---
title: Approximate Trisqueezed States
url: https://www.emergentmind.com/topics/approximate-trisqueezed-states
type: topic
---

# Approximate Trisqueezed States

Searching arXiv for the cited trisqueezed-state papers and closely related context.
Approximate trisqueezed states are finite-energy realizations or approximations of the ideal single-mode states generated by a third-order squeezing-like unitary,
\[
\ket{t}=e^{i(t^*\hat a^3+t\hat a^{\dagger 3})}\ket{0},
\]
or, in the generalized-squeezing notation,
\[
\ket{\zeta_3}=\exp\!\Bigl[-i\frac{\zeta_3}{2}(\hat a^3+\hat a^{\dagger 3})\Bigr]\ket{0}.
\]
They are non-Gaussian resource states whose structure is set by cubic bosonic nonlinearities and whose experimentally relevant and resource-theoretic significance has been analyzed from two complementary perspectives: deterministic Gaussian conversion into other non-Gaussian resources, especially cubic-phase states [2204.03373], and direct experimental generation of trisqueezed states and their superpositions in a trapped-ion harmonic oscillator [2409.03482]. In both settings, “approximate” refers not to a different formal definition, but to finite-parameter, finite-energy, or noisy realizations that emulate either the ideal trisqueezed state itself or another target resource state only up to finite fidelity.

## 1. Formal definition and state structure

The trisqueezed state is defined in the single-mode bosonic Hilbert space by
\[
\ket{t}=e^{i(t^* \hat{a}^3 + t \hat{a}^{\dagger 3})}\ket{0},
\]
where \(t\in\mathbb C\) is the triplicity and \(\hat a,\hat a^\dagger\) are the annihilation and creation operators [2204.03373]. In the generalized-squeezing formulation, the relevant interaction is generated by
\[
\hat H_3 \propto \hat\sigma_z(\hat a^3 e^{-i\phi}+\hat a^{\dagger 3}e^{i\phi}),
\]
or, more generally,
\[
\hat{H}_k = \frac{\hbar \Omega_k}{2}\,\hat{\sigma}_z\Bigl(\hat{a}^k e^{-i\phi} + (\hat{a}^\dagger)^k e^{i\phi}\Bigr),
\]
with trisqueezing corresponding to \(k=3\) [2409.03482].

Both descriptions encode the same essential feature: the state is produced by a third-order nonlinear bosonic generator built from \(\hat a^3\) and \(\hat a^{\dagger 3}\). Since \(\hat a^3\) lowers photon or phonon number by \(3\), while \(\hat a^{\dagger 3}\) raises it by \(3\), the vacuum evolves only into Fock sectors compatible with this selection rule. A formal expansion therefore has the form
\[
\ket{t}=\sum_{n=0}^\infty c_{3n}(t)\ket{3n},
\]
and numerical simulations in the trapped-ion experiment likewise show Fock populations supported only on levels with occupation numbers spaced by \(3\) for a single trisqueezed state [2204.03373, 2409.03482].

A notable technical point is the absence of a general closed analytic Fock expansion. The trapped-ion study states that for the non-Gaussian constituents, including trisqueezed states, “it is unclear if closed-form solutions exist,” so practical work proceeds numerically in truncated Fock space [2409.03482]. The Gaussian-conversion study similarly handles the state numerically via characteristic functions and Wigner functions rather than closed-form coefficients [2204.03373].

## 2. Non-Gaussian resource character

Within continuous-variable resource theories, trisqueezed states are non-Gaussian resources because their Wigner functions are non-Gaussian, exhibit negativity, and cannot be generated from Gaussian states by Gaussian CPTP maps alone [2204.03373]. The relevant free operations are single-mode deterministic Gaussian maps, including Gaussian unitaries, noisy Gaussian channels, and more general one-mode Gaussian CPTP transformations characterized on the symmetric characteristic function by
\[
\chi_{\Phi(\hat{\rho})}(\vec{r}) = e^{-\frac{1}{4} \vec{r}^{T} \Omega^{T} Y \Omega \vec{r} + i\,\vec{l}^{T}\Omega\vec{r}}\,
\chi_{\hat{\rho}\!\left(\Omega^{T} X^{T} \Omega \vec{r}\right),
\]
subject to
\[
Y \pm i\bigl(\Omega - X\Omega X^T\bigr)\ge 0
\]
[2204.03373].

A central resource quantifier in both works is Wigner logarithmic negativity. In the Gaussian-conversion study it is defined as
\[
M(\hat{\rho})=\log\!\left(\int d\vec r\,\bigl|W_{\hat\rho}(\vec r)\bigr|\right),
\]
while the trapped-ion study uses the equivalent notation
\[
\mathbf{W}(\rho)=\log\left(\int dx\,dp\,\bigl|W_\rho(x,p)\bigr|\right)
\]
[2204.03373, 2409.03482]. In one case WLN is used to match trisqueezed and cubic-phase states at comparable non-Gaussian “strength” before attempting Gaussian conversion [2204.03373]; in the other, it is used to compare experimentally generated generalized-squeezed superpositions against other nonclassical resources, with the reported conclusion that “The superpositions created in this work exhibit larger WLN than Fock or cat states for the same \(\bar n\)” [2409.03482].

Phase-space structure reinforces this resource interpretation. The Gaussian-conversion study reports that the Wigner function of a trisqueezed state with \(t=0.1\) shows strongly non-Gaussian features and negativity, with interference patterns reminiscent of threefold rotational structures [2204.03373]. The trapped-ion experiment likewise reports discrete three-fold symmetry for \(\ket{\zeta_3}\) and sixfold patterns for superpositions \(\ket{\zeta_3}\pm\ket{-\zeta_3}\), together with multiple lobes and interference fringes [2409.03482]. This suggests that approximate trisqueezed states are naturally characterized not only by finite fidelity to an ideal target vector but also by preservation of symmetry, Wigner negativity, and higher-order interference structure.

## 3. Approximation via deterministic Gaussian conversion

A major sense in which approximate trisqueezed states arise is as finite-resource inputs for conversion into approximate cubic-phase states under deterministic Gaussian processing. The target cubic-phase state is
\[
\ket{c}=e^{ic\hat q^3}\hat S(\xi)\ket 0,
\]
with cubicity \(c\) and Gaussian squeezing \(\xi\) [2204.03373]. The conversion task is to maximize the fidelity
\[
\mathcal F(\hat\rho,\ket c\bra c)=\bra c\hat\rho\ket c
\]
over all one-mode Gaussian CPTP maps, equivalently
\[
\max_{X,Y,\vec l}\ \mathcal F\bigl(\Phi(\ket t\bra t),\ket c\bra c\bigr)
\quad\text{subject to}\quad
Y\pm i(\Omega-X\Omega X^T)\ge 0
\]
[2204.03373].

The parameter choice is not arbitrary. For each cubicity \(c\) and cubic-phase squeezing \(\xi\), the triplicity \(t\) of the trisqueezed input is chosen so that the Wigner logarithmic negativity of the input matches that of the target cubic-phase state [2204.03373]. The study examines
\[
c\in\{0.04,0.06,0.08,0.10,0.12,0.14,0.16\},
\qquad
\xi\in\{0,0.25,-0.5\},
\]
and reports the following matched triplicities [2204.03373]:

| \(\xi\) | \(c\) values | matched \(t\) values |
|---|---|---|
| \(0.25\) | \(0.04,0.06,0.08,0.10,0.12,0.14,0.16\) | \(0.048,0.063,0.075,0.084,0.0922,0.0988,0.104\) |
| \(0\) | \(0.04,0.06,0.08,0.10,0.12,0.14,0.16\) | \(0.027,0.037,0.0464,0.0543,0.061,0.067,0.073\) |
| \(-0.5\) | \(0.04,0.06,0.08,0.10,0.12,0.14,0.16\) | \(0.078,0.095,0.107,0.116,0.124,0.130,0.136\) |

The reported behavior is qualitative but clear. For small cubicities, especially \(c\lesssim 0.08\), optimized Gaussian processing yields relatively high fidelities, and the trisqueezed state functions as a good approximate cubic-phase state [2204.03373]. As cubicity increases, fidelity decreases monotonically because the cubic-phase target develops more complex oscillatory and negativity structure than a single trisqueezed resource can reproduce under deterministic Gaussian processing [2204.03373].

The study emphasizes that this improves on earlier schemes by scanning a broader parameter range and optimizing over the full Gaussian CPTP class rather than pre-chosen Gaussian unitaries alone [2204.03373]. At the same time, it explicitly does not claim exact equivalence between trisqueezed and cubic-phase states in any asymptotic sense within the finite-energy deterministic Gaussian framework. Approximation remains finite-fidelity and parameter dependent [2204.03373].

## 4. Shape of the optimal Gaussian protocol

Although the optimization is formulated over the full Gaussian CPTP set, the numerically found optimum in the trisqueezed-to-cubic-phase task is essentially unitary. The Gaussian-conversion study states that “the optimal protocol consists of squeezing and small displacements along the \(p\)-axis” [2204.03373]. Noise matrices \(Y\) are not emphasized, and the discussion indicates that added Gaussian noise is not beneficial in the explored regime because it degrades the non-Gaussian structure one is trying to preserve [2204.03373].

This operationally important point narrows the practical meaning of “approximate trisqueezed state” in conversion settings. The approximation is not primarily achieved by dissipative reshaping or Gaussian noise engineering; rather, it is achieved by selecting a finite triplicity \(t\) with matched WLN and then applying a relatively simple Gaussian unitary correction, principally single-mode squeezing plus modest momentum displacement [2204.03373].

A plausible implication is that, in experimentally accessible finite-energy regimes, the dominant mismatch between trisqueezed and cubic-phase states is geometric in phase space rather than a defect that can be repaired by noisy Gaussian channels. This reading is consistent with the observation that the best protocols are close to unitary Gaussian conversions [2204.03373].

The same work places these results in comparative context. Photon-added and photon-subtracted squeezed states plus Gaussian maps can approximate cat states extremely well, with fidelities up to \(F\approx 0.995\) for certain even cat states, whereas trisqueezed-to-cubic-phase conversion is described as good but not perfect and confined to a limited low-cubicity regime [2204.03373]. Approximate trisqueezed states therefore occupy a more constrained position among non-Gaussian resources: useful, but not universally interchangeable with canonical cubic-phase resources.

## 5. Experimental realization as approximate physical states

A second, distinct sense of approximation concerns laboratory generation. In the trapped-ion experiment, trisqueezed states are produced in the motion of a single \(^{88}\mathrm{Sr}^+\) ion in a 3D Paul trap, with the axial mode at \(\omega_z/2\pi=1.2\,\mathrm{MHz}\) serving as the harmonic oscillator [2409.03482]. The oscillator is cooled to near vacuum with mean occupation \(\bar n=0.1\), and nonlinear generalized squeezing interactions are engineered from two spin-dependent forces with noncommuting spin conditionings [2409.03482].

The effective generalized squeezing Hamiltonian is obtained from
\[
\hat H=\frac{\hbar\Omega_\alpha}{2}\hat\sigma_\alpha(\hat a e^{-i\Delta t}+\hat a^\dagger e^{i\Delta t})
+\frac{\hbar\Omega_{\alpha'}}{2}\hat\sigma_{\alpha'}\bigl(\hat a e^{-i(m\Delta t+\phi)}+\hat a^\dagger e^{i(m\Delta t+\phi)}\bigr),
\]
leading, after rotating-wave and Magnus analysis, to
\[
\hat{H}_k = \frac{\hbar \Omega_k}{2}\,\hat{\sigma}_\beta\bigl(\hat a^k e^{-i\phi} + \hat a^{\dagger k} e^{i\phi}\bigr),
\]
with
\[
\Omega_{2,3,4}
=
\left\{
\frac{\Omega_{\alpha'}\Omega_\alpha}{\Delta},
\frac{\Omega_{\alpha'}\Omega_\alpha^2}{2\Delta^2},
\frac{\Omega_{\alpha'}\Omega_\alpha^3}{8\Delta^3}
\right\}
\sin\theta_{\alpha,\alpha'}
\]
[2409.03482]. For \(k=3\), preparing the spin in an eigenstate of \(\hat\sigma_\beta\) yields the oscillator evolution
\[
\hat U_3(\zeta_3)=
\exp\!\Bigl[-i\frac{\zeta_3}{2}\bigl(\hat a^3 e^{-i\phi}+\hat a^{\dagger 3}e^{i\phi}\bigr)\Bigr],
\qquad
\zeta_3=\Omega_3 e^{i\phi}t
\]
[2409.03482].

A representative reported realization uses \(P=1\) mW per SDF, \(\Omega_{\alpha,\alpha'}\simeq 2\pi\times(0.85\text{–}0.86)\,\mathrm{kHz}\), \(\Delta/2\pi=25\,\mathrm{kHz}\), and interaction time \(t=500\,\mu\mathrm{s}\), producing \(|\zeta_3|\approx 0.74\) [2409.03482]. The experiment also generates even and odd superpositions \(\ket{\zeta_3}\pm\ket{-\zeta_3}\) by entangling the motion with the spin, applying a second spin rotation, and performing a mid-circuit spin measurement [2409.03482].

These laboratory states are approximate trisqueezed states because several nonidealities prevent exact realization of the pure ideal state. The paper explicitly identifies finite interaction time and limited coupling, initial thermal occupation, motional heating at \(\dot{\bar n}\approx 300\,\mathrm{quanta/s}\), finite detection fidelity \(0.993\), and finite-grid tomography artifacts [2409.03482]. The result is a mixed or slightly impure state rather than a pure trisqueezed vacuum. Nonetheless, the work reports very good qualitative agreement between numerical and experimental Wigner functions for \(|\zeta_3|\lesssim 0.7\), and states that the experimentally generated states are high-fidelity approximations to the ideal trisqueezed states in this regime [2409.03482].

## 6. Symmetry, superpositions, and practical significance

Approximate trisqueezed states are particularly notable for the symmetry and support structure inherited from third-order generalized squeezing. For a single trisqueezed state, Fock support is concentrated on occupations \(n\equiv 0\pmod 3\); for the even or odd superpositions \(\ket{\zeta_3}\pm\ket{-\zeta_3}\), the non-vanishing Fock occupations are spaced by \(2k=6\) [2409.03482]. The trapped-ion work emphasizes that, in general, “the non-vanishing Fock state occupations of the superpositions are spaced by \(2k\), where \(k\) is the order of the interaction” [2409.03482]. This discrete support is reflected in rotation symmetry of the Wigner function and is one reason these states are relevant to rotation-symmetric bosonic codes [2409.03482].

The experimental work goes beyond isolated trisqueezed states by generating arbitrary superpositions between different generalized-squeezed resources, including squeezed, trisqueezed, and quadsqueezed states, using a qutrit hiding and unhiding protocol [2409.03482]. For example, it reports an even superposition \(\ket{\zeta_2}+\ket{\zeta_3}\) with \(|\zeta_2|\approx 1.12\) and \(|\zeta_3|\approx 0.25\) [2409.03482]. This demonstrates that approximate trisqueezed states can be coherently embedded in larger non-Gaussian state engineering programs rather than treated only as standalone states.

Their significance in continuous-variable quantum information follows two lines. First, as the Gaussian-conversion study shows, finite-triplicity trisqueezed states can serve as approximate cubic-phase resources after Gaussian processing, especially in the low-cubicity regime [2204.03373]. Second, as the trapped-ion experiment shows, direct generation and coherent superposition of trisqueezed states is experimentally feasible, with strong Wigner negativity and structured phase-space interference [2409.03482]. The latter work further notes applications to continuous-variable quantum computation, hybrid processors, rotation-symmetric bosonic codes, and quantum-enhanced metrology [2409.03482].

A recurring caution concerns evaluation criteria. The Gaussian-conversion study remarks that fidelity alone can be misleading, since some conversions may score high fidelity while failing to reproduce important Wigner features or symmetry structure [2204.03373]. This suggests that approximate trisqueezed states should often be assessed by a combination of metrics: fidelity to an ideal target, Wigner logarithmic negativity, qualitative phase-space structure, and symmetry-preserving Fock support.

## 7. Limits and open directions

Current results delimit both the promise and the constraints of approximate trisqueezed states. In deterministic Gaussian conversion, the notable success is the trisqueezed-to-cubic-phase direction, but only for modest cubicity; no evidence is presented that increasing triplicity within the same finite-energy, single-mode, deterministic Gaussian framework yields arbitrarily accurate cubic-phase approximation [2204.03373]. The study explicitly focuses on realistic finite energies and does not analyze any \(t\to\infty\) scaling regime [2204.03373].

In direct state generation, the main restrictions come from effective coupling strengths, heating, and detection imperfections. The trapped-ion work identifies \(|\zeta_3|\approx 0.75\) as the rough upper scale achieved in the reported data, with larger values requiring longer interactions and therefore more heating and decoherence [2409.03482]. It also states that for moderate \(|\zeta_3|\sim 0.7\) and typical phonon numbers of a few, the rotating-wave approximation and finite-basis truncation are very accurate [2409.03482].

Several research directions are explicitly indicated. The Gaussian-conversion study suggests exploring multi-mode distillation and probabilistic protocols beyond deterministic Gaussian maps, as well as more refined resource measures and task-specific figures of merit [2204.03373]. It also leaves open the systematic study of fidelity scaling with energy and the effect on logical gate performance when converted trisqueezed states are used as gate resources [2204.03373]. The trapped-ion work, for its part, indicates practical improvement routes by increasing \(\Omega_3\), reducing heating and leakage light, improving initial cooling, and optimizing mid-circuit measurement [2409.03482].

Taken together, these results establish approximate trisqueezed states as a technically precise category rather than a loose descriptive label. They are either finite-energy trisqueezed resources used as approximate surrogates for other non-Gaussian states under Gaussian processing [2204.03373], or experimentally realized noisy approximations to the ideal third-order generalized-squeezed vacuum and its superpositions [2409.03482]. In both senses, their defining features are cubic bosonic generation, non-Gaussian phase-space structure, discrete rotational symmetry, and a role as experimentally motivated non-Gaussian resources whose utility is substantial but sharply regime dependent.

Source: https://www.emergentmind.com/topics/approximate-trisqueezed-states