On the structure of the sets of mutually unbiased bases for N qubits

Abstract

For a system of N qubits, spanning a Hilbert space of dimension d=2^N, it is known that there exists d+1 mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases with different properties as far as separability is concerned. Here we derive the four sets of nine bases for three qubits, and show how they are unitarily related. We also briefly discuss the four-qubit case, give the entanglement structure of sixteen sets of bases,and show some of them, and their interrelations, as examples. The extension of the method to the general case of N qubits is outlined.

16 pages, 10 tables, 1 figure

References in corpus (21)

• Persistent entanglement in arrays of interacting particles
• A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound
• Multi-party entanglement in graph states
• Quantum Error Correction and Orthogonal Geometry
• A new proof for the existence of mutually unbiased bases
• Quantum cryptography with 3-state systems
• Discrete phase space based on finite fields
• Maximum-likelihood estimation of the density matrix
• Operationally Invariant Information in Quantum Measurements
• Universal Algorithm for Optimal Estimation of Quantum States from Finite Ensembles
• Mutually unbiased binary observable sets on N qubits
• The mean king's problem: Prime degrees of freedom
• Mutually Unbiased Bases and Finite Projective Planes
• Multicomplementary operators via finite Fourier transform
• Qubits in phase space: Wigner function approach to quantum error correction and the mean king problem
• Minimum Decision Cost for Quantum Ensembles
• Aspects of mutually unbiased bases in odd prime power dimensions
• Mean king's problem with mutually unbiased bases and orthogonal Latin squares
• Mutually Unbiased Bases and Trinary Operator Sets for N Qutrits
• The mean king's problem: Spin 1
• An expectation value expansion of Hermitian operators in a discrete Hilbert space

Cited by in corpus (48)

• On mutually unbiased bases
• Quantum Process Tomography: Resource Analysis of Different Strategies
• Experimental Realization of a Measurement-Induced Entanglement Phase Transition on a Superconducting Quantum Processor
• Quantum key distribution session with 16-dimensional photonic states
• Experimental quantum tomography of photonic qudits via mutually unbiased basis
• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Quantum process reconstruction based on mutually unbiased basis
• Geometrical approach to mutually unbiased bases
• A SU(2) recipe for mutually unbiased bases
• A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements
• Mutually Unbiased Bases for Continuous Variables
• Optimal quantum state reconstruction for cold trapped ions
• The Limited Role of Mutually Unbiased Product Bases in Dimension Six
• Mutually unbiased bases and discrete Wigner functions
• Discrete phase-space structure of $n$-qubit mutually unbiased bases
• Classification of GHZ-type, W-type and GHZ-W-type multiqubit entanglements
• On the Impossibility to Extend Triples of Mutually Unbiased Product Bases in Dimension Six
• Iso-entangled mutually unbiased bases, symmetric quantum measurements and mixed-state designs
• Entanglement Patterns in Mutually Unbiased Basis Sets for N Prime-state Particles
• Graph-state formalism for mutually unbiased bases
• The Frobenius formalism in Galois quantum systems
• Maximally Entangled States of Four Nonbinary Particles
• Five open problems in quantum information
• Quantum Key Distribution Highly Sensitive to Eavesdropping
• Detecting three-qubit bound MUB diagonal entangled states via Nonlinear optimal entanglement witnesses
• Optimal reconstruction of states in qutrits system
• Sizing up entanglement in mutually unbiased bases with Fisher information
• Weak mutually unbiased bases
• Maximum information gain in weak or continuous measurements of qudits: complementarity is not enough
• Orbits of mutually unbiased bases
• Quantum Polarimetry
• Structure of the sets of mutually unbiased bases with cyclic symmetry
• Mutually unbiased bases with free parameters
• A new method of construction of all sets of mutually unbiased bases for two-qubit systems
• Partially Unbiased Entangled Bases
• Invariant information and complementarity in high-dimensional states
• SU(2) nonstandard bases: the case of mutually unbiased bases
• Certainty relations between local and nonlocal observables
• Cyclic Mutually Unbiased Bases and Quantum Public-Key Encryption
• Machine learning of quantum channels on NISQ devices
• An analytic function approach to weak mutually unbiased bases
• Entanglement for any definition of two subsystems
• Invariant Off-Diagonality: SICs as Equicoherent Quantum States
• MUB-like structures and tomographic reconstruction for N-ququart systems
• Practical implementation of mutually unbiased bases using quantum circuits
• The correspondence between mutually unbiased bases and mutually orthogonal extraordinary supersquares
• Shot-noise reduction for lattice Hamiltonians
• Discrete phase-space approach to mutually orthogonal Latin squares