Approximate Trisqueezed States

• Approximate trisqueezed states are finite-energy realizations of third-order squeezed vacua, generated via cubic bosonic nonlinearity and exhibiting Fock support in multiples of three.
• They serve as non-Gaussian resources in continuous-variable quantum information, with pronounced Wigner negativity that enables deterministic Gaussian conversion into cubic-phase states.
• Experimental implementations in trapped-ion systems validate these states despite finite-energy constraints, paving the way for applications in quantum computation and bosonic error correction.

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 (Hahn et al., 2022), and direct experimental generation of trisqueezed states and their superpositions in a trapped-ion harmonic oscillator (Saner et al., 2024). 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 (Hahn et al., 2022). 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 $k=3$ (Saner et al., 2024).

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 $$\ket{\zeta_3}=\exp\!\Bigl[-i\frac{\zeta_3}{2}(\hat a^3+\hat a^{\dagger 3})\Bigr]\ket{0}.$$0 lowers photon or phonon number by $$\ket{\zeta_3}=\exp\!\Bigl[-i\frac{\zeta_3}{2}(\hat a^3+\hat a^{\dagger 3})\Bigr]\ket{0}.$$1, while $$\ket{\zeta_3}=\exp\!\Bigl[-i\frac{\zeta_3}{2}(\hat a^3+\hat a^{\dagger 3})\Bigr]\ket{0}.$$2 raises it by $$\ket{\zeta_3}=\exp\!\Bigl[-i\frac{\zeta_3}{2}(\hat a^3+\hat a^{\dagger 3})\Bigr]\ket{0}.$$3, the vacuum evolves only into Fock sectors compatible with this selection rule. A formal expansion therefore has the form

$$\ket{\zeta_3}=\exp\!\Bigl[-i\frac{\zeta_3}{2}(\hat a^3+\hat a^{\dagger 3})\Bigr]\ket{0}.$$4

and numerical simulations in the trapped-ion experiment likewise show Fock populations supported only on levels with occupation numbers spaced by $$\ket{\zeta_3}=\exp\!\Bigl[-i\frac{\zeta_3}{2}(\hat a^3+\hat a^{\dagger 3})\Bigr]\ket{0}.$$5 for a single trisqueezed state (Hahn et al., 2022, Saner et al., 2024).

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 (Saner et al., 2024). The Gaussian-conversion study similarly handles the state numerically via characteristic functions and Wigner functions rather than closed-form coefficients (Hahn et al., 2022).

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 (Hahn et al., 2022). 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

$$\ket{\zeta_3}=\exp\!\Bigl[-i\frac{\zeta_3}{2}(\hat a^3+\hat a^{\dagger 3})\Bigr]\ket{0}.$$6

subject to

$$\ket{\zeta_3}=\exp\!\Bigl[-i\frac{\zeta_3}{2}(\hat a^3+\hat a^{\dagger 3})\Bigr]\ket{0}.$$7

(Hahn et al., 2022).

A central resource quantifier in both works is Wigner logarithmic negativity. In the Gaussian-conversion study it is defined as

$$\ket{\zeta_3}=\exp\!\Bigl[-i\frac{\zeta_3}{2}(\hat a^3+\hat a^{\dagger 3})\Bigr]\ket{0}.$$8

while the trapped-ion study uses the equivalent notation

$$\ket{\zeta_3}=\exp\!\Bigl[-i\frac{\zeta_3}{2}(\hat a^3+\hat a^{\dagger 3})\Bigr]\ket{0}.$$9

(Hahn et al., 2022, Saner et al., 2024). In one case WLN is used to match trisqueezed and cubic-phase states at comparable non-Gaussian “strength” before attempting Gaussian conversion (Hahn et al., 2022); 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 $$\ket{t}=e^{i(t^* \hat{a}^3 + t \hat{a}^{\dagger 3})}\ket{0},$$0” (Saner et al., 2024).

Phase-space structure reinforces this resource interpretation. The Gaussian-conversion study reports that the Wigner function of a trisqueezed state with $$\ket{t}=e^{i(t^* \hat{a}^3 + t \hat{a}^{\dagger 3})}\ket{0},$$1 shows strongly non-Gaussian features and negativity, with interference patterns reminiscent of threefold rotational structures (Hahn et al., 2022). The trapped-ion experiment likewise reports discrete three-fold symmetry for $$\ket{t}=e^{i(t^* \hat{a}^3 + t \hat{a}^{\dagger 3})}\ket{0},$$2 and sixfold patterns for superpositions $$\ket{t}=e^{i(t^* \hat{a}^3 + t \hat{a}^{\dagger 3})}\ket{0},$$3, together with multiple lobes and interference fringes (Saner et al., 2024). 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

$$\ket{t}=e^{i(t^* \hat{a}^3 + t \hat{a}^{\dagger 3})}\ket{0},$$4

with cubicity $$\ket{t}=e^{i(t^* \hat{a}^3 + t \hat{a}^{\dagger 3})}\ket{0},$$5 and Gaussian squeezing $$\ket{t}=e^{i(t^* \hat{a}^3 + t \hat{a}^{\dagger 3})}\ket{0},$$6 (Hahn et al., 2022). The conversion task is to maximize the fidelity

$$\ket{t}=e^{i(t^* \hat{a}^3 + t \hat{a}^{\dagger 3})}\ket{0},$$7

over all one-mode Gaussian CPTP maps, equivalently

$$\ket{t}=e^{i(t^* \hat{a}^3 + t \hat{a}^{\dagger 3})}\ket{0},$$8

(Hahn et al., 2022).

The parameter choice is not arbitrary. For each cubicity $$\ket{t}=e^{i(t^* \hat{a}^3 + t \hat{a}^{\dagger 3})}\ket{0},$$9 and cubic-phase squeezing $t\in\mathbb C$0, the triplicity $t\in\mathbb C$1 of the trisqueezed input is chosen so that the Wigner logarithmic negativity of the input matches that of the target cubic-phase state (Hahn et al., 2022). The study examines

$t\in\mathbb C$2

and reports the following matched triplicities (Hahn et al., 2022):

$t\in\mathbb C$3	$t\in\mathbb C$4 values	matched $t\in\mathbb C$5 values
$t\in\mathbb C$6	$t\in\mathbb C$7	$t\in\mathbb C$8
$t\in\mathbb C$9	$\hat a,\hat a^\dagger$0	$\hat a,\hat a^\dagger$1
$\hat a,\hat a^\dagger$2	$\hat a,\hat a^\dagger$3	$\hat a,\hat a^\dagger$4

The reported behavior is qualitative but clear. For small cubicities, especially $\hat a,\hat a^\dagger$5, optimized Gaussian processing yields relatively high fidelities, and the trisqueezed state functions as a good approximate cubic-phase state (Hahn et al., 2022). 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 (Hahn et al., 2022).

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 (Hahn et al., 2022). 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 (Hahn et al., 2022).

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 $\hat a,\hat a^\dagger$6-axis” (Hahn et al., 2022). Noise matrices $\hat a,\hat a^\dagger$7 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 (Hahn et al., 2022).

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 $\hat a,\hat a^\dagger$8 with matched WLN and then applying a relatively simple Gaussian unitary correction, principally single-mode squeezing plus modest momentum displacement (Hahn et al., 2022).

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 (Hahn et al., 2022).

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 $\hat a,\hat a^\dagger$9 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 (Hahn et al., 2022). 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 $$\hat H_3 \propto \hat\sigma_z(\hat a^3 e^{-i\phi}+\hat a^{\dagger 3}e^{i\phi}),$$0 ion in a 3D Paul trap, with the axial mode at $$\hat H_3 \propto \hat\sigma_z(\hat a^3 e^{-i\phi}+\hat a^{\dagger 3}e^{i\phi}),$$1 serving as the harmonic oscillator (Saner et al., 2024). The oscillator is cooled to near vacuum with mean occupation $$\hat H_3 \propto \hat\sigma_z(\hat a^3 e^{-i\phi}+\hat a^{\dagger 3}e^{i\phi}),$$2, and nonlinear generalized squeezing interactions are engineered from two spin-dependent forces with noncommuting spin conditionings (Saner et al., 2024).

The effective generalized squeezing Hamiltonian is obtained from

$$\hat H_3 \propto \hat\sigma_z(\hat a^3 e^{-i\phi}+\hat a^{\dagger 3}e^{i\phi}),$$3

leading, after rotating-wave and Magnus analysis, to

$$\hat H_3 \propto \hat\sigma_z(\hat a^3 e^{-i\phi}+\hat a^{\dagger 3}e^{i\phi}),$$4

with

$$\hat H_3 \propto \hat\sigma_z(\hat a^3 e^{-i\phi}+\hat a^{\dagger 3}e^{i\phi}),$$5

(Saner et al., 2024). For $$\hat H_3 \propto \hat\sigma_z(\hat a^3 e^{-i\phi}+\hat a^{\dagger 3}e^{i\phi}),$$6, preparing the spin in an eigenstate of $$\hat H_3 \propto \hat\sigma_z(\hat a^3 e^{-i\phi}+\hat a^{\dagger 3}e^{i\phi}),$$7 yields the oscillator evolution

$$\hat H_3 \propto \hat\sigma_z(\hat a^3 e^{-i\phi}+\hat a^{\dagger 3}e^{i\phi}),$$8

(Saner et al., 2024).

A representative reported realization uses $$\hat H_3 \propto \hat\sigma_z(\hat a^3 e^{-i\phi}+\hat a^{\dagger 3}e^{i\phi}),$$9 mW per SDF, $$\hat{H}_k = \frac{\hbar \Omega_k}{2}\,\hat{\sigma}_z\Bigl(\hat{a}^k e^{-i\phi} + (\hat{a}^\dagger)^k e^{i\phi}\Bigr),$$0, $$\hat{H}_k = \frac{\hbar \Omega_k}{2}\,\hat{\sigma}_z\Bigl(\hat{a}^k e^{-i\phi} + (\hat{a}^\dagger)^k e^{i\phi}\Bigr),$$1, and interaction time $$\hat{H}_k = \frac{\hbar \Omega_k}{2}\,\hat{\sigma}_z\Bigl(\hat{a}^k e^{-i\phi} + (\hat{a}^\dagger)^k e^{i\phi}\Bigr),$$2, producing $$\hat{H}_k = \frac{\hbar \Omega_k}{2}\,\hat{\sigma}_z\Bigl(\hat{a}^k e^{-i\phi} + (\hat{a}^\dagger)^k e^{i\phi}\Bigr),$$3 (Saner et al., 2024). The experiment also generates even and odd superpositions $$\hat{H}_k = \frac{\hbar \Omega_k}{2}\,\hat{\sigma}_z\Bigl(\hat{a}^k e^{-i\phi} + (\hat{a}^\dagger)^k e^{i\phi}\Bigr),$$4 by entangling the motion with the spin, applying a second spin rotation, and performing a mid-circuit spin measurement (Saner et al., 2024).

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 $$\hat{H}_k = \frac{\hbar \Omega_k}{2}\,\hat{\sigma}_z\Bigl(\hat{a}^k e^{-i\phi} + (\hat{a}^\dagger)^k e^{i\phi}\Bigr),$$5, finite detection fidelity $$\hat{H}_k = \frac{\hbar \Omega_k}{2}\,\hat{\sigma}_z\Bigl(\hat{a}^k e^{-i\phi} + (\hat{a}^\dagger)^k e^{i\phi}\Bigr),$$6, and finite-grid tomography artifacts (Saner et al., 2024). 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 $$\hat{H}_k = \frac{\hbar \Omega_k}{2}\,\hat{\sigma}_z\Bigl(\hat{a}^k e^{-i\phi} + (\hat{a}^\dagger)^k e^{i\phi}\Bigr),$$7, and states that the experimentally generated states are high-fidelity approximations to the ideal trisqueezed states in this regime (Saner et al., 2024).

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 $$\hat{H}_k = \frac{\hbar \Omega_k}{2}\,\hat{\sigma}_z\Bigl(\hat{a}^k e^{-i\phi} + (\hat{a}^\dagger)^k e^{i\phi}\Bigr),$$8; for the even or odd superpositions $$\hat{H}_k = \frac{\hbar \Omega_k}{2}\,\hat{\sigma}_z\Bigl(\hat{a}^k e^{-i\phi} + (\hat{a}^\dagger)^k e^{i\phi}\Bigr),$$9, the non-vanishing Fock occupations are spaced by $k=3$0 (Saner et al., 2024). The trapped-ion work emphasizes that, in general, “the non-vanishing Fock state occupations of the superpositions are spaced by $k=3$1, where $k=3$2 is the order of the interaction” (Saner et al., 2024). 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 (Saner et al., 2024).

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 (Saner et al., 2024). For example, it reports an even superposition $k=3$3 with $k=3$4 and $k=3$5 (Saner et al., 2024). 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 (Hahn et al., 2022). 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 (Saner et al., 2024). The latter work further notes applications to continuous-variable quantum computation, hybrid processors, rotation-symmetric bosonic codes, and quantum-enhanced metrology (Saner et al., 2024).

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 (Hahn et al., 2022). 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 (Hahn et al., 2022). The study explicitly focuses on realistic finite energies and does not analyze any $k=3$6 scaling regime (Hahn et al., 2022).

In direct state generation, the main restrictions come from effective coupling strengths, heating, and detection imperfections. The trapped-ion work identifies $k=3$7 as the rough upper scale achieved in the reported data, with larger values requiring longer interactions and therefore more heating and decoherence (Saner et al., 2024). It also states that for moderate $k=3$8 and typical phonon numbers of a few, the rotating-wave approximation and finite-basis truncation are very accurate (Saner et al., 2024).

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 (Hahn et al., 2022). 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 (Hahn et al., 2022). The trapped-ion work, for its part, indicates practical improvement routes by increasing $k=3$9, reducing heating and leakage light, improving initial cooling, and optimizing mid-circuit measurement (Saner et al., 2024).

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 (Hahn et al., 2022), or experimentally realized noisy approximations to the ideal third-order generalized-squeezed vacuum and its superpositions (Saner et al., 2024). 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.