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<p>We initiate a systematic study of triplets of mutually unbiased bases (MUBs). We show that each MUB-triplet in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {C}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> is characterized by a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d\times d\times d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>×</mo> <mi>d</mi> <mo>×</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation> object that we call a <i>Hadamard cube</i>. We describe the basic properties of Hadamard cubes and show how a MUB-triplet can be reconstructed from such a cube, up to unitary equivalence. We also present an algebraic identity which is conjectured to hold for all MUB-triplets in dimension 6. If true, it would imply the long-standing conjecture of Zauner that the maximum number of MUBs in dimension 6 is three.</p>

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Triplets of mutually unbiased bases

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月4日

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  • Máte Matolcsi,
  • Ákos K. Matszangosz,
  • Dániel Varga,
  • Mihály Weiner

摘要

We initiate a systematic study of triplets of mutually unbiased bases (MUBs). We show that each MUB-triplet in \(\mathbb {C}^d\) C d is characterized by a \(d\times d\times d\) d × d × d object that we call a Hadamard cube. We describe the basic properties of Hadamard cubes and show how a MUB-triplet can be reconstructed from such a cube, up to unitary equivalence. We also present an algebraic identity which is conjectured to hold for all MUB-triplets in dimension 6. If true, it would imply the long-standing conjecture of Zauner that the maximum number of MUBs in dimension 6 is three.