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Connections between Mutually Unbiased Bases and Quantum Random Access Codes.

Edgar A Aguilar1, Jakub J Borkała1, Piotr Mironowicz1,2

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Summary
This summary is machine-generated.

We introduce a new quantum communication protocol to measure unbiasedness for Hilbert spaces. This allows us to determine the existence of mutually unbiased bases using semidefinite programming.

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Area of Science:

  • Quantum information science
  • Quantum communication complexity

Background:

  • Understanding the structure of quantum states and their properties is crucial in quantum information science.
  • The concept of mutually unbiased bases (MUBs) is fundamental in quantum measurement theory and has applications in quantum cryptography and quantum computing.

Purpose of the Study:

  • To introduce a novel quantum communication complexity protocol, the promise-quantum random access code (P-QRAC).
  • To define and operationalize a new measure of unbiasedness for bases in Hilbert spaces.
  • To provide a method for investigating the existence of a specific number of MUBs in a given dimension.

Main Methods:

  • Development of the promise-quantum random access code (P-QRAC) protocol.
  • Introduction of a new measure quantifying the unbiasedness of bases.
  • Application of semidefinite programming techniques to analyze the existence of MUBs.

Main Results:

  • The proposed measure of unbiasedness has a clear operational meaning.
  • The P-QRAC protocol provides a tool to study MUBs.
  • Semidefinite programming can be used to determine the existence of MUBs in a given dimension.

Conclusions:

  • The new measure and protocol offer novel insights into the structure of Hilbert spaces.
  • The findings contribute to the understanding of MUBs and their limitations.
  • This work opens avenues for further research in quantum information theory and its applications.