A complementarity-based approach to phase in finite-dimensional quantum systems

Abstract

We develop a comprehensive theory of phase for finite-dimensional quantum systems. The only physical requirement we impose is that phase is complementary to amplitude. To implement this complementarity we use the notion of mutually unbiased bases, which exist for dimensions that are powers of a prime. For a d-dimensional system (qudit) we explicitly construct d+1 classes of maximally commuting operators, each one consisting of d-1 operators. One of this class consists of diagonal operators that represent amplitudes (or inversions). By the finite Fourier transform, it is mapped onto ladder operators that can be appropriately interpreted as phase variables. We discuss the examples of qubits and qutrits, and show how these results generalize previous approaches.

6 pages, no figures

References in corpus (23)

• Encoding a qubit in an oscillator
• A new proof for the existence of mutually unbiased bases
• Information and Computation: Classical and Quantum Aspects
• Discrete phase space based on finite fields
• Fundamentals of Quantum Information Theory
• Iterative maximum-likelihood reconstruction in quantum homodyne tomography
• Phase covariant quantum cloning
• Quantum Encodings in Spin Systems and Harmonic Oscillators
• Mutually unbiased binary observable sets on N qubits
• A generalized Pancharatnam geometric phase formula for three level systems
• The mean king's problem: Prime degrees of freedom
• Quantum lithography, entanglement and Heisenberg-limited parameter estimation
• Mutually Unbiased Bases and Finite Projective Planes
• Multicomplementary operators via finite Fourier transform
• Aspects of mutually unbiased bases in odd prime power dimensions
• Operationally Invariant Measure of the Distance between Quantum States by Complementary Measurements
• Optimal estimation of multiple phases
• Quantum phases of a qutrit
• Generation of eigenstates using the phase-estimation algorithm
• Ascertaining the Values of ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$ of a Polarization Qubit
• Optimal quantum estimation of the coupling between two bosonic modes
• A description of entanglement in terms of quantum phase
• Testing operational phase concepts in quantum optics

Cited by in corpus (5)

• Multi-parameter Quantum Metrology
• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Variations on a theme of Heisenberg, Pauli and Weyl
• Discrete squeezed states for finite-dimensional spaces
• The Frobenius formalism in Galois quantum systems