Mutually Unbiased Bases and Complementary Spin 1 Observables

TL;DR

This paper investigates the properties of mutually unbiased bases (MUBs) for spin 1 systems, exploring their physical meaning, state squeezing, and measurement implications, extending the concept beyond spin 1/2.

Contribution

It analyzes spin 1 MUBs, introduces analogs of position and momentum operators, and studies their generation and measurement via spin squeezing, extending MUB concepts to higher spins.

Findings

Spin 1 MUB states must be squeezed.

Analog position and momentum operators are defined for spin 1.

Fourier-like transform used to generate and measure MUB states.

Abstract

The two observables are complementary if they cannot be measured simultaneously, however they become maximally complementary if their eigenstates are mutually unbiased. Only then the measurement of one observable gives no information about the other observable. The spin projection operators onto three mutually orthogonal directions are maximally complementary only for the spin 1/2. For the higher spin numbers they are no longer unbiased. In this work we examine the properties of spin 1 Mutually Unbiased Bases (MUBs) and look for the physical meaning of the corresponding operators. We show that if the computational basis is chosen to be the eigenbasis of the spin projection operator onto some direction z, the states of the other MUBs have to be squeezed. Then, we introduce the analogs of momentum and position operators and interpret what information about the spin vector the observer gains while measuring them. Finally, we study the generation and the measurement of MUBs states by introducing the Fourier like transform through spin squeezing. The higher spin numbers are also considered.