States that "look the same" with respect to every basis in a mutually unbiased set

Abstract

A complete set of mutually unbiased bases in a Hilbert space of dimension $d$ defines a set of $d+1$ orthogonal measurements. Relative to such a set, we define a "MUB-balanced state" to be a pure state for which the list of probabilities of the $d$ outcomes of one of these measurements is independent of the choice of measurement, up to permutations. In this paper we explicitly construct a MUB-balanced state for each prime power dimension $d$ for which $d=3$ (mod 4). These states have already been constructed by Appleby in unpublished notes, but our presentation here is different in that both the expression for the states themselves and the proof of MUB-balancedness are given in terms of the discrete Wigner function, rather than the density matrix or state vector. The discrete Wigner functions of these states are "rotationally symmetric" in a sense roughly analogous to the rotational symmetry of the energy eigenstates of a harmonic oscillator in the continuous two-dimensional phase space. Upon converting the Wigner function to a density matrix, we find that the states are expressible as real state vectors in the standard basis. We observe numerically that when $d$ is large (and not a power of 3), a histogram of the components of such a state vector appears to form a semicircular distribution.

arXiv comment24 pages; minor notational change in v2; simplified notation and a note added on inequivalent MUBs in v3

References in corpus (29)

• Security of quantum key distribution using d-level systems
• Optimal eavesdropping in quantum cryptography with six states
• On mutually unbiased bases
• A new proof for the existence of mutually unbiased bases
• Quantum cryptography with 3-state systems
• The Resource Theory of Stabilizer Computation
• Discrete phase space based on finite fields
• Negative Quasi-Probability as a Resource for Quantum Computation
• Higher-dimensional orbital angular momentum based quantum key distribution with mutually unbiased bases
• Hudson's Theorem for finite-dimensional quantum systems
• Quasi-probability representations of quantum theory with applications to quantum information science
• Entropic uncertainty relations and locking: tight bounds for mutually unbiased bases
• Maximal Sets of Mutually Unbiased Quantum States in Dimension Six
• Quantum Secret Sharing with Multi-level Mutually (Un-)Biased Bases
• Mutually unbiased bases in dimension six: The four most distant bases
• Unconditionally Secure Key Distribution In Higher Dimensions By Depolarization
• Numerical evidence for the maximum number of mutually unbiased bases in dimension six
• Non-negative Wigner functions in prime dimensions
• Generation of mutually unbiased bases as powers of a unitary matrix in 2-power dimensions
• Properties of the extended Clifford group with applications to SIC-POVMs and MUBs
• On the Impossibility to Extend Triples of Mutually Unbiased Product Bases in Dimension Six
• Extrema of discrete Wigner functions and applications
• Discrete coherent and squeezed states of many-qudit systems
• An operational link between MUBs and SICs
• Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension six
• Unbiased bases (Hadamards) for 6-level systems: Four ways from Fourier
• Quantum Key Distribution Highly Sensitive to Eavesdropping
• Complete sets of cyclic mutually unbiased bases in even prime power dimensions
• Cyclic mutually unbiased bases, Fibonacci polynomials and Wiedemann's conjecture

Cited by in corpus (5)

• Maximum nonlocality and minimum uncertainty using magic states
• Mutually unbiased bases as minimal Clifford covariant 2-designs
• Product states and Schmidt rank of mutually unbiased bases in dimension six
• States that are far from being stabilizer states
• Cyclic measurements and simplified quantum state tomography