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

doi:10.12677/APP.2019.96037

Applied Physics
Vol. 09 No. 06 ( 2019 ), Article ID: 30931 , 14 pages
10.12677/APP.2019.96037

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

Wanqing Li¹, Shenghuo Liu², Jiamei Wang¹

●How to Cite this Article

¹School of Mathematics & Physics, Anhui University of Technology, Ma’anshan Anhui

²Tianjiabing Middle School, Anqing Anhui

Received: Jun. 5^th, 2019; accepted: Jun. 18^th, 2019; published: Jun. 25^th, 2019

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.

Keywords:Unified (r, s)-Relative Differential Entropy, Diagonal Element, Random Density Matrix

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

李婉晴¹，刘胜火²，汪加梅¹

¹安徽工业大学，数理科学与工程学院，安徽马鞍山

²田家炳中学，安徽安庆

收稿日期：2019年6月5日；录用日期：2019年6月18日；发布日期：2019年6月25日

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。

关键词 :统一(r, s)相对微分熵，对角元，随机密度矩阵

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

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 $Re (z) > 0$ 上的伽马函数为 $Γ (z) = \int_{0}^{\infty} t^{z - 1} e^{- t} d t$ ， $Γ (x), ψ (x)$ 分别代表Gamma 函数和Digamma函数。

$ψ (x) = \frac{d \ln Γ (x)}{d x} = \frac{Γ^{'} (x)}{Γ (x)}$ ， $ψ (k + 1) = H_{k} - γ$ ，其中 $H_{k} = \sum_{j = 1}^{k} \frac{1}{j}$ ， $γ \approx 0.57721$ 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 $σ^{2}$ 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 $(μ_{j} \in [0, \infty), j = 1, 2, \dots, m)$ 的联合概率函数为

$q (μ_{1}, \dots, μ_{m}) = C_{q} \exp (- \sum_{j = 1}^{m} μ_{j}) \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2} \prod_{j = 1}^{m} μ_{j}^{n - m}$ ，其中 $\sum_{j = 1}^{m} μ_{j} = 1$ (1)

$C_{q} = 1 / \prod_{1 \leq j \leq m} [j! (n - j)!]$ (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 $(λ_{j} \in [0, 1], j = 1, 2, \dots, m)$ 的联合概率函数为

$p (λ_{1}, \dots, λ_{m}) = C_{p} \prod_{1 \leq i < j \leq m} {(λ_{i} - λ_{j})}^{2} \prod_{j = 1}^{m} λ_{j}^{n - m}$ ，其中 $\sum_{j = 1}^{m} λ_{j} = 1$ (3)

$C_{p} = Γ (m n) C_{q}$ (4)

4) 在Haar分布的双体纯态上其对角元的联合概率函数为

$ψ (ρ_{11}, ρ_{22}, \dots, ρ_{m m}) = C_{ψ} \prod_{j = 1}^{m} ρ_{j j}^{n - 1}$ ，其中 $\sum_{j = 1}^{m} ρ_{j j} = 1$ (5)

$C_{Ψ} = \frac{Γ (m n)}{Γ {(n)}^{m}}$ (6)

5) $I_{m} (α, r) = \int_{0}^{\infty} \cdot \cdot \cdot \int_{0}^{\infty} \exp (- \sum_{j = 1}^{m} μ_{j}) \prod_{1 \leq i < j \leq m} {| μ_{i} - μ_{j} |}^{2 r} \prod_{k = 1}^{m} (μ_{k}^{α} d μ_{k}) = \prod_{k = 0}^{m - 1} \frac{Γ (α + 1 + k r) Γ (1 + (k + 1) r)}{Γ (1 + r)}$ (7)

$I_{m} (n - m, 1) = \frac{1}{C_{q}}$ (8)

$I_{m} (n - 1, 0) = Γ^{m} (n)$ (9)

因此，我们可以得到

$\begin{matrix} \frac{\partial}{\partial r} \partial I_{m} (α (r), β (r)) = \frac{\partial I_{m} (α (r), β (r))}{\partial α} \times \frac{d α}{d r} + \frac{\partial I_{m} (α (r), β (r))}{\partial β} \times \frac{d β}{d r} \\ = I_{m} (α (r), β (r)) {α^{'} (r) \times \sum_{k = 0}^{m - 1} ψ (α (r) + 1 + k β (r)) \\ + β^{'} (r) \times [\sum_{k = 1}^{m - 1} k ψ (α (r) + 1 + k β (r)) \\ + \sum_{k = 1}^{m} k ψ (1 + k β (r)) - m ψ (1 + β (r))]} \end{matrix}$ (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

$E_{r}^{s} (p | | ψ) = {\begin{array}{l} h_{r}^{s} (p | | ψ), r \neq 1, s \neq 0 \\ h_{r} (p | | ψ), r \neq 1, s = 0 \\ h^{r} (p | | ψ), r \neq 1, s = 1 \\ {}_{r}h (p | | ψ), r \neq 1, s = r^{- 1} \\ h (p | | ψ), r = 1 \end{array}$

其中

$h_{r}^{s} (p | | ψ) = - {[(1 - r) s]}^{- 1} [{(\int p (λ) {(p (λ) / ψ (λ))}^{r - 1} d λ)}^{s} - 1], r > 0, r \neq 1, s \neq 0$

$h_{r} (p | | ψ) = - {(1 - r)}^{- 1} \ln (\int p (λ) {(p (λ) / ψ (λ))}^{r - 1} d λ), r > 0, r \neq 1$

$h^{r} (p | | ψ) = - {(1 - r)}^{- 1} (\int p (λ) {(p (λ) / ψ (λ))}^{r - 1} d λ - 1), r > 0, r \neq 1$

${}_{r}h (p | | ψ) = - {(r - 1)}^{- 1} [{(\int p (λ) {(p (λ) / ψ (λ))}^{1 / r - 1} d λ)}^{r} - 1], r > 0, r \neq 1$

$h (p | | ψ) = \int p (λ) \ln (p (λ) / ψ (λ)) d λ$

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

$E_{r}^{s} (ψ | | p) = {\begin{array}{l} h_{r}^{s} (ψ | | p), r \neq 1, s \neq 0 \\ h_{r} (ψ | | p), r \neq 1, s = 0 \\ h^{r} (ψ | | p), r \neq 1, s = 1 \\ {}_{r}h (ψ | | p), r \neq 1, s = r^{- 1} \\ h (ψ | | p), r = 1 \end{array}$

其中

$h_{r}^{s} (ψ | | p) = - {[(1 - r) s]}^{- 1} [{(\int ψ (λ) {(ψ (λ) / p (λ))}^{r - 1} d λ)}^{s} - 1], r > 0, r \neq 1, s \neq 0$

$h_{r} (ψ | | p) = - {(1 - r)}^{- 1} \ln (\int ψ (λ) {(ψ (λ) / p (λ))}^{r - 1} d λ), r > 0, r \neq 1$

$h^{r} (ψ | | p) = - {(1 - r)}^{- 1} (\int ψ (λ) {(ψ (λ) / p (λ))}^{r - 1} d λ - 1), r > 0, r \neq 1$

${}_{r}h (ψ | | p) = - {(r - 1)}^{- 1} [{(\int ψ (λ) {(ψ (λ) / p (λ))}^{1 / r - 1} d λ)}^{r} - 1], r > 0, r \neq 1$

$h (ψ | | p) = \int ψ (λ) \ln (ψ (λ) / p (λ)) d λ$

3. 统一(r, s)相对微分熵的计算

定理3.1：当 $r > 0, r \neq 1, s = 0$ 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

$h_{r} (p | | ψ) = {(r - 1)}^{- 1} [r \ln C_{p} + (1 - r) \ln C_{ψ} + \ln I_{m} ((1 - m) r + n - 1, r) - \ln Γ (m n)]$

证明：

$\begin{matrix} h_{r} (p | | ψ) = - {(1 - r)}^{- 1} \ln (\int_{0}^{\infty} \dots \int_{0}^{\infty} p (λ_{1}, \dots, λ_{m}) {(p (λ_{1}, \dots, λ_{m}) / ψ (λ_{1}, \dots, λ_{m}))}^{r - 1} \prod_{k = 1}^{m} d λ_{k}) \\ = {(r - 1)}^{- 1} \ln (\int_{0}^{\infty} \dots \int_{0}^{\infty} p^{r} (λ_{1}, \dots, λ_{m}) ψ^{1 - r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k}) \end{matrix}$

令

$F_{r} (t) = {(r - 1)}^{- 1} \ln (\int_{0}^{\infty} \dots \int_{0}^{\infty} δ (t - \sum_{j = 1}^{m} λ_{j}) p^{r} (λ_{1}, \dots, λ_{m}) ψ^{1 - r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k})$

对 $F_{r} (t)$ 应用Laplace变换 $(t \to s)$ ，则有

${\tilde{F}}_{r} (s) = {(r - 1)}^{- 1} \ln (\int_{0}^{\infty} \dots \int_{0}^{\infty} \exp (- s \sum_{j = 1}^{m} λ_{j}) p^{r} (λ_{1}, \dots, λ_{m}) ψ^{1 - r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k})$

令 $λ_{k} = \frac{μ_{k}}{s}, k = 1, 2, \dots, m$

$\frac{C_{p}}{C_{q}} \exp (\sum_{i = 1}^{m} μ_{m}) q (μ_{1}, \dots μ_{m}) = p (μ_{1}, \dots μ_{m})$ ， $p (\frac{μ_{1}}{s}, \dots, \frac{μ_{m}}{s}) = s^{m - n m} p (μ_{1}, \dots, μ_{m})$

再由式(1)和式(3)联立得到

$p (λ_{1}, \dots, λ_{m}) = s^{m - m n} C_{p} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2} \prod_{k = 1}^{m} μ_{k}^{n - m}$ (11)

$ψ (λ_{1}, \dots, λ_{m}) = s^{m - m n} C_{ψ} \prod_{j = 1}^{m} μ_{j}^{n - 1}$ (12)

则

$\begin{matrix} {\tilde{F}}_{r} (s) = {(r - 1)}^{- 1} \ln (\int_{0}^{\infty} \dots \int_{0}^{\infty} \exp (- s \sum_{j = 1}^{m} λ_{j}) {(s^{m - m n} C_{p} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2} \prod_{k = 1}^{m} μ_{k}^{n - m})}^{r} {(s^{m - m n} C_{ψ} \prod_{j = 1}^{m} μ_{j}^{n - 1})}^{1 - r} \prod_{k = 1}^{m} \frac{d μ_{k}}{s}) \\ = {(r - 1)}^{- 1} \ln (\int_{0}^{\infty} \dots \int_{0}^{\infty} s^{- m n} C_{p}^{r} C_{ψ}^{1 - r} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2 r} \exp (- \sum_{j = 1}^{m} μ_{j}) \prod_{k = 1}^{m} μ_{k}^{(1 - m) r + n - 1} d μ_{k}) \\ = {(r - 1)}^{- 1} \ln (s^{- m n} C_{p}^{r} C_{ψ}^{1 - r} I_{m} ((1 - m) r + n - 1, r)) \end{matrix}$

由于

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

所以

$L^{- 1} {s^{- m n}} (t) = \frac{t^{m n - 1}}{Γ (m n)}$ (14)

$F_{r} (t) = {(r - 1)}^{- 1} \ln (\frac{t^{m n - 1}}{Γ (m n)} C_{p}^{r} C_{ψ}^{1 - r} I_{m} ((1 - m) r + n - 1, r))$

那么得到

$\begin{matrix} h_{r} (p | | ψ) = F_{r} (1) = {(r - 1)}^{- 1} \ln (\frac{C_{p}^{r} C_{ψ}^{1 - r}}{Γ (m n)} I_{m} ((1 - m) r + n - 1, r)) \\ = {(r - 1)}^{- 1} [r \ln C_{p} + (1 - r) \ln C_{ψ} + \ln I_{m} ((1 - m) r + n - 1, r) - \ln Γ (m n)] \end{matrix}$ (15)

注1：当 $r \to 1$ 时，

$\begin{matrix} \lim_{r \to 1} h_{r} (p | | ψ) = - \ln (\prod_{k = 1}^{m} k! (n - k)!) - m ψ (2) + m \ln Γ (n) \\ - \sum_{k = n - m}^{n - 1} (n - k - 1) ψ (k + 1) + \sum_{k = 1}^{m} k ψ (k + 1) \end{matrix}$

证明：

由式(10)得到

$\begin{array}{l} {\frac{\partial \ln I_{m} ((1 - m) r + n - 1, r)}{\partial r} |}_{r = 1} = \frac{1}{I_{m} ((1 - m) r + n - 1, r)} {\frac{\partial I_{m} ((1 - m) r + n - 1, r)}{\partial r} |}_{r = 1} \\ = (1 - m) \times \sum_{k = 0}^{m - 1} ψ ((1 - m) r + n + k r) - m ψ (1 + r) \\ + \sum_{k = 1}^{m - 1} k ψ ((1 - m) r + n + k r) + {\sum_{k = 1}^{m} k ψ (1 + k r) |}_{r = 1} \\ = \sum_{k = 0}^{m - 1} (1 - m + n + k) \times ψ (1 - m + n + k) - m ψ (2) + \sum_{k = 1}^{m} k ψ (1 + k) \end{array}$ (16)

把式(2)、式(4)、式(6)、式(8)、式(16)代入式(15)得到

$\begin{matrix} \lim_{r \to 1} h_{r} (p | | ψ) = {(r - 1)}^{- 1} [r \ln C_{p} + (1 - r) \ln C_{ψ} + \ln I_{m} ((1 - m) r + n - 1, r) - \ln Γ (m n)] \\ = \ln C_{p} - \ln C_{ψ} + {\frac{\partial \ln I_{m} ((1 - m) r + n - 1, r)}{\partial r} |}_{r = 1} \\ = - \ln (\prod_{k = 1}^{m} k! (n - k)!) - m ψ (2) + m \ln Γ (n) \\ - \sum_{k = n - m}^{n - 1} (n - k - 1) ψ (k + 1) + \sum_{k = 1}^{m} k ψ (k + 1) \end{matrix}$

定理3.2：当 $r > 0, r \neq 1, s = 1$ 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

$h^{r} (p | | ψ) = {(r - 1)}^{- 1} (\frac{C_{p}^{r} C_{ψ}^{1 - r}}{Γ (m n)} I_{m} ((1 - m) r + n - 1, r) - 1)$

证明：

$\begin{matrix} h^{r} (p | | ψ) = - {(1 - r)}^{- 1} (\int_{0}^{\infty} \dots \int_{0}^{\infty} p (λ_{1}, \dots, λ_{m}) {(p (λ_{1}, \dots, λ_{m}) / ψ (λ_{1}, \dots, λ_{m}))}^{r - 1} \prod_{k = 1}^{m} d λ_{k} - 1) \\ = {(r - 1)}^{- 1} (\int_{0}^{\infty} \dots \int_{0}^{\infty} p^{r} (λ_{1}, \dots, λ_{m}) ψ^{1 - r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k} - 1) \end{matrix}$

令

$F^{r} (t) = {(r - 1)}^{- 1} (\int_{0}^{\infty} \dots \int_{0}^{\infty} δ (t - \sum_{j = 1}^{m} λ_{j}) p^{r} (λ_{1}, \dots, λ_{m}) ψ^{1 - r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k} - 1)$

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

${\tilde{F}}^{r} (s) = {(r - 1)}^{- 1} (\int_{0}^{\infty} \dots \int_{0}^{\infty} \exp (- s \sum_{j = 1}^{m} λ_{j}) p^{r} (λ_{1}, \dots, λ_{m}) ψ^{1 - r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k} - 1)$

把式(7)、式(11)、式(12)代入上式得到

$\begin{matrix} {\tilde{F}}^{r} (s) = {(r - 1)}^{- 1} (\int_{0}^{\infty} \dots \int_{0}^{\infty} \exp (- s \sum_{j = 1}^{m} λ_{j}) {(s^{m - m n} C_{p} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2} \prod_{k = 1}^{m} μ_{k}^{n - m})}^{r} {(s^{m - m n} C_{ψ} \prod_{j = 1}^{m} μ_{j}^{n - 1})}^{1 - r} \prod_{k = 1}^{m} \frac{d μ_{k}}{s} - 1) \\ = {(r - 1)}^{- 1} (\int_{0}^{\infty} \dots \int_{0}^{\infty} s^{- m n} C_{p}^{r} C_{ψ}^{1 - r} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2 r} \exp (- \sum_{j = 1}^{m} μ_{j}) \prod_{k = 1}^{m} μ_{k}^{(1 - m) r + n - 1} d μ_{k} - 1) \\ = {(r - 1)}^{- 1} (s^{- m n} C_{p}^{r} C_{ψ}^{1 - r} I_{m} ((1 - m) r + n - 1, r) - 1) \end{matrix}$

把式(14)代入上式得到

$F^{r} (t) = {(r - 1)}^{- 1} (\frac{t^{m n - 1}}{Γ (m n)} C_{p}^{r} C_{ψ}^{1 - r} I_{m} ((1 - m) r + n - 1, r) - 1)$

那么

$h^{r} (p | | ψ) = F^{r} (1) = {(r - 1)}^{- 1} (\frac{C_{p}^{r} C_{ψ}^{1 - r}}{Γ (m n)} I_{m} ((1 - m) r + n - 1, r) - 1)$ (17)

注2：当 $r \to 1$ 时，

$\begin{matrix} \lim_{r \to 1} h^{r} (p | | ψ) = - \ln (\prod_{k = 1}^{m} k! (n - k)!) - m ψ (2) + m \ln Γ (n) \\ - \sum_{k = n - m}^{n - 1} (n - k - 1) ψ (k + 1) + \sum_{k = 1}^{m} k ψ (k + 1) \end{matrix}$

证明：

由式(10)，我们得到

$\begin{array}{l} {\frac{\partial I_{m} ((1 - m) r + n - 1, r)}{\partial r} |}_{r = 1} \\ = I_{m} ((1 - m) r + n - 1, r) [(1 - m) \times \sum_{k = 0}^{m - 1} ψ ((1 - m) r + n + k r) - m ψ (1 + r) \\ + \sum_{k = 1}^{m - 1} k ψ ((1 - m) r + n + k r) + {\sum_{k = 1}^{m} k ψ (1 + k r)] |}_{r = 1} \\ = I_{m} (n - m, 1) [\sum_{k = 0}^{m - 1} (1 - m + n + k) \times ψ (1 - m + n + k) - m ψ (2) + \sum_{k = 1}^{m} k ψ (1 + k)] \end{array}$ (18)

把式(2)、式(4)、式(6)、式(8)、式(18)代入式(17)得到

$\begin{matrix} \lim_{r \to 1} h^{r} (p | | ψ) = [\ln C_{p} C_{p}^{r} C_{ψ}^{1 - r} \frac{I_{m} ((1 - m) r + n - 1, r)}{Γ (m n)} - \ln C_{ψ} C_{p}^{r} C_{ψ}^{1 - r} \frac{I_{m} ((1 - m) r + n - 1, r)}{Γ (m n)} \\ + {\frac{C_{p}^{r} C_{ψ}^{1 - r}}{Γ (m n)} \frac{\partial I_{m} ((1 - m) r + n - 1, r)}{\partial r}] |}_{r = 1} \\ = \ln C_{p} - \ln C_{ψ} + \frac{C_{p}}{Γ (m n)} {\frac{\partial I_{m} ((1 - m) r + n - 1, r)}{\partial r} |}_{r = 1} \\ = - \ln (\prod_{k = 1}^{m} k! (n - k)!) - m ψ (2) + m \ln Γ (n) - \sum_{k = n - m}^{n - 1} (n - k - 1) ψ (k + 1) + \sum_{k = 1}^{m} k ψ (k + 1) \end{matrix}$

定理3.3：当 $r > 0, r \neq 1, s = 1 / r$ 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

${}_{r}h (p | | ψ) = {(1 - r)}^{- 1} [\frac{C_{p} C_{ψ}^{r - 1}}{Γ (m n r)} {(I_{m} (n - 1 - \frac{m - 1}{r}, \frac{1}{r}))}^{r} - 1]$

证明：

$\begin{matrix} {}_{r}h (p | | ψ) = - {(r - 1)}^{- 1} [{(\int_{0}^{\infty} \dots \int_{0}^{\infty} p (λ_{1}, \dots, λ_{m}) {(p (λ_{1}, \dots, λ_{m}) / ψ (λ_{1}, \dots, λ_{m}))}^{1 / r - 1} \prod_{k = 1}^{m} d λ_{k})}^{r} - 1] \\ = {(1 - r)}^{- 1} [{(\int_{0}^{\infty} \dots \int_{0}^{\infty} p^{1 / r} (λ_{1}, \dots, λ_{m}) ψ^{1 - 1 / r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k})}^{r} - 1] \end{matrix}$

令

${}_{r}F (t) = {(1 - r)}^{- 1} [{(\int_{0}^{\infty} \dots \int_{0}^{\infty} δ (t - \sum_{j = 1}^{m} λ_{j}) p^{1 / r} (λ_{1}, \dots, λ_{m}) ψ^{1 - 1 / r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k})}^{r} - 1]$

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

${}_{r}{\tilde{F}} (s) = {(1 - r)}^{- 1} [{(\int_{0}^{\infty} \dots \int_{0}^{\infty} \exp (- s \sum_{j = 1}^{m} λ_{j}) p^{1 / r} (λ_{1}, \dots, λ_{m}) ψ^{1 - 1 / r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k})}^{r} - 1]$

把式(7)、式(11)、式(12) 代入上式得到

$\begin{matrix} {}_{r}{\tilde{F}} (s) = {(1 - r)}^{- 1} [{(\int_{0}^{\infty} \dots \int_{0}^{\infty} \exp (- s \sum_{j = 1}^{m} λ_{j}) {(s^{m - m n} C_{p} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2} \prod_{k = 1}^{m} μ_{k}^{n - m})}^{1 / r} {(s^{m - m n} C_{ψ} \prod_{j = 1}^{m} μ_{j}^{n - 1})}^{1 - 1 / r} \prod_{k = 1}^{m} \frac{d μ_{k}}{s})}^{r} - 1] \\ = {(1 - r)}^{- 1} [{(\int_{0}^{\infty} \dots \int_{0}^{\infty} s^{- m n} C_{p}^{1 / r} C_{ψ}^{1 - 1 / r} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2 / r} \exp (- \sum_{j = 1}^{m} μ_{j}) \prod_{k = 1}^{m} μ_{k}^{n - 1 + (1 - m) / r} d μ_{k})}^{r} - 1] \\ = {(1 - r)}^{- 1} [s^{- m n r} C_{p} C_{ψ}^{r - 1} {(I_{m} (n - 1 + (1 - m) / r, 1 / r))}^{r} - 1] \end{matrix}$

由式(13)得到

$L^{- 1} {s^{- m n r}} (t) = \frac{t^{m n r - 1}}{Γ (m n r)}$ (19)

${}_{r}F (t) = {(1 - r)}^{- 1} [\frac{t^{m n r - 1}}{Γ (m n r)} C_{p} C_{ψ}^{r - 1} {(I_{m} (n - 1 + (1 - m) / r, 1 / r))}^{r} - 1]$

那么

${}_{r}h (p | | ψ) = {}_{r}F (1) = {(1 - r)}^{- 1} [\frac{C_{p} C_{ψ}^{r - 1}}{Γ (m n r)} {(I_{m} (n - 1 + (1 - m) / r, 1 / r))}^{r} - 1]$ (20)

注3：当 $r \to 1$ 时，

$\begin{matrix} \lim_{r \to 1} {}_{r}h (p | | ψ) = - \ln (\prod_{k = 1}^{m} k! (n - k)!) - m ψ (2) + m \ln Γ (n) - \ln Γ (m n) - m n ψ (m n) \\ - \sum_{k = n - m}^{n - 1} (n - k - 1) ψ (k + 1) + \sum_{k = 1}^{m} k ψ (k + 1) \end{matrix}$

证明：

令

$I_{m} {(α (r), β (r))}^{r} = y$

那么对等式两边取以e为底的对数得到

$r \ln I_{m} (α (r), β (r)) = \ln y$

再对r求导得到

$\begin{array}{l} \ln I_{m} (α (r), β (r)) + \frac{r}{I_{m} (α (r), β (r))} \frac{\partial I_{m} (α (r), β (r))}{\partial r} = \frac{y^{'}}{y} \\ y^{'} = y [\ln I_{m} (α (r), β (r)) + \frac{r}{I_{m} (α (r), β (r))} \frac{\partial I_{m} (α (r), β (r))}{\partial r}] \end{array}$ (21)

由式(21)得到

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

由式(10)得到

$\begin{array}{l} {\frac{\partial I_{m} (n - 1 + (1 - m) / r, 1 / r)}{\partial r} |}_{r = 1} \\ = I_{m} (n - 1 + (1 - m) / r, 1 / r) [\frac{m - 1}{r^{2}} \sum_{k = 0}^{m - 1} ψ (n + (1 - m) / r + k / r) \\ - \frac{1}{r^{2}} \sum_{k = 1}^{m - 1} k ψ (n + (1 - m) / r + k / r) - \frac{1}{r^{2}} \sum_{k = 1}^{m} k ψ (1 + k / r) + {\frac{1}{r^{2}} m ψ (1 + 1 / r)] |}_{r = 1} \\ = I_{m} (n - m, 1) [\sum_{k = n - m}^{n - 1} (1 + k - n) ψ (k + 1) - \sum_{k = 1}^{m} k ψ (1 + k) + m ψ (2)] \end{array}$ (23)

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

$\begin{matrix} \lim_{r \to 1} {}_{r}h (p | | ψ) = - [\frac{C_{p} C_{ψ}^{r - 1} \ln C_{ψ}}{Γ (m n r)} {(I_{m} (n - 1 + (1 - m) / r, 1 / r))}^{r} \\ + \frac{C_{p} C_{ψ}^{r - 1} ψ (m n r) m n}{Γ (m n r)} {(I_{m} (n - 1 + (1 - m) / r, 1 / r))}^{r} \\ + {\frac{C_{p} C_{ψ}^{r - 1}}{Γ (m n r)} \frac{\partial {(I_{m} (n - 1 + (1 - m) / r, 1 / r))}^{r}}{\partial r}] |}_{r = 1} \end{matrix}$

$\begin{array}{l} = - {[\ln C_{p} - \ln C_{ψ} + \frac{C_{p}}{Γ (m n)} \frac{\partial {(I_{m} (n - 1 + (1 - m) / r, 1 / r))}^{r}}{\partial r}] |}_{r = 1} \\ = - \ln (\prod_{k = 1}^{m} k! (n - k)!) - m ψ (2) + m \ln Γ (n) - \ln Γ (m n) - m n ψ (m n) \\ - \sum_{k = n - m}^{n - 1} (n - k - 1) ψ (k + 1) + \sum_{k = 1}^{m} k ψ (k + 1) \end{array}$

定理3.4：当 $r > 0, r \neq 1, s = 0$ 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

$h_{r} (ψ | | p) = {(r - 1)}^{- 1} [r \ln C_{ψ} + (1 - r) \ln C_{p} + \ln I_{m} ((m - 1) r - m + n, 1 - r) - \ln Γ (m n)]$

证明：

$\begin{matrix} h_{r} (ψ | | p) = - {(1 - r)}^{- 1} \ln (\int_{0}^{\infty} \dots \int_{0}^{\infty} ψ (λ_{1}, \dots, λ_{m}) {(ψ (λ_{1}, \dots, λ_{m}) / p (λ_{1}, \dots, λ_{m}))}^{r - 1} \prod_{k = 1}^{m} d λ_{k}) \\ = {(r - 1)}^{- 1} \ln (\int_{0}^{\infty} \dots \int_{0}^{\infty} ψ^{r} (λ_{1}, \dots, λ_{m}) p^{1 - r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k}) \end{matrix}$

令

${F^{'}}_{r} (t) = {(r - 1)}^{- 1} \ln (\int_{0}^{\infty} \dots \int_{0}^{\infty} δ (t - \sum_{j = 1}^{m} λ_{j}) ψ^{r} (λ_{1}, \dots, λ_{m}) p^{1 - r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k})$

对 ${F^{'}}_{r} (t)$ 应用Laplace变换 $(t \to s)$ ，则有

${\tilde{F^{'}}}_{r} (s) = {(r - 1)}^{- 1} \ln (\int_{0}^{\infty} \dots \int_{0}^{\infty} \exp (- s \sum_{j = 1}^{m} λ_{j}) ψ^{r} (λ_{1}, \dots, λ_{m}) p^{1 - r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k})$

把式(7)、式(11)、式(12)代入上式得到

$\begin{matrix} {\tilde{F^{'}}}_{r} (s) = {(r - 1)}^{- 1} \ln (\int_{0}^{\infty} \dots \int_{0}^{\infty} \exp (- s \sum_{j = 1}^{m} λ_{j}) {(s^{m - m n} C_{p} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2} \prod_{k = 1}^{m} μ_{k}^{n - m})}^{1 - r} {(s^{m - m n} C_{ψ} \prod_{j = 1}^{m} μ_{j}^{n - 1})}^{r} \prod_{k = 1}^{m} \frac{d μ_{k}}{s}) \\ = {(r - 1)}^{- 1} \ln (\int_{0}^{\infty} \dots \int_{0}^{\infty} s^{- m n} C_{ψ}^{r} C_{p}^{1 - r} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2 (1 - r)} \exp (- \sum_{j = 1}^{m} μ_{j}) \prod_{k = 1}^{m} μ_{k}^{(m - 1) r - m + n} d μ_{k}) \\ = {(r - 1)}^{- 1} \ln (s^{- m n} C_{ψ}^{r} C_{p}^{1 - r} I_{m} ((m - 1) r - m + n, 1 - r)) \end{matrix}$

把式(14)代入上式得到

${F^{'}}_{r} (t) = {(r - 1)}^{- 1} \ln (\frac{t^{m n - 1}}{Γ (m n)} C_{ψ}^{r} C_{p}^{1 - r} I_{m} ((m - 1) r + n - m, 1 - r))$

那么

$\begin{matrix} h_{r} (ψ | | p) = {F^{'}}_{r} (1) = {(r - 1)}^{- 1} \ln (\frac{C_{ψ}^{r} C_{p}^{1 - r}}{Γ (m n)} I_{m} ((m - 1) r + n - m, 1 - r)) \\ = {(r - 1)}^{- 1} [r \ln C_{ψ} + (1 - r) \ln C_{p} + \ln I_{m} ((m - 1) r + n - m, 1 - r) - \ln Γ (m n)] \end{matrix}$ (24)

注4：当 $r \to 1$ 时，

$\lim_{r \to 1} h_{r} (ψ | | p) = \ln (\prod_{k = 1}^{m} k! (n - k)!) - m \ln Γ (n) + \frac{m^{2} - m}{2} ψ (n) + \frac{m - m^{2}}{2} ψ (1)$

证明：

由式(10)，我们得到

$\begin{array}{l} {\frac{\partial \ln I_{m} ((m - 1) r + n - m, 1 - r)}{\partial r} |}_{r = 1} = \frac{1}{I_{m} ((m - 1) r + n - m, 1 - r)} {\frac{\partial I_{m} ((m - 1) r + n - m, 1 - r)}{\partial r} |}_{r = 1} \\ = (m - 1) \times \sum_{k = 0}^{m - 1} ψ ((m - 1) r + n - m + 1 + k (1 - r)) + m ψ (1 + (1 - r)) \\ - \sum_{k = 1}^{m - 1} k ψ ((m - 1) r + n - m + 1 + k (1 - r)) - {\sum_{k = 1}^{m} k ψ (1 + k (1 - r)) |}_{r = 1} \\ = (m - 1) \times m ψ (n) - \frac{m^{2} + m}{2} ψ (1) + m ψ (1) \end{array}$ (25)

把式(2)、式(4)、式(6)、式(8)、式(25)代入式(24)得到

$\begin{matrix} \lim_{r \to 1} h_{r} (ψ | | p) = {(r - 1)}^{- 1} [r \ln C_{p} + (1 - r) \ln C_{ψ} + \ln I_{m} ((m - 1) r + n - m, 1 - r) - \ln Γ (m n)] \\ = \ln C_{ψ} - \ln C_{p} + {\frac{\partial \ln I_{m} ((m - 1) r + n - m, 1 - r)}{\partial r} |}_{r = 1} \\ = \ln (\prod_{k = 1}^{m} k! (n - k)!) - m \ln Γ (n) + \frac{m^{2} - m}{2} ψ (n) + \frac{m - m^{2}}{2} ψ (1) \end{matrix}$

定理3.5：当 $r > 0, r \neq 1, s = 1$ 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

$h^{r} (ψ | | p) = {(r - 1)}^{- 1} [\frac{C_{ψ}^{r} C_{p}^{1 - r}}{Γ (m n)} I_{m} ((m - 1) r + n - m, 1 - r) - 1]$

证明：

$\begin{matrix} h^{r} (ψ | | p) = - {(1 - r)}^{- 1} [\int_{0}^{\infty} \dots \int_{0}^{\infty} ψ (λ_{1}, \dots, λ_{m}) {(ψ (λ_{1}, \dots, λ_{m}) / p (λ_{1}, \dots, λ_{m}))}^{r - 1} \prod_{k = 1}^{m} d λ_{k} - 1] \\ = {(r - 1)}^{- 1} [\int_{0}^{\infty} \dots \int_{0}^{\infty} ψ^{r} (λ_{1}, \dots, λ_{m}) p^{1 - r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k} - 1] \end{matrix}$

令

$F^{r}^{'} (t) = {(r - 1)}^{- 1} [\int_{0}^{\infty} \dots \int_{0}^{\infty} δ (t - \sum_{j = 1}^{m} λ_{j}) ψ^{r} (λ_{1}, \dots, λ_{m}) p^{1 - r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k} - 1]$

对 $F^{r}^{'} (t)$ 应用Laplace变换 $(t \to s)$ ，则有

${\tilde{F}}^{r}^{'} (s) = {(r - 1)}^{- 1} [\int_{0}^{\infty} \dots \int_{0}^{\infty} \exp (- s \sum_{j = 1}^{m} λ_{j}) ψ^{r} (λ_{1}, \dots, λ_{m}) p^{1 - r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k} - 1]$

把式(7)、式(11)、式(12) 代入上式得到

$\begin{matrix} {\tilde{F}}^{r}^{'} (s) = {(r - 1)}^{- 1} [\int_{0}^{\infty} \dots \int_{0}^{\infty} \exp (- s \sum_{j = 1}^{m} λ_{j}) {(s^{m - m n} C_{p} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2} \prod_{k = 1}^{m} μ_{k}^{n - m})}^{1 - r} {(s^{m - m n} C_{ψ} \prod_{j = 1}^{m} μ_{j}^{n - 1})}^{r} \prod_{k = 1}^{m} \frac{d μ_{k}}{s} - 1] \\ = {(r - 1)}^{- 1} [\int_{0}^{\infty} \dots \int_{0}^{\infty} s^{- m n} C_{ψ}^{r} C_{p}^{1 - r} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2 (1 - r)} \exp (- \sum_{j = 1}^{m} μ_{j}) \prod_{k = 1}^{m} μ_{k}^{(m - 1) r + n - m} d μ_{k} - 1] \\ = {(r - 1)}^{- 1} [s^{- m n} C_{ψ}^{r} C_{p}^{1 - r} I_{m} ((m - 1) r + n - m, 1 - r) - 1] \end{matrix}$

把式(14)代入上式得到

$F^{r}^{'} (t) = {(r - 1)}^{- 1} [\frac{t^{m n - 1}}{Γ (m n)} C_{ψ}^{r} C_{p}^{1 - r} I_{m} ((m - 1) r + n - m, 1 - r) - 1]$

那么

$h^{r}^{'} (ψ | | p) = F^{r}^{'} (1) = {(r - 1)}^{- 1} [\frac{C_{ψ}^{r} C_{p}^{1 - r}}{Γ (m n)} I_{m} ((m - 1) r + n - m, 1 - r) - 1]$

注5：当 $r \to 1$ 时，

$\lim_{r \to 1} h^{r} (ψ | | p) = \ln (\prod_{k = 1}^{m} k! (n - k)!) - m \ln Γ (n) + \frac{m^{2} - m}{2} ψ (n) + \frac{m - m^{2}}{2} ψ (1)$ (26)

证明：

由式(10)，我们得到

$\begin{array}{l} {\frac{\partial I_{m} ((m - 1) r + n - m, 1 - r)}{\partial r} |}_{r = 1} \\ = I_{m} ((m - 1) r + n - m, 1 - r) [(m - 1) \times \sum_{k = 0}^{m - 1} ψ ((m - 1) r + n - m + 1 + k (1 - r)) + m ψ (2 - r) \\ - \sum_{k = 1}^{m - 1} k ψ ((m - 1) r + n - m + 1 + k (1 - r)) - {\sum_{k = 1}^{m} k ψ (1 + k (1 - r))] |}_{r = 1} \\ = I_{m} (n - 1, 0) [\sum_{k = 0}^{m - 1} (m - 1 - k) \times ψ (n) + m ψ (1) - \sum_{k = 1}^{m} k (1)] \end{array}$ (27)

把式(2)、式(4)、式(6)、式(9)、式(27)代入式(26)得到

$\begin{matrix} \lim_{r \to 1} h^{r} (ψ | | p) = [\ln C_{ψ} C_{ψ}^{r} C_{p}^{1 - r} \frac{I_{m} ((m - 1) r + n - m, 1 - r)}{Γ (m n)} - \ln C_{p} C_{ψ}^{r} C_{p}^{1 - r} \frac{I_{m} ((m - 1) r + n - m, 1 - r)}{Γ (m n)} \\ + {\frac{C_{ψ}^{r} C_{p}^{1 - r}}{Γ (m n)} \frac{\partial I_{m} ((m - 1) r + n - m, 1 - r)}{\partial r}] |}_{r = 1} \\ = \frac{\ln C_{ψ} I_{m} (n - 1, 0)}{Γ^{m} (n)} - \frac{\ln C_{p} I_{m} (n - 1, 0)}{Γ^{m} (n)} + {\frac{I_{m} (n - 1, 0)}{Γ^{m} (n)} \frac{\partial I_{m} ((m - 1) r + n - m, 1 - r)}{\partial r} |}_{r = 1} \\ = \ln (\prod_{k = 1}^{m} k! (n - k)!) - m \ln Γ (n) + \frac{m^{2} - m}{2} ψ (n) + \frac{m - m^{2}}{2} ψ (1) \end{matrix}$

定理3.6：当 $r > 0, r \neq 1, s = 1 / r$ 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

${}_{r}h (ψ | | p) = {(1 - r)}^{- 1} [\frac{C_{ψ} C_{p}^{r - 1}}{Γ (m n r)} {(I_{m} (n - m + \frac{m - 1}{r}, 1 - \frac{1}{r}))}^{r} - 1]$

证明：

$\begin{matrix} {}_{r}h (ψ | | p) = - {(r - 1)}^{- 1} [{(\int_{0}^{\infty} \dots \int_{0}^{\infty} ψ (λ_{1}, \dots, λ_{m}) {(ψ (λ_{1}, \dots, λ_{m}) / p (λ_{1}, \dots, λ_{m}))}^{1 / r - 1} \prod_{k = 1}^{m} d λ_{k})}^{r} - 1] \\ = {(1 - r)}^{- 1} [{(\int_{0}^{\infty} \dots \int_{0}^{\infty} ψ^{1 / r} (λ_{1}, \dots, λ_{m}) p^{1 - 1 / r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k})}^{r} - 1] \end{matrix}$

令

${}_{r}{F^{'}} (t) = {(1 - r)}^{- 1} [{(\int_{0}^{\infty} \dots \int_{0}^{\infty} δ (t - \sum_{j = 1}^{m} λ_{j}) ψ^{1 / r} (λ_{1}, \dots, λ_{m}) p^{1 - 1 / r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k})}^{r} - 1]$

对 ${}_{r}{F^{'}} (t)$ 应用Laplace变换 $(t \to s)$ ，则有

${}_{r}{\tilde{F^{'}}} (s) = {(1 - r)}^{- 1} [{(\int_{0}^{\infty} \dots \int_{0}^{\infty} \exp (- s \sum_{j = 1}^{m} λ_{j}) ψ^{1 / r} (λ_{1}, \dots, λ_{m}) p^{1 - 1 / r} (λ_{1}, \dots, λ_{m}) \prod_{k = 1}^{m} d λ_{k})}^{r} - 1]$

把式(7)、式(11)、式(12)代入上式得到

$\begin{matrix} {}_{r}{\tilde{F^{'}}} (s) = {(1 - r)}^{- 1} [{(\int_{0}^{\infty} \dots \int_{0}^{\infty} \exp (- s \sum_{j = 1}^{m} λ_{j}) {(s^{m - m n} C_{p} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2} \prod_{k = 1}^{m} μ_{k}^{n - m})}^{1 - 1 / r} {(s^{m - m n} C_{ψ} \prod_{j = 1}^{m} μ_{j}^{n - 1})}^{1 / r} \prod_{k = 1}^{m} \frac{d μ_{k}}{s})}^{r} - 1] \\ = {(1 - r)}^{- 1} [{(\int_{0}^{\infty} \dots \int_{0}^{\infty} s^{- m n} C_{ψ}^{1 / r} C_{p}^{1 - 1 / r} \prod_{1 \leq i < j \leq m} {(μ_{i} - μ_{j})}^{2 (1 - 1 / r)} \exp (- \sum_{j = 1}^{m} μ_{j}) \prod_{k = 1}^{m} μ_{j}^{n - m + (m - 1) / r} d μ_{k})}^{r} - 1] \\ = {(1 - r)}^{- 1} [s^{- m n r} C_{ψ} C_{p}^{r - 1} {(I_{m} (n - m + (m - 1) / r, 1 - 1 / r))}^{r} - 1] \end{matrix}$

把式(19)代入上式得到

${}_{r}{F^{'}} (t) = {(1 - r)}^{- 1} [\frac{t^{m n r - 1}}{Γ (m n r)} C_{ψ} C_{p}^{r - 1} {(I_{m} (n - m + (m - 1) / r, 1 - 1 / r))}^{r} - 1]$

那么

${}_{r}h (ψ | | p) = {}_{r}{F^{'}} (1) = {(1 - r)}^{- 1} [\frac{C_{ψ} C_{p}^{r - 1}}{Γ (m n r)} {(I_{m} (n - m + (m - 1) / r, 1 - 1 / r))}^{r} - 1]$ (28)

注6：当 $r \to 1$ 时，

$\lim_{r \to 1} {}_{r}h (ψ | | p) = \ln (\prod_{k = 1}^{m} k! (n - k)!) - m \ln Γ (n) - \ln Γ (m n) - m n ψ (m n) + \frac{m^{2} - m}{2} ψ (n) + \frac{m - m^{2}}{2} ψ (1)$

证明：

由式(21)得到

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

由式(10)得到

${\frac{\partial I_{m} (n - m + (m - 1) / r, 1 - 1 / r)}{\partial r} |}_{r = 1} = I_{m} (n - 1, 0) [(1 - m) \sum_{k = 0}^{m - 1} ψ (n) + \sum_{k = 1}^{m} k ψ (n) + \sum_{k = 1}^{m} k ψ (1) - m ψ (1)]$ (30)

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

$\begin{matrix} \lim_{r \to 1} {}_{r}h (ψ | | p) = - [\frac{C_{ψ} C_{p}^{r - 1} \ln C_{p}}{Γ (m n r)} {(I_{m} (n - m + (m - 1) / r, 1 - 1 / r))}^{r} \\ + \frac{C_{ψ} C_{p}^{r - 1} ψ (m n r) m n}{Γ (m n r)} {(I_{m} (n - m + (m - 1) / r, 1 - 1 / r))}^{r} \\ + \frac{C_{ψ} C_{p}^{r - 1}}{Γ (m n r)} {\frac{\partial {(I_{m} (n - m + (m - 1) / r, 1 - 1 / r))}^{r}}{\partial r}] |}_{r = 1} \\ = - \ln (\prod_{k = 1}^{m} k! (n - k)!) - m ψ (2) + m \ln Γ (n) - \ln Γ (m n) - m n ψ (m n) \\ - \sum_{k = n - m}^{n - 1} (n - k - 1) ψ (k + 1) + \sum_{k = 1}^{m} k ψ (k + 1) \end{matrix}$

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)。

文章引用

李婉晴,刘胜火,汪加梅. 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[J]. 应用物理, 2019, 09(06): 305-318. https://doi.org/10.12677/APP.2019.96037

参考文献

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2. Furuichi, S., Yanagi, K. and Kuriyama, K. (2004) Funda-mental Properties of Tsallis Relative Entropy. Journal of Mathematical Physics, 45, 4868-4877. https://doi.org/10.1063/1.1805729

3. Hu, X.H. and Ye, Z.X. (2006) Generalized Quantum Entropy. Journal of Mathematical Physics, 47, 109-120. https://doi.org/10.1063/1.2165794

4. Wang, J.M., Wu, J.D. and Cho, M. (2011) Unified (r,s)-Relative Entropy. International Journal of Theoretical Physics, 50, 1282-1295. https://doi.org/10.1007/s10773-010-0583-z

5. Luo, L.Z., Wang, J.M., Zhang, L., et al. (2016) The Differential Entropy of the Joint Distribution of Eigenvalues of Random Density Matrices. MDPI, 18, 342. https://doi.org/10.3390/e18090342

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