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<p>We show that a symmetric informationally complete positive operator-valued measure exists in a given dimension <i>d</i> if and only if there exists a <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="23_2025_1633_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(d^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>d</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-dimensional operator system satisfying certain order-theoretic conditions. We also describe a method of constructing such an operator system and demonstrate that the first step of this construction can be carried out successfully. We obtain analogous results for the existence of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="23_2025_1633_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(d+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> mutually unbiased bases in a given dimension.</p>

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Operator System Characterizations of SIC-POVMs and Mutually Unbiased Bases

  • 期刊论文

  • 发表日期：

    2025年10月9日

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  • Travis B. Russell

摘要

We show that a symmetric informationally complete positive operator-valued measure exists in a given dimension d if and only if there exists a \(d^2\) d 2 -dimensional operator system satisfying certain order-theoretic conditions. We also describe a method of constructing such an operator system and demonstrate that the first step of this construction can be carried out successfully. We obtain analogous results for the existence of \(d+1\) d + 1 mutually unbiased bases in a given dimension.