Mutually Unbiased Product Bases for Multiple Qudits

TL;DR

This paper explores the structure and limitations of mutually unbiased product bases in multi-qudit systems, revealing conditions for mutual unbiasedness and classifying bases for qubits and qutrits, with implications for entanglement.

Contribution

It provides a complete classification of mutually unbiased product bases for multiple qubits or qutrits and establishes conditions for mutual unbiasedness involving entanglement.

Findings

Mutually unbiased product states require each factor to be mutually unbiased in its subsystem.

There is a tight limit on the number of mutually unbiased product bases supported.

Only maximally entangled states can be mutually unbiased to a maximal set of product bases.

Abstract

We investigate the interplay between mutual unbiasedness and product bases for multiple qudits of possibly different dimensions. A product state of such a system is shown to be mutually unbiased to a product basis only if each of its factors is mutually unbiased to all the states which occur in the corresponding factors of the product basis. This result implies both a tight limit on the number of mutually unbiased product bases which the system can support and a complete classification of mutually unbiased product bases for multiple qubits or qutrits. In addition, only maximally entangled states can be mutually unbiased to a maximal set of mutually unbiased product bases.