Average entropy of a subsystem over a global unitary orbit of a mixed bipartite state

We investigate the average entropy of a subsystem within a global unitary orbit of a given mixed bipartite state in the finite-dimensional space. Without working out the closed-form expression of such average entropy for the mixed state case, we provide an analytical lower bound for this average entropy. In deriving this analytical lower bound, we get some useful by-products of independent interest. We also apply these results to estimate average correlation along a global unitary orbit of a given mixed bipartite state. When the notion of von Neumann entropy is replaced by linear entropy, the similar problem can be considered also, and moreover the exact average linear entropy formula is derived for a subsystem over a global unitary orbit of a mixed bipartite state.

L. Zhang is supported by Natural Science Foundation of Zhejiang Province of China (LY17A010027) and also by National Natural Science Foundation of China (Nos.11301124 and 61673145). H. Xiang is supported by the National Natural Science Foundation of China (Nos.11571265 and 11471253). Michael Walter is also acknowledged for his comments on this manuscript.

Authors and Affiliations

1. Institute of Mathematics, Hangzhou Dianzi University, Hangzhou, 310018, China

   Lin Zhang

2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

   Hua Xiang

Corresponding author

Correspondence to Lin Zhang.

Appendix 1: The proof of Lemma 2.1

In this paper, we will utilize some notion of matrix integral [4, 5, 22]. The formula in Lemma 2.1 is given firstly. A detailed reasoning is presented here.

The proof of Lemma 2.1

Firstly, we note that

Then we have:

In what follows, we compute this value step by step.

1. (1).

   If \((\pi ,\sigma )=((1),(1))\), then

   $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_1\rangle \langle i_2 | i'_2\rangle \langle i_3 | i'_3\rangle \langle j_1 | j'_1\rangle \langle j_2 | j'_2\rangle \langle j_3 | j'_3\rangle \end{aligned}$$

   (4.4)
   $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( AD\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( BC\right) \langle i_1 | i'_{1}\rangle . \end{aligned}$$

   (4.5)
2. (2).

   If \((\pi ,\sigma )=((1),(12))\), then

   $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_1\rangle \langle i_2 | i'_2\rangle \langle i_3 | i'_3\rangle \langle j_1 | j'_2\rangle \langle j_2 | j'_1\rangle \langle j_3 | j'_3\rangle \end{aligned}$$

   (4.6)
   $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( BC\right) \langle i_1 | i'_{1}\rangle . \end{aligned}$$

   (4.7)
3. (3).

   If \((\pi ,\sigma )=((1),(13))\), then

   $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_1\rangle \langle i_2 | i'_2\rangle \langle i_3 | i'_3\rangle \langle j_1 | j'_3\rangle \langle j_2 | j'_2\rangle \langle j_3 | j'_1\rangle \end{aligned}$$

   (4.8)
   $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( ADX\right) {{\mathrm{Tr}}}_{}\left( BC\right) \langle i_1 | i'_1\rangle . \end{aligned}$$

   (4.9)
4. (4).

   If \((\pi ,\sigma )=((1),(23))\), then

   $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_1\rangle \langle i_2 | i'_2\rangle \langle i_3 | i'_3\rangle \langle j_1 | j'_1\rangle \langle j_2 | j'_3\rangle \langle j_3 | j'_2\rangle \end{aligned}$$

   (4.10)
   $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( DAX\right) {{\mathrm{Tr}}}_{}\left( BC\right) \langle i_1 | i'_1\rangle . \end{aligned}$$

   (4.11)
5. (5).

   If \((\pi ,\sigma )=((1),(123))\), then

   $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_1\rangle \langle i_2 | i'_2\rangle \langle i_3 | i'_3\rangle \langle j_1 | j'_2\rangle \langle j_2 | j'_3\rangle \langle j_3 | j'_1\rangle \end{aligned}$$

   (4.12)
   $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( DX\right) {{\mathrm{Tr}}}_{}\left( BC\right) \langle i_1 | i'_1\rangle . \end{aligned}$$

   (4.13)
6. (6).

   If \((\pi ,\sigma )=((1),(132))\), then

   $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_1\rangle \langle i_2 | i'_2\rangle \langle i_3 | i'_3\rangle \langle j_1 | j'_3\rangle \langle j_2 | j'_1\rangle \langle j_3 | j'_2\rangle \end{aligned}$$

   (4.14)
   $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( AX\right) {{\mathrm{Tr}}}_{}\left( BC\right) \langle i_1 | i'_1\rangle . \end{aligned}$$

   (4.15)
7. (7).

   If \((\pi ,\sigma )=((12),(1))\), then

   $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_2\rangle \langle i_2 | i'_1\rangle \langle i_3 | i'_3\rangle \langle j_1 | j'_1\rangle \langle j_2 | j'_2\rangle \langle j_3 | j'_3\rangle \end{aligned}$$

   (4.16)
   $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( AD\right) {{\mathrm{Tr}}}_{}\left( X\right) \left\langle i_1 \left| BC \right| i'_1 \right\rangle . \end{aligned}$$

   (4.17)
8. (8).

   If \((\pi ,\sigma )=((12),(12))\), then

   $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_2\rangle \langle i_2 | i'_1\rangle \langle i_3 | i'_3\rangle \langle j_1 | j'_2\rangle \langle j_2 | j'_1\rangle \langle j_3 | j'_3\rangle \end{aligned}$$

   (4.18)
   $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( X\right) \left\langle i_1 \left| BC \right| i'_1 \right\rangle . \end{aligned}$$

   (4.19)
9. (9).

   If \((\pi ,\sigma )=((12),(13))\), then

   $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_2\rangle \langle i_2 | i'_1\rangle \langle i_3 | i'_3\rangle \langle j_1 | j'_3\rangle \langle j_2 | j'_2\rangle \langle j_3 | j'_1\rangle \end{aligned}$$

   (4.20)
   $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( ADX\right) \left\langle i_1 \left| BC \right| i'_1 \right\rangle . \end{aligned}$$

   (4.21)
10. (10).

    If \((\pi ,\sigma )=((12),(23))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_2\rangle \langle i_2 | i'_1\rangle \langle i_3 | i'_3\rangle \langle j_1 | j'_1\rangle \langle j_2 | j'_3\rangle \langle j_3 | j'_2\rangle \end{aligned}$$

    (4.22)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( DAX\right) \left\langle i_1 \left| BC \right| i'_1 \right\rangle . \end{aligned}$$

    (4.23)
11. (11).

    If \((\pi ,\sigma )=((12),(123))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_2\rangle \langle i_2 | i'_1\rangle \langle i_3 | i'_3\rangle \langle j_1 | j'_2\rangle \langle j_2 | j'_3\rangle \langle j_3 | j'_1\rangle \end{aligned}$$

    (4.24)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( DX\right) \left\langle i_1 \left| BC \right| i'_1 \right\rangle . \end{aligned}$$

    (4.25)
12. (12).

    If \((\pi ,\sigma )=((12),(132))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_2\rangle \langle i_2 | i'_1\rangle \langle i_3 | i'_3\rangle \langle j_1 | j'_3\rangle \langle j_2 | j'_1\rangle \langle j_3 | j'_2\rangle \end{aligned}$$

    (4.26)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( AX\right) \left\langle i_1 \left| BC \right| i'_1 \right\rangle . \end{aligned}$$

    (4.27)
13. (13).

    If \((\pi ,\sigma )=((13),(1))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_3\rangle \langle i_2 | i'_2\rangle \langle i_3 | i'_1\rangle \langle j_1 | j'_1\rangle \langle j_2 | j'_2\rangle \langle j_3 | j'_3\rangle \end{aligned}$$

    (4.28)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( AD\right) {{\mathrm{Tr}}}_{}\left( X\right) \left\langle i_1 \left| CB \right| i'_1 \right\rangle . \end{aligned}$$

    (4.29)
14. (14).

    If \((\pi ,\sigma )=((13),(12))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_3\rangle \langle i_2 | i'_2\rangle \langle i_3 | i'_1\rangle \langle j_1 | j'_2\rangle \langle j_2 | j'_1\rangle \langle j_3 | j'_3\rangle \end{aligned}$$

    (4.30)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( X\right) \left\langle i_1 \left| CB \right| i'_1 \right\rangle . \end{aligned}$$

    (4.31)
15. (15).

    If \((\pi ,\sigma )=((13),(13))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_3\rangle \langle i_2 | i'_2\rangle \langle i_3 | i'_1\rangle \langle j_1 | j'_3\rangle \langle j_2 | j'_2\rangle \langle j_3 | j'_1\rangle \end{aligned}$$

    (4.32)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( ADX\right) \left\langle i_1 \left| CB \right| i'_1 \right\rangle . \end{aligned}$$

    (4.33)
16. (16).

    If \((\pi ,\sigma )=((13),(23))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_3\rangle \langle i_2 | i'_2\rangle \langle i_3 | i'_1\rangle \langle j_1 | j'_1\rangle \langle j_2 | j'_3\rangle \langle j_3 | j'_2\rangle \end{aligned}$$

    (4.34)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( DAX\right) \left\langle i_1 \left| CB \right| i'_1 \right\rangle . \end{aligned}$$

    (4.35)
17. (17).

    If \((\pi ,\sigma )=((13),(123))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_3\rangle \langle i_2 | i'_2\rangle \langle i_3 | i'_1\rangle \langle j_1 | j'_2\rangle \langle j_2 | j'_3\rangle \langle j_3 | j'_1\rangle \end{aligned}$$

    (4.36)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( DX\right) \left\langle i_1 \left| CB \right| i'_1 \right\rangle . \end{aligned}$$

    (4.37)
18. (18).

    If \((\pi ,\sigma )=((13),(132))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_3\rangle \langle i_2 | i'_2\rangle \langle i_3 | i'_1\rangle \langle j_1 | j'_3\rangle \langle j_2 | j'_1\rangle \langle j_3 | j'_2\rangle \end{aligned}$$

    (4.38)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( AX\right) \left\langle i_1 \left| CB \right| i'_1 \right\rangle . \end{aligned}$$

    (4.39)
19. (19).

    If \((\pi ,\sigma )=((23),(1))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_1\rangle \langle i_2 | i'_3\rangle \langle i_3 | i'_2\rangle \langle j_1 | j'_1\rangle \langle j_2 | j'_2\rangle \langle j_3 | j'_3\rangle \end{aligned}$$

    (4.40)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( AD\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( B\right) {{\mathrm{Tr}}}_{}\left( C\right) \langle i_1 | i'_{1}\rangle . \end{aligned}$$

    (4.41)
20. (20).

    If \((\pi ,\sigma )=((23),(12))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_1\rangle \langle i_2 | i'_3\rangle \langle i_3 | i'_2\rangle \langle j_1 | j'_2\rangle \langle j_2 | j'_1\rangle \langle j_3 | j'_3\rangle \end{aligned}$$

    (4.42)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( B\right) {{\mathrm{Tr}}}_{}\left( C\right) \langle i_1 | i'_{1}\rangle . \end{aligned}$$

    (4.43)
21. (21).

    If \((\pi ,\sigma )=((23),(13))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_1\rangle \langle i_2 | i'_3\rangle \langle i_3 | i'_2\rangle \langle j_1 | j'_3\rangle \langle j_2 | j'_2\rangle \langle j_3 | j'_1\rangle \end{aligned}$$

    (4.44)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( ADX\right) {{\mathrm{Tr}}}_{}\left( B\right) {{\mathrm{Tr}}}_{}\left( C\right) \langle i_1 | i'_1\rangle . \end{aligned}$$

    (4.45)
22. (22).

    If \((\pi ,\sigma )=((23),(23))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_1\rangle \langle i_2 | i'_3\rangle \langle i_3 | i'_2\rangle \langle j_1 | j'_1\rangle \langle j_2 | j'_3\rangle \langle j_3 | j'_2\rangle \end{aligned}$$

    (4.46)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( DAX\right) {{\mathrm{Tr}}}_{}\left( B\right) {{\mathrm{Tr}}}_{}\left( C\right) \langle i_1 | i'_1\rangle . \end{aligned}$$

    (4.47)
23. (23).

    If \((\pi ,\sigma )=((23),(123))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_1\rangle \langle i_2 | i'_3\rangle \langle i_3 | i'_2\rangle \langle j_1 | j'_2\rangle \langle j_2 | j'_3\rangle \langle j_3 | j'_1\rangle \end{aligned}$$

    (4.48)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( DX\right) {{\mathrm{Tr}}}_{}\left( B\right) {{\mathrm{Tr}}}_{}\left( C\right) \langle i_1 | i'_1\rangle . \end{aligned}$$

    (4.49)
24. (24).

    If \((\pi ,\sigma )=((23),(132))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_1\rangle \langle i_2 | i'_3\rangle \langle i_3 | i'_2\rangle \langle j_1 | j'_3\rangle \langle j_2 | j'_1\rangle \langle j_3 | j'_2\rangle \end{aligned}$$

    (4.50)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( AX\right) {{\mathrm{Tr}}}_{}\left( B\right) {{\mathrm{Tr}}}_{}\left( C\right) \langle i_1 | i'_1\rangle . \end{aligned}$$

    (4.51)
25. (25).

    If \((\pi ,\sigma )=((123),(1))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_2\rangle \langle i_2 | i'_3\rangle \langle i_3 | i'_1\rangle \langle j_1 | j'_1\rangle \langle j_2 | j'_2\rangle \langle j_3 | j'_3\rangle \end{aligned}$$

    (4.52)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( AD\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( C\right) \left\langle i_1 \left| B \right| i'_1 \right\rangle . \end{aligned}$$

    (4.53)
26. (26).

    If \((\pi ,\sigma )=((123),(12))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_2\rangle \langle i_2 | i'_3\rangle \langle i_3 | i'_1\rangle \langle j_1 | j'_2\rangle \langle j_2 | j'_1\rangle \langle j_3 | j'_3\rangle \end{aligned}$$

    (4.54)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( C\right) \left\langle i_1 \left| B \right| i'_1 \right\rangle . \end{aligned}$$

    (4.55)
27. (27).

    If \((\pi ,\sigma )=((123),(13))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_2\rangle \langle i_2 | i'_3\rangle \langle i_3 | i'_1\rangle \langle j_1 | j'_3\rangle \langle j_2 | j'_2\rangle \langle j_3 | j'_1\rangle \end{aligned}$$

    (4.56)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( ADX\right) {{\mathrm{Tr}}}_{}\left( C\right) \left\langle i_1 \left| B \right| i'_1 \right\rangle . \end{aligned}$$

    (4.57)
28. (28).

    If \((\pi ,\sigma )=((123),(23))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_2\rangle \langle i_2 | i'_3\rangle \langle i_3 | i'_1\rangle \langle j_1 | j'_1\rangle \langle j_2 | j'_3\rangle \langle j_3 | j'_2\rangle \end{aligned}$$

    (4.58)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( DAX\right) {{\mathrm{Tr}}}_{}\left( C\right) \left\langle i_1 \left| B \right| i'_1 \right\rangle . \end{aligned}$$

    (4.59)
29. (29).

    If \((\pi ,\sigma )=((123),(123))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_2\rangle \langle i_2 | i'_3\rangle \langle i_3 | i'_1\rangle \langle j_1 | j'_2\rangle \langle j_2 | j'_3\rangle \langle j_3 | j'_1\rangle \end{aligned}$$

    (4.60)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( DX\right) {{\mathrm{Tr}}}_{}\left( C\right) \left\langle i_1 \left| B \right| i'_1 \right\rangle . \end{aligned}$$

    (4.61)
30. (30).

    If \((\pi ,\sigma )=((123),(132))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_2\rangle \langle i_2 | i'_3\rangle \langle i_3 | i'_1\rangle \langle j_1 | j'_3\rangle \langle j_2 | j'_1\rangle \langle j_3 | j'_2\rangle \end{aligned}$$

    (4.62)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( AX\right) {{\mathrm{Tr}}}_{}\left( C\right) \left\langle i_1 \left| B \right| i'_1 \right\rangle . \end{aligned}$$

    (4.63)
31. (31).

    If \((\pi ,\sigma )=((132),(1))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_3\rangle \langle i_2 | i'_1\rangle \langle i_3 | i'_2\rangle \langle j_1 | j'_1\rangle \langle j_2 | j'_2\rangle \langle j_3 | j'_3\rangle \end{aligned}$$

    (4.64)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( AD\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( B\right) \left\langle i_1 \left| C \right| i'_1 \right\rangle . \end{aligned}$$

    (4.65)
32. (32).

    If \((\pi ,\sigma )=((132),(12))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_3\rangle \langle i_2 | i'_1\rangle \langle i_3 | i'_2\rangle \langle j_1 | j'_2\rangle \langle j_2 | j'_1\rangle \langle j_3 | j'_3\rangle \end{aligned}$$

    (4.66)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( B\right) \left\langle i_1 \left| C \right| i'_1 \right\rangle . \end{aligned}$$

    (4.67)
33. (33).

    If \((\pi ,\sigma )=((132),(13))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_3\rangle \langle i_2 | i'_1\rangle \langle i_3 | i'_2\rangle \langle j_1 | j'_3\rangle \langle j_2 | j'_2\rangle \langle j_3 | j'_1\rangle \end{aligned}$$

    (4.68)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( ADX\right) {{\mathrm{Tr}}}_{}\left( B\right) \left\langle i_1 \left| C \right| i'_1 \right\rangle . \end{aligned}$$

    (4.69)
34. (34).

    If \((\pi ,\sigma )=((132),(23))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_3\rangle \langle i_2 | i'_1\rangle \langle i_3 | i'_2\rangle \langle j_1 | j'_1\rangle \langle j_2 | j'_3\rangle \langle j_3 | j'_2\rangle \end{aligned}$$

    (4.70)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( DAX\right) {{\mathrm{Tr}}}_{}\left( B\right) \left\langle i_1 \left| C \right| i'_1 \right\rangle . \end{aligned}$$

    (4.71)
35. (35).

    If \((\pi ,\sigma )=((132),(123))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_3\rangle \langle i_2 | i'_1\rangle \langle i_3 | i'_2\rangle \langle j_1 | j'_2\rangle \langle j_2 | j'_3\rangle \langle j_3 | j'_1\rangle \end{aligned}$$

    (4.72)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( DX\right) {{\mathrm{Tr}}}_{}\left( B\right) \left\langle i_1 \left| C \right| i'_1 \right\rangle . \end{aligned}$$

    (4.73)
36. (36).

    If \((\pi ,\sigma )=((132),(132))\), then

    $$\begin{aligned}&\sum _{i_2,i_3,j_1,j_2,j_3,i'_2,i'_3,j'_1,j'_2,j'_3} A_{j_1,j'_2}B_{i'_2,i_3}X_{j_3,j'_3}C_{i'_3,i_2}D_{j_2,j'_1} \nonumber \\&\quad \langle i_1 | i'_3\rangle \langle i_2 | i'_1\rangle \langle i_3 | i'_2\rangle \langle j_1 | j'_3\rangle \langle j_2 | j'_1\rangle \langle j_3 | j'_2\rangle \end{aligned}$$

    (4.74)
    $$\begin{aligned}&\quad = {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( AX\right) {{\mathrm{Tr}}}_{}\left( B\right) \left\langle i_1 \left| C \right| i'_1 \right\rangle . \end{aligned}$$

    (4.75)

    Combing the 36 cases together gives the desired conclusion:

    $$\begin{aligned}&\int UAU^\dagger BUXU^\dagger CUDU^\dagger \mathrm {d}\mu (U)= \mu _1\cdot \mathbb {1}_d+\mu _2\cdot BC+\mu _3\cdot CB \nonumber \\&\quad +\,\mu _4\cdot B + \mu _5\cdot C, \end{aligned}$$

    (4.76)

    where the coefficients \(\mu _j(j=1,\ldots ,5)\) are given below:

    $$\begin{aligned} \mu _1:= & {} \mathrm {Wg}(1,1,1){{\mathrm{Tr}}}_{}\left( AD\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( BC\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( BC\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( ADX\right) {{\mathrm{Tr}}}_{}\left( BC\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( DAX\right) {{\mathrm{Tr}}}_{}\left( BC\right) \nonumber \\&+\,\mathrm {Wg}(3){{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( DX\right) {{\mathrm{Tr}}}_{}\left( BC\right) \nonumber \\&+\,\mathrm {Wg}(3){{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( AX\right) {{\mathrm{Tr}}}_{}\left( BC\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( AD\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( B\right) {{\mathrm{Tr}}}_{}\left( C\right) \nonumber \\&+\,\mathrm {Wg}(3){{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( B\right) {{\mathrm{Tr}}}_{}\left( C\right) \nonumber \\&+\,\mathrm {Wg}(3){{\mathrm{Tr}}}_{}\left( ADX\right) {{\mathrm{Tr}}}_{}\left( B\right) {{\mathrm{Tr}}}_{}\left( C\right) \nonumber \\&+\,\mathrm {Wg}(1,1,1){{\mathrm{Tr}}}_{}\left( DAX\right) {{\mathrm{Tr}}}_{}\left( B\right) {{\mathrm{Tr}}}_{}\left( C\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( DX\right) {{\mathrm{Tr}}}_{}\left( B\right) {{\mathrm{Tr}}}_{}\left( C\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( AX\right) {{\mathrm{Tr}}}_{}\left( B\right) {{\mathrm{Tr}}}_{}\left( C\right) , \end{aligned}$$

    (4.77)
    $$\begin{aligned} \mu _2:= & {} \mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( AD\right) {{\mathrm{Tr}}}_{}\left( X\right) \nonumber \\&+\,\mathrm {Wg}(1,1,1){{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( X\right) +\mathrm {Wg}(3){{\mathrm{Tr}}}_{}\left( ADX\right) \nonumber \\&+\,\mathrm {Wg}(3){{\mathrm{Tr}}}_{}\left( DAX\right) +\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( DX\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( AX\right) , \end{aligned}$$

    (4.78)
    $$\begin{aligned} \mu _3:= & {} \mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( AD\right) {{\mathrm{Tr}}}_{}\left( X\right) \nonumber \\&+\,\mathrm {Wg}(3){{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( X\right) +\mathrm {Wg}(1,1,1){{\mathrm{Tr}}}_{}\left( ADX\right) \nonumber \\&+\,\mathrm {Wg}(3){{\mathrm{Tr}}}_{}\left( DAX\right) +\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( DX\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( AX\right) , \end{aligned}$$

    (4.79)
    $$\begin{aligned} \mu _4:= & {} \mathrm {Wg}(3){{\mathrm{Tr}}}_{}\left( AD\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( C\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( C\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( ADX\right) {{\mathrm{Tr}}}_{}\left( C\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( DAX\right) {{\mathrm{Tr}}}_{}\left( C\right) \nonumber \\&+\,\mathrm {Wg}(1,1,1){{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( DX\right) {{\mathrm{Tr}}}_{}\left( C\right) \nonumber \\&+\,\mathrm {Wg}(3){{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( AX\right) {{\mathrm{Tr}}}_{}\left( C\right) , \end{aligned}$$

    (4.80)
    $$\begin{aligned} \mu _5:= & {} \mathrm {Wg}(3){{\mathrm{Tr}}}_{}\left( AD\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( B\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( X\right) {{\mathrm{Tr}}}_{}\left( B\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( ADX\right) {{\mathrm{Tr}}}_{}\left( B\right) \nonumber \\&+\,\mathrm {Wg}(2,1){{\mathrm{Tr}}}_{}\left( DAX\right) {{\mathrm{Tr}}}_{}\left( B\right) \nonumber \\&+\,\mathrm {Wg}(3){{\mathrm{Tr}}}_{}\left( A\right) {{\mathrm{Tr}}}_{}\left( DX\right) {{\mathrm{Tr}}}_{}\left( B\right) \nonumber \\&+\,\mathrm {Wg}(1,1,1){{\mathrm{Tr}}}_{}\left( D\right) {{\mathrm{Tr}}}_{}\left( AX\right) {{\mathrm{Tr}}}_{}\left( B\right) . \end{aligned}$$

    (4.81)

We are done. \(\square \)

Remark 4.1

\(\mathrm {Wg}:=\frac{1}{(k!)^2}\sum _{\lambda \vdash k}\frac{\dim (\mathbf {P}_\lambda )^2}{\dim (\mathbf {Q}_\lambda )}\chi _\lambda \) is called the Weingarten function [4]. In particular, for \(\lambda \vdash 3\), we have:

where \(N_d=(d^2-1)(d^2-4)\). With these coefficients, we then have

Appendix 2: The proof of Theorem 2.2

By Lemma 2.1,

where

Note that the meaning of the notation \(\mathrm {Wg}(*)\) can be found in Appendix. Thus for \(f:=\sum ^{d_B}_{i,j,l=1}f(i,j,l)\), \(g:=\sum ^{d_B}_{i,j,l=1}g(i,j,l)\), and \(h:=\sum ^{d_B}_{i,j,l=1}h(i,j,l)\), we have

Hence, for \(T=\mathbb {1}_A\otimes \mathbb {1}_B/d_B - \rho _{AB}\), \({{\mathrm{Tr}}}_{}\left( T\right) =d_A-1\) and \({{\mathrm{Tr}}}_{}\left( T^2\right) =\frac{d_A-2}{d_B}+{{\mathrm{Tr}}}_{}\left( \rho ^2_{AB}\right) \), then

Therefore,

implying that

We make further analysis of the term \(a_n\), although we have already known the fact that \(\lim _{n\rightarrow \infty }a_n=0\). Let \(\varphi _\rho (X)={{\mathrm{Tr}}}_{}\left( \rho X\right) \). Apparently, \(\varphi _\rho \) is a positive unital linear mapping (in fact, it is a positive unital linear functional from the set of \(n\times n\) Hermitian matrices to \(\mathbb {R}\)). It is easily seen that \(f(x)=x^n\) is a convex function from \(\mathbb {R}\) to \(\mathbb {R}\). By using [10, Theorem 4.15, pp 147], we see that

By (2.4), we have

Thus

Then

where \(\mathrm {S}_L(\rho _{AB})\in \left[ 0,1-\frac{1}{d}\right] \). Clearly

Now,

We see from the above lower bound, i.e., \(-\ln (1-a_1)\), that when the purity of \(\rho _{AB}\) decreases, \(a_1\) increases. Hence such lower bound will be tighter.

Appendix 3: The proof of Proposition 3.2

Clearly, the first inequality is easily obtained

In order to show the second inequality, note that, for any two density matrices \(\rho \) and \(\sigma \),

Then, for \(\rho '_{AB}=U\rho _{AB}U^\dagger \), we have

implying that

By the concavity of fidelity, we have

Therefore,

This completes the proof.

Appendix 4: The proof of Theorem 3.4

Note that \(\rho '_{AB}=U\rho _{AB}U^\dagger \). We see from (3.8) that

Since \(I(A:B)_{\rho '} = \mathrm {S}(\rho '_A)+\mathrm {S}(\rho '_B)-\mathrm {S}(\rho _{AB})\), it follows that

That is,

This confirms the first inequality. Besides, by Eq. (3.4), we get

This confirms the second inequality. Therefore, we have

This is equivalent to the following

Next, we show that \(\mathrm {S}\left( c_0\cdot \mathbb {1}_d + c_1\cdot \rho _{AB} + c_2\cdot \rho ^2_{AB}\right) = \mathrm {S}(\rho _{AB})\) if and only if \(\rho _{AB}\) is maximally mixed state. Clearly, if \(\rho _{AB}\) is maximally mixed state, i.e., \(\mathrm {S}(\rho _{AB})=\ln (d)\), since \(\mathrm {S}\left( c_0\cdot \mathbb {1}_d + c_1\cdot \rho _{AB} + c_2\cdot \rho ^2_{AB}\right) - \mathrm {S}(\rho _{AB})\geqslant 0\), then \(\mathrm {S}\left( c_0\cdot \mathbb {1}_d + c_1\cdot \rho _{AB} + c_2\cdot \rho ^2_{AB}\right) \geqslant \ln (d)\), apparently \(\mathrm {S}\left( c_0\cdot \mathbb {1}_d + c_1\cdot \rho _{AB} + c_2\cdot \rho ^2_{AB}\right) \leqslant \ln (d)\), thus \(\mathrm {S}(\rho _{AB})=\mathrm {S}\left( c_0\cdot \mathbb {1}_d + c_1\cdot \rho _{AB} + c_2\cdot \rho ^2_{AB}\right) =\ln (d)\), the maximum of von Neuman entropy. Reversely, if \(\mathrm {S}\left( c_0\cdot \mathbb {1}_d + c_1\cdot \rho _{AB} + c_2\cdot \rho ^2_{AB}\right) = \mathrm {S}(\rho _{AB})\), then by the obtained inequality, we have

which holds if and only if \(\rho _{AB} = c_0\cdot \mathbb {1}_d + c_1\cdot \rho _{AB} + c_2\cdot \rho ^2_{AB}\). This means that for any eigenvalue \(\lambda (\geqslant 0)\) of \(\rho _{AB}\) must satisfy that

Solving this equation, we get

Note that we have dropped another root being larger than one. Thus \(\rho _{AB}\) is maximally mixed state. In fact, we get that \(\rho _{AB} = c_0\cdot \mathbb {1}_d + c_1\cdot \rho _{AB} + c_2\cdot \rho ^2_{AB}\) if and only if \(\rho _{AB}\) is maximally mixed state.

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