Squeezing the Quantum State: Multiplets and Phase Space

by

Author: Denis Avetisyan

This review explores the unique properties of squeezed quantum multiplets and their representation in phase space, paving the way for more robust quantum technologies.

The characteristic function of a squeezed multiplet in three dimensions, examined with a radius of 1.2, decomposes into distinct direct and interference terms - visualized as panels (a)-(f) - which, when combined, fully define the function itself, highlighting the interplay between these components in constructing the overall mathematical description.
The characteristic function of a squeezed multiplet in three dimensions, examined with a radius of 1.2, decomposes into distinct direct and interference terms – visualized as panels (a)-(f) – which, when combined, fully define the function itself, highlighting the interplay between these components in constructing the overall mathematical description.

The article details the Wigner function representation of squeezed quantum multiplets and their potential applications in quantum error correction and information processing.

While robust quantum information storage demands states resilient to phase-space perturbations, conventional approaches often lack detailed characterization of multi-photon entanglement. This paper, ‘Squeezed quantum multiplets: properties and phase space representation’, introduces and analyzes a class of states – squeezed quantum multiplets – formed by superpositions of squeezed states along multiple quadrature directions. We demonstrate that these multiplets exhibit unique phase-space distributions, expressible analytically via Wigner and characteristic functions, and reveal sensitivities to perturbations distinct from single-squeezed states. Could these properties unlock novel strategies for quantum error correction and enhance the precision of quantum metrology protocols?

The Illusion of Control: Foundations of Continuous Variable Computation

The potential of quantum computation lies in its promise to solve problems intractable for classical computers, offering exponential speedups for specific tasks. However, many proposed quantum architectures rely on manipulating discrete quantum states – akin to flipping switches – demanding precise control over individual qubits and complex, scalable control systems. This presents significant engineering challenges as the number of qubits increases, as maintaining coherence and minimizing errors becomes exponentially more difficult. These discrete approaches, while conceptually powerful, often necessitate intricate fabrication and control mechanisms that limit practical implementation. Consequently, researchers are exploring alternative paradigms that may circumvent these hurdles, seeking pathways to quantum advantage with more readily achievable technologies.

Unlike many quantum computing approaches that rely on discrete variables like the spin of an electron, Continuous Variable (CV) quantum computation utilizes continuous degrees of freedom to encode quantum information. This means qubits aren’t represented by 0 or 1, but by continuous properties of a system, such as the amplitude and phase of light – known as light quadrature. This approach offers a fundamentally different paradigm, potentially simplifying the construction of quantum hardware and allowing for compatibility with existing technologies used in classical optics and communication. By treating quantum states as continuous variables, researchers can leverage well-established tools from classical physics, like phase-space representations, to analyze and manipulate quantum information. This offers a pathway towards building scalable quantum computers that harness the continuous nature of the physical world, potentially bypassing some of the challenges associated with discrete qubit systems.

Quantum states in continuous variable (CV) systems, unlike those defined by discrete qubits, are described by continuous degrees of freedom, demanding a different mathematical toolkit for their representation. The Wigner function, a quasi-probability distribution in phase space, offers a way to visualize and analyze these states, though it can take on negative values, a hallmark of non-classicality. Complementary to the Wigner function is the characteristic function, a Fourier transform of the Wigner function, which provides a convenient means for calculating moments and expectation values of quantum observables. These functions effectively map quantum information onto a classical phase space, allowing researchers to leverage existing classical tools for analyzing and manipulating CV quantum states – a crucial step in harnessing the potential of this promising quantum computation paradigm. The relationship between the two is defined as W(x,p) = \frac{1}{\pi} \int_{-\infty}^{\infty} \chi(x, \tau) e^{i x \tau} d\tau.

Realizing the potential of continuous-variable quantum computation hinges on the ability to meticulously control and measure quantum states. Current research focuses on leveraging established quantum technologies to achieve this precision. Ion traps, utilizing the quantized motion of individual ions, provide a highly stable environment for encoding and manipulating continuous variables, offering long coherence times and high fidelity operations. Simultaneously, circuit quantum electrodynamics (Circuit QED) presents a complementary approach, employing superconducting circuits to create and control microwave photons – the carriers of continuous variable qubits. Through careful design and fabrication of these circuits, researchers can achieve strong coupling between photons and artificial atoms, enabling precise measurements and gate operations. Both platforms face unique challenges in scaling and maintaining coherence, but represent leading avenues for translating the theoretical advantages of continuous variable quantum computation into tangible quantum processors.

Beyond the Gaussian Mirror: The Quest for Non-Gaussian States

Universal quantum computation using continuous-variable (CV) systems is fundamentally limited by the resources available within Gaussian states. While Gaussian states can efficiently represent and manipulate quantum information for certain tasks, they are insufficient for implementing a complete set of quantum gates required for universal computation. Specifically, Gaussian states cannot generate the necessary entanglement to perform non-Gaussian operations, such as parity measurements or Einstein-Podolsky-Rosen (EPR) steering, which are proven to be universal resources for CV quantum computation. Therefore, the creation and precise control of Non-Gaussian states – quantum states with probability distributions that are not Gaussian – are essential to unlock the full potential of CV systems and achieve scalable, fault-tolerant quantum computation. These states allow for the realization of gates and measurements beyond the scope of purely Gaussian operations, enabling a complete and versatile quantum computational framework.

Non-Gaussian quantum states represent a departure from states fully characterized by Gaussian probability distributions; this distinction is fundamental to their enhanced computational capabilities. Gaussian states, while efficiently described by first and second statistical moments, are limited in their ability to represent all possible quantum states and thus restrict the types of quantum operations achievable. Non-Gaussian states, possessing higher-order correlations not captured by Gaussian descriptions, allow for quantum information processing tasks demonstrably beyond the reach of classical computation. Specifically, these states are required for universal quantum computation with continuous-variable (CV) systems, enabling operations such as \sqrt{SWAP} which are essential for quantum error correction and scalable quantum computing, and cannot be efficiently implemented using only Gaussian resources.

The generation of non-Gaussian states in continuous-variable (CV) quantum systems fundamentally relies on introducing nonlinearity into the system’s Hamiltonian. Linear systems, governed by quadratic Hamiltonians, can only produce Gaussian states, irrespective of initial conditions or input states. Nonlinearity, typically achieved through interactions like Kerr nonlinearities or parametric down-conversion, allows for the creation of terms beyond the quadratic, enabling state preparation and manipulation that are impossible with Gaussian states alone. These nonlinear interactions facilitate the creation of entanglement between photons and allow for the generation of superposition and squeezing beyond the limits imposed by Gaussian states, ultimately unlocking the potential for universal quantum computation with CV systems. The degree of nonlinearity directly impacts the complexity and type of non-Gaussian states that can be created, influencing the fidelity and efficiency of subsequent quantum operations.

Controlled Displacement and Controlled Squeezing are established techniques for generating non-Gaussian quantum states in continuous-variable (CV) systems. Controlled Displacement involves conditionally shifting the quadrature of a squeezed state based on the state of a control qubit, effectively superposing different displacement values. Controlled Squeezing, conversely, modulates the amount of squeezing applied to a mode conditioned on a control qubit. Both methods leverage the interaction between optical modes and utilize ancillary qubits to achieve deterministic state engineering. Precision is attained through careful calibration of control pulses and optimization of interaction parameters, allowing for the creation of states with tailored non-Gaussian features, such as higher-order correlations and non-classicality, crucial for advanced quantum information processing tasks.

Encoding Reality: Squeezed Multiplets and the GKP Framework

The GKP approach to continuous-variable (CV) quantum information processing encodes a qubit-a unit of quantum information-not in discrete energy levels, but in the amplitude and phase of a coherent state. Specifically, the |0\rangle and |1\rangle logical qubit states are defined as superpositions of coherent states centered at \alpha and i\alpha in the complex amplitude plane, where \alpha is a real number. This encoding scheme allows for the creation of quantum states with non-classical properties, and importantly, provides a pathway to universal quantum computation within a CV framework. The robustness of this encoding relies on the ability to reliably prepare and measure these superpositions of coherent states, and is a primary area of research in CV quantum computing.

Rotationally Invariant States (RIS) form the basis for encoding quantum information in continuous-variable (CV) systems due to their symmetry properties, simplifying analysis and reducing sensitivity to certain types of noise. However, the performance of RIS-based encoding can be substantially improved by utilizing Squeezed Multiplets. These multiplets are constructed from multiple, correlated squeezed states, and their utilization effectively reduces the quantum Fisher information, leading to enhanced precision in parameter estimation and state discrimination. The degree of enhancement is directly related to the number of squeezed states comprising the multiplet and the amount of squeezing applied; higher dimensionality and increased squeezing generally yield greater improvements in encoding fidelity and resilience against decoherence.

Squeezed multiplets, utilized in continuous-variable quantum information processing, are constructed from combinations of squeezed states to enhance the robustness of encoded qubits against noise and improve the precision of state determination. These multiplets exhibit a Wigner function that can be analytically characterized, allowing for a detailed understanding of their quantum properties and noise susceptibility. The analytical form of the Wigner function reveals that squeezing reduces the quadrature noise in one observable at the expense of increased noise in the conjugate observable, effectively concentrating the quantum information and making the state more resistant to certain types of decoherence. This noise reduction is directly correlated to the squeezing parameter r, with larger values of r indicating greater squeezing and improved resilience, but also potentially increased sensitivity to other noise sources. Characterization via the Wigner function is crucial for optimizing the design of these multiplets and predicting their performance in practical quantum systems.

Encoded qubit states within the GKP framework can be efficiently measured using parity measurements, which determine the displacement of the quantum state. However, super-parity measurements offer improved robustness against noise and imperfections. The performance of both measurement schemes is directly influenced by the squeezing parameter r, which dictates the degree of noise reduction, and the dimensionality D of the squeezed multiplet used for encoding; higher values of r and D generally lead to increased measurement fidelity but also greater experimental complexity. Specifically, the success probability of these measurements scales with the squeezing and multiplet size, impacting the achievable error rates in qubit readout.

Expanding the Quantum Palette: The Promise of Higher-Order Squeezed States

Standard squeezed states, valuable tools in quantum information science, exhibit reduced noise in one quadrature of the electromagnetic field at the expense of increased noise in the other. Higher-order squeezed states build upon this concept, offering a significantly expanded capacity for state preparation by manipulating not just a single quadrature, but multiple – effectively creating ‘ripples’ in the quantum field’s probability distribution. These advanced states are defined as linear combinations of Fock states where the photon numbers are multiples of p(m+nD), allowing for precise control over the quantum field’s characteristics. This increased flexibility isn’t merely a refinement; it enables the creation of complex, multi-mode entangled states crucial for implementing sophisticated quantum algorithms and enhancing the resilience of quantum computations against errors. The ability to engineer these states opens doors to novel approaches in quantum communication and sensing, promising a substantial leap forward in harnessing the power of quantum mechanics.

Higher-order squeezed states represent a significant advancement in the creation of complex quantum states, moving beyond simple superpositions to harness linear combinations of Fock states. These states are characterized by photon numbers that are multiples of p(m+nD), where ‘p’ dictates the order of the squeezing, ‘m’ represents an integer, and ‘n’ is a positive integer, allowing for finely-tuned control over quantum fluctuations. This nuanced control is particularly valuable for advanced quantum algorithms, as it enables the construction of robust states less susceptible to decoherence and noise – major hurdles in quantum computation. By engineering these intricate quantum superpositions, researchers aim to create more reliable qubits and improve the performance of complex quantum calculations, potentially unlocking solutions to problems currently intractable for classical computers.

The precise control afforded by manipulation and measurement of higher-order squeezed states represents a significant step towards realizing practical, fault-tolerant quantum computation. Unlike traditional qubits which are highly susceptible to environmental noise, these states distribute quantum information across multiple photon number levels, creating inherent redundancy. This distribution allows for the detection and correction of errors without collapsing the quantum state – a crucial requirement for complex algorithms. Furthermore, the ability to engineer correlations between these non-classical states – described by |\psi\rangle = \sum_{n} c_n |n, n, ..., n\rangle – provides a pathway to implement error-correcting codes that are resilient to photon loss, a dominant source of error in continuous-variable quantum systems. This enhanced robustness promises to unlock the potential for scaling quantum computations to levels previously unattainable, paving the way for truly useful quantum technologies.

The pursuit of fully realized continuous-variable (CV) quantum computation hinges on the development and control of increasingly sophisticated quantum states, and continued research into higher-order squeezed states represents a crucial frontier. These states, going beyond traditional squeezed light, offer a richer landscape for encoding and manipulating quantum information, potentially circumventing limitations inherent in simpler schemes. By enabling more complex correlations and enhanced resilience against noise, these advanced states could pave the way for fault-tolerant quantum algorithms capable of tackling problems currently intractable for classical computers. Investigations are focusing on refining techniques for both generating and accurately characterizing these states, alongside exploring novel measurement strategies that can fully exploit their unique properties, ultimately promising a significant leap forward in the capabilities of CV quantum information processing.

The exploration of squeezed quantum multiplets, as detailed in this work, reveals a delicate interplay between theoretical prediction and observational data – a humbling reminder of the limitations inherent in any model. Indeed, as Niels Bohr observed, “Prediction is very difficult, especially about the future.” This sentiment resonates deeply with the challenges of representing non-Gaussian states, like these squeezed multiplets, within the phase-space formalism using the Wigner function. The comparison of theoretical predictions with simulated data, crucial for assessing the viability of these states for quantum error correction, underscores that even the most sophisticated calibrations cannot fully escape the uncertainties that lie beyond the horizon of our current understanding.

What Lies Beyond?

The investigation into squeezed quantum multiplets, and their phase-space representation via the Wigner function, serves as a rigorous exercise in mathematical possibility. Current quantum gravity theories suggest that inside the event horizon spacetime ceases to have classical structure; therefore, even the notion of a ‘phase-space representation’ may be fundamentally altered, or even meaningless. The demonstrated potential for robust quantum information storage, while mathematically sound, remains experimentally unverified, a familiar refrain in this pursuit.

A critical limitation resides in the assumption of rotational invariance. Should this symmetry be broken at a fundamental level-as some models propose-the carefully constructed properties of these multiplets would require substantial revision. Furthermore, the efficacy of these states in quantum error correction relies on precise control and isolation, conditions rarely, if ever, achieved in any real-world system. The elegance of the formalism should not be mistaken for proximity to reality.

The next logical step necessitates exploring the behavior of these squeezed states in increasingly complex, and ultimately, relativistic environments. Whether such investigation will reveal genuinely novel physical insights, or merely confirm the limitations of the current framework, remains to be seen. Everything discussed is mathematically rigorous but experimentally unverified; the horizon, as always, awaits.

Original article: https://arxiv.org/pdf/2512.21229.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

See also:

2025-12-26 00:28