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<p>In quantum information theory, symmetric informationally complete positive operator-valued measures (SIC-POVMs) are related to quantum state tomography, quantum cryptography and foundational studies. It is difficult to construct SIC-POVMs and remains unknown whether SIC-POVMs exist for infinitely dimension. Therefore, researchers have proposed the solution to construct approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs). This paper proposes a construction of ASIC-POVMs using character sums over Galois rings. The dimension of this ASIC-POVMs is <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4656_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(q-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where <i>q</i> is a prime power. Also, our constructions include partial results about ASIC-POVMs constructed by X. Cao et al (X. Cao, J. Mi and S. Xu: Two constructions of approximately symmetric informationally complete positive operator-valued measures. J. Math. Phys., 58(6), 062201 (2017)).</p>

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New constructions of approximately symmetric informationally complete positive operator-valued measures by character sums over Galois rings

  • 期刊论文

  • 发表日期：

    2025年2月14日

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  • Gang Wang,
  • You Gao,
  • Mingyue Xie,
  • Minyao Niu

摘要

In quantum information theory, symmetric informationally complete positive operator-valued measures (SIC-POVMs) are related to quantum state tomography, quantum cryptography and foundational studies. It is difficult to construct SIC-POVMs and remains unknown whether SIC-POVMs exist for infinitely dimension. Therefore, researchers have proposed the solution to construct approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs). This paper proposes a construction of ASIC-POVMs using character sums over Galois rings. The dimension of this ASIC-POVMs is \(q-1\) q - 1 , where q is a prime power. Also, our constructions include partial results about ASIC-POVMs constructed by X. Cao et al (X. Cao, J. Mi and S. Xu: Two constructions of approximately symmetric informationally complete positive operator-valued measures. J. Math. Phys., 58(6), 062201 (2017)).