Overview of the phase space formulation of quantum mechanics with application to quantum technologies

Abstract

The phase-space formulation of quantum mechanics has recently seen increased use in testing quantum technologies, including metho ds of tomography for state verification and device validation. Here, an overview of quantum mechanics in phase space is presented. The formulation to generate a generalized phase-space function for any arbitrary quantum system is shown, such as the Wigner and Weyl functions along with the asso ciated Q and P functions. Examples of how these different formulations have b een used in quantum technologies are provided, with a focus on discrete quantum systems, qubits in particular. Also provided are some results that, to the authors' knowledge, have not been published elsewhere. These results provide insight into the relation between different representations of phase space and how the phase-space representation is a powerful tool in understanding quantum information and quantum technologies.

42 pages, 4 figures

References in corpus (15)

• Multi-party entanglement in graph states
• Reconstruction of non-classical cavity field states with snapshots of their decoherence
• Direct Fidelity Estimation from Few Pauli Measurements
• Negativity and contextuality are equivalent notions of nonclassicality
• Toolbox for entanglement detection and fidelity estimation
• Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations
• Observation of Entanglement Between Itinerant Microwave Photons and a Superconducting Qubit
• Framed Hilbert space: hanging the quasi-probability pictures of quantum theory
• Visualizing operators of coupled spin systems
• Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects
• Wigner tomography of multispin quantum states
• Atomic nonclassicality quasiprobabilities
• Measurement of complete and continuous Wigner functions for discrete atomic systems
• $SU(1,1)$ covariant $s$-parametrized maps
• Some Applications for an Euler Angle Parameterization of SU(N) and U(N)

Cited by in corpus (26)

• New Phase Space Formulations and Quantum Dynamics Approaches
• Robust and Deterministic Preparation of Bosonic Logical States in a Trapped Ion
• Sub-Planck phase-space structure and sensitivity for SU(1,1) compass states
• Sub-Planck structures and sensitivity of the superposed photon-added or photon-subtracted squeezed-vacuum states
• Simple, reliable and noise-resilient continuous-variable quantum state tomography with convex optimization
• Single-Atom Verification of the Optimal Trade-Off between Speed and Cost in Shortcuts to Adiabaticity
• Quantum speed limits in arbitrary phase spaces
• Mixed discrete variable Gaussian states
• Relating the Glauber-Sudarshan, Wigner and Husimi quasiprobability distributions operationally through the quantum limited amplifier and attenuator channels
• Quantum and Relativistic corrections to Maxwell-Boltzmann ideal gas model from a Quantum Phase Space approach
• Polytopes of Absolutely Wigner Bounded Spin States
• Navigating the phase diagram of quantum many-body systems in phase space
• Stochastic correction to the Maxwell-Bloch equations via the positive $P$ representation
• Parity symmetry breaking of spin-$j$ coherent state superpositions in Gaussian noise channel
• Grand Unification of All Discrete Wigner Functions on $d\times d$ Phase Space
• Frame representations of qudit quantum mechanics
• Covariant operator bases for continuous variables
• Symmetries and Wigner representations of operational theories
• Intensity correlations in the Wigner representation
• Enhancing wave-particle duality
• Quantum-to-Classical Transition via Single-Shot Generalized Measurements
• Benchmarking quantum phase-space methods for near-resonant light propagation
• Efficient verification and fidelity estimation of discrete bipartite squeezed states
• Understanding Squeezed States of Light Through Wigner's Phase-Space
• Correspondence Between the Energy Equipartition Theorem in Classical Mechanics and its Phase-Space Formulation in Quantum Mechanics
• Third quantization with Hartree approximation for open-system bosonic transport