Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p_0,p_1<p_2< \cdots <p_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo><</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> be positive real numbers and r be any integer. We obtain the determinants of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n\times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> Kraus matrix <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K_r(p_0:p_1,\dots ,p_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mi>r</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>:</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> whose (i, j) entry is equal to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \frac{1}{p_i-p_j}{\left( \frac{p_i^{r}-p_0^{r}}{p_i-p_0}-\frac{p_j^{r}-p_0^{r}}{p_j-p_0} \right) }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mrow> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>p</mi> <mi>j</mi> </msub> </mrow> </mfrac> <mfenced close=")" open="("> <mfrac> <mrow> <msubsup> <mi>p</mi> <mi>i</mi> <mi>r</mi> </msubsup> <mo>-</mo> <msubsup> <mi>p</mi> <mn>0</mn> <mi>r</mi> </msubsup> </mrow> <mrow> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <msubsup> <mi>p</mi> <mi>j</mi> <mi>r</mi> </msubsup> <mo>-</mo> <msubsup> <mi>p</mi> <mn>0</mn> <mi>r</mi> </msubsup> </mrow> <mrow> <msub> <mi>p</mi> <mi>j</mi> </msub> <mo>-</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> </mrow> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, in the terms of Schur polynomials.

Determinants of Kraus matrices associated with power functions

期刊论文
发表日期：

2025年10月9日

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Bhumika Choudhary,
Yogesh Kapil,
Mandeep Singh

摘要

Let $p_0,p_1<p_2< \cdots <p_n$ $p_{0}, p_{1} < p_{2} < \dots < p_{n}$ be positive real numbers and r be any integer. We obtain the determinants of $n\times n$ $n \times n$ Kraus matrix $K_r(p_0:p_1,\dots ,p_n)$ $K_{r} (p_{0} : p_{1}, \dots, p_{n})$ whose (i, j) entry is equal to $ \frac{1}{p_i-p_j}{\left( \frac{p_i^{r}-p_0^{r}}{p_i-p_0}-\frac{p_j^{r}-p_0^{r}}{p_j-p_0} \right) }$ $\frac{1}{p_{i} - p_{j}} (\frac{p_{i}^{r} - p_{0}^{r}}{p_{i} - p_{0}} - \frac{p_{j}^{r} - p_{0}^{r}}{p_{j} - p_{0}})$ , in the terms of Schur polynomials.