46, 1351 (1981)

1981
[11]

R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys.21, 467 (1982)

1982
[12]

I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys.86, 153 (2014)

2014
[13]

Altman, K

E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Dem- ler, C. Chin, B. DeMarco, S. E. Economou, M. A. Eriks- son, K.-M. C. Fu,et al., Quantum Simulators: Archi- tectures and Opportunities, PRX Quantum2, 017003 (2021)

2021
[14]

C. W. Bauer, Z. Davoudi, A. B. Balantekin, T. Bhat- tacharya, M. Carena, W. A. de Jong, P. Draper, A. El- Khadra, N. Gemelke, M. Hanada,et al., Quantum Simu- lation for High-Energy Physics, PRX Quantum4, 027001 (2023)

2023
[15]

J. C. Halimeh, N. Mueller, J. Knolle, Z. Papić, and Z. Davoudi, Quantum simulation of out-of-equilibrium dynamics in gauge theories, arXiv:2509.03586 (2025)

2025 arXiv
[16]

Bloch, J

I. Bloch, J. Dalibard, and S. Nascimbène, Quantum simulations with ultracold quantum gases, Nat. Phys.8, 267 (2012)

2012
[17]

Periwal, E

A. Periwal, E. S. Cooper, P. Kunkel, J. F. Wienand, E. J. Davis, and M. Schleier-Smith, Programmable in- teractions and emergent geometry in an array of atom clouds, Nature600, 630 (2021)

2021
[18]

Viermann, M

C. Viermann, M. Sparn, N. Liebster, M. Hans, E. Kath, A. Parra-López, M. Tolosa-Simeón, N. Sánchez-Kuntz, T. Haas, H. Strobel, S. Floerchinger, and M. K. Oberthaler, Quantum field simulator for dynamics in 23 curved spacetime, Nature611, 260 (2022)

2022
[19]

Tajik, M

M. Tajik, M. Gluza, N. Sebe, P. Schüttelkopf, F. Catal- dini, J. Sabino, F. Møller, S.-C. Ji, S. Erne, G. Guarnieri, S. Sotiriadis, J. Eisert, and J. Schmiedmayer, Experi- mental observation of curved light-cones in a quantum field simulator, Proc. Natl. Acad. Sci.120, e230128...

2023
[20]

Blatt and C

R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. Phys.8, 277 (2012)

2012
[21]

Brydges, A

T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Probing Rényi entanglement entropy via random- ized measurements, Science364, 260–263 (2019)

2019
[22]

Monroe, W

C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys.93, 025001 (2021)

2021
[23]

Browaeys and T

A. Browaeys and T. Lahaye, Many-body physics with individually controlled Rydberg atoms, Nat. Phys.16, 132 (2020)

2020
[24]

Bluvstein, A

D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Se- meghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho,et al., Controlling quantum many-body dynamics in driven Rydberg atom arrays, Science 371, 1355 (2021)

2021
[25]

González-Cuadra, M

D. González-Cuadra, M. Hamdan, T. V. Zache, B. Braverman, M. Kornjača, A. Lukin, S. H. Cantú, F. Liu, S.-T. Wang, A. Keesling, M. D. Lukin, P. Zoller, and A. Bylinskii, Observation of string breaking on a (2 + 1)D Rydberg quantum simulator, Nature642, 321 (2025)

2025
[26]

Aspuru-Guzik and P

A. Aspuru-Guzik and P. Walther, Photonic quantum simulators, Nat. Phys.8, 285 (2012)

2012
[27]

Noh and D

C. Noh and D. G. Angelakis, Quantum simulations and many-body physics with light, Rep. Prog. Phys. 80, 016401 (2016)

2016
[28]

F. H. B. Somhorst, R. van der Meer, M. Correa Anguita, R. Schadow, H. J. Snijders, M. de Goede, B. Kassen- berg, P. Venderbosch, C. Taballione, J. P. Epping,et al., Quantum simulation of thermodynamics in an integrated quantum photonic processor, Nat. Commun.14, 3895 (2023)

2023
[29]

Knill, R

E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature 409, 46 (2001)

2001
[30]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear optical quantum computing with photonic qubits, Rev. Mod. Phys.79, 135 (2007)

2007
[31]

Romero and G

J. Romero and G. Milburn, Photonic Quantum Comput- ing (2025)

2025
[32]

Aaronson and A

S. Aaronson and A. Arkhipov, The Computational Com- plexity of Linear Optics, Theo. Comp.9, 143 (2013)

2013
[33]

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

2017
[34]

Kruse, C

R. Kruse, C. S. Hamilton, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Detailed study of Gaussian boson sampling, Phys. Rev. A100, 032326 (2019)

2019
[35]

Hangleiter and J

D. Hangleiter and J. Eisert, Computational advantage of quantum random sampling, Rev. Mod. Phys.95, 035001 (2023)

2023
[36]

Zhong, H

H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.- C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, et al., Quantum computational advantage using photons, Science 370, 1460 (2020)

2020
[37]

Zhong, Y.-H

H.-S. Zhong, Y.-H. Deng, J. Qin, H. Wang, M.-C. Chen, L.-C. Peng, Y.-H. Luo, D. Wu, S.-Q. Gong, H. Su,et al., Phase-Programmable Gaussian Boson Sampling Using Stimulated Squeezed Light, Phys. Rev. Lett.127, 180502 (2021)

2021
[38]

Deng, Y.-C

Y.-H. Deng, Y.-C. Gu, H.-L. Liu, S.-Q. Gong, H. Su, Z.- J. Zhang, H.-Y. Tang, M.-H. Jia, J.-M. Xu, M.-C. Chen, et al., Gaussian Boson Sampling with Pseudo-Photon- Number-Resolving Detectors and Quantum Computa- tional Advantage, Phys. Rev. Lett.131, 150601 (2023)

2023
[39]

H.-L. Liu, H. Su, S.-Q. Gong, Y.-C. Gu, H.-Y. Tang, M.-H. Jia, Q. Wei, Y. Song, D. Wang, M. Zheng, et al., Robust quantum computational advantage with programmable 3050-photon Gaussian boson sampling, arXiv:2508.09092 (2025)

2025 arXiv
[40]

L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins,et al., Quantum computational advantage with a programmable photonic processor, Na- ture 606, 75 (2022)

2022
[41]

Stefszky, K.-H

M. Stefszky, K.-H. Luo, J.-L. Eickmann, S. Atzeni, F. Lütkewitte, C. Pandey, F. Schlue, J. Lammers, M. Roiz, T. Schapeler,et al., Benchmarking Gaussian and non-Gaussian input states with a hybrid sampling platform, arXiv:2512.08433 (2025)

2025
[42]

C. Oh, Y. Lim, Y. Wong, B. Fefferman, and L. Jiang, Quantum-inspired classical algorithms for molecular vi- bronic spectra, Nat. Phys.20, 225–231 (2024)

2024
[43]

C. Oh, M. Liu, Y. Alexeev, B. Fefferman, and L. Jiang, Classical algorithm for simulating experimental Gaussian boson sampling, Nat. Phys.20, 1461–1468 (2024)

2024
[44]

Pelucchi, G

E. Pelucchi, G. Fagas, I. Aharonovich, D. Englund, E. Figueroa, Q. Gong, H. Hannes, J. Liu, C.-Y. Lu, N. Matsuda, et al., The potential and global outlook of integrated photonics for quantum technologies, Nat. Rev. Phys.4, 194 (2022)

2022
[45]

H. Wang, T. C. Ralph, J. J. Renema, C.-Y. Lu, and J.-W. Pan, Scalable photonic quantum technologies, Nat. Rev. Phys.24, 1883 (2025)

2025
[46]

Aghaee Rad, T

H. Aghaee Rad, T. Ainsworth, R. N. Alexander, B. Al- tieri, M. F. Askarani, R. Baby, L. Banchi, B. Q. Bara- giola, J. E. Bourassa, R. S. Chadwick,et al., Scaling and networking a modular photonic quantum computer, Nature 638, 912 (2025)

2025
[47]

PsiQuantum, A manufacturable platform for photonic quantum computing, Nature641, 876 (2025)

2025
[48]

S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum Algorithms for Quantum Field Theories, Science336, 1130 (2012)

2012
[49]

S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum computation of scattering in scalar quantum field theo- ries, Quantum Info. Comput.14, 1014–1080 (2014)

2014
[50]

Klco and M

N. Klco and M. J. Savage, Digitization of scalar fields for quantum computing, Phys. Rev. A99, 052335 (2019)

2019
[51]

Barceló, S

C. Barceló, S. Liberati, and M. Visser, Analogue Gravity, Living Rev. Relativ.14, 3 (2011)

2011
[52]

L.Braunstein, M.Faizal, L.M

S. L.Braunstein, M.Faizal, L.M. Krauss, F. Marino,and N. A. Shah, Analogue simulations of quantum gravity with fluids, Nat. Rev. Phys.5, 612 (2023). 24

2023
[53]

Eckel, A

S. Eckel, A. Kumar, T. Jacobson, I. B. Spielman, and G. K. Campbell, A Rapidly Expanding Bose-Einstein Condensate: An Expanding Universe in the Lab, Phys. Rev. X8, 021021 (2018)

2018
[54]

Steinhauer, Observation of self-amplifying Hawking radiation in an analogue black-hole laser, Nat

J. Steinhauer, Observation of self-amplifying Hawking radiation in an analogue black-hole laser, Nat. Phys.10, 864–869 (2014)

2014
[55]

J. R. Muñoz de Nova, K. Golubkov, V. I. Kolobov, and J. Steinhauer, Observation of thermal Hawking radiation and its temperature in an analogue black hole, Nature 569, 688–691 (2019)

2019
[56]

J. Hu, L. Feng, Z. Zhang, and C. Chin, Quantum simu- lation of Unruh radiation, Nat. Phys.15, 785 (2019)

2019
[57]

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, Fiber-Optical Analog of the Event Horizon, Science319, 1367–1370 (2008)

2008
[58]

Steinhauer, M

J. Steinhauer, M. Abuzarli, T. Aladjidi, T. Bienaimé, C. Piekarski, W. Liu, E. Giacobino, A. Bramati, and Q. Glorieux, Analogue cosmological particle creation in an ultracold quantum fluid of light, Nat. Commun.13, 2890 (2022)

2022
[59]

Švančara, P

P. Švančara, P. Smaniotto, L. Solidoro, J. F. MacDonald, S. Patrick, R. Gregory, C. F. Barenghi, and S. Weinfurt- ner, Rotating curved spacetime signatures from a giant quantum vortex, Nature628, 66 (2024)

2024
[60]

Marshall, R

K. Marshall, R. Pooser, G. Siopsis, and C. Weedbrook, Quantum simulation of quantum field theory using con- tinuous variables, Phys. Rev. A92, 063825 (2015)

2015
[61]

Kalajdzievski, C

T. Kalajdzievski, C. Weedbrook, and P. Rebentrost, Continuous-variable gate decomposition for the Bose- Hubbard model, Phys. Rev. A97, 062311 (2018)

2018
[62]

S. Abel, M. Spannowsky, and S. Williams, Simulating quantum field theories on continuous-variable quantum computers, Phys. Rev. A110, 012607 (2024)

2024
[63]

Houde, W

M. Houde, W. McCutcheon, and N. Quesada, Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson, Can. J. Phys. 102, 497–507 (2024)

2024
[64]

Bloch and A

C. Bloch and A. Messiah, The canonical form of an antisymmetric tensor and its application to the theory of superconductivity, Nucl. Phys.39, 95 (1962)

1962
[65]

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Experimental realization of any discrete unitary operator, Phys. Rev. Lett.73, 58 (1994)

1994
[66]

W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, Optimal design for uni- versal multiport interferometers, Optica3, 1460 (2016)

2016
[67]

Taballione, M

C. Taballione, M. C. Anguita, M. de Goede, P. Vender- bosch, B. Kassenberg, H. Snijders, N. Kannan, W. L. Vleeshouwers, D. Smith, J. P. Epping, R. van der Meer, P. W. H. Pinkse, H. van den Vlekkert, and J. J. Ren- ema, 20-Mode Universal Quantum Photonic Processor, Quantum 7, 1...

2023
[68]

Fyrillas, O

A. Fyrillas, O. Faure, N. Maring, J. Senellart, and N. Be- labas, Scalable machine learning-assisted clear-box char- acterization for optimally controlled photonic circuits, Optica 11, 427 (2024)

2024
[69]

M. B. Plenio and S. Virmani, An Introduction to En- tanglement Measures, Quantum Inf. Comput.7, 1–51 (2007)

2007
[70]

Adesso, D

G. Adesso, D. Girolami, and A. Serafini, Measuring Gaussian Quantum Information and Correlations Using the Rényi Entropy of Order 2, Phys. Rev. Lett.109, 190502 (2012)

2012
[71]

Amico, R

L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entan- glement in many-body systems, Rev. Mod. Phys.80, 517 (2008)

2008
[72]

Eisert, M

J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010)

2010
[73]

Calabrese and J

P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech.: The- ory Exp.2005 (04), P04010

2005
[74]

Calabrese, Entanglement spreading in non-equilibrium integrable systems, SciPost Phys

P. Calabrese, Entanglement spreading in non-equilibrium integrable systems, SciPost Phys. Lect. Notes , 20 (2020)

2020
[75]

Modak, V

R. Modak, V. Alba, and P. Calabrese, Entanglement re- vivals as a probe of scrambling in finite quantum systems, J. Stat. Mech.: Theory Exp.2020 (8), 083110

2020
[76]

Weedbrook, S

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

2012
[77]

Serafini,Quantum Continuous Variables(CRC Press, 2017)

A. Serafini,Quantum Continuous Variables(CRC Press, 2017)

2017
[78]

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. Math. 58, 141 (1936)

1936
[79]

N. T. Son and T. Stykel, Symplectic eigenvalues of positive-semidefinite matrices and the trace minimiza- tion theorem, Electron. J. Lin. Algebra38, 607 (2022)

2022
[80]

Nicacio, Williamson theorem in classical, quantum, and statistical physics, Am

F. Nicacio, Williamson theorem in classical, quantum, and statistical physics, Am. J. Phys. 89, 1139–1151 (2021)

2021
[81]

M. A. de Gosson,Symplectic Geometry and Quantum Mechanics, 2006th ed. (Birkhäuser Basel, 2006)

2006
[82]

N. T. Son, P.-A. Absil, B. Gao, and T. Stykel, Comput- ing Symplectic Eigenpairs of Symmetric Positive-Definite Matrices via Trace Minimization and Riemannian Op- timization, SIAM J. Matrix Anal. Appl.42, 1732–1757 (2021)

2021
[83]

D’Achille, M

M. D’Achille, M. Gärttner, and T. Haas, Optical Time Algorithm, https://github.com/mauro-dachille/OTA (2025)

2025
[84]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010)

2010
[85]

Marsili, V

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, Detecting single infrared photons with 93% system efficiency, Nat. Phot.7, 210 (2013)

2013
[86]

D. V. Reddy, R. R. Nerem, S. W. Nam, R. P. Mirin, and V. B. Verma, Superconducting nanowire single- photon detectors with 98% system detection efficiency at 1550nm, Optica7, 1649 (2020)

2020
[87]

Cheng, Y

R. Cheng, Y. Zhou, S. Wang, M. Shen, T. Taher, and H. X. Tang, A 100-pixel photon-number-resolving detec- tor unveiling photon statistics, Nat. Photonics17, 112 (2023)

2023
[88]

Björklund, B

A. Björklund, B. Gupt, and N. Quesada, A faster hafnian formula for complex matrices and its benchmarking on a supercomputer, ACM J. Exp. Algo.24, 1 (2019)

2019
[89]

Thekkadath, S

G. Thekkadath, S. Sempere-Llagostera, B. Bell, R. Patel, M. Kim, and I. Walmsley, Experimental Demonstration of Gaussian Boson Sampling with Displacement, PRX Quantum 3, 020336 (2022)

2022
[90]

Bertini, P

B. Bertini, P. Calabrese, M. Collura, K. Klobas, and C. Rylands, Nonequilibrium Full Counting Statistics and Symmetry-Resolved Entanglement from Space-Time 25 Duality, Phys. Rev. Lett.131, 140401 (2023)

2023
[91]

L. K. Joshi, F. Ares, M. K. Joshi, C. F. Roos, and P. Calabrese, Measuring Full Counting Statistics in a Trapped-Ion Quantum Simulator, Phys. Rev. Lett.135, 160601 (2025)

2025
[92]

F. Ares, S. Murciano, and P. Calabrese, Entanglement asymmetry as a probe of symmetry breaking, Nat. Com- mun. 14, 2036 (2023)

2023
[93]

Rylands, K

C. Rylands, K. Klobas, F. Ares, P. Calabrese, S. Mur- ciano,andB.Bertini,MicroscopicOriginoftheQuantum Mpemba Effect in Integrable Systems, Phys. Rev. Lett. 133, 010401 (2024)

2024
[94]

Pitaevskii and S

L. Pitaevskii and S. Stringari,Bose-Einstein Condensa- tion and Superfluidity(Oxford University Press, Oxford, 2016)

2016
[95]

Coleman, Quantum sine-Gordon equation as the mas- sive Thirring model, Phys

S. Coleman, Quantum sine-Gordon equation as the mas- sive Thirring model, Phys. Rev. D11, 2088 (1975)

1975
[96]

Blumenstein, J

C. Blumenstein, J. Schäfer, S. Mietke, S. Meyer, A. Dollinger, M. Lochner, X. Y. Cui, L. Patthey, R. Matz- dorf, and R. Claessen, Atomically controlled quantum chains hosting a Tomonaga–Luttinger liquid, Nat. Phys. 7, 776 (2011)

2011
[97]

F. D. M. Haldane, ’Luttinger liquid theory’ of one- dimensional quantum fluids. I. Properties of the Lut- tinger model and their extension to the general 1D inter- acting spinless Fermi gas, J. Phys. C: Solid State Phys. 14, 2585 (1981)

1981
[98]

Tajik, I

M. Tajik, I. Kukuljan, S. Sotiriadis, B. Rauer, T. Schwei- gler, F. Cataldini, J. Sabino, F. Møller, P. Schüttelkopf, S.-C. Ji, D. Sels, E. Demler, and J. Schmiedmayer, Ver- ification of the area law of mutual information in a quantum field simulator, Nat. Phys.19, 1022 (2023)

2023
[99]

Defenu, T

N. Defenu, T. Donner, T. Macrì, G. Pagano, S. Ruffo, and A. Trombettoni, Long-range interacting quantum systems, Rev. Mod. Phys.95, 035002 (2023)

2023
[100]

Schachenmayer, B

J. Schachenmayer, B. P. Lanyon, C. F. Roos, and A. J. Daley, Entanglement Growth in Quench Dynamics with Variable Range Interactions, Phys. Rev. X3, 031015 (2013)

2013
[101]

Richerme, Z.-X

P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A. V. Gorshkov, and C. Mon- roe, Non-local propagation of correlations in quantum systems with long-range interactions, Nature511, 198 (2014)

2014
[102]

Jurcevic, B

P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos, Quasiparticle engineering and entanglement propagation in a quantum many-body system, Nature511, 202 (2014)

2014
[103]

Gärttner, J

M. Gärttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey, Measuring out-of-time- order correlations and multiple quantum spectra in a trapped-ion quantum magnet, Nat. Phys.13, 781 (2017)

2017
[104]

Zhang, G

J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z. X. Gong, and C. Monroe, Observation of a many-body dynami- cal phase transition with a 53-qubit quantum simulator, Nature 551, 601 (2017)

2017
[105]

M. K. Joshi, A. Elben, B. Vermersch, T. Brydges, C. Maier, P. Zoller, R. Blatt, and C. F. Roos, Quantum Information Scrambling in a Trapped-Ion Quantum Sim- ulator with Tunable Range Interactions, Phys. Rev. Lett. 124, 240505 (2020)

2020
[106]

M. G. Nezhadhaghighi and M. A. Rajabpour, Quantum entanglement entropy and classical mutual information in long-range harmonic oscillators, Phys. Rev. B88, 045426 (2013)

2013
[107]

M. G. Nezhadhaghighi and M. A. Rajabpour, Entan- glement dynamics in short- and long-range harmonic oscillators, Phys. Rev. B90, 205438 (2014)

2014
[108]

M. A. Rajabpour and S. Sotiriadis, Quantum quench in long-range field theories, Phys. Rev. B91, 045131 (2015)

2015
[109]

B. Basa, G. La Nave, and P. W. Phillips, Classification of nonlocal actions: Area versus volume entanglement entropy, Phys. Rev. D101, 106006 (2020)

2020
[110]

Lischke, G

A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth, and G. E. Karniadakis, What is the frac- tional Laplacian? A comparative review with new results, J. Comp. Phys.404, 109009 (2020)

2020
[111]

Daoud and E

M. Daoud and E. H. Laamri, Fractional laplacians: A short survey, Discrete Contin. Dyn. Syst. - S15, 95 (2022)

2022
[112]

Roy, Aspects of entanglement in non-local field the- ories with fractional Laplacian, J

P. Roy, Aspects of entanglement in non-local field the- ories with fractional Laplacian, J. High Energy Phys. 2022, 101

2022
[113]

G. W. Gibbons, C. N. Pope, and S. Solodukhin, Higher derivative scalar quantum field theory in curved space- time, Phys. Rev. D100, 105008 (2019)

2019
[114]

Bambi, L

C. Bambi, L. Modesto, and I. Shapiro, eds.,Handbook of Quantum Gravity(Springer Singapore, 2024)

2024
[115]

Shiba and T

N. Shiba and T. Takayanagi, Volume law for the en- tanglement entropy in non-local QFTs, J. High Energy Phys. 2014 (2)

2014
[116]

Jaksch, C

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Cold Bosonic Atoms in Optical Lattices, Phys. Rev. Lett.81, 3108 (1998)

1998
[117]

Greiner, O

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature 415, 39 (2002)

2002
[118]

N. D. Birrell and P. C. W. Davies,Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 1982)

1982
[119]

Mukhanov and S

V. Mukhanov and S. Winitzki,Introduction to Quantum Effects in Gravity(Cambridge University Press, Cam- bridge, 2007)

2007
[120]

P. Jain, S. Weinfurtner, M. Visser, and C. W. Gardiner, AnalogmodelofaFriedmann-Robertson-Walkeruniverse inBose-Einsteincondensates: Applicationoftheclassical field method, Phys. Rev. A76, 033616 (2007)

2007
[121]

Tolosa-Simeón, Á

M. Tolosa-Simeón, Á. Parra-López, N. Sánchez-Kuntz, T. Haas, C. Viermann, M. Sparn, N. Liebster, M. Hans, E. Kath, H. Strobel, M. K. Oberthaler, and S. Flo- erchinger, Curved and expanding spacetime geometries in Bose-Einstein condensates, Phys. Rev. A106, 033313 (2022)

2022
[122]

Sánchez-Kuntz, Á

N. Sánchez-Kuntz, Á. Parra-López, M. Tolosa-Simeón, T. Haas, and S. Floerchinger, Scalar quantum fields in cosmologies with 2 + 1spacetime dimensions, Phys. Rev. D 105, 105020 (2022)

2022
[123]

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Sonic Analog of Gravitational Black Holes in Bose-Einstein Condensates, Phys. Rev. Lett.85, 4643 (2000)

2000
[124]

Weinfurtner, S

S. Weinfurtner, S. Liberati, and M. Visser, Analogue Space-time Based on 2-Component Bose-Einstein Con- densates, inQuantum Analogues: From Phase Transi- tions to Black Holes and Cosmology, edited by W. G. 26 Unruh and R. Schützhold (Springer Berlin Heidelberg, Berlin, Heidelb...

2007
[125]

C. W. Misner, K. Thorne, and J. Wheeler,Gravitation (Princeton University Press, 2017)

2017
[126]

Weinberg,Cosmology (Oxford University Press, Ox- ford, 2008)

S. Weinberg,Cosmology (Oxford University Press, Ox- ford, 2008)

2008
[127]

Ammon and J

M. Ammon and J. Erdmenger,Gauge/Gravity Duality: Foundations and Applications (Cambridge University Press, 2015)

2015
[128]

D’Auria, S

V. D’Auria, S. Fornaro, A. Porzio, S. Solimeno, S. Oli- vares, and M. G. A. Paris, Full Characterization of Gaus- sian Bipartite Entangled States by a Single Homodyne Detector, Phys. Rev. Lett.102, 020502 (2009)

2009
[129]

Ferrini, J

G. Ferrini, J. P. Gazeau, T. Coudreau, C. Fabre, and N. Treps, Compact Gaussian quantum computation by multi-pixel homodyne detection, New J. Phys.15, 093015 (2013)

2013
[130]

Armstrong, J.-F

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, Programmable multimode quantum networks, Nat. Commun.3, 1026 (2012)

2012
[131]

B. S. Harms, B. E. Anthony, N. T. Holte, H. A. Das- sonville, and A. M. C. Dawes, Multimode quantum state tomography using unbalanced array detection, Phys. Rev. A 90, 053818 (2014)

2014
[132]

Y. Cai, J. Roslund, G. Ferrini, F. Arzani, X. Xu, C. Fabre, and N. Treps, Multimode entanglement in re- configurable graph states using optical frequency combs, Nat. Commun.8, 15645 (2017)

2017
[133]

X. Jia, C. Zhai, X. Zhu, C. You, Y. Cao, X. Zhang, Y. Zheng, Z. Fu, J. Mao, T. Dai, L. Chang, X. Su, Q. Gong, and J. Wang, Continuous-variable multipartite entanglement in an integrated microcomb, Nature639, 329 (2025)

2025
[134]

Pechukas, Reduced Dynamics Need Not Be Com- pletely Positive, Phys

P. Pechukas, Reduced Dynamics Need Not Be Com- pletely Positive, Phys. Rev. Lett.73, 1060 (1994)

1994
[135]

Haas, Area laws from classical entropies, arXiv:2404.12320 (2024)

T. Haas, Area laws from classical entropies, arXiv:2404.12320 (2024)

2024 arXiv
[136]

Deller, M

Y. Deller, M. Gärttner, T. Haas, M. K. Oberthaler, M. Reh, and H. Strobel, Area laws and thermalization from classical entropies in a Bose-Einstein condensate, Phys. Rev. A112, L011303 (2025)

2025
[137]

Deller, M

Y. Deller, M. Gärttner, T. Haas, M. K. Oberthaler, M. Reh, and H. Strobel, Entropy estimation in a spin- 1 Bose-Einstein condensate, New J. Phys.27, 043004 (2025)

2025
[138]

Calabrese and J

P. Calabrese and J. Cardy, Quantum quenches in ex- tended systems, J. Stat. Mech.: Theory Exp.2007 (06), P06008

2007
[139]

Calabrese and J

P. Calabrese and J. Cardy, Quantum quenches in 1+1 dimensional conformal field theories, J. Stat. Mech.: The- ory Exp.2016 (6), 064003

2016
[140]

Calabrese, Entanglement and thermodynamics in non- equilibrium isolated quantum systems, Phys

P. Calabrese, Entanglement and thermodynamics in non- equilibrium isolated quantum systems, Phys. A: Stat. Mech. Appl.504, 31 (2018), lecture Notes of the 14th International Summer School on Fundamental Problems in Statistical Physics

2018
[141]

Alba and P

V. Alba and P. Calabrese, Entanglement dynamics after quantum quenches in generic integrable systems, SciPost Phys. 4, 017 (2018)

2018
[142]

Langen, S

T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, Experimental observation of a generalized Gibbs ensemble, Science348, 207 (2015)

2015
[143]

Vidmar and M

L. Vidmar and M. Rigol, Generalized Gibbs ensemble in integrable lattice models, J. Stat. Mech.: Theory Exp. 2016 (6), 064007

2016
[144]

Vahlbruch, M

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schn- abel, Detection of 15 dB Squeezed States of Light and their Application for the Absolute Calibration of Pho- toelectric Quantum Efficiency, Phys. Rev. Lett.117, 110801 (2016)

2016
[145]

E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Comm. Math. Phys.28, 251 (1972)

1972
[146]

Cheneau, P

M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß, T. Fukuhara, C. Gross, I. Bloch, C. Kollath, and S. Kuhr, Light-cone-like spreading of correlations in a quantum many-body system, Nature481, 484 (2012)

2012
[147]

M. C. Tran, A. Y. Guo, C. L. Baldwin, A. Ehrenberg, A. V. Gorshkov, and A. Lucas, Lieb-Robinson Light Cone for Power-Law Interactions, Phys. Rev. Lett.127, 160401 (2021)

2021
[148]

(Anthony) Chen, A

C.-F. (Anthony) Chen, A. Lucas, and C. Yin, Speed limits and locality in many-body quantum dynamics, Rep. Prog. Phys.86, 116001 (2023)

2023
[149]

Slussarenko and G

S. Slussarenko and G. J. Pryde, Photonic quantum in- formation processing: A concise review, App. Phys. Rev. 6, 041303 (2019)

2019
[150]

Parýzková, C

M. Parýzková, C. S. Hamilton, I. Jex, M. Stefszky, and C. Silberhorn, Input phase noise in Gaussian Boson sampling, arXiv:2512.19596 (2025)

2025 arXiv
[151]

J. F. F. Bulmer, B. A. Bell, R. S. Chadwick, A. E. Jones, D. Moise, A. Rigazzi, J. Thorbecke, U.-U. Haus, T. V. Vaerenbergh, R. B. Patel, I. A. Walmsley, and A. Laing, The boundary for quantum advantage in Gaussian boson sampling, Sci. Adv.8, eabl9236 (2022)

2022
[152]

Cilluffo, M

D. Cilluffo, M. Kost, N. Lorenzoni, and M. B. Plenio, Operator-basis Matrix Product State formalism for opti- cal circuits, arXiv:2502.01737 (2025)

2025 arXiv
[153]

Naumann and R

A. Naumann and R. Strahlendorf, Lossy Gaussian Inter- ferometer Computation - LoGIC,https://github.com/ alexn2002/LoGIC (2025)

2025
[154]

Walschaers, Non-Gaussian Quantum States and Where to Find Them, PRX Quantum2, 030204 (2021)

M. Walschaers, Non-Gaussian Quantum States and Where to Find Them, PRX Quantum2, 030204 (2021)

2021
[155]

Chabaud and M

U. Chabaud and M. Walschaers, Resources for Bosonic Quantum Computational Advantage, Phys. Rev. Lett. 130, 090602 (2023)

2023
[156]

F. A. Mele, A. A. Mele, L. Bittel, J. Eisert, V. Giovan- netti, L. Lami, L. Leone, and S. F. E. Oliviero, Learn- ing quantum states of continuous-variable systems, Nat. Phys. 21, 2002 (2025)

2025
[157]

Lukin, M

A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kauf- man, S. Choi, V. Khemani, J. Léonard, and M. Greiner, Probing entanglement in a many-body–localized system, Science 364, 256 (2019)

2019
[158]

Goldstein and E

M. Goldstein and E. Sela, Symmetry-Resolved Entan- glement in Many-Body Systems, Phys. Rev. Lett.120, 200602 (2018)

2018
[159]

M. J. Hartmann, Quantum simulation with interacting photons, J. Opt.18, 104005 (2016)

2016
[160]

D. E. Chang, V. Vuletić, and M. D. Lukin, Quantum nonlinear optics-photon by photon, Nat. Photonics8, 685 (2014)

2014
[161]

Walther, B

H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, Cavity quantum electrodynamics, Rep. Prog. Phys. 69, 1325 (2006). 27

2006
[162]

Bajcsy, S

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, Efficient All-Optical Switching Using Slow Light within a Hollow Fiber, Phys. Rev. Lett.102, 203902 (2009)

2009
[163]

Peyronel, O

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, Quantum nonlinear optics with single photons enabled by strongly interacting atoms, Nature488, 57 (2012)

2012
[164]

Firstenberg, C

O. Firstenberg, C. S. Adams, and S. Hofferberth, Nonlin- ear quantum optics mediated by Rydberg interactions, J. Phys. B: At. Mol. Opt. Phys.49, 152003 (2016)

2016
[165]

J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson, Integrated photonic quantum technologies, Nat. Photon- ics 14, 3895 (2020)

2020
[166]

Brandt, M

F. Brandt, M. Hiekkamäki, F. Bouchard, M. Huber, and R. Fickler, High-dimensional quantum gates using full-field spatial modes of photons, Optica7, 98 (2020)

2020
[167]

S. Goel, S. Leedumrongwatthanakun, N. H. Valencia, W. McCutcheon, A. Tavakoli, C. Conti, P. W. H. Pinkse, and M. Malik, Inverse design of high-dimensional quan- tum optical circuits in a complex medium, Nat. Phys. 20, 232 (2024)

2024
[168]

Q. Wang, J. Liu, D. Lyu, and J. Wang, Ultrahigh-fidelity spatial mode quantum gates in high-dimensional space by diffractive deep neural networks, Light Sci. Appl.13, 10 (2024)

2024
[169]

Bouchard, K

F. Bouchard, K. Fenwick, K. Bonsma-Fisher, D. Eng- land, P. J. Bustard, K. Heshami, and B. Sussman, Programmable Photonic Quantum Circuits with Ultra- fast Time-Bin Encoding, Phys. Rev. Lett.133, 090601 (2024)

2024
[170]

Folge, M

P. Folge, M. Stefszky, B. Brecht, and C. Silberhorn, A Framework for Fully Programmable Frequency-Encoded Quantum Networks Harnessing Multioutput Quantum Pulse Gates, PRX Quantum5, 040329 (2024)

2024
[171]

Blais, A

A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys.93, 025005 (2021)

2021
[172]

Gross and I

C. Gross and I. Bloch, Quantum simulations with ultra- cold atoms in optical lattices, Science357, 995 (2017)

2017
[173]

Schäfer, T

F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Tools for quantum simulation with ultra- cold atoms in optical lattices, Nat. Rev. Phys.2, 411 (2020)

2020
[174]

Cilluffo, Commentary on the decomposition of univer- sal multiport interferometers: how it works in practice, arXiv:2412.11955 (2024)

D. Cilluffo, Commentary on the decomposition of univer- sal multiport interferometers: how it works in practice, arXiv:2412.11955 (2024)

2024 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Entanglement generation We consider a spatial subregion[0, ℓ] with ℓ ≤ L, see inset of Fig. 4(a). As the modes are coupled, the degrees of freedom in [0, ℓ] become entangled with the ones in the complementary region(ℓ, L]. Thus, while the global state ρ(t) remains pure during the unitary evolution, the local state ρ(t, l) = T rL−ℓ{ρ(t)} follows the evolut...

[2] [2]

Correlation spreading We now study how two equally sized spatial regions of length ℓ separated by a distanced >0 become correlated after the quench, as shown in the inset of Fig. 4(b). To this end, we consider the Gaussian Rényi-2mutual information I2(t, ℓ, d) = 2S2(t, ℓ) − S2(t, ℓ∪ ℓ) = 1 2 ln det σ2(t, ℓ) det σ(t, ℓ∪ ℓ) , (53) where S2(t, ℓ∪ ℓ) and σ(t,...

[3] [3]

(14) The isolation of the time parameter into single-mode phase shifters is achieved by generalizing Williamson’s theorem for the Hamiltonian matrixH = S⊺ SDDSSD [see Eq

Proof of Eq. (14) The isolation of the time parameter into single-mode phase shifters is achieved by generalizing Williamson’s theorem for the Hamiltonian matrixH = S⊺ SDDSSD [see Eq. (8)] to the symplectic matrixS(t) describing time evolution. After introducing the corresponding unitary transformation USD defined via SSDr = U † SDrUSD, the Hamiltonian be...

[4] [4]

Eigensystems of H ϕ and H π For the OTA, we consider the class of block diagonal Hamiltonians with commuting blocksH ϕ and H π

Proof of the OTA a. Eigensystems of H ϕ and H π For the OTA, we consider the class of block diagonal Hamiltonians with commuting blocksH ϕ and H π. Given Eq. (A3), our goal is to further decompose the SD matrix by means of optical transformations. We make the ansatz SSD = ON AN BN ON , (A4) with AN and BN being further constrained asSSD must diagonalize t...

[5] [5]

Starting once again from Williamson’s theorem(8), i.e., H(t) = S⊺ SD(t)D(t)SSD(t), and using the polar decomposition SSD = SSq(t)P [see Eqs

Time-dependent case We generalize the OTA to time-dependent Hamiltonian matrices H = H(t) for time-independent eigenvectors P ̸= P (t), which ensures [H(t1), H(t2)] = 0. Starting once again from Williamson’s theorem(8), i.e., H(t) = S⊺ SD(t)D(t)SSD(t), and using the polar decomposition SSD = SSq(t)P [see Eqs. (A10) and (A12)] results in H(t) = 1 2 UIr⊺S⊺ ...

[6] [6]

IIIB2) in the more general setting of coherent inputs by proving(25) and a generalization of (26)

Coherent inputs Here, we show that the OTA dramatically simplifies (as discussed in Sec. IIIB2) in the more general setting of coherent inputs by proving(25) and a generalization of (26). We define coherent states as displaced vacuum states |α⟩ = D(α) |0⟩, where D(α) = eαa†−α∗a with a = ( a1, . . . ,aN )⊺ (analogously for the vector of com- plex phases α ...

[7] [7]

We discuss the two most prominent choices: periodic and open boundaries

Finite size effects In a finite volumeL <∞, quasi-particles have discrete momenta and their dynamics are influenced by boundary conditions. We discuss the two most prominent choices: periodic and open boundaries. a. Periodic boundary conditions (PBCs) For periodic boundaries, quasi-particles reenter from the left (right) when traveling to the right (left)...

[8] [8]

To computenk, we first determine the eigenmodes of the relativistic theory(31)

Entropy density of the GGE and mode populations The entropy densitys2,k follows from computing the Rényi-2 entanglement entropy(51) of the GGEρ(∞, l) ∝ exp(− P k yknk) in the eigenbasis of the quasi-particle number operator, leading to [65, 132] S2(∞, ℓ) = S2,GGE = X k∈K s2,k, (D7) with coefficients s2,k = ln n2 k − (1 − nk)2 , (D8) where nk = ⟨0|nk|0⟩ sp...