基于随机密度矩阵特征值联合分布的统一(r, s)相对微分熵

The Unified (r, s)-Relative Differential Entropy Based on Joint Distribution of Random Density Matrix

摘要: 采用Laplace变换和Laplace逆变换研究随机密度矩阵特征值联合分布的统一(r, s)相对微分熵。一方面，定义了在Haar分布的双体纯态上取部分迹所诱导的随机密度矩阵的特征值的联合分布相对于其对角元的联合分布(其对角元的联合分布相对于取部分迹所诱导的随机密度矩阵的特征值的联合分布)的统一(r, s)相对微分熵。另一方面，计算三种情形下的统一(r, s)相对微分熵，推广了微分熵的范围。

Abstract: The unified (r, s)-relative differential entropy of the joint distribution of eigenvalues of random density matrices is studied by Laplace transform and Laplace inverse transform. On the one hand, the unified (r, s)-relative differential entropy of the joint distribution of the eigenvalues to diagonal entries of random density matrices induced by partial tracing (the diagonal entries of random density matrices induced by partial tracing to joint distribution of the eigenvalues) over Haar-distributed bipartite pure states is defined. On the other hand, the unified (r, s)-relative differential entropy in the three cases is calculated. The range of differential entropy is generalized.

1. 引言

熵在统计物理学和信息论中都起着重要作用。在1991年，Rathie和Taneja引进了以下统一(r, s)熵，其中包括(r, s)熵，Renyi有序熵 [1] 。在2004年，Furuichi研究了量子Tsallis相对熵的数学性质 [2] 。在2006年，胡和叶介绍了经典统一(r, s)熵的量子版本 [3] 。在2011年，汪等人定义了相应的统一(r, s)相对熵并研究了它的性质 [4] 。在2016年，罗等人研究了随机密度矩阵特征值联合分布的微分熵 [5] 。在本文中，我们将定义和研究随机密度矩阵特征值联合分布的统一(r, s)相对微分熵。

2. 统一(r, s)相对微分熵的定义

由 [5] 中的定义，我们知道：

1) 在 $\mathrm{Re}\left(z\right)>0$ 上的伽马函数为 $\Gamma \left(z\right)={\displaystyle {\int }_0^{\infty }{t}^{z-1}{\text{e}}^{-t}\text{d}t}$ ， $\Gamma \left(x\right),\psi \left(x\right)$ 分别代表Gamma 函数和Digamma函数。

$\psi \left(x\right)=\frac{\text{d}\ln \Gamma \left(x\right)}{\text{d}x}=\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}$ ， $\psi \left(k+1\right)=H_k-\gamma$ ，其中 $H_k={\displaystyle {\sum }_{j=1}^k\frac{1}{j}}$ ， $\gamma \approx 0.57721$ 为欧拉常数。

2) 若Wishart矩阵W的构成矩阵Z是复的均值为0，方差为 ${\sigma }^{\text{2}}$ 的独立同分布的高斯变量， $Z{Z}^{+}$ 联合概率密度的相应特征值 $\left({\mu }_j\in \left[0,\infty \right),j=1,2,\cdots ,m\right)$ 的联合概率函数为

$q\left({\mu }_1,\cdots ,{\mu }_m\right)=C_q\exp \left(-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\mu }_j}\right){\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^2}{\displaystyle \underset{j=1}{\overset{m}{\prod }}{\mu }_j^{n-m}}$ ，其中 ${\displaystyle \underset{j=1}{\overset{m}{\sum }}{\mu }_j}=1$ (1)

$C_{\text{q}}=1/{\displaystyle \underset{1\le j\le m}{\prod }\left[j!\left(n-j\right)!\right]}$ (2)

3) 在Haar分布的双体纯态上取部分迹所诱导的随机密度矩阵的特征值 $\left({\lambda }_j\in \left[0,1\right],j=1,2,\cdots ,m\right)$ 的联合概率函数为

$p\left({\lambda }_1,\cdots ,{\lambda }_m\right)=C_p{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\lambda }_i-{\lambda }_j\right)}^2}{\displaystyle \underset{j=1}{\overset{m}{\prod }}{\lambda }_j^{n-m}}$ ，其中 ${\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}=1$ (3)

$C_p=\Gamma \left(mn\right)C_q$ (4)

4) 在Haar分布的双体纯态上其对角元的联合概率函数为

$\psi \left({\rho }_{11},{\rho }_{22},\cdots ,{\rho }_{mm}\right)=C_{\psi }{\displaystyle \underset{j=1}{\overset{m}{\prod }}{\rho }_{jj}^{n-1}}$ ，其中 ${\displaystyle \underset{j=1}{\overset{m}{\sum }}{\rho }_{jj}}=1$ (5)

$C_{\Psi }=\frac{\Gamma \left(mn\right)}{\Gamma {\left(n\right)}^m}$ (6)

5) $I_m\left(\alpha ,r\right)={\displaystyle {\int }_0^{\infty }\cdot \cdot \cdot {\displaystyle {\int }_0^{\infty }\exp \left(-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\mu }_j}\right){\displaystyle \underset{1\le i<j\le m}{\prod }{\left|{\mu }_i-{\mu }_j\right|}^{2r}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\left({\mu }_k^{\alpha }\text{d}{\mu }_k\right)}}}}={\displaystyle \underset{\text{k}=0}{\overset{m-1}{\prod }}\frac{\Gamma \left(\alpha +1+kr\right)\Gamma \left(1+\left(k+1\right)r\right)}{\Gamma \left(1+r\right)}}$ (7)

$I_m\left(n-m,1\right)=\frac{\text{1}}{C_q}$ (8)

$I_m\left(n-\text{1},\text{0}\right)={\Gamma }^m\left(n\right)$ (9)

因此，我们可以得到

$\begin{array}{c}\frac{\partial }{\partial r}\partial I_m\left(\alpha \left(r\right),\beta \left(r\right)\right)=\frac{\partial I_m\left(\alpha \left(r\right),\beta \left(r\right)\right)}{\partial \alpha }\times \frac{\text{d}\alpha }{\text{d}r}+\frac{\partial I_m\left(\alpha \left(r\right),\beta \left(r\right)\right)}{\partial \beta }\times \frac{\text{d}\beta }{\text{d}r}\\ =I_m\left(\alpha \left(r\right),\beta \left(r\right)\right)\{{\alpha }^{\prime }\left(r\right)\times {\displaystyle \underset{k=0}{\overset{m-1}{\sum }}\psi \left(\alpha \left(r\right)+1+k\beta \left(r\right)\right)}\\ +{\beta }^{\prime }\left(r\right)\times [{\displaystyle \underset{k=1}{\overset{m-1}{\sum }}k\psi \left(\alpha \left(r\right)+1+k\beta \left(r\right)\right)}\\ +{\displaystyle \underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k\beta \left(r\right)\right)}-m\psi \left(1+\beta \left(r\right)\right)]\}\end{array}$ (10)

定义2.1：对任意的 $r>0$ 和实数s，在Haar分布的双体纯态上取部分迹所诱导的随机密度矩阵的特征值的联合分布相对于其对角元的联合分布的统一(r, s)相对微分熵为

$E_r^s\left(p\left|\right|\psi \right)=\{\begin{array}{l}{h}_r^s\left(p\left|\right|\psi \right),r\ne 1,s\ne 0\hfill \\ h_r\left(p\left|\right|\psi \right),r\ne 1,s=0\hfill \\ h^r\left(p\left|\right|\psi \right),r\ne 1,s=1\hfill \\ {}_{r}h\left(p\left|\right|\psi \right),r\ne 1,s=r^{-1}\hfill \\ h\left(p\left|\right|\psi \right),r=1\hfill \end{array}$

其中

$h_r^s\left(p\left|\right|\psi \right)=-{\left[\left(1-r\right)s\right]}^{-1}\left[{\left({\displaystyle \int p\left(\lambda \right)}{\left(p\left(\lambda \right)/\psi \left(\lambda \right)\right)}^{r-1}\text{d}\lambda \right)}^s-1\right],r>0,r\ne 1,s\ne 0$

$h_r\left(p\left|\right|\psi \right)=-{\left(1-r\right)}^{-1}\ln \left({\displaystyle \int p\left(\lambda \right)}{\left(p\left(\lambda \right)/\psi \left(\lambda \right)\right)}^{r-1}\text{d}\lambda \right),r>0,r\ne 1$

$h^r\left(p\left|\right|\psi \right)=-{\left(1-r\right)}^{-1}\left({\displaystyle \int p\left(\lambda \right)}{\left(p\left(\lambda \right)/\psi \left(\lambda \right)\right)}^{r-1}\text{d}\lambda -1\right),r>0,r\ne 1$

${}_{r}h\left(p\left|\right|\psi \right)=-{\left(r-1\right)}^{-1}\left[{\left({\displaystyle \int p\left(\lambda \right)}{\left(p\left(\lambda \right)/\psi \left(\lambda \right)\right)}^{1/r-1}\text{d}\lambda \right)}^r-1\right],r>0,r\ne 1$

$h\left(p\left|\right|\psi \right)={\displaystyle \int p\left(\lambda \right)}\ln \left(p\left(\lambda \right)/\psi \left(\lambda \right)\right)\text{d}\lambda$

定义2.2：对任意的 $r>0$ 和实数s，在Haar分布的双体纯态上其对角元的联合分布相对于取部分迹所诱导的随机密度矩阵的特征值的联合分布的统一(r, s)相对微分熵为

$E_r^s\left(\psi \left|\right|p\right)=\{\begin{array}{l}{h}_r^s\left(\psi \left|\right|p\right),r\ne 1,s\ne 0\hfill \\ h_r\left(\psi \left|\right|p\right),r\ne 1,s=0\hfill \\ h^r\left(\psi \left|\right|p\right),r\ne 1,s=1\hfill \\ {}_{r}h\left(\psi \left|\right|p\right),r\ne 1,s=r^{-1}\hfill \\ h\left(\psi \left|\right|p\right),r=1\hfill \end{array}$

其中

$h_r^s\left(\psi \left|\right|p\right)=-{\left[\left(1-r\right)s\right]}^{-1}\left[{\left({\displaystyle \int \psi \left(\lambda \right)}{\left(\psi \left(\lambda \right)/p\left(\lambda \right)\right)}^{r-1}\text{d}\lambda \right)}^s-1\right],r>0,r\ne 1,s\ne 0$

$h_r\left(\psi \left|\right|p\right)=-{\left(1-r\right)}^{-1}\ln \left({\displaystyle \int \psi \left(\lambda \right)}{\left(\psi \left(\lambda \right)/p\left(\lambda \right)\right)}^{r-1}\text{d}\lambda \right),r>0,r\ne 1$

$h^r\left(\psi \left|\right|p\right)=-{\left(1-r\right)}^{-1}\left({\displaystyle \int \psi \left(\lambda \right)}{\left(\psi \left(\lambda \right)/p\left(\lambda \right)\right)}^{r-1}\text{d}\lambda -1\right),r>0,r\ne 1$

${}_{r}h\left(\psi \left|\right|p\right)=-{\left(r-1\right)}^{-1}\left[{\left({\displaystyle \int \psi \left(\lambda \right)}{\left(\psi \left(\lambda \right)/p\left(\lambda \right)\right)}^{1/r-1}\text{d}\lambda \right)}^r-1\right],r>0,r\ne 1$

$h\left(\psi \left|\right|p\right)={\displaystyle \int \psi \left(\lambda \right)}\ln \left(\psi \left(\lambda \right)/p\left(\lambda \right)\right)\text{d}\lambda$

3. 统一(r, s)相对微分熵的计算

定理3.1：当 $r>0,r\ne 1,s=0$ 时，在Haar分布的双体纯态上取部分迹所诱导的随机密度矩阵的特征值的联合分布相对于其对角元的联合分布的统一(r, s)相对微分熵为

$h_r\left(p\left|\right|\psi \right)={\left(r-1\right)}^{-1}\left[r\ln C_p+\left(1-r\right)\ln C_{\psi }+\ln I_m\left(\left(1-m\right)r+n-1,r\right)-\ln \Gamma \left(mn\right)\right]$

证明：

$\begin{array}{c}{h}_r\left(p\left|\right|\psi \right)=-{\left(1-r\right)}^{-1}\ln \left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }p\left({\lambda }_1,\cdots ,{\lambda }_m\right)}}{\left(p\left({\lambda }_1,\cdots ,{\lambda }_m\right)/\psi \left({\lambda }_1,\cdots ,{\lambda }_m\right)\right)}^{r-1}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}\right)\\ ={\left(r-1\right)}^{-1}\ln \left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }{p}^r\left({\lambda }_1,\cdots ,{\lambda }_m\right){\psi }^{1-r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}}}\right)\end{array}$

令

$F_r\left(t\right)={\left(r-1\right)}^{-1}\ln \left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\delta \left(t-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right)p^r\left({\lambda }_1,\cdots ,{\lambda }_m\right){\psi }^{1-r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}}}\right)$

对 $F_r\left(t\right)$ 应用Laplace变换 $\left(t\to s\right)$ ，则有

${\tilde{F}}_r\left(s\right)={\left(r-1\right)}^{-1}\ln \left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\exp \left(-s{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right)p^r\left({\lambda }_1,\cdots ,{\lambda }_m\right){\psi }^{1-r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}}}\right)$

令 ${\lambda }_k=\frac{{\mu }_k}{s},k=1,2,\cdots ,m$

$\frac{C_p}{C_q}\exp \left({\displaystyle \underset{i=1}{\overset{m}{\sum }}{\mu }_m}\right)q\left({\mu }_1,\cdots {\mu }_m\right)=p\left({\mu }_1,\cdots {\mu }_m\right)$ ， $p\left(\frac{{\mu }_1}{s},\cdots ,\frac{{\mu }_m}{s}\right)=s^{m-nm}p\left({\mu }_1,\cdots ,{\mu }_m\right)$

再由式(1)和式(3)联立得到

$p\left({\lambda }_1,\cdots ,{\lambda }_m\right)=s^{m-mn}{C}_p{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^2}{\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_k^{n-m}}$ (11)

$\psi \left({\lambda }_1,\cdots ,{\lambda }_m\right)=s^{m-mn}{C}_{\psi }{\displaystyle \underset{j=1}{\overset{m}{\prod }}{\mu }_j^{n-1}}$ (12)

则

$\begin{array}{c}{\tilde{F}}_r\left(s\right)={\left(r-1\right)}^{-1}\ln \left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\exp \left(-s{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right){\left(s^{m-mn}{C}_p{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^2}{\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_k^{n-m}}\right)}^r{\left(s^{m-mn}{C}_{\psi }{\displaystyle \underset{j=1}{\overset{m}{\prod }}{\mu }_j^{n-1}}\right)}^{1-r}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\frac{\text{d}{\mu }_k}{s}}}}\right)\\ ={\left(r-1\right)}^{-1}\ln \left({\displaystyle {\int }_0^{\infty }\cdots }{\displaystyle {\int }_0^{\infty }{s}^{-mn}{C}_p^r{C}_{\psi }^{1-r}}{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^{2r}}\exp \left(-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\mu }_j}\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_k^{\left(1-m\right)r+n-1}\text{d}{\mu }_k}\right)\\ ={\left(r-1\right)}^{-1}\ln \left(s^{-mn}{C}_p^r{C}_{\psi }^{1-r}{I}_m\left(\left(1-m\right)r+n-1,r\right)\right)\end{array}$

由于

$L^{-1}\left\{s^{-m}\right\}\left(t\right)=\frac{t^{m-1}}{\Gamma \left(m\right)}$ (13)

所以

$L^{-1}\left\{s^{-mn}\right\}\left(t\right)=\frac{t^{mn-1}}{\Gamma \left(mn\right)}$ (14)

$F_r\left(t\right)={\left(r-1\right)}^{-1}\ln \left(\frac{t^{mn-1}}{\Gamma \left(mn\right)}{C}_p^r{C}_{\psi }^{1-r}{I}_m\left(\left(1-m\right)r+n-1,r\right)\right)$

那么得到

$\begin{array}{c}{h}_r\left(p\left|\right|\psi \right)=F_r\left(1\right)={\left(r-1\right)}^{-1}\ln \left(\frac{C_p^r{C}_{\psi }^{1-r}}{\Gamma \left(mn\right)}{I}_m\left(\left(1-m\right)r+n-1,r\right)\right)\\ ={\left(r-1\right)}^{-1}\left[r\ln C_p+\left(1-r\right)\ln C_{\psi }+\ln I_m\left(\left(1-m\right)r+n-1,r\right)-\ln \Gamma \left(mn\right)\right]\end{array}$ (15)

注1：当 $r\to 1$ 时，

$\begin{array}{c}\underset{r\to 1}{\lim }{h}_r\left(p\left|\right|\psi \right)=-\ln \left({\displaystyle \underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!}\right)-m\psi \left(2\right)+m\ln \Gamma \left(n\right)\\ -{\displaystyle \underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)}+{\displaystyle \underset{k=1}{\overset{m}{\sum }}k}\psi \left(k+1\right)\end{array}$

证明：

由式(10)得到

$\begin{array}{l}{\frac{\partial \ln I_m\left(\left(1-m\right)r+n-1,r\right)}{\partial r}|}_{r=1}=\frac{1}{I_m\left(\left(1-m\right)r+n-1,r\right)}{\frac{\partial I_m\left(\left(1-m\right)r+n-1,r\right)}{\partial r}|}_{r=1}\\ =\left(1-m\right)\times {\displaystyle \underset{k=0}{\overset{m-1}{\sum }}\psi \left(\left(1-m\right)r+n+kr\right)}-m\psi \left(1+r\right)\\ +{\displaystyle \underset{k=1}{\overset{m-1}{\sum }}k\psi \left(\left(1-m\right)r+n+kr\right)}+{{\displaystyle \underset{k=1}{\overset{m}{\sum }}k\psi \left(1+kr\right)}|}_{r=1}\\ ={\displaystyle \underset{k=0}{\overset{m-1}{\sum }}\left(1-m+n+k\right)\times \psi \left(1-m+n+k\right)}-m\psi \left(2\right)+{\displaystyle \underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k\right)}\end{array}$ (16)

把式(2)、式(4)、式(6)、式(8)、式(16)代入式(15)得到

$\begin{array}{c}\underset{r\to 1}{\lim }{h}_r\left(p\left|\right|\psi \right)={\left(r-1\right)}^{-1}\left[r\ln C_p+\left(1-r\right)\ln C_{\psi }+\ln I_m\left(\left(1-m\right)r+n-1,r\right)-\ln \Gamma \left(mn\right)\right]\\ =\ln C_p-\ln C_{\psi }+{\frac{\partial \ln I_m\left(\left(1-m\right)r+n-1,r\right)}{\partial r}|}_{r=1}\\ =-\ln \left({\displaystyle \underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!}\right)-m\psi \left(2\right)+m\ln \Gamma \left(n\right)\\ -{\displaystyle \underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)}+{\displaystyle \underset{k=1}{\overset{m}{\sum }}k}\psi \left(k+1\right)\end{array}$

定理3.2：当 $r>0,r\ne 1,s=1$ 时，在Haar分布的双体纯态上取部分迹所诱导的随机密度矩阵的特征值的联合分布相对于其对角元的联合分布的统一(r, s)相对微分熵为

$h^r\left(p\left|\right|\psi \right)={\left(r-1\right)}^{-1}\left(\frac{C_p^r{C}_{\psi }^{1-r}}{\Gamma \left(mn\right)}{I}_m\left(\left(1-m\right)r+n-1,r\right)-1\right)$

证明：

$\begin{array}{c}{h}^r\left(p\left|\right|\psi \right)=-{\left(1-r\right)}^{-1}\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }p\left({\lambda }_1,\cdots ,{\lambda }_m\right)}}{\left(p\left({\lambda }_1,\cdots ,{\lambda }_m\right)/\psi \left({\lambda }_1,\cdots ,{\lambda }_m\right)\right)}^{r-1}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}-1\right)\\ ={\left(r-1\right)}^{-1}\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }{p}^r\left({\lambda }_1,\cdots ,{\lambda }_m\right){\psi }^{1-r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k-1}}}\right)\end{array}$

令

$F^r\left(t\right)={\left(r-1\right)}^{-1}\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\delta \left(t-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right)p^r\left({\lambda }_1,\cdots ,{\lambda }_m\right){\psi }^{1-r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k-1}}}\right)$

对 $F^r\left(t\right)$ 应用Laplace变换 $\left(t\to s\right)$ ，则有

${\tilde{F}}^r\left(s\right)={\left(r-1\right)}^{-1}\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\exp \left(-s{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right)p^r\left({\lambda }_1,\cdots ,{\lambda }_m\right){\psi }^{1-r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k-1}}}\right)$

把式(7)、式(11)、式(12)代入上式得到

$\begin{array}{c}{\tilde{F}}^r\left(s\right)={\left(r-1\right)}^{-1}\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\exp \left(-s{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right){\left(s^{m-mn}{C}_p{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^2}{\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_k^{n-m}}\right)}^r{\left(s^{m-mn}{C}_{\psi }{\displaystyle \underset{j=1}{\overset{m}{\prod }}{\mu }_j^{n-1}}\right)}^{1-r}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\frac{\text{d}{\mu }_k}{s}-1}}}\right)\\ ={\left(r-1\right)}^{-1}\left({\displaystyle {\int }_0^{\infty }\cdots }{\displaystyle {\int }_0^{\infty }{s}^{-mn}{C}_p^r{C}_{\psi }^{1-r}}{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^{2r}}\exp \left(-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\mu }_j}\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_k^{\left(1-m\right)r+n-1}\text{d}{\mu }_k}-1\right)\\ ={\left(r-1\right)}^{-1}\left(s^{-mn}{C}_p^r{C}_{\psi }^{1-r}{I}_m\left(\left(1-m\right)r+n-1,r\right)-1\right)\end{array}$

把式(14)代入上式得到

$F^r\left(t\right)={\left(r-1\right)}^{-1}\left(\frac{t^{mn-1}}{\Gamma \left(mn\right)}{C}_p^r{C}_{\psi }^{1-r}{I}_m\left(\left(1-m\right)r+n-1,r\right)-1\right)$

那么

$h^r\left(p\left|\right|\psi \right)=F^r\left(1\right)={\left(r-1\right)}^{-1}\left(\frac{C_p^r{C}_{\psi }^{1-r}}{\Gamma \left(mn\right)}{I}_m\left(\left(1-m\right)r+n-1,r\right)-1\right)$ (17)

注2：当 $r\to 1$ 时，

$\begin{array}{c}\underset{r\to 1}{\lim }{h}^r\left(p\left|\right|\psi \right)=-\ln \left({\displaystyle \underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!}\right)-m\psi \left(2\right)+m\ln \Gamma \left(n\right)\\ -{\displaystyle \underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)}+{\displaystyle \underset{k=1}{\overset{m}{\sum }}k}\psi \left(k+1\right)\end{array}$

证明：

由式(10)，我们得到

$\begin{array}{l}{\frac{\partial I_m\left(\left(1-m\right)r+n-1,r\right)}{\partial r}|}_{r=1}\\ =I_m\left(\left(1-m\right)r+n-1,r\right)[\left(1-m\right)\times {\displaystyle \underset{k=0}{\overset{m-1}{\sum }}\psi \left(\left(1-m\right)r+n+kr\right)}-m\psi \left(1+r\right)\\ +{\displaystyle \underset{k=1}{\overset{m-1}{\sum }}k\psi \left(\left(1-m\right)r+n+kr\right)}+{{\displaystyle \underset{k=1}{\overset{m}{\sum }}k\psi \left(1+kr\right)}]|}_{r=1}\\ =I_m\left(n-m,1\right)\left[{\displaystyle \underset{k=0}{\overset{m-1}{\sum }}\left(1-m+n+k\right)\times \psi \left(1-m+n+k\right)}-m\psi \left(2\right)+{\displaystyle \underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k\right)}\right]\end{array}$ (18)

把式(2)、式(4)、式(6)、式(8)、式(18)代入式(17)得到

$\begin{array}{c}\underset{r\to 1}{\lim }{h}^r\left(p\left|\right|\psi \right)=[\ln C_p{C}_p^r{C}_{\psi }^{1-r}\frac{I_m\left(\left(1-m\right)r+n-1,r\right)}{\Gamma \left(mn\right)}-\ln C_{\psi }{C}_p^r{C}_{\psi }^{1-r}\frac{I_m\left(\left(1-m\right)r+n-1,r\right)}{\Gamma \left(mn\right)}\\ +{\frac{C_p^r{C}_{\psi }^{1-r}}{\Gamma \left(mn\right)}\frac{\partial I_m\left(\left(1-m\right)r+n-1,r\right)}{\partial r}]|}_{r=1}\\ =\ln C_p-\ln C_{\psi }+\frac{C_p}{\Gamma \left(mn\right)}{\frac{\partial I_m\left(\left(1-m\right)r+n-1,r\right)}{\partial r}|}_{r=1}\\ =-\ln \left({\displaystyle \underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!}\right)-m\psi \left(2\right)+m\ln \Gamma \left(n\right)-{\displaystyle \underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)}+{\displaystyle \underset{k=1}{\overset{m}{\sum }}k}\psi \left(k+1\right)\end{array}$

定理3.3：当 $r>0,r\ne 1,s=1/r$ 时，在Haar分布的双体纯态上取部分迹所诱导的随机密度矩阵的特征值的联合分布相对于其对角元的联合分布的统一(r, s)相对微分熵为

${}_{r}h\left(p\left|\right|\psi \right)={\left(1-r\right)}^{-1}\left[\frac{C_p{C}_{\psi }^{r-1}}{\Gamma \left(mnr\right)}{\left(I_m\left(n-1-\frac{m-1}{r},\frac{1}{r}\right)\right)}^r-1\right]$

证明：

$\begin{array}{c}{}_{r}h\left(p\left|\right|\psi \right)=-{\left(r-1\right)}^{-1}\left[{\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }p\left({\lambda }_1,\cdots ,{\lambda }_m\right)}}{\left(p\left({\lambda }_1,\cdots ,{\lambda }_m\right)/\psi \left({\lambda }_1,\cdots ,{\lambda }_m\right)\right)}^{1/r-1}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}\right)}^r-1\right]\\ ={\left(1-r\right)}^{-1}\left[{\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }{p}^{1/r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\psi }^{1-1/r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}}}\right)}^r-1\right]\end{array}$

令

${}_{r}F\left(t\right)={\left(1-r\right)}^{-1}\left[{\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\delta \left(t-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right)p^{1/r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\psi }^{1-1/r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}}}\right)}^r-1\right]$

对 ${}_{r}F\left(t\right)$ 应用Laplace变换 $\left(t\to s\right)$ ，则有

${}_r\tilde{F}\left(s\right)={\left(1-r\right)}^{-1}\left[{\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\exp \left(-s{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right)p^{1/r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\psi }^{1-1/r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}}}\right)}^r-1\right]$

把式(7)、式(11)、式(12) 代入上式得到

$\begin{array}{c}{}_r\tilde{F}\left(s\right)={\left(1-r\right)}^{-1}\left[{\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\exp \left(-s{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right){\left(s^{m-mn}{C}_p{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^2}{\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_k^{n-m}}\right)}^{1/r}{\left(s^{m-mn}{C}_{\psi }{\displaystyle \underset{j=1}{\overset{m}{\prod }}{\mu }_j^{n-1}}\right)}^{1-1/r}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\frac{\text{d}{\mu }_k}{s}}}}\right)}^r-1\right]\\ ={\left(1-r\right)}^{-1}\left[{\left({\displaystyle {\int }_0^{\infty }\cdots }{\displaystyle {\int }_0^{\infty }{s}^{-mn}{C}_p^{1/r}{C}_{\psi }^{1-1/r}}{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^{2/r}}\exp \left(-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\mu }_j}\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_k^{n-1+\left(1-m\right)/r}\text{d}{\mu }_k}\right)}^r-1\right]\\ ={\left(1-r\right)}^{-1}\left[s^{-mnr}{C}_p{C}_{\psi }^{r-1}{\left(I_m\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^r-1\right]\end{array}$

由式(13)得到

$L^{-1}\left\{s^{-mnr}\right\}\left(t\right)=\frac{t^{mnr-1}}{\Gamma \left(mnr\right)}$ (19)

${}_{r}F\left(t\right)={\left(1-r\right)}^{-1}\left[\frac{t^{mnr-1}}{\Gamma \left(mnr\right)}{C}_p{C}_{\psi }^{r-1}{\left(I_m\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^r-1\right]$

那么

${}_{r}h\left(p\left|\right|\psi \right)={}_{r}F\left(1\right)={\left(1-r\right)}^{-1}\left[\frac{C_p{C}_{\psi }^{r-1}}{\Gamma (mnr)}{\left(I_m\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^r-1\right]$ (20)

注3：当 $r\to 1$ 时，

$\begin{array}{c}\underset{r\to 1}{\lim }{}_{r}h\left(p\left|\right|\psi \right)=-\ln \left({\displaystyle \underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!}\right)-m\psi \left(2\right)+m\ln \Gamma \left(n\right)-\ln \Gamma \left(mn\right)-mn\psi \left(mn\right)\\ -{\displaystyle \underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)}+{\displaystyle \underset{k=1}{\overset{m}{\sum }}k}\psi \left(k+1\right)\end{array}$

证明：

令

$I_m{\left(\alpha \left(r\right),\beta \left(r\right)\right)}^r=y$

那么对等式两边取以e为底的对数得到

$r\ln I_m\left(\alpha \left(r\right),\beta \left(r\right)\right)=\ln y$

再对r求导得到

$\begin{array}{l}\ln I_m\left(\alpha \left(r\right),\beta \left(r\right)\right)+\frac{r}{I_m\left(\alpha \left(r\right),\beta \left(r\right)\right)}\frac{\partial I_m\left(\alpha \left(r\right),\beta \left(r\right)\right)}{\partial r}=\frac{y^{\prime }}{y}\\ y^{\prime }=y\left[\ln I_m\left(\alpha \left(r\right),\beta \left(r\right)\right)+\frac{r}{I_m\left(\alpha \left(r\right),\beta \left(r\right)\right)}\frac{\partial I_m\left(\alpha \left(r\right),\beta \left(r\right)\right)}{\partial r}\right]\end{array}$ (21)

由式(21)得到

$\begin{array}{l}\frac{\partial {\left(I_m\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^r}{\partial r}={\left(I_m\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^r\\ [\ln I_m\left(n-1+\left(1-m\right)/r,1/r\right)\begin{array}{c}\\ \end{array}+\frac{r}{I_m\left(n-1+\left(1-m\right)/r,1/r\right)}\times \frac{\partial I_m\left(n-1+\left(1-m\right)/r,1/r\right)}{\partial r}]\end{array}$ (22)

由式(10)得到

$\begin{array}{l}{\frac{\partial I_m\left(n-1+\left(1-m\right)/r,1/r\right)}{\partial r}|}_{r=1}\\ =I_m\left(n-1+\left(1-m\right)/r,1/r\right)[\frac{m-1}{r^2}{\displaystyle \underset{k=0}{\overset{m-1}{\sum }}\psi \left(n+\left(1-m\right)/r+k/r\right)}\\ -\frac{1}{r^2}{\displaystyle \underset{k=1}{\overset{m-1}{\sum }}k\psi \left(n+\left(1-m\right)/r+k/r\right)}-\frac{1}{r^2}{\displaystyle \underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k/r\right)}+{\frac{1}{r^2}m\psi \left(1+1/r\right)]|}_{r=1}\\ =I_m\left(n-m,1\right)\left[{\displaystyle \underset{k=n-m}{\overset{n-1}{\sum }}\left(1+k-n\right)\psi \left(k+1\right)}-{\displaystyle \underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k\right)}+m\psi \left(2\right)\right]\end{array}$ (23)

把式(2)、式(4)、式(6)、式(8)、式(22)、式(23)代入式(20)得到

$\begin{array}{c}\underset{r\to 1}{\lim }{}_{r}h\left(p\left|\right|\psi \right)=-[\frac{C_p{C}_{\psi }^{r-1}\ln C_{\psi }}{\Gamma \left(mnr\right)}{\left(I_m\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^r\\ +\frac{C_p{C}_{\psi }^{r-1}\psi \left(mnr\right)mn}{\Gamma \left(mnr\right)}{\left(I_m\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^r\\ +{\frac{C_p{C}_{\psi }^{r-1}}{\Gamma \left(mnr\right)}\frac{\partial {\left(I_m\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^r}{\partial r}]|}_{r=1}\end{array}$

$\begin{array}{l}=-{\left[\ln C_p-\ln C_{\psi }+\frac{C_p}{\Gamma \left(mn\right)}\frac{\partial {\left(I_m\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^r}{\partial r}\right]|}_{r=1}\\ =-\ln \left({\displaystyle \underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!}\right)-m\psi \left(2\right)+m\ln \Gamma \left(n\right)-\ln \Gamma \left(mn\right)-mn\psi \left(mn\right)\\ -{\displaystyle \underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)}+{\displaystyle \underset{k=1}{\overset{m}{\sum }}k}\psi \left(k+1\right)\end{array}$

定理3.4：当 $r>0,r\ne 1,s=0$ 时，在Haar分布的双体纯态上其对角元的联合分布相对于取部分迹所诱导的随机密度矩阵的联合分布的统一(r, s)相对微分熵为

$h_r\left(\psi \left|\right|p\right)={\left(r-1\right)}^{-1}\left[r\ln C_{\psi }+\left(1-r\right)\ln C_p+\ln I_m\left(\left(m-1\right)r-m+n,1-r\right)-\ln \Gamma \left(mn\right)\right]$

证明：

$\begin{array}{c}{h}_r\left(\psi \left|\right|p\right)=-{\left(1-r\right)}^{-1}\ln \left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\psi \left({\lambda }_1,\cdots ,{\lambda }_m\right)}}{\left(\psi \left({\lambda }_1,\cdots ,{\lambda }_m\right)/p\left({\lambda }_1,\cdots ,{\lambda }_m\right)\right)}^{r-1}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}\right)\\ ={\left(r-1\right)}^{-1}\ln \left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }{\psi }^r\left({\lambda }_1,\cdots ,{\lambda }_m\right)p^{1-r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}}}\right)\end{array}$

令

${F^{\prime }}_r\left(t\right)={\left(r-1\right)}^{-1}\ln \left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\delta \left(t-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right){\psi }^r\left({\lambda }_1,\cdots ,{\lambda }_m\right)p^{1-r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}}}\right)$

对 ${F^{\prime }}_r\left(t\right)$ 应用Laplace变换 $\left(t\to s\right)$ ，则有

${\widetilde{F^{\prime }}}_r\left(s\right)={\left(r-1\right)}^{-1}\ln \left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\exp \left(-s{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right){\psi }^r\left({\lambda }_1,\cdots ,{\lambda }_m\right)p^{1-r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}}}\right)$

把式(7)、式(11)、式(12)代入上式得到

$\begin{array}{c}{\widetilde{F^{\prime }}}_r\left(s\right)={\left(r-1\right)}^{-1}\ln \left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\exp \left(-s{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right){\left(s^{m-mn}{C}_p{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^2}{\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_k^{n-m}}\right)}^{1-r}{\left(s^{m-mn}{C}_{\psi }{\displaystyle \underset{j=1}{\overset{m}{\prod }}{\mu }_j^{n-\text{1}}}\right)}^r{\displaystyle \underset{k=1}{\overset{m}{\prod }}\frac{\text{d}{\mu }_k}{s}}}}\right)\\ ={\left(r-1\right)}^{-1}\ln \left({\displaystyle {\int }_0^{\infty }\cdots }{\displaystyle {\int }_0^{\infty }{s}^{-mn}{C}_{\psi }^r{C}_p^{1-r}}{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^{2\left(1-r\right)}}\exp \left(-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\mu }_j}\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_k^{\left(m-1\right)r-m+n}\text{d}{\mu }_k}\right)\\ ={\left(r-1\right)}^{-1}\ln \left(s^{-mn}{C}_{\psi }^r{C}_p^{1-r}{I}_m\left(\left(m-1\right)r-m+n,1-r\right)\right)\end{array}$

把式(14)代入上式得到

${F^{\prime }}_r\left(t\right)={\left(r-1\right)}^{-1}\ln \left(\frac{t^{mn-1}}{\Gamma \left(mn\right)}{C}_{\psi }^r{C}_p^{1-r}{I}_m\left(\left(m-1\right)r+n-m,1-r\right)\right)$

那么

$\begin{array}{c}{h}_r\left(\psi \left|\right|p\right)={F^{\prime }}_r\left(1\right)={\left(r-1\right)}^{-1}\ln \left(\frac{C_{\psi }^r{C}_p^{1-r}}{\Gamma \left(mn\right)}{I}_m\left(\left(m-1\right)r+n-m,1-r\right)\right)\\ ={\left(r-1\right)}^{-1}\left[r\ln C_{\psi }+\left(1-r\right)\ln C_p+\ln I_m\left(\left(m-1\right)r+n-m,1-r\right)-\ln \Gamma \left(mn\right)\right]\end{array}$ (24)

注4：当 $r\to 1$ 时，

$\underset{r\to 1}{\lim }{h}_r\left(\psi \left|\right|p\right)=\ln \left({\displaystyle \underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!}\right)-m\ln \Gamma \left(n\right)+\frac{m^2-m}{2}\psi \left(n\right)+\frac{m-m^2}{2}\psi (1)$

证明：

由式(10)，我们得到

$\begin{array}{l}{\frac{\partial \ln I_m\left(\left(m-1\right)r+n-m,1-r\right)}{\partial r}|}_{r=1}=\frac{1}{I_m\left(\left(m-1\right)r+n-m,1-r\right)}{\frac{\partial I_m\left(\left(m-1\right)r+n-m,1-r\right)}{\partial r}|}_{r=1}\\ =\left(m-1\right)\times {\displaystyle \underset{k=0}{\overset{m-1}{\sum }}\psi \left(\left(m-1\right)r+n-m+1+k\left(1-r\right)\right)}+m\psi \left(1+\left(1-r\right)\right)\\ -{\displaystyle \underset{k=1}{\overset{m-1}{\sum }}k\psi \left(\left(m-1\right)r+n-m+1+k\left(1-r\right)\right)}-{{\displaystyle \underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k\left(1-r\right)\right)}|}_{r=1}\\ =\left(m-1\right)\times m\psi \left(n\right)-\frac{m^2+m}{2}\psi \left(1\right)+m\psi \left(1\right)\end{array}$ (25)

把式(2)、式(4)、式(6)、式(8)、式(25)代入式(24)得到

$\begin{array}{c}\underset{r\to 1}{\lim }{h}_r\left(\psi \left|\right|p\right)={\left(r-1\right)}^{-1}\left[r\ln C_p+\left(1-r\right)\ln C_{\psi }+\ln I_m\left(\left(m-1\right)r+n-m,1-r\right)-\ln \Gamma \left(mn\right)\right]\\ =\ln C_{\psi }-\ln C_p+{\frac{\partial \ln I_m\left(\left(m-1\right)r+n-m,1-r\right)}{\partial r}|}_{r=1}\\ =\ln \left({\displaystyle \underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!}\right)-m\ln \Gamma \left(n\right)+\frac{m^2-m}{2}\psi \left(n\right)+\frac{m-m^2}{2}\psi (1)\end{array}$

定理3.5：当 $r>0,r\ne 1,s=1$ 时，在Haar分布的双体纯态上其对角元的联合分布相对于取部分迹所诱导的随机密度矩阵的联合分布的统一(r, s)相对微分熵为

$h^r\left(\psi \left|\right|p\right)={\left(r-1\right)}^{-1}\left[\frac{C_{\psi }^r{C}_p^{1-r}}{\Gamma \left(mn\right)}{I}_m\left(\left(m-1\right)r+n-m,1-r\right)-1\right]$

证明：

$\begin{array}{c}{h}^r\left(\psi \left|\right|p\right)=-{\left(1-r\right)}^{-1}\left[{\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\psi \left({\lambda }_1,\cdots ,{\lambda }_m\right)}}{\left(\psi \left({\lambda }_1,\cdots ,{\lambda }_m\right)/p\left({\lambda }_1,\cdots ,{\lambda }_m\right)\right)}^{r-1}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}-1\right]\\ ={\left(r-1\right)}^{-1}\left[{\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }{\psi }^r\left({\lambda }_1,\cdots ,{\lambda }_m\right)p^{1-r}\left({\lambda }_1,\cdots ,{\lambda }_m\right)}}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k-1}\right]\end{array}$

令

$F^r{}^{\prime }\left(t\right)={\left(r-1\right)}^{-1}\left[{\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\delta \left(t-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right){\psi }^r\left({\lambda }_1,\cdots ,{\lambda }_m\right)p^{1-r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}}}-1\right]$

对 $F^r{}^{\prime }\left(t\right)$ 应用Laplace变换 $\left(t\to s\right)$ ，则有

${\tilde{F}}^r{}^{\prime }\left(s\right)={\left(r-1\right)}^{-1}\left[{\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\exp \left(-s{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right){\psi }^r\left({\lambda }_1,\cdots ,{\lambda }_m\right)p^{1-r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k-1}}}\right]$

把式(7)、式(11)、式(12) 代入上式得到

$\begin{array}{c}{\tilde{F}}^r{}^{\prime }\left(s\right)={\left(r-1\right)}^{-1}\left[{\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\exp \left(-s{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right){\left(s^{m-mn}{C}_p{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^2}{\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_k^{n-m}}\right)}^{1-r}{\left(s^{m-mn}{C}_{\psi }{\displaystyle \underset{j=1}{\overset{m}{\prod }}{\mu }_j^{n-\text{1}}}\right)}^r{\displaystyle \underset{k=1}{\overset{m}{\prod }}\frac{\text{d}{\mu }_k}{s}-\text{1}}}}\right]\\ ={\left(r-1\right)}^{-1}\left[{\displaystyle {\int }_0^{\infty }\cdots }{\displaystyle {\int }_0^{\infty }{s}^{-mn}{C}_{\psi }^r{C}_p^{1-r}}{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^{2\left(1-r\right)}}\exp \left(-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\mu }_j}\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_k^{\left(m-1\right)r+n-m}\text{d}{\mu }_k}-\text{1}\right]\\ ={\left(r-1\right)}^{-1}\left[s^{-mn}{C}_{\psi }^r{C}_p^{1-r}{I}_m\left(\left(m-1\right)r+n-m,1-r\right)-1\right]\end{array}$

把式(14)代入上式得到

$F^r{}^{\prime }\left(t\right)={\left(r-1\right)}^{-1}\left[\frac{t^{mn-1}}{\Gamma \left(mn\right)}{C}_{\psi }^r{C}_p^{1-r}{I}_m\left(\left(m-1\right)r+n-m,1-r\right)-1\right]$

那么

$h^r{}^{\prime }\left(\psi \left|\right|p\right)=F^r{}^{\prime }\left(1\right)={\left(r-1\right)}^{-1}\left[\frac{C_{\psi }^r{C}_p^{1-r}}{\Gamma \left(mn\right)}{I}_m\left(\left(m-1\right)r+n-m,1-r\right)-1\right]$

注5：当 $r\to 1$ 时，

$\underset{r\to 1}{\lim }{h}^r\left(\psi \left|\right|p\right)=\ln \left({\displaystyle \underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!}\right)-m\ln \Gamma \left(n\right)+\frac{m^2-m}{2}\psi \left(n\right)+\frac{m-m^2}{2}\psi \left(1\right)$ (26)

证明：

由式(10)，我们得到

$\begin{array}{l}{\frac{\partial I_m\left(\left(m-1\right)r+n-m,1-r\right)}{\partial r}|}_{r=1}\\ =I_m\left(\left(m-1\right)r+n-m,1-r\right)[\left(m-1\right)\times {\displaystyle \underset{k=0}{\overset{m-1}{\sum }}\psi \left(\left(m-1\right)r+n-m+1+k\left(1-r\right)\right)}+m\psi \left(2-r\right)\\ -{\displaystyle \underset{k=1}{\overset{m-1}{\sum }}k\psi \left(\left(m-1\right)r+n-m+1+k\left(1-r\right)\right)}-{{\displaystyle \underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k\left(1-r\right)\right)}]|}_{r=1}\\ =I_m\left(n-1,0\right)\left[{\displaystyle \underset{k=0}{\overset{m-1}{\sum }}\left(m-1-k\right)\times \psi \left(n\right)}+m\psi \left(1\right)-{\displaystyle \underset{k=1}{\overset{m}{\sum }}k\left(1\right)}\right]\end{array}$ (27)

把式(2)、式(4)、式(6)、式(9)、式(27)代入式(26)得到

$\begin{array}{c}\underset{r\to 1}{\lim }{h}^r\left(\psi \left|\right|p\right)=[\ln C_{\psi }{C}_{\psi }^r{C}_p^{1-r}\frac{I_m\left(\left(m-1\right)r+n-m,1-r\right)}{\Gamma \left(mn\right)}-\ln C_p{C}_{\psi }^r{C}_p^{1-r}\frac{I_m\left(\left(m-1\right)r+n-m,1-r\right)}{\Gamma \left(mn\right)}\\ +{\frac{C_{\psi }^r{C}_p^{1-r}}{\Gamma \left(mn\right)}\frac{\partial I_m\left(\left(m-1\right)r+n-m,1-r\right)}{\partial r}]|}_{r=1}\\ =\frac{\ln C_{\psi }{I}_m\left(n-1,0\right)}{{\Gamma }^m\left(n\right)}-\frac{\ln C_p{I}_m\left(n-1,0\right)}{{\Gamma }^m\left(n\right)}+{\frac{I_m\left(n-1,0\right)}{{\Gamma }^m\left(n\right)}\frac{\partial I_m\left(\left(m-1\right)r+n-m,1-r\right)}{\partial r}|}_{r=1}\\ =\ln \left({\displaystyle \underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!}\right)-m\ln \Gamma \left(n\right)+\frac{m^2-m}{2}\psi \left(n\right)+\frac{m-m^2}{2}\psi (1)\end{array}$

定理3.6：当 $r>0,r\ne 1,s=1/r$ 时，在Haar分布的双体纯态上取部分迹所诱导的随机密度矩阵的特征值的联合分布相对于其对角元的联合分布的统一(r, s)相对微分熵为

${}_{r}h\left(\psi \left|\right|p\right)={\left(1-r\right)}^{-1}\left[\frac{C_{\psi }{C}_p^{r-1}}{\Gamma \left(mnr\right)}{\left(I_m\left(n-m+\frac{m-1}{r},1-\frac{1}{r}\right)\right)}^r-1\right]$

证明：

$\begin{array}{c}{}_{r}h\left(\psi \left|\right|p\right)=-{\left(r-1\right)}^{-1}\left[{\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\psi \left({\lambda }_1,\cdots ,{\lambda }_m\right)}}{\left(\psi \left({\lambda }_1,\cdots ,{\lambda }_m\right)/p\left({\lambda }_1,\cdots ,{\lambda }_m\right)\right)}^{1/r-1}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}\right)}^r-1\right]\\ ={\left(1-r\right)}^{-1}\left[{\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }{\psi }^{1/r}\left({\lambda }_1,\cdots ,{\lambda }_m\right)p^{1-1/r}\left({\lambda }_1,\cdots ,{\lambda }_m\right)}}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}\right)}^r-1\right]\end{array}$

令

${}_r{F}^{\prime }\left(t\right)={\left(1-r\right)}^{-1}\left[{\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\delta \left(t-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right){\psi }^{1/r}\left({\lambda }_1,\cdots ,{\lambda }_m\right)p^{1-1/r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}}}\right)}^r-1\right]$

对 ${}_r{F}^{\prime }\left(t\right)$ 应用Laplace变换 $\left(t\to s\right)$ ，则有

${}_r\widetilde{F^{\prime }}\left(s\right)={\left(1-r\right)}^{-1}\left[{\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\exp \left(-s{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right){\psi }^{1/r}\left({\lambda }_1,\cdots ,{\lambda }_m\right)p^{1-1/r}\left({\lambda }_1,\cdots ,{\lambda }_m\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_k}}}\right)}^r-1\right]$

把式(7)、式(11)、式(12)代入上式得到

$\begin{array}{c}{}_r\widetilde{F^{\prime }}\left(s\right)={\left(1-r\right)}^{-1}\left[{\left({\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }\exp \left(-s{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\lambda }_j}\right){\left(s^{m-mn}{C}_p{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^2}{\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_k^{n-m}}\right)}^{1-1/r}{\left(s^{m-mn}{C}_{\psi }{\displaystyle \underset{j=1}{\overset{m}{\prod }}{\mu }_j^{n-1}}\right)}^{1/r}{\displaystyle \underset{k=1}{\overset{m}{\prod }}\frac{\text{d}{\mu }_k}{s}}}}\right)}^r-1\right]\\ ={\left(1-r\right)}^{-1}\left[{\left({\displaystyle {\int }_0^{\infty }\cdots }{\displaystyle {\int }_0^{\infty }{s}^{-mn}{C}_{\psi }^{1/r}{C}_p^{1-1/r}}{\displaystyle \underset{1\le i<j\le m}{\prod }{\left({\mu }_i-{\mu }_j\right)}^{2\left(1-1/r\right)}}\exp \left(-{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\mu }_j}\right){\displaystyle \underset{k=1}{\overset{m}{\prod }}{\mu }_j^{n-m+\left(m-1\right)/r}\text{d}{\mu }_k}\right)}^r-1\right]\\ ={\left(1-r\right)}^{-1}\left[s^{-mnr}{C}_{\psi }{C}_p^{r-1}{\left(I_m\left(n-m+(m-1)/r,1-1/r\right)\right)}^r-1\right]\end{array}$

把式(19)代入上式得到

${}_r{F}^{\prime }\left(t\right)={\left(1-r\right)}^{-1}\left[\frac{t^{mnr-1}}{\Gamma \left(mnr\right)}{C}_{\psi }{C}_p^{r-1}{\left(I_m\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^r-1\right]$

那么

${}_{r}h\left(\psi \left|\right|p\right)={}_r{F}^{\prime }\left(1\right)={\left(1-r\right)}^{-1}\left[\frac{C_{\psi }{C}_p^{r-1}}{\Gamma \left(mnr\right)}{\left(I_m\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^r-1\right]$ (28)

注6：当 $r\to 1$ 时，

$\underset{r\to 1}{\lim }{}_{r}h\left(\psi \left|\right|p\right)=\ln \left({\displaystyle \underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!}\right)-m\ln \Gamma \left(n\right)-\ln \Gamma \left(mn\right)-mn\psi \left(mn\right)+\frac{m^2-m}{2}\psi \left(n\right)+\frac{m-m^2}{2}\psi (1)$

证明：

由式(21)得到

$\begin{array}{l}\frac{\partial {\left(I_m\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^r}{\partial r}={\left(I_m\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^r\\ [\ln I_m\left(n-m+\left(m-1\right)/r,1-1/r\right)\begin{array}{c}\\ \end{array}+\frac{r}{I_m\left(n-m+\left(m-1\right)/r,1-1/r\right)}\times \frac{\partial I_m\left(n-m+\left(m-1\right)/r,1-1/r\right)}{\partial r}]\end{array}$ (29)

由式(10)得到

${\frac{\partial I_m\left(n-m+\left(m-1\right)/r,1-1/r\right)}{\partial r}|}_{r=1}=I_m\left(n-1,0\right)\left[\left(1-m\right){\displaystyle \underset{k=0}{\overset{m-1}{\sum }}\psi \left(n\right)}+{\displaystyle \underset{k=1}{\overset{m}{\sum }}k\psi \left(n\right)}+{\displaystyle \underset{k=1}{\overset{m}{\sum }}k\psi \left(1\right)}-m\psi \left(1\right)\right]$ (30)

把式(2)、式(4)、式(6)、式(9)、式(22)、式(29)、式(30)代入式(28)得到

$\begin{array}{c}\underset{r\to 1}{\lim }{}_{r}h\left(\psi \left|\right|p\right)=-[\frac{C_{\psi }{C}_p^{r-1}\ln C_p}{\Gamma \left(mnr\right)}{\left(I_m\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^r\\ +\frac{C_{\psi }{C}_p^{r-1}\psi \left(mnr\right)mn}{\Gamma \left(mnr\right)}{\left(I_m\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^r\\ +\frac{C_{\psi }{C}_p^{r-1}}{\Gamma \left(mnr\right)}{\frac{\partial {\left(I_m\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^r}{\partial r}]|}_{r=1}\\ =-\ln \left({\displaystyle \underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!}\right)-m\psi \left(2\right)+m\ln \Gamma \left(n\right)-\ln \Gamma \left(mn\right)-mn\psi \left(mn\right)\\ -{\displaystyle \underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)}+{\displaystyle \underset{k=1}{\overset{m}{\sum }}k}\psi \left(k+1\right)\end{array}$

4. 总结

本文定义了在Haar分布的双体纯态上取部分迹所诱导的随机密度矩阵的特征值的联合分布相对于其对角元的联合分布(其对角元的联合分布相对于取部分迹所诱导的随机密度矩阵的特征值的联合分布)统一(r, s)相对微分熵，计算了三种情形下的统一(r, s)相对微分熵，取极限后的结果表明三种情形下的结果基本相等，推广了随机密度矩阵特征值联合分布的微分熵的范围。

基金项目

国家自然科学基金(11401007)；安徽自然科学基金(KJ2017A042)。