S. N. Bose National Centre for Basic Sciences: Researchers Quantify Quantum Advantage Using Wigner Function Negativity

A new framework detects Wigner negativity, a key indicator of quantum advantage, for any quantum state. Sudip Chakrabarty and colleagues at S. N. Bose National Centre for Basic Sciences and Indian Institute of Technology Goa present a method to identify this negativity from a limited number of state copies, avoiding complete phase-space tomography. Their approach uses experimentally measurable moments of the Wigner function and links them to parity-based observables, offering a scalable route for quantifying nonclassical resources in continuous-variable quantum systems and extending to the identification of multipartite entanglement. Wigner moments are a flexible set of tools for assessing quantum states and their potential for advanced computation.

Wigner moments characterise nonclassicality and entanglement in continuous-variable quantum systems

Numerical simulations of randomised-measurement and classical-shadow protocols were performed, extending the framework to identifying nonclassical resources such as bipartite and multipartite entanglement. These results establish Wigner moments as a flexible set of tools for scalable detection and quantification of nonclassical resources in continuous-variable quantum systems. Continuous-variable (CV) quantum systems, particularly optical platforms utilising degrees of freedom like quadrature amplitudes of light, represent a central architecture for modern quantum technologies due to their experimental accessibility, high-bandwidth operation, and intrinsic scalability. The ability to manipulate and measure these continuous variables offers advantages in certain quantum information processing tasks.

Quantum states in such systems are naturally described by phase-space quasiprobability distributions. The Wigner function plays a distinguished role, providing a complete phase-space representation of quantum states while retaining many features of a classical probability distribution. However, unlike classical probability distributions, the Wigner function can take on negative values, a characteristic that signifies nonclassical behaviour. Its importance is clarified by its connection to the Gottesman-Knill theorem, which establishes that quantum circuits restricted to Clifford operations admit efficient classical simulation. Clifford circuits are a limited subset of all possible quantum circuits, and their efficient simulability on classical computers provides a benchmark for assessing the difficulty of simulating more general quantum computations.

Efficient simulability generally breaks down when Wigner negativity is present, and is therefore widely regarded as a necessary resource for quantum computational advantage. States exhibiting negative Wigner functions demonstrate quantum advantages in a variety of information tasks, including quantum computational speedup, quantum error correction, and quantum state distillation. The presence of negativity indicates that the quantum state cannot be efficiently described by classical means, suggesting a potential for surpassing the capabilities of classical algorithms. Significant effort has been devoted to its detection and quantification, driven by this fundamental role in quantum information processing. Quantifying the degree of Wigner negativity provides a measure of the ‘quantumness’ of a state and its potential for achieving a computational advantage.

Existing methods largely rely on quantum-state tomography and subsequent reconstruction of the Wigner function. These procedures require a number of state copies that grows rapidly with system size, scaling exponentially with the number of degrees of freedom, rendering the detection of Wigner negativity increasingly demanding for large-scale and arbitrary quantum states. The need for numerous state copies is a significant limitation, as obtaining and maintaining quantum states is often resource-intensive and prone to decoherence. Consequently, researchers are motivated to search for methods that can detect and quantify Wigner negativity using only few copies, without full phase-space reconstruction. Reducing the number of required state copies is crucial for enabling the practical implementation of quantum information processing protocols.

This letter addresses this challenge by developing an efficient framework for the detection and quantification of Wigner negativity. The approach is based on moments of the Wigner function, a family of global phase-space quantities obtained by integrating powers of the distribution. Specifically, the $n$-th moment is calculated by integrating xⁿ multiplied by the Wigner function over all phase space. Positivity of the Wigner function imposes non-trivial constraints on these moments, and this structure is exploited to derive complementary hierarchies of negativity detection criteria. These criteria are based on mathcalL_p-norm inequalities, which provide a mathematical framework for quantifying the deviation of the Wigner function from positivity. Certifiable lower bounds on logarithmic Wigner negativity, measured in decibels, are also obtained, providing a quantitative measure of the degree of nonclassicality.

Wigner function moments offer a practical route towards demonstrating quantum computational advantage

Methods to prove quantum advantage, a potential speed boost over conventional computers, are being refined by identifying states exhibiting Wigner negativity, a key sign of this advantage. The new framework focuses on measuring moments of the Wigner function, statistical properties of a quantum state, rather than fully reconstructing the entire state. This is achieved by relating the Wigner function moments to measurable parity-based observables, which can be directly accessed through quantum measurements. Simulations demonstrate success, but the durability of this approach when confronted with imperfections inherent in actual quantum devices remains a key question. Real-world quantum devices are subject to noise, loss, and other imperfections that can degrade the performance of quantum information processing protocols.

This work provides a valuable step forward in verifying quantum advantage, acknowledging that real quantum devices introduce imperfections which may limit the practical application of these negativity criteria. The proposed method demonstrates robustness against experimental noise and imperfections, an important area for future research. It is particularly important as it allows for scalable detection of nonclassical resources, including entanglement, even with imperfect hardware and limited state copies. The ability to detect entanglement, a crucial resource for many quantum applications, is particularly significant. S. N. Bose National Centre for Basic Sciences and Indian Institute of Technology Goa have developed a new method to identify Wigner negativity without needing to fully map a quantum state.

This framework utilises measurable properties of quantum states, termed ‘moments’ of the Wigner function, offering a more efficient pathway than previous techniques which demanded extensive data collection. By linking these moments to parity-based observables, the approach provides a scalable way to assess quantum states and their potential for advanced computation. The parity operator measures the eigenvalues of the system, providing information about the symmetry of the quantum state. This represents a significant improvement over existing methods, which often require extensive resources and are less robust to experimental noise. The reduction in required resources and increased robustness are crucial for enabling the practical implementation of quantum technologies.

The research successfully detected Wigner negativity, a key indicator of quantum advantage, using experimentally accessible moments of the Wigner function. This is important because it offers a more efficient method for verifying nonclassical resources than techniques requiring full state mapping. Researchers established a link between Wigner moments and parity-based observables, enabling scalable assessment of quantum states with fewer resources. The authors demonstrated the method’s performance through numerical simulations, noting that further research should address the impact of imperfections in real quantum devices.