Qubits in phase space: Wigner function approach to quantum error correction and the mean king problem

Abstract

We analyze and further develop a new method to represent the quantum state of a system of $n$ qubits in a phase space grid of $N\times N$ points (where $N=2^n$). The method, which was recently proposed by Wootters and co--workers (Gibbons {\it et al.}, quant-ph/0401155), is based on the use of the elements of the finite field $GF(2^n)$ to label the phase space axes. We present a self--contained overview of the method, we give new insights on some of its features and we apply it to investigate problems which are of interest for quantum information theory: We analyze the phase space representation of stabilizer states and quantum error correction codes and present a phase space solution to the so--called ``mean king problem''.

18 pages, 16 figures; typos fixed, some minor corrections, figures of the circuits were changed

References in corpus (15)

• Improved Simulation of Stabilizer Circuits
• A new proof for the existence of mutually unbiased bases
• Discrete phase space based on finite fields
• Mutually unbiased binary observable sets on N qubits
• Quantum computers in phase space
• Discrete Wigner functions and quantum computational speedup
• The mean king's problem: Prime degrees of freedom
• Discrete Wigner functions and the phase space representation of quantum teleportation
• Aspects of mutually unbiased bases in odd prime power dimensions
• Phase-space approach to the study of decoherence in quantum walks
• Wigner-function description of quantum teleportation in arbitrary dimensions and continuous limit
• Decoherence for classically chaotic quantum maps
• Quantum Computing of Quantum Chaos in the Kicked Rotator Model
• Discrete Wigner functions and the phase space representation of quantum computers
• Quantum algorithms for phase space tomography

Cited by in corpus (39)

• On mutually unbiased bases
• Quasi-probability representations of quantum theory with applications to quantum information science
• Classicality in discrete Wigner functions
• On the structure of the sets of mutually unbiased bases for N qubits
• Wigner tomography of two qubit states and quantum cryptography
• Overview of the phase space formulation of quantum mechanics with application to quantum technologies
• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects
• Geometrical approach to mutually unbiased bases
• Selective and Efficient Quantum Process Tomography
• New Phase Space Formulations and Quantum Dynamics Approaches
• Discrete phase-space structure of $n$-qubit mutually unbiased bases
• Extrema of discrete Wigner functions and applications
• Extended Cahill and Glauber formalism for finite dimensional spaces II. Applications in quantum tomography and quantum teleportation
• Wigner function for a particle in an infinite lattice
• Mutually unbiased bases and generalized Bell states
• Spectra of phase point operators in odd prime dimensions and the extended Clifford group
• Quantum correlation exists in any non-product state
• Discrete Wigner distribution for two qubits: a characterization of entanglement properties
• Generation of all sets of mutually unbiased bases for three-qubit systems
• Quantum Fourier transform, Heisenberg groups and quasiprobability distributions
• Macroscopic features of quantum fluctuations in large N qubit system
• Harnessing quantumness of states using discrete Wigner functions under (non)-Markovian quantum channels
• Structure of the sets of mutually unbiased bases with cyclic symmetry
• Interference in discrete Wigner functions
• Minimal sets of dequantizers and quantizers for finite-dimensional quantum systems
• A new method of construction of all sets of mutually unbiased bases for two-qubit systems
• Discrete phase-space mappings, tomographic condition and permutation invariance
• Relation between phase space coverage and entanglement for spin-1/2 systems
• Transition probabilities and transition rates in discrete phase space
• Stokes vector and its relationship to Discrete Wigner Functions of multiphoton states
• Quantum phase uncertainty in mutually unbiased measurements and Gauss sums
• Matrix reduction and Lagrangian submodules
• Wigner formalism for a particle on an infinite lattice: dynamics and spin
• Macroscopic approach to N-qudit systems
• Spin flip of multiqubit states in discrete phase space
• Bases for spin systems and qudits from angular momentum theory
• Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three
• Phase-space descriptions of operators and the Wigner distribution in quantum mechanics II. The finite dimensional case