Pith. sign in

REVIEW 2 major objections 5 minor 160 references

Detecting genuine non-Gaussian entanglement

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read Third-order partial-transpose moment criteria detect NOON entanglement for every photon number $N$ via three-copy Fourier interferometry.

desk verdict The multicopy DFT readout of PT-moments is a genuinely useful new CV entanglement detection tool, but the noisy-copy robustness analysis in Sec. VB1 is flawed because it evaluates criteria on unequal copies. read the letter →

arxiv 2504.15831 v1 pith:ZH52ADAO submitted 2025-04-22 quant-ph

Serge Deside , Tobias Haas , Nicolas J. Cerf This is my paper · ORCID
classification quant-ph PACS 03.67.Mn42.50.Ex
keywords partialtransposemomentscontinuousvariablesentanglementdetectionNOONstatesmulticopymethodFourierinterferometerphoton-number-resolvingnon-Gaussian
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

This paper establishes that entanglement tests based on moments of the partially transposed state, previously limited to discrete-variable systems, work for continuous-variable optical and atomic systems. The core claim is that any separable state must satisfy the two third-order inequalities $p_3 \ge p_2^2$ and $p_3 \ge (3p_2-1)/2$, where $p_n = \mathrm{Tr}\{\tilde{\rho}^n\}$ are moments of the partial transpose; violating either one certifies entanglement. The authors propose a concrete readout of $p_2$ and $p_3$ from just two or three copies of the state, passive linear optics, and particle-number-resolving detectors. This detects genuine non-Gaussian entanglement in families that homodyne, entropic, and quadrature-moment criteria miss, including every NOON state for every mode number $N$. They also model losses, copy-to-copy noise, and finite statistics, and simulate a full experiment showing that a Bell-type NOON state is certified within about $10^3$ samples even at 40% loss.

What carries the argument

The central object is the partial-transpose moment $p_n = \mathrm{Tr}\{\rho^{\otimes n}\, (\vec{\Pi}^A_n \otimes \overleftarrow{\Pi}^B_n)\}$, where the shift operators permute the $n$ copies on each side. Because the shift operator is circulant, the $n$-mode discrete Fourier transform $F(n)$ diagonalizes it into a phase observable $D_n$ that depends only on the output particle numbers, so $p_n$ becomes a weighted average of $n$th roots of unity on the measured photon counts. For $n=2$ this is just a balanced beam splitter followed by parity measurements; for $n=3$ it is a three-beam-splitter, three-phase-shift interferometer for both parties, with Bob measuring the inverse phase weights. The inequalities themselves come from two hierarchies: Hankel-matrix positivity of the Stieltjes moment problem, and Newton's identities combined with Descartes' rule of signs on the characteristic polynomial of $\tilde{\rho}$.

What would settle it

Search over separable continuous-variable states for one whose third elementary symmetric polynomial $e_3$, computed from the PT-moments via Newton's identities, is negative while $p_3 \ge p_2^2$ and $p_3 \ge (3p_2-1)/2$ both hold; finding such a state would show that the Descartes-rule extension to infinite dimensions is invalid. A concrete numerical route is to take truncated Fock-space approximations of a candidate separable state and check whether the violation persists as the cutoff grows.

Watch

Extended reading notes

Core claim

The paper's central claim is that the $p_n$-PPT criteria extend to continuous variables, where $\tilde{\rho} = (1 \otimes T_B)\rho$ is the partial transpose of a bipartite state on an infinite-dimensional Hilbert space. For third order, separability implies the two conditions in Eq. (6), and for pure states these conditions are also sufficient, since $p_3 = \sum_i |c_i|^6$ in the Schmidt basis equals $1$ only for product states. For NOON states $|\psi\rangle = \alpha|N,0\rangle + \beta|0,N\rangle$, the paper computes $p_3 = |\alpha|^6 + |\beta|^6$, independent of $N$, so every nontrivial NOON state violates the criterion; the violation is largest, $-3/4$, at balanced amplitudes. The argument also shows that the noisy, lossy, finite-statistics version remains a valid witness, with explicit error models and simulated experimental runs.

Load-bearing premise

The paper assumes, without proof, that a classical sign-counting rule for polynomials still works for the infinitely many eigenvalues, some possibly negative, of the partially transposed state; if that assumption is wrong, the linear criterion can produce false-positive entanglement verdicts.

Editorial extensions

If this is right

  • Every pure entangled continuous-variable state is certified by the third-order conditions, because pure-state entanglement is equivalent to $p_3 < 1$.
  • All NOON states with both amplitudes nonzero are detected, independent of the photon number $N$, closing a gap left by second-moment, entropic, and existing multicopy witnesses.
  • Third-order PT-moment criteria are robust: for NOON states up to $N=10$, losses up to roughly 20% still permit certification, and no certification is possible beyond 50% loss.
  • The readout cost is fixed by the copy number rather than the Hilbert-space dimension: two copies for $p_2$ and three for $p_3$, with small sample counts required in the simulated Bell-state experiment.
  • The method applies to settings without phase-stable local oscillators, such as high-harmonic-generation sources, where homodyne or heterodyne quadrature readout is unavailable.

Reading between the lines

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

  • A direct extension not pursued here would apply the same Fourier readout to moments of the realignment map; if it works, bound-entangled Gaussian states could be certified with the same two-or-three-copy resources.
  • The infinite-dimensional validity question could be tested numerically by truncating the Fock basis at increasing cutoff $d$: if a separable state with negative $e_3$ emerges while the $p_3$ inequalities hold for all cutoffs, the Descartes-rule extension would be refuted.
  • Because the PT-moment inequalities are basis-independent, one testable prediction is that they certify entanglement for high-$N$ NOON states with a sample budget growing only mildly with $N$ under fixed loss, which would make the method useful for metrology-oriented photonic experiments.
  • A multimode or multipartite generalization, which the authors sketch, would make the same observable a candidate for detecting genuine multipartite non-Gaussian entanglement in many-body bosonic systems; this is my inference, not a result of the paper.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper extends recently introduced partial-transpose moment (PT-moment) entanglement criteria to continuous-variable systems and proposes a multicopy interferometric readout of the first nontrivial moments p2 and p3 using discrete Fourier transforms and particle-number-resolving measurements. It derives third-order criteria, benchmarks them on Gaussian states, mixed cat states, high-harmonic-generation states, and NOON states, and analyzes robustness to loss, noisy copies, and finite statistics. The central claim is that the criteria certify genuine non-Gaussian entanglement, including all NOON states for all mode numbers N, and remain robust under realistic experimental constraints.

Significance. If the claims hold, this is a significant contribution: PT-moment criteria provide an experimentally concrete route to certifying entanglement in states that escape Gaussian second-moment and entropic criteria. The analytic NOON result for arbitrary N is striking, and the proposed Fourier-interferometer readout is feasible with current technology. The paper contains no fitted free parameters: the benchmark values are computed analytically from the stated state families, and the finite-statistics error model is explicit and testable. The core third-order inequalities (6)-(7) are valid for separable CV states, as they follow from elementary moment inequalities for the positive semidefinite partial transpose, and the multicopy implementation is a natural extension of known purity and Rényi-entropy measurements. However, the noisy-copy robustness analysis in Sec. VB1 evaluates p2 and p3 on unequal copies, for which no separability bound exists, so the robustness claim in its current form is not established.

major comments (2)
  1. [Sec. VB1, Fig. 3b, criteria (6)-(7)] The noisy-copy analysis computes p2 from copies 1 and 2 and p3 from copies 1, 2, and 3 with independently varying parameters. The inequalities (6)-(7) are only valid for p_n = Tr{ρ̃^n} of a single state ρ; for unequal copies no such separability bound exists. This is not a harmless modeling choice: take Bob's state fixed and Alice's three single-mode states |0>, -1/2|0>+√3/2|1>, and -1/2|0>-√3/2|1>. Each copy is a bipartite product state and hence separable, yet the two-copy overlap is 1/4 and the three-copy cyclic overlap is -1/8, so p3 = -1/8 < p2^2 = 1/16, meaning the quadratic criterion would flag a fully separable ensemble as entangled. The 'detected entanglement' regions in Fig. 3b can therefore contain false positives, and the claimed robustness to noisy copies is not established by this analysis. The same clarification is needed for the full simulation in Sec. VB2 and Fig. 4: if the phase and loss parameters are drawn independently for each copy within a run, the identical-copy assumption of the criterion is violated.
  2. [Sec. IIA2, Eq. (5)] The extension of Newton's identities and Descartes' rule of signs to the infinite-dimensional partial transpose is asserted in one sentence and is not a standard result: a characteristic polynomial is not generally defined for a trace-class operator with countably many eigenvalues. The third-order criteria (6) and (7) are nevertheless correct for separable CV states, because ρ̃ is positive semidefinite and p_n are moments of a probability distribution. For example, p3 ≥ p2^2 follows from Cauchy-Schwarz, and p3 ≥ (3p2 - 1)/2 follows from λ^3 - (3/2)λ^2 + 1/2 = (λ-1)(λ^2 - λ/2 - 1/2) ≥ 0 for λ ∈ [0,1]. The authors should replace the Descartes-rule justification with such an elementary derivation, or provide a rigorous treatment of the infinite-dimensional case.
minor comments (5)
  1. [Sec. IVD1 / Abstract] The phrase 'detecting genuine non-Gaussian entanglement' could be read as claiming that the criteria certify non-Gaussianity; in fact the same third-order criteria also detect Gaussian entangled states, as shown in Sec. IVB. Please clarify that the criteria certify entanglement, and that the non-Gaussian character is a property of the target states being benchmarked.
  2. [Eq. (45)] The Hermite-Gauss wavefunction is typeset ambiguously; please add explicit parentheses and define the parameters σ+ and σ- in the text.
  3. [Fig. 3b] The caption should state precisely which quantities are computed analytically and which are sampled, and it should define the boundary and color conventions for the 'detected entanglement' regions.
  4. [References] References [88] and [89] appear to share the same journal volume and article number; please verify that these are two distinct papers with correct bibliographic data.
  5. [Sec. IVA1] The comparison between the linear p3-PPT criterion and the Shchukin-Vogel criterion relies on an implicitly defined matrix c via f2(c) = p2(ρ) - (3/2)ρ + 1/2; please state explicitly whether this map is shown to be well-defined for all states or only for the examples considered.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity found; the CV extension of PT-moment criteria is analytically self-contained, with the unproved infinite-dimensional Descartes step and the unequal-copy robustness test being correctness risks rather than circular reasoning.

full rationale

The central derivation chain is not circular. The third-order criteria in Eq. (6) follow from the external PPT-moment framework of Refs. [82-84]: for any separable state the partial transpose is positive semidefinite, so the Hankel matrix [[1,p2],[p2,p3]] is positive semidefinite (giving p3 >= p2^2) and the elementary symmetric polynomial e3 = (1 - 3p2 + 2p3)/6 is nonnegative (giving p3 >= (3p2 - 1)/2). No parameter is fitted to the benchmark states; p2 and p3 are computed analytically for cat states, HHG states, and NOON states, and the criteria are then compared with independent standard methods. The multicopy interferometric readout in Sec. III is a direct algebraic consequence of diagonalizing the cyclic-shift operators by discrete Fourier transforms, not a restatement of the target entanglement conclusion. The paper does cite several works by overlapping authors [11,12,86-89], but these support the experimental multicopy toolbox and prior applications, not the validity of the separability inequalities themselves, and no uniqueness theorem is imported from the authors' own work. Two non-circular weaknesses should nevertheless be noted. First, Sec. IIA2 asserts that Descartes' rule of signs extends to the characteristic polynomial of an infinite-dimensional matrix, which is not proved; this is a mathematical-presentation gap, and the necessary direction used in the paper is in fact safe without that assertion, so it does not make the derivation circular. Second, the noisy-copy analysis in Sec. VB1 and Fig. 3b evaluates p2 and p3 on unequal copies of a state, whereas the separability bounds (6)-(7) are derived for identical copies; this can produce false positives and undermines the robustness claim for that specific analysis, but it is a correctness issue, not a circularity of the claimed derivation. The i.i.d. simulation in Fig. 4 is better posed. Overall, the paper's predictions are not equivalent to its inputs by construction, and no fitted parameter is relabeled as a prediction.

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

The central claim introduces no fitted parameters and no new physical entities. It borrows PT-moment criteria from prior discrete-variable results and assumes standard multicopy and DFT tools. The main load-bearing axiom is the unproved applicability of Newton's identities and Descartes' rule to infinite-dimensional partial transposes, which is flagged in the report.

assumptions (5)
  • standard math Peres-Horodecki PPT criterion: separable states have positive partial transpose.
    Foundation of all PT-moment criteria; invoked in Sec. IIA via moments of rho-tilde.
  • standard math Stieltjes moment problem: a non-negative measure with moments p_n implies positivity of Hankel matrices P_n.
    Used to derive Eq. (2); validity for countably infinite supports is assumed.
  • domain assumption Newton's identities and Descartes' rule of signs apply to the eigenvalue list and characteristic polynomial of the infinite-dimensional partial transpose rho-tilde.
    Assumed in Sec. IIA2 to obtain the linear criterion in Eq. (6); stated without proof.
  • standard math Multicopy identity p_n = Tr{rho^otimes n Pi_A tensor Pi_B^-1} and its DFT diagonalization.
    Standard trace-permutation identity used in Sec. IIIB to convert PT-moments to particle-number measurements.
  • domain assumption n copies of the target state can be produced identically and interfered with passive linear optics.
    Experimental premise of the readout; motivates the multicopy method in Sec. IIIA.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Detecting genuine non-Gaussian entanglement." pith.science (2026). https://pith.science/paper/ZH52ADAO

@misc{pith2026250415831,
  author       = {Pith},
  title        = {Pith review of: Detecting genuine non-Gaussian entanglement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZH52ADAO}},
  note         = {Machine review of arXiv:2504.15831}
}
read the original abstract

Efficiently certifying non-Gaussian entanglement in continuous-variable quantum systems is a central challenge for advancing quantum information processing, photonic quantum computing, and metrology. Here, we put forward continuous-variable counterparts of the recently introduced entanglement criteria based on moments of the partially transposed state, together with simple readout schemes that require only a few replicas of the state, passive linear optics, and particle-number measurements. Our multicopy method enables the detection of genuine non-Gaussian entanglement for various relevant state families overlooked by standard approaches, which includes the entire class of NOON states. Further, it is robust against realistic experimental constraints (losses, noise, and finite statistics), which we demonstrate by extensive numerical simulations.

Figures

Figures reproduced from arXiv: 2504.15831 by the authors.

Figure 1
Figure 1. a) Measurement routine for detecting the nth PT-moment pn. After preparing n replicas, that is, independent and identical copies, of the bipartite state of interest, Alice and Bob separately assess their subsystems A (blue) and B (red), respectively. Both perform a discrete Fourier transform (DFT) F(n) of order n, see (13), and then measure the particle numbers on their output modes 2 to n. They obtain pn from the m… view at source ↗
Figure 2
Figure 2. a) Detected Gaussian entanglement by PT-moment-based criteria of order three (shaded regions) in terms of the symplectic eigenvalues ν˜1, ν˜2. The set of physical Gaussian states constrained by ν˜1ν˜2 ≥ 1 is divided into separable (light gray) and entangled (dark gray) regions by Simon’s criterion (30) (black straight lines). The performance among the criteria is dictated by the purity p2, with the tipping point cor… view at source ↗
Figure 3
Figure 3. The three types of imperfections discussed in Sec. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Simulated experiment for detecting the entanglement [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Gaussian entanglement detected by the pn-PPT criteria (2) of order n = 3 (blue), n = 5 (petrol) and n = 7 (red); analogous to Fig. 2a). More mixed entangled states are detected for increasing n, indicating convergence towards Simon’s criterion in the limit n → ∞. Appen…
Figure 6
Figure 6. Figure 6: Particle-number output distributions for the balanced ( [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Numerical validation of the statistical error models for the three [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Complementary analysis to Fig [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

160 extracted references · 58 canonical work pages

  1. [1]

    It is well known that the two-mode DFT is nothing but a 50:50 beam splitter operation, i.e., F(2) = 1√ 2 ( 1 1 1 −1 )

    Purity p2 The protocol for obtaining the state’s purity using two copies is arguably the most prominent application of the multicopy method [85, 87, 88, 99] and has been successfully implemented in cold-atom systems [11, 12], and integrated photonic devices [106]. It is well known that the two-mode DFT is nothing but a 50:50 beam splitter operation, i.e.,...

  2. [2]

    Third-order PT-moment p3 Following (13), the third-order DFT evaluates to F(3) = 1√ 3   1 1 1 1 e−i 2π 3 ei 2π 3 1 ei 2π 3 e−i 2π 3  , (21) whose implementation requires a combination of three beam splitters and phase shifts each, namely, F(3) =R3 (π 6 ) R2 ( −π 6 ) B23 (1 2 ) ×R3 (π 2 ) B13 (2 3 ) B12 (1 2 ) , (22) see Fig. 1c). However, as we conduc...

  3. [3]

    In contrast to PT-moment-based criteria, they involve moments of mode operators

    Shchukin–Vogel criteria The well-known Shchukin–Vogel hierarchy constitutes another complete set of criteria testing for PPT [71–73]. In contrast to PT-moment-based criteria, they involve moments of mode operators. The central idea is that the partially transposed state ˜ρ is non-negative if and only if T r{ ˜ρf†f}≥ 0 for all normally orderedf. It is conv...

  4. [4]

    Criteria based on non-local variables In another common approach, PPT is assessed via the non-local quadrature operatorsx± =xA±xB and p± =pA±pB from the EPR argument [1]. By noting that taking the partial transpose in subsystemB translates intopB→−pB in phase space [57], physicality (meaning non-negativity) of ˜ρ implies that uncertainty relations must be...

  5. [5]

    (28) Then, the canonical commutation relations take the form [χj,χj′] = iΩjj′, where Ω = 12⊗ (iσ2) denotes the symplectic metric

    Symplectic formalism and Simon’s criterion It is convenient to group the two pairs of canonical operators into a single vector in phase space as χ = (X1,P1,X2,P2)T. (28) Then, the canonical commutation relations take the form [χj,χj′] = iΩjj′, where Ω = 12⊗ (iσ2) denotes the symplectic metric. A partially transposed Gaussian state is fully characterized b...

  6. [6]

    For the partially transposed state, it reads pn = 2∏ j=1 2n (˜νj + 1)n− (˜νj− 1)n

    pn-PPT criteria for Gaussian states Aclosedformulaforthe nthmomentofaGaussianstate in terms of its symplectic eigenvalues has been derived in [43, 108]. For the partially transposed state, it reads pn = 2∏ j=1 2n (˜νj + 1)n− (˜νj− 1)n. (31) Forn = 2 and n = 3, the formula evaluates to p2 = 1 ˜ν1˜ν2 = 1√det ˜γ, p 3 = 16 (1 + 3˜ν2

  7. [7]

    Hence, a Gaussian state is pure if and only if the product of symplectic eigenvalues equals unity, i.e., ˜ν1˜ν2 = 1

    (1 + 3˜ν2 2), (32) respectively. Hence, a Gaussian state is pure if and only if the product of symplectic eigenvalues equals unity, i.e., ˜ν1˜ν2 = 1. This implies that all states have to respect the physicality condition ˜ν1˜ν2≥ 1 (see dashed black curves), whichisequivalentto ˜γ beingnon-negative( ˜ν1˜ν2 =ν1ν2 is ensured by symplectic matrices having uni...

  8. [8]

    How- ever, note that this criterion is biased in the sense that it only holds under the assumption that the underlying state is of Gaussian form

    Optimal PT-moment-based criterion for Gaussian entanglement When expressing Simon’s criterion(30) through the two lowest-order PT-momentsp2 and p3, we find that a Gaussian state is separable if and only if p3≥ 4p2 2 3 +p2 2 , (35) which constitutes the optimal third-order criterion for Gaussian entanglement; see straight black curves. How- ever, note that...

Show all 160 references
  1. [9]

    Mixed entangled cat states Non-Gaussianity naturally arises from mixtures of Gaussian states, while coherent superpositions are re- quired to generate genuine non-Gaussian states, for which Schrödinger cat states display an important exam- ple [78, 109]. In the following, we c...

  2. [10]

    High Harmonic Generation (HHG) Assessing entanglement by counting photons is par- ticularly relevant for applications where other standard readout methods are inapplicable. A prime example of re- cent interest is the process of High Harmonic Generation (HHG), in which intense ...

  3. [11]

    NOON states Entanglement detection is hardest when considering finite superpositions in Fock space. Of particular impor- tance is the class of bosonic NOON states [80] |ψ⟩ =α|N, 0⟩ +β|0,N⟩, (41) with mode population N ∈ N+ and phases α,β ∈ C constrained by|α|2 +|β|2 = 1 due to...

  4. [12]

    In contrast, higher-order NOON states with N > 2 were generated probabilistically by addi- tionally invoking post-selection, see [112–118]

    States of the form(41) have successfully been prepared deter- ministically forN≤ 2, e.g., using Hong-Ou-Mandel in- terferometry. In contrast, higher-order NOON states with N > 2 were generated probabilistically by addi- tionally invoking post-selection, see [112–118]. Although...

  5. [13]

    Some of them are typical bench- marks for entanglement criteria, while others are of par- ticular experimental relevance

    Other pure-state families In the following, we list a few important pure entangled states that belong to the class of genuine non-Gaussian states, which are all automatically witnessed by any third- order PPT-criterion. Some of them are typical bench- marks for entanglement cr...

  6. [14]

    Losses Particle losses are inevitable in the preparation pro- cess of a given input stateρ, the optical circuit, and the detection of single particles. With advances in supercon- ducting photon-number-resolving detectors, efficiencies in photonic setups are now on the order of...

  7. [15]

    The resulting variations among the state copies and subsequent experimental runs are commonly referred to as noise

    Noisy copies In real-world experiments, input-state and circuit pa- rameters can vary from one experimental realization to another. The resulting variations among the state copies and subsequent experimental runs are commonly referred to as noise. We account for both effects b...

  8. [16]

    Finite statistics In every experiment, an observable of interest is ulti- mately estimated from a finite number of repetitions of 11 the experiment, which can greatly influence the statistical significance of the expected outcome. In the context of entanglement detection, wher...

  9. [17]

    To this end, we first incorporate losses following(47), which renders the underlying state mixed, thereby impeding entanglement detection

    Analyses of individual errors We analyze the influence of all considered experimental imperfections on the prospects of witnessing higher-order NOON-state entanglement with our proposed method. To this end, we first incorporate losses following(47), which renders the underlyin...

  10. [18]

    NOON states up toN = 10 in Fig. 3a). As N increases, fewer losses are tolerable for the state’s entanglement to remain certified (see colored rings for the zero crossings). However, even for high occupations (N = 10), our method detects entanglement up to∼ 20% loss (note that ...

  11. [19]

    Full simulation To complete the picture, we combine all sources of imperfections by sampling from the photon-number- output distribution of the lossy and noisy Bell state (N = 1,α = 1/ √ 2). For each individual copy and each experimental run, we draw the phasesα and the losses...

  12. [20]

    From yellow to blue, the NOON state’s mode population numberN ranges from 1 to 10

    NOON states.a) Optimal p3-PPT criterion as a function of the loss parameterτ of the lossy channelLτ. From yellow to blue, the NOON state’s mode population numberN ranges from 1 to 10. Colored rings are placed at points where the criterion first flags entanglement, with a toler...

  13. [21]

    Two-mode squeezed state at finite temperature To illustrate the breakdown of the various PT-moment- based criteria for sufficiently mixed Gaussian states, we consider a two-mode squeezed state at finite tempera- ture. Starting from a product of two identical thermal states ρ¯n...

  14. [22]

    Higher-order criteria We analyze higher-order pn-PPT criteria from the Stieltjes approach [cf.(2)] for Gaussian states in Fig. 5. For the two-mode squeezed state at finite temperature considered in App. A1, thepn-PPT criteria forn = 5 (green curve) andn = 7 (red curve) flag en...

  15. [23]

    To this end, we normal order (41), for which we use the identity |0⟩⟨ 0| =:e−a†a :=∑ ja†jaj(−1)j/(j!)

    Mode-operator moments We compute the set of constraint mode-operator mo- ments for the class of NOON states. To this end, we normal order (41), for which we use the identity |0⟩⟨ 0| =:e−a†a :=∑ ja†jaj(−1)j/(j!). We end up with ρijkl =|α|2 (−1)i+k−N N! (i−N)!k!δijδklΘ(i−N) +|β|...

  16. [24]

    Then, the lossy NOON state reads ρ = (LτA⊗LτB) (|ψ⟩⟨ψ|), (D5) where|ψ⟩ is the lossless NOON state as defined in(41)

    Lossy NOON states Since we account for noisy copies by fluctuating state parameters, we consider in general different losses for Alice and Bob. Then, the lossy NOON state reads ρ = (LτA⊗LτB) (|ψ⟩⟨ψ|), (D5) where|ψ⟩ is the lossless NOON state as defined in(41). Direct calculati...

  17. [25]

    1b)] and three [see Fig

    Particle-number output distributions We show the particle-number output distributions f2(NA 2 ,N B 2 ) and f3(NA 2 ,N B 2 ,N A 3 ,N B 3 ) of the Fourier in- terferometers of orders two [see Fig. 1b)] and three [see Fig. 1c)] in Tab. I and Tab. II, respectively. Again, we prese...

  18. [26]

    a) showsf2 from the purity circuit, see Tab

    NOON state of orderN = 1 for various lossesτ. a) showsf2 from the purity circuit, see Tab. I for analytic expressions.b) and c) depict f3 from thep3-circuit for subsystemA and in the second mode, respectively, see Tab. II. When the total particle number at the output exceeds i...

  19. [27]

    NOON state with mode populationN = 1, considered for the three typical lossesτ = 0.9, 0.75, 0.6 (blue, red, green) together with the lossless caseτ = 1 (black). See Fig. 4 for reference. linear quadratic optimal 0 0.2 0.4 0.6 0.8 1 0.01 0.02 0.03 0.04 τ 1σ Figure 8. Complement...

  20. [28]

    VA3 and the numerical sampling of the third-order criteria for the noiseless and balanced (α = 1/ √

    Statistical error models We validate the consistency between the statistical er- ror models discussed in Sec. VA3 and the numerical sampling of the third-order criteria for the noiseless and balanced (α = 1/ √

  21. [29]

    NOON state of orderN = 1 in Fig. 7. As a complementary check, we plot the standard deviations obtained in(52) relative to the lossτ in Fig. 8. The numerical data aligns well with the theoretical pre- dictions for the linear and quadratic criteria. The optimal criterion matches...

  22. [30]

    Einstein, B

    A. Einstein, B. Podolsky, and N. Rosen, Can Quantum- Mechanical Description of Physical Reality Be Consid- ered Complete?, Phys. Rev.47, 777 (1935)

    1935

  23. [31]

    J. S. Bell, On the Einstein Podolsky Rosen paradox, Phys. Phys. Fiz.1, 195 (1964)

    1964

  24. [32]

    J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed Experiment to Test Local Hidden-Variable Theories, Phys. Rev. Lett.23, 880 (1969)

    1969

  25. [33]

    Aspect, J

    A. Aspect, J. Dalibard, and G. Roger, Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett.49, 1804 (1982)

    1982

  26. [34]

    Weihs, T

    G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Violation of Bell’s Inequality under Strict Einstein Locality Conditions, Phys. Rev. Lett.81, 5039 (1998)

    1998

  27. [35]

    B. G. Christensen, K. T. McCusker, J. B. Altepeter, B. Calkins, T. Gerrits, A. E. Lita, A. Miller, L. K. Shalm, Y. Zhang, S. W. Nam, N. Brunner, C. C. W. Lim, N. Gisin, and P. G. Kwiat, Detection-Loophole-Free Test of Quantum Nonlocality, and Applications, Phys. Rev. Lett.111,...

    2013

  28. [36]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. Theo. Exp.2004, P06002 (2004)

    2004

  29. [37]

    Casini and M

    H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A Math. Theo.42, 504007 (2009)

    2009

  30. [38]

    Eisert, M

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

    2010

  31. [39]

    D. A. Abanin and E. Demler, Measuring Entanglement Entropy of a Generic Many-Body System with a Quan- tum Switch, Phys. Rev. Lett.109, 020504 (2012)

    2012

  32. [40]

    Islam, R

    R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukinand, M. Rispoli, and M. Greiner, Measuring entanglement entropy in a quantum many-body system, Nature528, 77 (2015)

    2015

  33. [41]

    A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Quantum ther- malization through entanglement in an isolated many- body system, Science353, 794 (2016)

    2016

  34. [42]

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

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

    2024 arXiv

  35. [43]

    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, arXiv:2404.12321 (2024)

    2024 arXiv

  36. [44]

    Deller, M

    Y. Deller, M. Gärttner, T. Haas, M. K. Oberthaler, M. Reh, and H. Strobel, Area laws for classical entropies in a spin-1 Bose-Einstein condensate, arXiv:2404.12323 (2024)

    2024 arXiv

  37. [45]

    Bombelli, R

    L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, Quantum source of entropy for black holes, Phys. Rev. D 34, 373 (1986)

    1986

  38. [46]

    Srednicki, Entropy and area, Phys

    M. Srednicki, Entropy and area, Phys. Rev. Lett.71, 666 (1993)

    1993

  39. [47]

    Callan and F

    C. Callan and F. Wilczek, On geometric entropy, Phys. Lett. B333, 55 (1994)

    1994

  40. [48]

    S. N. Solodukhin, Entanglement Entropy of Black Holes, Living Rev. Relativ.14, 8 (2011)

    2011

  41. [49]

    Vedral, M

    V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Quantifying Entanglement, Phys. Rev. Lett.78, 2275 (1997)

    1997

  42. [50]

    M. B. Plenio and S. Virmani, An Introduction to Entan- glement Measures, Quantum Inf. Comput.7, 1 (2007)

    2007

  43. [51]

    Chitambar and G

    E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys.91, 025001 (2019)

    2019

  44. [52]

    Jozsa and N

    R. Jozsa and N. Linden, On the Role of Entanglement in Quantum-Computational Speed-Up, Proc. Math. Phys. Eng. Sci.459, 2011 (2003)

    2003

  45. [53]

    Knill, R

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

    2001

  46. [54]

    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

  47. [55]

    Slussarenko and G

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

    2019

  48. [56]

    Moody, V

    G. Moody, V. J. Sorger, D. J. Blumenthal, P. W. Juodawlkis, W. Loh, C. Sorace-Agaskar, A. E. Jones, K. C. Balram, J. C. F. Matthews, A. Laing, M. Da- vanco, L. Chang, J. E. Bowers, N. Quack, C. Gal- land, I. Aharonovich, M. A. Wolff, C. Schuck, N. Sin- clair, M. Lončar, T. Kom...

    2022

  49. [57]

    Romero and G

    J. Romero and G. Milburn, Photonic Quantum Comput- ing, arXiv:2404.03367 (2024)

    2024

  50. [58]

    Bloch, J

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

    2012

  51. [59]

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

    2014

  52. [60]

    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, M. Greiner, K. R. Hazzard, R. G. Hulet, A. J. Kollár, B. L. Lev, M. D. Lukin, R. Ma, X. Mi, S. Misra, C. Monroe, K. Murch, Z. Nazario, K.-K. Ni, A. C...

    2021

  53. [61]

    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 curved spacetime, Nature611, 260 (2022)

    2022

  54. [62]

    Tolosa-Simeón, A

    M. Tolosa-Simeón, A. 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- 19 erchinger, Curved and expanding spacetime geometries in Bose-Einstein condensates, Phys. Rev. A106, 033313 (2022)

    2022

  55. [63]

    G. M. D’Ariano, P. Lo Presti, and M. G. A. Paris, Us- ing Entanglement Improves the Precision of Quantum Measurements, Phys. Rev. Lett.87, 270404 (2001)

    2001

  56. [64]

    M. G. A. Paris, Quantum estimation for quantum tech- nology, Int. J. Quantum Inf.07, 125 (2009)

    2009

  57. [65]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nat. Photon.5, 222 (2011)

    2011

  58. [66]

    Pezzè, A

    L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys.90, 035005 (2018)

    2018

  59. [67]

    Horodecki, P

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

    2009

  60. [68]

    Gühne and G

    O. Gühne and G. Tóth, Entanglement detection, Phys. Rep. 474, 1 (2009)

    2009

  61. [69]

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

    2005

  62. [70]

    N. J. Cerf, G. Leuchs, and E. S. Polzik,Quantum Infor- mation with Continuous Variables of Atoms and Light (Imperial College Press, 2007)

    2007

  63. [71]

    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

  64. [72]

    Serafini,Quantum Continuous Variables(CRC Press, 2017)

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

    2017

  65. [73]

    R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, Experimental entangle- ment distillation of mesoscopic quantum states, Nat. Phys. 4, 919 (2008)

    2008

  66. [74]

    Schneeloch, C

    J. Schneeloch, C. C. Tison, M. L. Fanto, P. M. Alsing, and G. A. Howland, Quantifying entanglement in a 68- billion-dimensional quantum state space, Nat. Commun. 10, 1 (2019)

    2019

  67. [75]

    Asavanant, Y

    W. Asavanant, Y. Shiozawa, S. Yokoyama, B. Charoen- sombutamon, H. Emura, R. Alexander, S. Takeda, J.-i. Yoshikawa, N. Menicucci, H. Yonezawa, and A. Fu- rusawa, Generation of time-domain-multiplexed two- dimensional cluster state, Science366, 373 (2019)

    2019

  68. [76]

    Z. Qin, M. Gessner, Z. Ren, X. Deng, D. Han, W. Li, X. Su, A. Smerzi, and K. Peng, Characterizing the multi- partite continuous-variable entanglement structure from squeezing coefficients and the Fisher information, npj Quantum Inf.5, 3 (2019)

    2019

  69. [77]

    Gross, H

    C. Gross, H. Strobel, E. Nicklas, T. Zibold, N. Bar-Gill, G. Kurizki, and M. K. Oberthaler, Atomic homodyne detection of continuous-variable entangled twin-atom states, Nature480, 219 (2011)

    2011

  70. [78]

    Strobel, W

    H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B. Hume, L. Pezzè, A. Smerzi, and M. K. Oberthaler, Fisher information and entanglement of non-Gaussian spin states, Science345, 424 (2014)

    2014

  71. [79]

    Peise, I

    J. Peise, I. Kruse, K. Lange, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, K. Hammerer, L. Santos, A. Smerzi, and C. Klempt, Satisfying the Einstein-Podolsky-Rosen cri- terion with massive particles, Nat. Commun.6, 8984 (2015)

    2015

  72. [80]

    Kunkel, M

    P. Kunkel, M. Prüfer, H. Strobel, D. Linnemann, A. Frölian, T. Gasenzer, M. Gärttner, and M. K. Oberthaler, Spatially distributed multipartite entangle- ment enables EPR steering of atomic clouds, Science 360, 413 (2018)

    2018

  73. [81]

    Lange, J

    K. Lange, J. Peise, B. Lücke, I. Kruse, G. Vitagliano, I. Apellaniz, M. Kleinmann, G. Tóth, and C. Klempt, Entanglement between two spatially separated atomic modes, Science360, 416 (2018)

    2018

  74. [82]

    Fadel, T

    M. Fadel, T. Zibold, B. Décamps, and P. Treutlein, Spatial entanglement patterns and Einstein-Podolsky- Rosen steering in Bose-Einstein condensates, Science 360, 409 (2018)

    2018

  75. [83]

    Peres, Separability Criterion for Density Matrices, Phys

    A. Peres, Separability Criterion for Density Matrices, Phys. Rev. Lett.77, 1413 (1996)

    1996

  76. [84]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Separa- bility of mixed states: necessary and sufficient conditions, Phys. Lett. A223, 1 (1996)

    1996

  77. [85]

    L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, In- separability Criterion for Continuous Variable Systems, Phys. Rev. Lett.84, 2722 (2000)

    2000

  78. [86]

    Simon, Peres-Horodecki Separability Criterion for Continuous Variable Systems, Phys

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

    2000

  79. [87]

    R. F. Werner and M. M. Wolf, Bound Entangled Gaus- sian States, Phys. Rev. Lett.86, 3658 (2001)

    2001

  80. [88]

    Mancini, V

    S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, Entangling Macroscopic Oscillators Exploiting Radiation Pressure, Phys. Rev. Lett.88, 120401 (2002)

    2002

  81. [89]

    Giovannetti, S

    V. Giovannetti, S. Mancini, D. Vitali, and P. Tombesi, Characterizing the entanglement of bipartite quantum systems, Phys. Rev. A67, 022320 (2003)

    2003

  82. [90]

    L. Lami, A. Serafini, and G. Adesso, Gaussian entangle- ment revisited, New J. Phys.20, 023030 (2018)

    2018

  83. [91]

    S. P. Walborn, B. G. Taketani, A. Salles, F. Toscano, and R. L. de Matos Filho, Entropic Entanglement Criteria for Continuous Variables, Phys. Rev. Lett.103, 160505 (2009)

    2009

  84. [92]

    S. P. Walborn, A. Salles, R. M. Gomes, F. Toscano, and P. H. S. Ribeiro, Revealing Hidden Einstein-Podolsky- Rosen Nonlocality, Phys. Rev. Lett.106, 130402 (2011)

    2011

  85. [93]

    Saboia, F

    A. Saboia, F. Toscano, and S. P. Walborn, Family of continuous-variable entanglement criteria using general entropy functions, Phys. Rev. A83, 032307 (2011)

    2011

  86. [94]

    D. S. Tasca, Ł. Rudnicki, R. M. Gomes, F. Toscano, and S. P. Walborn, Reliable Entanglement Detection under Coarse-Grained Measurements, Phys. Rev. Lett.110, 210502 (2013)

    2013

  87. [95]

    Schneeloch and G

    J. Schneeloch and G. A. Howland, Quantifying high- dimensional entanglement with Einstein-Podolsky-Rosen correlations, Phys. Rev. A97, 042338 (2018)

    2018

  88. [96]

    Floerchinger, T

    S. Floerchinger, T. Haas, and H. Müller-Groeling, Wehrl entropy, entropic uncertainty relations, and entangle- ment, Phys. Rev. A103, 062222 (2021)

    2021

  89. [97]

    Floerchinger, M

    S. Floerchinger, M. Gärttner, T. Haas, and O. R. Stock- dale, Entropic entanglement criteria in phase space, Phys. Rev. A105, 012409 (2022)

    2022

  90. [98]

    Gärttner, T

    M. Gärttner, T. Haas, and J. Noll, Detecting continuous- variable entanglement in phase space with theQ distri- bution, Phys. Rev. A108, 042410 (2023)

    2023

  91. [99]

    Gärttner, T

    M. Gärttner, T. Haas, and J. Noll, General Class of Continuous Variable Entanglement Criteria, Phys. Rev. Lett. 131, 150201 (2023)

    2023

  92. [100]

    Shchukin and W

    E. Shchukin and W. Vogel, Inseparability Criteria for Continuous Bipartite Quantum States, Phys. Rev. Lett. 95, 230502 (2005). 20

    2005

  93. [101]

    Inseparabil- ity Criteria for Continuous Bipartite Quantum States

    A. Miranowicz and M. Piani, Comment on “Inseparabil- ity Criteria for Continuous Bipartite Quantum States”, Phys. Rev. Lett.97, 058901 (2006)

    2006

  94. [102]

    Miranowicz, M

    A. Miranowicz, M. Piani, P. Horodecki, and R. Horodecki, Inseparability criteria based on matrices of moments, Phys. Rev. A80, 052303 (2009)

    2009

  95. [103]

    Eisert, S

    J. Eisert, S. Scheel, and M. B. Plenio, Distilling Gaussian states with Gaussian operations is impossible, Phys. Rev. Lett. 89, 137903 (2002)

    2002

  96. [104]

    Fiurášek, Gaussian transformations and distillation of entangled Gaussian states, Phys

    J. Fiurášek, Gaussian transformations and distillation of entangled Gaussian states, Phys. Rev. Lett.89, 137904 (2002)

    2002

  97. [105]

    Giedke and J

    G. Giedke and J. Ignacio Cirac, Characterization of Gaussian operations and distillation of Gaussian states, Phys. Rev. A66, 032316 (2002)

    2002

  98. [106]

    Niset, J

    J. Niset, J. Fiurášek, and N. J. Cerf, No-go theorem for Gaussian quantum error correction, Phys. Rev. Lett. 102, 120501 (2009)

    2009

  99. [107]

    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

  100. [108]

    Chabaud and M

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

    2023

  101. [109]

    B. C. Sanders, Quantum dynamics of the nonlinear rota- tor and the effects of continual spin measurement, Phys. Rev. A40, 2417 (1989)

    1989

  102. [110]

    J. P. Dowling, Quantum optical metrology - the lowdown on high-N00N states, Contemp. Phys.49, 125 (2008)

    2008

  103. [111]

    Elben, R

    A. Elben, R. Kueng, H.-Y. R. Huang, R. van Bij- nen, C. Kokail, M. Dalmonte, P. Calabrese, B. Kraus, J. Preskill, P. Zoller, and B. Vermersch, Mixed-State Entanglement from Local Randomized Measurements, Phys. Rev. Lett.125, 200501 (2020)

    2020

  104. [112]

    Neven, J

    A. Neven, J. Carrasco, V. Vitale, C. Kokail, A. Elben, M. Dalmonte, P. Calabrese, P. Zoller, B. Vermersch, R. Kueng, and B. Kraus, Symmetry-resolved entangle- ment detection using partial transpose moments, npj Quantum Inf.7, 152 (2021)

    2021

  105. [113]

    X.-D. Yu, S. Imai, and O. Gühne, Optimal entanglement certification from moments of the partial transpose, Phys. Rev. Lett.127, 060504 (2021)

    2021

  106. [114]

    T. A. Brun, Measuring polynomial functions of states, Quant. Inf. Comp.4, 401 (2004)

    2004

  107. [115]

    Hertz, O

    A. Hertz, O. Oreshkov, and N. J. Cerf, Multicopy uncer- tainty observable inducing a symplectic-invariant uncer- tainty relation in position and momentum phase space, Phys. Rev. A100, 052112 (2019)

    2019

  108. [116]

    Arnhem, C

    M. Arnhem, C. Griffet, and N. J. Cerf, Multicopy observ- ables for the detection of optically nonclassical states, Phys. Rev. A106, 043705 (2022)

    2022

  109. [117]

    Griffet, M

    C. Griffet, M. Arnhem, S. De Bièvre, and N. J. Cerf, In- terferometric measurement of the quadrature coherence scale using two replicas of a quantum optical state, Phys. Rev. A108, 022421 (2023)

    2023

  110. [118]

    Griffet, T

    C. Griffet, T. Haas, and N. J. Cerf, Accessing continuous- variable entanglement witnesses with multimode spin observables, Phys. Rev. A108, 022421 (2023)

    2023

  111. [119]

    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

  112. [120]

    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

  113. [121]

    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, Nature17, 112 (2023)

    2023

  114. [122]

    W. Bakr, J. Gillen, A. Peng, S. Fölling, and M. Greiner, A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice, Nature462, 74 (2009)

    2009

  115. [123]

    Sherson, C

    J. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and K. S., Single-atom-resolved fluorescence imaging of an atomic Mott insulator, Nature467, 68 (2010)

    2010

  116. [124]

    D. B. Hume, I. Stroescu, M. Joos, W. Muessel, H. Stro- bel, and M. K. Oberthaler, Accurate Atom Counting in Mesoscopic Ensembles, Phys. Rev. Lett.111, 253001 (2013)

    2013

  117. [125]

    Gross and W

    C. Gross and W. S. Bakr, Quantum gas microscopy for single atom and spin detection, Nat. Phys.17, 1316 (2021)

    2021

  118. [126]

    A. K. Ekert, C. M. Alves, D. K. L. Oi, M. Horodecki, P. Horodecki, and L. C. Kwek, Direct Estimations of Linear and Nonlinear Functionals of a Quantum State, Phys. Rev. Lett.88, 217901 (2002)

    2002

  119. [127]

    C. M. Alves, P. Horodecki, D. K. L. Oi, L. C. Kwek, and A. K. Ekert, Direct estimation of functionals of density operators by local operations and classical communica- tion, Phys. Rev. A68, 032306 (2003)

    2003

  120. [128]

    A. J. Daley, H. Pichler, J. Schachenmayer, and P. Zoller, Measuring Entanglement Growth in Quench Dynamics of Bosons in an Optical Lattice, Phys. Rev. Lett.109, 020505 (2012)

    2012

  121. [129]

    J. Gray, L. Banchi, A. Bayat, and S. Bose, Machine- Learning-Assisted Many-Body Entanglement Measure- ment, Phys. Rev. Lett.121, 150503 (2018)

    2018

  122. [130]

    Fuchs, J

    S. Fuchs, J. J. Abel, J. Nathanael, J. Reinhard, F. Wies- ner, M. Wünsche, S. Skruszewicz, C. Rödel, D. Born, H. Schmidt, and G. G. Paulus, Photon counting of ex- treme ultraviolet high harmonics using a superconduct- ing nanowire single-photon detector, Appl. Phys. B128, 26 (2022)

    2022

  123. [131]

    Hu, X.-M

    M.-J. Hu, X.-M. Hu, and Y.-S. Zhang, Are observables necessarily Hermitian?, Quantum Stud.: Math. Found. 4, 243 (2017)

    2017

  124. [132]

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

    1994

  125. [133]

    N. J. Cerf, C. Adami, and P. G. Kwiat, Optical simula- tion of quantum logic, Phys. Rev. A57, R1477 (1998)

    1998

  126. [134]

    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

  127. [135]

    A. Z. Goldberg, G. S. Thekkadath, and K. Heshami, Measuring the quadrature coherence scale on a cloud quantum computer, Phys. Rev. A107, 042610 (2023)

    2023

  128. [136]

    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

  129. [137]

    K. P. Seshadreesan, L. Lami, and M. M. Wilde, Rényi relative entropies of quantum Gaussian states, J. Math. Phys. 59, 072204 (2018)

    2018

  130. [138]

    Yurke and D

    B. Yurke and D. Stoler, Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion, Phys. Rev. Lett.57, 13 (1986). 21

    1986

  131. [139]

    W.Peters,andG.Weinreich, Generation of Optical Harmonics, Phys

    P.A.Franken, A.E.Hill, C. W.Peters,andG.Weinreich, Generation of Optical Harmonics, Phys. Rev. Lett.7, 118 (1961)

    1961

  132. [140]

    Stammer, J

    P. Stammer, J. Rivera-Dean, T. Lamprou, E. Pisanty, M. F. Ciappina, P. Tzallas, and M. Lewenstein, High Photon Number Entangled States and Coherent State Superposition from the Extreme Ultraviolet to the Far Infrared, Phys. Rev. Lett.128, 123603 (2022)

    2022

  133. [141]

    Walther, J.-W

    P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gas- paroni, and A. Zeilinger, De Broglie wavelength of a non-local four-photon state, Nature429, 158 (2004)

    2004

  134. [142]

    M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, Super-resolving phase measurements with a multiphoton entangled state, Nature429, 161 (2004)

    2004

  135. [143]

    K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, Time- Reversal and Super-Resolving Phase Measurements, Phys. Rev. Lett.98, 223601 (2007)

    2007

  136. [144]

    Nagata, R

    T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, Beating the Standard Quantum Limit with Four-Entangled Photons, Science316, 726 (2007)

    2007

  137. [145]

    I. Afek, O. Ambar, and Y. Silberberg, High-NOON States by Mixing Quantum and Classical Light, Science 328, 879 (2010)

    2010

  138. [146]

    Israel, I

    Y. Israel, I. Afek, S. Rosen, O. Ambar, and Y. Silberberg, Experimental tomography of NOON states with large photon numbers, Phys. Rev. A85, 022115 (2012)

    2012

  139. [147]

    Slussarenko, M

    S. Slussarenko, M. M. Weston, H. M. Chrzanowski, L. K. Shalm, V. B. Verma, S. W. Nam, and G. J. Pryde, Un- conditional violation of the shot-noise limit in photonic quantum metrology, Nat. Photonics11, 700 (2017)

    2017

  140. [148]

    Callus, M

    E. Callus, M. Gärttner, and T. Haas, Interferometric en- tanglement detection using two states, arXiv:2502.14700 (2025)

    2025 arXiv

  141. [149]

    C. Rodó, G. Adesso, and A. Sanpera, Operational Quan- tification of Continuous-Variable Correlations, Phys. Rev. Lett. 100, 110505 (2008)

    2008

  142. [150]

    Hertz, E

    A. Hertz, E. Karpov, A. Mandilara, and N. J. Cerf, Detection of non-Gaussian entangled states with an im- proved continuous-variable separability criterion, Phys. Rev. A93, 032330 (2016)

    2016

  143. [151]

    Aaronson and A

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

    2013

  144. [152]

    Rudolph, Further Results on the Cross Norm Cri- terion for Separability, Quantum Inf

    O. Rudolph, Further Results on the Cross Norm Cri- terion for Separability, Quantum Inf. Process.4, 219 (2005)

    2005

  145. [153]

    Zhang, S

    C. Zhang, S. Yu, Q. Chen, and C. H. Oh, Detecting and estimating continuous-variable entanglement by local orthogonal observables, Phys. Rev. Lett.111, 190501 (2013)

    2013

  146. [154]

    Hertz, M

    A. Hertz, M. Arnhem, A. Asadian, and N. J. Cerf, Re- alignment separability criterion assisted with filtration for detecting continuous-variable entanglement, Phys. Rev. A104, 022427 (2021)

    2021

  147. [155]

    Aggarwal and S

    S. Aggarwal and S. Adhikari, Theoretical proposal for the experimental realization of realignment operation, Quantum Inf. Process.23, 223 (2024)

    2024

  148. [156]

    Horodecki, J

    P. Horodecki, J. I. Cirac, and M. Lewenstein, Bound entanglement for continuous variables is a rare phe- nomenon, in Quantum Information with Continuous Variables, edited by S. L. Braunstein and A. K. Pati (Springer Netherlands, Dordrecht, 2003) pp. 211–228

    2003

  149. [157]

    Jungnitsch, T

    B. Jungnitsch, T. Moroder, and O. Gühne, Taming Mul- tiparticle Entanglement, Phys. Rev. Lett.106, 190502 (2011)

    2011

  150. [158]

    Dubus, T

    B. Dubus, T. Haas, and N. J. Cerf, From bosons and fermions to spins: A multi-mode extension of the Jordan- Schwinger map, arXiv:2411.04918 (2024)

    2024 arXiv

  151. [159]

    Gandhari, V

    S. Gandhari, V. V. Albert, T. Gerrits, J. M. Taylor, and M. J. Gullans, Precision Bounds on Continuous- Variable State Tomography Using Classical Shadows, PRX Quantum5, 010346 (2024)

    2024

  152. [160]

    Becker, N

    S. Becker, N. Datta, L. Lami, and C. Rouze, Classical Shadow Tomography for Continuous Variables Quantum Systems, IEEE Trans. Inf. Theory70, 3427 (2024)

    2024

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.