Isospectral oscillators as a resource for quantum information processing

Abstract: We address quantum systems isospectral to the harmonic oscillator, as those found within the framework of supersymmetric quantum mechanics, as potential resources for continuous variable quantum information. These deformed oscillator potentials share the equally spaced energy levels of the shifted harmonic oscillator but differ significantly in that they are non-harmonic. Consequently, their ground states and thermal equilibrium states are no longer Gaussian and exhibit non-classical properties. We quantify their non-Gaussianity and evaluate their non-classicality using various measures, including quadrature squeezing, photon number squeezing, Wigner function negativity, and quadrature coherence scale. Additionally, we employ quantum estimation theory to identify optimal measurement strategies and establish ultimate precision bounds for inferring the deformation parameter. Our findings prove that quantum systems isospectral to the harmonic oscillator may represent promising platforms for quantum information with continuous variables. In turn, non-Gaussian and non-classical stationary states may be obtained and these features persist at non-zero temperature.

• The paper demonstrates that SUSYQM-engineered isospectral oscillators produce non-Gaussian, non-classical states with equally spaced energy levels, offering a new quantum resource.
• It utilizes quantum relative entropy, squeezing measures, and quantum Fisher information to quantify non-Gaussianity and optimal parameter estimation in both ground and thermal states.
• The study highlights practical potential for implementing deformed oscillator potentials in quantum simulators to enhance metrology, cryptography, and computation.

Isospectral Oscillators as a Resource for Quantum Information Processing

Introduction and Motivation

The paper "Isospectral oscillators as a resource for quantum information processing" (2504.02444) investigates a class of quantum systems—deformed oscillator potentials constructed via supersymmetric quantum mechanics (SUSYQM)—that are strictly isospectral to the shifted harmonic oscillator (SHO). Unlike the canonical SHO whose eigenstates are Gaussian and well-characterized, these isospectral systems possess non-harmonic potentials while retaining the SHO's equally spaced energy spectrum. Their stationary states (both ground and thermal) deviate from Gaussianity and exhibit enhanced non-classical features, making them attractive candidates for continuous-variable (CV) quantum information tasks where standard Gaussian resources are limited.

A central motivation is to provide inexpensive, robust sources of non-Gaussian, non-classical states without relying on complex nonlinear processes or ancillary systems. Non-Gaussian states are vital for quantum simulation, computation, and cryptographic protocols but are usually challenging to generate. The approach proposed leverages the isospectrality induced by SUSY transformations to yield such states directly as the stationary states of the deformed potentials.

Formal Construction of Isospectral Potentials

The isospectrality is engineered through the SUSYQM/Darboux transformation framework, which constructs a continuous family of potentials $\widehat{V}^{(1)}(\lambda, x)$ specified by a deformation parameter $\lambda$, all sharing the spectral properties of the SHO. The ground state energy is set to zero via a shift, allowing the SUSY factorization and the subsequent construction of deformed potentials. These strictly isospectral deformations introduce strong nonlinearity, with the original SHO recovered in the limit $\lambda\rightarrow 0$.

Figure 1: Isospectral SHO potentials $\widehat V^{(1)}(\lambda, x)$ for representative values of $\lambda$, with corresponding ground and first excited state wavefunctions illustrating non-Gaussian localization and distortion.

Non-Gaussianity and Non-Classicality in Stationary States

Non-Gaussianity Quantification

The non-Gaussianity (nonG) of the ground and thermal states was quantified using quantum relative entropy (QRE) to distinguish them from their reference Gaussian states, matching first moments and covariances. Numerical evaluation indicates a monotonic increase in nonG with $\lambda$ for low values, peaking before a slow decline at high $\lambda$. Notably, for thermal states, nonG increases both with $\lambda$ and temperature, contrasting with the typical decoherence-induced reduction in non-classicality, evidencing the persistence of non-Gaussian features even at non-zero temperature.

Figure 2: QRE-based nonG $\delta[\rho_\lambda]$ versus deformation parameter $\lambda$ for isospectral ground states, illustrating monotonic increase and saturation.

Non-Classicality: Squeezing and Wigner Negativity

Non-classicality was characterized through multiple criteria:

• Quadrature Squeezing: All ground states exhibited position quadrature squeezing, with the variance $\Delta X^2$ falling below the shot-noise limit for all $\lambda > 0$. The uncertainty product deviates from the minimum, indicating departure from coherent state statistics.

  Figure 3: Variances $\Delta X^2$ (red) and $\Delta P^2$ (blue) for ground states as a function of $\lambda$; squeezing emerges robustly with increasing deformation.

• Photon Number Squeezing: Sub-Poissonian statistics (Fano factor ${\mathcal F}[\rho]<1$) were observed only for sufficiently large $\lambda$, differentiating regimes of photon number statistics inaccessible to standard oscillators.

Figure 4: Fano factor $\mathcal{F}[\rho_\lambda]$ for ground states across $\lambda$; photon number squeezing becomes significant for $\lambda > 315$.

• Wigner Negativity and Quadrature Coherence Scale (QCS): Wigner function negativity and growth in the QCS both tracked non-Gaussianity, with increased deformation leading to greater quantum coherence and negative quasi-probability volume.

  Figure 5: QRE-based nonG $\delta[\rho_\lambda^{th}]$ for thermal states as a function of $\lambda$ and temperature, showing enhancement of non-Gaussianity with both deformation and thermal excitation.

Thermal States: Robustness of Quantum Features

Thermalization does not immediately suppress non-Gaussian and non-classical characteristics. Moderate temperature increases further enhance nonG, while quadrature squeezing and Wigner negativity remain present up to threshold temperatures, showing robustness of these features under realistic (non-zero temperature) environmental conditions.

Figure 6: Squeezing quadrature $(\Delta X)^2$ for thermal states at various temperatures, demonstrating preservation and eventual attenuation of squeezing at high $T$.

Quantum Parameter Estimation: Precision Bounds

The parameter $\lambda$ encodes the degree of nonlinearity; its precise inference is crucial for characterizing system properties. Quantum estimation theory was applied, computing the quantum Fisher information (QFI) for both pure ground and mixed thermal states. Explicit analytical results indicate:

• QFI for ground states scales as $H(\lambda) = \frac{2}{3}(1+ \sqrt{2}\lambda)^{-2}$, decreasing for large $\lambda$ but remaining bounded.
• Optimal precision is attained by position measurements, saturating the quantum Cramér-Rao bound, both for pure and thermal scenarios.
• At nonzero $T$, QFI slightly decreases but otherwise mirrors the pure-state behavior, demonstrating measurement robustness and feasibility.

  Figure 7: QFI for isospectral SHO thermal states $\rho_\lambda^{th}$ across $\lambda$ and increasing temperature; ground state QFI sets the upper bound.

Implications and Future Directions

This work demonstrates that the stationary states (ground and thermal) of SUSY-engineered isospectral oscillators systematically realize non-Gaussian, non-classical resources natively, without recourse to complex state preparation or non-linear optical elements. Their rich structure, including robust Wigner negativity, quadrature and photon number squeezing, and enhanced parameter-estimation bounds, positions them as versatile platforms for CV quantum information tasks.

From a practical standpoint, the ability to implement such isospectral potentials—potentially through engineered quantum-well or ion-trap Hamiltonians—could facilitate resource-efficient generation of quantum states tailored for teleportation, cryptography, or metrology. The theoretical implications extend to understanding noise tolerance, decoherence, and the interplay of SUSY transformations and quantum resource theory.

Further research may focus on:

• Realizing experimental quantum simulators of isospectral potentials;
• Extending analysis to multimode and entangled states;
• Leveraging coherent states of isospectral oscillators;
• Exploring error-correction and fault-tolerance properties unique to these systems.

Conclusion

The paper establishes that isospectral oscillator systems defined via SUSYQM present a rich, tunable source of non-Gaussian and non-classical states, with distinctive thermal robustness and measurement-optimality properties. These findings contribute important insights into both resource generation and characterization in CV quantum information, suggesting new avenues for exploiting non-harmonicity and spectral engineering in future quantum technologies.