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On phase-space representations of spin systems and their relations to infinite-dimensional quantum states

Bálint Koczor

Published Jan 1, 2019• 5 citations
OpenAlex

Abstract

Classical phase spaces have been widely applied in physics, engineering, economics or biology. In this thesis, phase spaces of quantum systems are considered, which have become a powerful tool for describing, analyzing, and tomographically reconstructing quantum states. We provide a complete phase-space description of (coupled) spin systems including their time evolution, tomography, large-spin approximations and their infinite-dimensional limit, which recovers the well-known case of quantum optics. Finally, Born-Jordan distributions of infinite-dimensional quantum systems are described.

Related papers

Continuous phase-space representations for finite-dimensional quantum states and their tomography

Bálint Koczor, Robert Zeier, Steffen J. Glaser

Continuous phase spaces have become a powerful tool for describing, analyzing, and tomographically reconstructing quantum states in quantum optics and beyond. A plethora of these phase-space techniques are known, however a thorough understanding of their relations is still lacking for finite-dimensional quantum states. We present a unified approach to continuous phase-space representations which highlights their relations and tomography. The infinite-dimensional case from quantum optics is then recovered in the large-spin limit.

arXiv:1711.07994•Physical Review A•Feb 18, 2020•51 citations

Time evolution of spin systems in a generalized Wigner representation

Bálint Koczor, Robert Zeier, Steffen J. Glaser

Phase-space representations as given by Wigner functions are a powerful tool for representing the quantum state and characterizing its time evolution in the case of infinite-dimensional quantum systems and have been widely used in quantum optics and beyond. However, much less is known for finite-dimensional quantum systems such as spin systems and we present a complete theory of Wigner functions for this finite-dimensional case. In particular, we provide a self-contained Wigner formalism for describing and predicting the time evolution of coupled spins which lends itself to visualizing the high-dimensional structure of multi-partite quantum states. We completely treat the case of an arbitrary number of spins 1/2, thereby establishing the equation of motion using Wigner functions. The explicit form of the time evolution is then calculated for up to three spins 1/2. The underlying physical principles of our Wigner representations are illustrated with multiple examples for coupled spin systems which are easily translatable to other experimental scenarios.

•Dec 20, 2016•1 citation

Continuous phase spaces and the time evolution of spins: star products and spin-weighted spherical harmonics

Bálint Koczor, Robert Zeier, Steffen J. Glaser

Abstract We study continuous phase spaces of single spins and develop a complete description of their time evolution. The time evolution is completely specified by so-called star products. We explicitly determine these star products for general spin numbers using a simplified approach which applies spin-weighted spherical harmonics. This approach naturally relates phase spaces of increasing spin number to their quantum-optical limit and allows for efficient approximations of the time evolution for large spin numbers. We also approximate phase-space representations of certain quantum states that are challenging to calculate for large spin numbers. All of these applications are explored in concrete examples and we outline extensions to coupled spin systems.

arXiv:1808.02697•Journal of Physics A Mathematical and Theoretical•Nov 22, 2018•17 citations

Phase Spaces, Parity Operators, and the Born–Jordan Distribution

Bálint Koczor, Frederik vom Ende, Maurice A. de Gosson +2 more

Abstract Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time–frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191–194, 1976. https://doi.org/10.1007/BF01617867 ), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781–786, 1966. https://doi.org/10.1063/1.1931206 ) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born–Jordan distribution which originates from the Born–Jordan quantization (Born and Jordan in Z Phys 34(1):858–888, 1925. https://doi.org/10.1007/BF01328531 ). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born–Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born–Jordan distribution.

arXiv:1811.05872•Annales Henri Poincaré•Aug 1, 2023•9 citations

Probability Representation of Quantum States

Olga V. Man’ko, Vladimir I. Man’ko

The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born's rule and recently suggested method of dequantizer-quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical-like equations for the probability distributions determining the quantum system states. Relations to phase-space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.

Entropy•Apr 29, 2021•43 citations

Wigner representation, its reconstruction, and optimal control of spin systems with applications in NMR spectroscopy, quantum technology, and beyond

David Leiner

In this thesis, we introduce new methodologies for the visualization, evaluation, and optimal control of spin systems based on four publications appended in Appendix A -D. Essential parts are established on the DROPS representation, a phase-space method which exploits generalized Wigner functions, denoted as droplets, to represent and visualize quantum systems.In the first publication "Wigner tomography of multispin quantum states" in Appendix A, a general sampling theorem for spherical functions is derived.This is evolved to a novel Wigner quantum state tomography: A technique to reconstruct the surface of the droplets representing quantum states by measuring expectation values of rotated axial tensor operators.This concept is extended in the second publication "Wigner process tomography: Visualization of spin propagators and their spinor properties" in Appendix B to experimentally recover the phase space representation of propagators by mapping these operators to density matrices of an augmented system with an additional ancillar qubit and by applying a modified form of the Wigner quantum state tomography.Both techniques are experimentally demonstrated for quantum states of systems up to three qubits and for standard 1 qubit gates using nuclear magnetic resonance spectroscopy.In doing so, the droplets are associated to measurable quantities, which were originally the mere result of an abstract mathematical mapping, and an interpretation in form of multipoles is provided.Those tomography schemes can be regarded as universally applicable validation methods for quantum systems.They are particularly well suited for the characterization of spin operators, pulse sequence elements, or entire pulse sequences in spectroscopy and in quantum information processing.In a third publication "Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems" in Appendix C, the concept of the DROPS representation, which is in its original form restricted to systems of up to four spins 1/2 due to computational challenges, is generalized to any number of coupled spins 1/2 and to two coupled spins with arbitrary spin numbers.These obstacles are circumvented by extending and introducing two different approaches based on explicit projection operators and coefficients of fractional parentage.This results in a powerful and versatilely applicable visualization tool for the analysis of highly diverse spin topologies and paves the way for the Wigner tomography schemes in such quantum systems.This thesis is complemented by proposing novel optimal control strategies, which, combined with the previous techniques, provide a rich toolbox for the control, analysis, and verification of spin systems.In a last publication "Linking the rotation of a rigid body to the Schrödinger equation: The quantum tennis racket effect and beyond" in Appendix D, efficient and robust control strategies are introduced by linking the free rotation of a classical rigid body to the control of two-level quantum systems using external electromagnetic fields.In the case of state to state transfer and gate implementation, a new class of optimal controls is derived.This relies on two free parameters and allows for a variable efficiency, time, and robustness of the transfer process.An experimental demonstration is presented using techniques of nuclear magnetic resonance.

•Jan 1, 2018•0 citations
Classical phase spaces have been widely applied in physics, engineering, economics or biology. In this thesis, phase spaces of quantum systems are considered, which have become a powerful tool for describing, analyzing, and tomographically reconstructing quantum states. We provide a complete phase-space description of (coupled) spin systems including their time evolution, tomography, large-spin approximations and their infinite-dimensional limit, which recovers the well-known case of quantum optics. Finally, Born-Jordan distributions of infinite-dimensional quantum systems are described.
Continuous phase spaces have become a powerful tool for describing, analyzing, and tomographically reconstructing quantum states in quantum optics and beyond. A plethora of these phase-space techniques are known, however a thorough understanding of their relations is still lacking for finite-dimensional quantum states. We present a unified approach to continuous phase-space representations which highlights their relations and tomography. The infinite-dimensional case from quantum optics is then recovered in the large-spin limit.
Phase-space representations as given by Wigner functions are a powerful tool for representing the quantum state and characterizing its time evolution in the case of infinite-dimensional quantum systems and have been widely used in quantum optics and beyond. However, much less is known for finite-dimensional quantum systems such as spin systems and we present a complete theory of Wigner functions for this finite-dimensional case. In particular, we provide a self-contained Wigner formalism for describing and predicting the time evolution of coupled spins which lends itself to visualizing the high-dimensional structure of multi-partite quantum states. We completely treat the case of an arbitrary number of spins 1/2, thereby establishing the equation of motion using Wigner functions. The explicit form of the time evolution is then calculated for up to three spins 1/2. The underlying physical principles of our Wigner representations are illustrated with multiple examples for coupled spin systems which are easily translatable to other experimental scenarios.
Abstract We study continuous phase spaces of single spins and develop a complete description of their time evolution. The time evolution is completely specified by so-called star products. We explicitly determine these star products for general spin numbers using a simplified approach which applies spin-weighted spherical harmonics. This approach naturally relates phase spaces of increasing spin number to their quantum-optical limit and allows for efficient approximations of the time evolution for large spin numbers. We also approximate phase-space representations of certain quantum states that are challenging to calculate for large spin numbers. All of these applications are explored in concrete examples and we outline extensions to coupled spin systems.
Abstract Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time–frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191–194, 1976. https://doi.org/10.1007/BF01617867 ), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781–786, 1966. https://doi.org/10.1063/1.1931206 ) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born–Jordan distribution which originates from the Born–Jordan quantization (Born and Jordan in Z Phys 34(1):858–888, 1925. https://doi.org/10.1007/BF01328531 ). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born–Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born–Jordan distribution.
The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born's rule and recently suggested method of dequantizer-quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical-like equations for the probability distributions determining the quantum system states. Relations to phase-space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.
In this thesis, we introduce new methodologies for the visualization, evaluation, and optimal control of spin systems based on four publications appended in Appendix A -D. Essential parts are established on the DROPS representation, a phase-space method which exploits generalized Wigner functions, denoted as droplets, to represent and visualize quantum systems.In the first publication "Wigner tomography of multispin quantum states" in Appendix A, a general sampling theorem for spherical functions is derived.This is evolved to a novel Wigner quantum state tomography: A technique to reconstruct the surface of the droplets representing quantum states by measuring expectation values of rotated axial tensor operators.This concept is extended in the second publication "Wigner process tomography: Visualization of spin propagators and their spinor properties" in Appendix B to experimentally recover the phase space representation of propagators by mapping these operators to density matrices of an augmented system with an additional ancillar qubit and by applying a modified form of the Wigner quantum state tomography.Both techniques are experimentally demonstrated for quantum states of systems up to three qubits and for standard 1 qubit gates using nuclear magnetic resonance spectroscopy.In doing so, the droplets are associated to measurable quantities, which were originally the mere result of an abstract mathematical mapping, and an interpretation in form of multipoles is provided.Those tomography schemes can be regarded as universally applicable validation methods for quantum systems.They are particularly well suited for the characterization of spin operators, pulse sequence elements, or entire pulse sequences in spectroscopy and in quantum information processing.In a third publication "Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems" in Appendix C, the concept of the DROPS representation, which is in its original form restricted to systems of up to four spins 1/2 due to computational challenges, is generalized to any number of coupled spins 1/2 and to two coupled spins with arbitrary spin numbers.These obstacles are circumvented by extending and introducing two different approaches based on explicit projection operators and coefficients of fractional parentage.This results in a powerful and versatilely applicable visualization tool for the analysis of highly diverse spin topologies and paves the way for the Wigner tomography schemes in such quantum systems.This thesis is complemented by proposing novel optimal control strategies, which, combined with the previous techniques, provide a rich toolbox for the control, analysis, and verification of spin systems.In a last publication "Linking the rotation of a rigid body to the Schrödinger equation: The quantum tennis racket effect and beyond" in Appendix D, efficient and robust control strategies are introduced by linking the free rotation of a classical rigid body to the control of two-level quantum systems using external electromagnetic fields.In the case of state to state transfer and gate implementation, a new class of optimal controls is derived.This relies on two free parameters and allows for a variable efficiency, time, and robustness of the transfer process.An experimental demonstration is presented using techniques of nuclear magnetic resonance.