Relations between the observational entropy and Rényi information measures

Abstract

Observational entropy is a generalization of Boltzmann entropy to quantum mechanics. Observational entropy based on coarse-grained measurements has a certain relations with other quantum information measures. We study the relations between observational entropy and Rényi information measures and give some examples to explain the rationality of these relations.

Appendix

Let $\rho =\frac{1}{4^2}(\hat{I}+\sum _{j=1}^3 c_j \sigma _j\otimes \sigma _j)$ be two-partite X-state with $\dim \mathcal {H}=16$, where $|c_j|\le 1$ and

$$\begin{aligned} \sigma _1=\left( \begin{array}{ccccc} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \\ \end{array} \right) , \sigma _{2}=\left( \begin{array}{ccccc} 0 &{} 0 &{} 0 &{} -i \\ 0 &{} 0 &{} i &{} 0 \\ 0 &{} -i &{} 0 &{} 0 \\ i &{} 0 &{} 0 &{} 0 \\ \end{array} \right) , \sigma _{3}=\left( \begin{array}{ccccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 \\ \end{array} \right) . \end{aligned}$$

We can verify that

$$\begin{aligned} \begin{aligned} S_{VN}(\rho )&=-\frac{1}{4}[(1-c_{1}-c_{2}-c_{3})\log _2 (1-c_{1}-c_{2}-c_{3})\\&+(1-c_{1}+c_{2}+c_{3})\log _2 (1-c_{1}+c_{2}+c_{3})\\&+(1+c_{1}+c_{2}-c_{3})\log _2 (1+c_{1}+c_{2}-c_{3})\\&+(1+c_{1}-c_{2}+c_{3})\log _2 (1+c_{1}-c_{2}+c_{3})]+4. \end{aligned} \end{aligned}$$

(40)

We choose a coarse-graining $\mathcal {C}_{k}=\{\hat{P}^{k}_{x}: \hat{P}^{k}_{x}=\sum _{i}^{m}|i \rangle \langle i|\otimes \sum _{j}^{n}|j \rangle \langle j|, \sum _{x}\hat{P}^{k}_{x}=\hat{I}_{16}, i\le m\le 3, j\le n\le 3, k\in N^{+} \}$, where $|i \rangle \langle i|$ $(i=0,1,2,3)$ and $|j \rangle \langle j|$ $(j=0,1,2,3)$ are standard orthogonal basis of 4-dimensional Hilbert space, and $\hat{I}_{16}$ stands for the corresponding identity operator of $\mathcal {H}$. We obtain the probabilities and volumes after $\mathcal {C}_{k}$ acting on $\rho$. One can verify that the probabilities is independent of $c_1$ and $c_2$. If the probabilities are related to $c_3$, we have $S_{O(\mathcal {C}_{k})}(\rho )=f(c_3)+4$, where $f(c_3)$ is a function of $c_3$ on $[-1,1]$ and $f(0)=0$. Otherwise, we have $S_{O(\mathcal {C}_{k})}(\rho )=4$. The former is less than $\log _2 \dim \mathcal {H}$, and the latter is equal to $\log _2 \dim \mathcal {H}$, where $\log _2 \dim \mathcal {H}$ is the maximal entropy of the total space.

For example, we choose the coarse-graining $\mathcal {C}_{1}=\{\hat{P}^{1}_{0}, \hat{P}^{1}_{1}, \hat{P}^{1}_{2}\}$ as follows.

$$\begin{aligned} \begin{aligned} \hat{P}^{1}_{0}&=|0 \rangle \langle 0|\otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|+ |2 \rangle \langle 2|),\\ \hat{P}^{1}_{1}&=(|1 \rangle \langle 1|+ |2 \rangle \langle 2|+ |3 \rangle \langle 3|)\otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|+ |2 \rangle \langle 2|),\\ \hat{P}^{1}_{2}&=\hat{I}\otimes |3 \rangle \langle 3|. \end{aligned} \end{aligned}$$

We perform the coarse-graining measurement on $\rho$ with probabilities $p_{1x}=tr(\hat{P}^{1}_{x}\rho \hat{P}^{1}_{x})$ and volumes $V_{1x}=tr(\hat{P}^{1}_{x})$, $x=0, 1, 2$. We can verify that

$$\begin{aligned} p_{10}=\frac{c_{3}+3}{16},~~V_{10}=3,~~ and~~ p_{11}=\frac{9-c_{3}}{16},~~V_{11}=9,~~ and~~ p_{12}=\frac{1}{4},~~V_{12}=4. \end{aligned}$$

According to the definition of observational entropy, we have

$$\begin{aligned} \begin{aligned} S_{O(\mathcal {C}_{1})}(\rho )&=-\sum ^{2}_{x=0}p_{1x}\log _{2}\frac{p_{1x}}{V_{1x}}\\&=-\frac{c_{3}+3}{16}\log _{2} (c_{3}+3)-\frac{9-c_{3}}{16}\log _{2} (9-c_{3})+\frac{21-c_{3}}{16}\log _{2} 3+4. \end{aligned} \end{aligned}$$

Since $|c_{3}|\le 1$, we have $S_{O(\mathcal {C}_{1})}(\rho )\le 4$ and $S_{O(\mathcal {C}_{1})}(\rho )= 4$ if $c_{3}=0$ (as described in Fig. 3).

On the other hand, we choose another coarse-graining $\mathcal {C}_{2}=\{\hat{P}^{2}_{0}, \hat{P}^{2}_{1}, \hat{P}^{2}_{2}\}$ as follows.

$$\begin{aligned} \begin{aligned} \hat{P}^{2}_{0}&=(|0 \rangle \langle 0|+ |1 \rangle \langle 1|)\otimes (|2 \rangle \langle 2|+ |3 \rangle \langle 3|),\\ \hat{P}^{2}_{1}&=(|0 \rangle \langle 0|+ |1 \rangle \langle 1|)\otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|),\\ \hat{P}^{2}_{2}&=(|2 \rangle \langle 2|+ |3 \rangle \langle 3|)\otimes \hat{I}. \end{aligned} \end{aligned}$$

We perform the coarse-graining measurement on $\rho$ with probabilities $p_{2x}=tr(\hat{P}^{2}_{x}\rho \hat{P}^{2}_{x})$ and volumes $V_{2x}=tr(\hat{P}^{2}_{x})$, $x=0, 1, 2$. We can verify that

$$\begin{aligned} p_{20}=\frac{1}{4},~~V_{20}=4,~~ and~~ p_{21}=\frac{1}{4},~~V_{21}=4,~~ and~~ p_{22}=\frac{1}{2},~~V_{22}=8. \end{aligned}$$

According to the definition of observational entropy, we have

$$\begin{aligned} \begin{aligned} S_{O(\mathcal {C}_{2})}(\rho )&=-\sum ^{2}_{x=0}p_{2x}\log _{2}\frac{p_{2x}}{V_{2x}}\\&=-\frac{1}{4}\log _{2} \frac{1}{16}-\frac{1}{4}\log _{2} \frac{1}{16}-\frac{1}{2}\log _{2} \frac{1}{16}=4. \end{aligned} \end{aligned}$$

The above result shows that if $p_{2x}$ is independent of $c_3$, the value of observational entropy is equal to the maximal entropy.

According to Lemma 3, we have $S_{VN}(\rho ) \le S_{O(\mathcal {C}_{1})}(\rho ) \le \log _{2} \dim \mathcal {H}$ (as described in Figs. 3 and 4), where $i=1, 2$ and $\log _{2}\dim \mathcal {H}=4$. Since $S_{O(\mathcal {C}_{2})}(\rho )=4$, we have $S_{O(\mathcal {C}_{1})}(\rho )\le S_{O(\mathcal {C}_{2})}(\rho )$.

We choose a multiple coarse-graining $(\mathcal {C}_{1}, \mathcal {C}_{2})=\{\hat{P}^{1}_{l}\cdot \hat{P}^{2}_{m}\}$, $l=0, 1, 2$ and $m=0, 1, 2$ as follows.

$$\begin{aligned} \begin{aligned} \hat{P}^{2}_{0}\hat{P}^{1}_{0}&=|0 \rangle \langle 0|\otimes |2 \rangle \langle 2|,~~\\ \hat{P}^{2}_{0}\hat{P}^{1}_{1}&=|1 \rangle \langle 1|\otimes |2 \rangle \langle 2|,~~\\ \hat{P}^{2}_{0}\hat{P}^{1}_{2}&=(|0 \rangle \langle 0|+ |1 \rangle \langle 1|)\otimes |3 \rangle \langle 3 |,\\ \hat{P}^{2}_{1}\hat{P}^{1}_{0}&=|0 \rangle \langle 0 |\otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|),~~\\ \hat{P}^{2}_{1}\hat{P}^{1}_{1}&=|1 \rangle \langle 1 |\otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|),~~\hat{P}^{2}_{1}\hat{P}^{1}_{2}=0,\\ \hat{P}^{2}_{2}\hat{P}^{1}_{0}&=0,~~\hat{P}^{2}_{2}\hat{P}^{1}_{1}=(|2 \rangle \langle 2 |+ |3 \rangle \langle 3 |) \otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|+ |2 \rangle \langle 2|),~~\\ \hat{P}^{2}_{2}\hat{P}^{1}_{2}&=(|2 \rangle \langle 2 |+ |3 \rangle \langle 3 |) \otimes |3 \rangle \langle 3|. \end{aligned} \end{aligned}$$

We perform the coarse-graining $(\mathcal {C}_{1}, \mathcal {C}_{2})$ on $\rho$ with probabilities $p_{lm}=tr(\hat{P}^{2}_{m}\hat{P}^{1}_{l}\rho \hat{P}^{1}_{l}\hat{P}^{2}_{m})$ and volumes $V_{lm}=tr(\hat{P}^{2}_{m}\hat{P}^{1}_{l})$ as Tables 1 and 2.

Table 1 Values of probabilities $p_{lm}$, l and $m=0, 1, 2$

Table 2 Values of volumes $V_{lm}$, l and $m=0, 1, 2$

According to Definition 2, we have

$$\begin{aligned} S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho )&=-\sum ^{2}_{l=0}\sum ^{2}_{m=0}p_{lm}\log _{2}\frac{p_{lm}}{V_{lm}}\\&=-\frac{1+c_{3}}{16}\log _{2} \frac{1+c_{3}}{16} -\frac{1}{8}\log _{2} \frac{1}{16} -\frac{1}{8}\log _{2} \frac{1}{16}\\&-\frac{1-c_{3}}{16}\log _{2} \frac{1-c_{3}}{16}-\frac{1}{8}\log _{2} \frac{1}{16} \\&-\frac{3}{8}\log _{2} \frac{1}{16} -\frac{1}{8}\log _{2} \frac{1}{16}\\&=-\frac{1+c_{3}}{16}\log _{2} (1+c_{3}) -\frac{1-c_{3}}{16}\log _{2} (1-c_{3}) +4. \end{aligned}$$

In the second equality, if probabilities and volumes are equal to zero, we denote $0\log _{2}\frac{0}{0}=0$.

According to Lemma 5, we have $S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho ) \le S_{O(\mathcal {C}_{1})}(\rho )$ (as described in Fig. 3).

Meanwhile, we can verify that $\hat{P}^{1}_{l}\cdot \hat{P}^{2}_{m}=\hat{P}^{2}_{m}\cdot \hat{P}^{1}_{l}$, where l and $m=0, 1, 2$. Thus, $\hat{P}^{1}_{l}$ and $\hat{P}^{2}_{m}$ are commutative. From Lemma 4, we can rewrite multiple coarse-graining $(\mathcal {C}_{1}, \mathcal {C}_{2})$ as a single coarse-graining $\mathcal {C}_{1, 2}$. From Definition 2, $S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho )$ can be called as observational entropy with a joint coarse-graining and denoted as $S_{O(C_{1, 2})}$. As described in Fig. 3, we obtain that observational entropy with a joint coarse-graining is not larger than observational entropy with a single coarse-graining, if the joint coarse-graining is made up of this single coarse-graining. On the other hand, we can verify that $\hat{P}^{2}_{m}\cdot \hat{P}^{1}_{l}\cdot \hat{P}^{1}_{l}=\hat{P}^{2}_{m}\cdot \hat{P}^{1}_{l}$, which means that multiple coarse-graining $(\mathcal {C}_{1}, \mathcal {C}_{2})$ is finer than coarse-graining $\mathcal {C}_{1}$ (Definition 6, [9]). According to Lemma 2 , we have $S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho ) \le S_{O(\mathcal {C}_{1})}(\rho )$ (as shown in Fig. 3). In fact, we have $\mathcal {C}_{1}\hookrightarrow (\mathcal {C}_{1}, \mathcal {C}_{2})$ for the same reason, which means $S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho ) \le S_{O(\mathcal {C}_{2})}(\rho )$ (as shown in Fig. 3).

Moreover, for the above X-states, we can calculate the observational entropy with local coarse-graining. Set $\mathcal {C}_{A}\otimes \mathcal {C}_{B}\equiv \{\hat{P}^A_l\otimes \hat{P}^B_m\}$ as a local coarse-graining acting on $\rho$, where $l=0, 1$ and $m=0, 1, 2$, respectively. Denote

$$\begin{aligned} \hat{P}^A_{0}=V_A(\Pi _0+\Pi _2)V^{\dagger }_A,~\hat{P}^A_{1}=V_A(\Pi _1+\Pi _3)V^{\dagger }_A, \end{aligned}$$

and

$$\begin{aligned} \hat{P}^B_0=V_B(\Pi _0+\Pi _2)V^{\dagger }_B,~\hat{P}^B_1=V_B\Pi _1V^{\dagger }_B,~\hat{P}^B_2=V_B\Pi _3 V^{\dagger }_B, \end{aligned}$$

where $V_{X}=t_{X0}\hat{I}+t_{X1}\sigma _1 i+t_{X2}\sigma _2 i+t_{X3}\sigma _3 i$ and $\sum ^{3}_{k=0} t^2_{Xk}=1$, $X\in \{A,B\}$. Denote

$$\begin{aligned}&m_{X1}=2(t_{X1}t_{X3}-t_{X2}t_{X0}),~m_{X2}=2(t_{X2}t_{X3}+t_{X1}t_{X0}),\\&m_{X3}=t^2_{X0}+t^2_{X3}-t^2_{X1}-t^2_{X2}, \end{aligned}$$

where $m^2_{X1}+m^2_{X2}+m^2_{X1}=1$. Denote

$$\begin{aligned} \alpha =c_1 m_{A1}m_{B1}+c_2 m_{A2}m_{B2}+c_3 m_{A3}m_{B3},~|\alpha |\le 1. \end{aligned}$$

After $\mathcal {C}_{A}\otimes \mathcal {C}_{B}$ acting on $\rho$, we get final states $\rho _{lm}=\frac{1}{p_{lm}}(\hat{P}^A_{l}\otimes \hat{P}^B_{m})\rho (\hat{P}^A_{l}\otimes \hat{P}^B_{m})$ with respect to probability $p_{lm}=tr[(\hat{P}^A_{l}\otimes \hat{P}^B_{m}\otimes )\rho (\hat{P}^A_{l}\otimes \hat{P}^B_{m})]$ as follows,

$$\begin{aligned} \rho _{00}=\frac{1}{p_{00}}(\hat{P}^A_{0}\otimes \hat{P}^B_{0})\rho (\hat{P}^A_{0}\otimes \hat{P}^B_{0})=\frac{1}{4}(\hat{P}^A_{0}\otimes \hat{P}^B_{0} ),~p_{00}=\frac{1}{4}(1+\alpha ), \\ \rho _{01}=\frac{1}{p_{01}}(\hat{P}^A_{0}\otimes \hat{P}^B_{1})\rho (\hat{P}^A_{0}\otimes \hat{P}^B_{1})=\frac{1}{2}(\hat{P}^A_{0}\otimes \hat{P}^B_{1}),~p_{01}=\frac{1}{8}(1-\alpha ), \\ \rho _{02}=\frac{1}{p_{02}}(\hat{P}^A_{0}\otimes \hat{P}^B_{2})\rho (\hat{P}^A_{0}\otimes \hat{P}^B_{2})=\frac{1}{2}(\hat{P}^A_{0}\otimes \hat{P}^B_{2} ),~p_{02}=\frac{1}{8}(1-\alpha ), \\ \rho _{10}=\frac{1}{p_{10}}(\hat{P}^A_{1}\otimes \hat{P}^B_{0})\rho (\hat{P}^A_{1}\otimes \hat{P}^B_{0})=\frac{1}{4}(\hat{P}^A_{1}\otimes \hat{P}^B_{0} ),~p_{10}=\frac{1}{4}(1-\alpha ), \\ \rho _{11}=\frac{1}{p_{11}}(\hat{P}^A_{1}\otimes \hat{P}^B_{1})\rho (\hat{P}^A_{1}\otimes \hat{P}^B_{1})=\frac{1}{2}(\hat{P}^A_{1}\otimes \hat{P}^B_{1} ),~p_{11}=\frac{1}{8}(1+\alpha ), \\ \rho _{12}=\frac{1}{p_{12}}(\hat{P}^A_{1}\otimes \hat{P}^B_{2})\rho (\hat{P}^A_{1}\otimes \hat{P}^B_{2})=\frac{1}{2}(\hat{P}^A_{1}\otimes \hat{P}^B_{2} ),~p_{12}=\frac{1}{8}(1+\alpha ), \end{aligned}$$

where $\sum \limits ^{1}_{l=0}\sum \limits ^{2}_{m=0}p_{lm}=1$.

According to Definition 3, we have

$$\begin{aligned} \begin{aligned} S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho )&=-\sum _{lm}p_{lm}\log _{2}\frac{p_{lm}}{V_{lm}}\\&=-\frac{1+\alpha }{4}\log _2\frac{1+\alpha }{16}-2\cdot \frac{1+\alpha }{8}\log _2\frac{1+\alpha }{16}-2\cdot \frac{1-\alpha }{8}\log _2\frac{1-\alpha }{16}\\&-\frac{1-\alpha }{4}\log _2\frac{1-\alpha }{16}\\&=-\frac{1+\alpha }{2}\log _2 (1+\alpha )-\frac{1-\alpha }{2}\log _2 (1-\alpha )+4, \end{aligned} \end{aligned}$$

where $V_{lm}=tr{(\hat{P}^A_{l}\otimes \hat{P}^B_{m})}$ and $\sum \limits ^{1}_{l=0}\sum \limits ^{2}_{m=0}V_{lm}=16$.

Fig. 3

This graph shows the values of observational entropy for state $\rho$ and $\tilde{\rho }$, where $\rho =\frac{1}{4^2}(\hat{I}+ 0.03\cdot \sigma _1\otimes \sigma _1- 0.5\cdot \sigma _2\otimes \sigma _2 + c_{3} \cdot \sigma _3\otimes \sigma _3 )$ and $\tilde{\rho }=\frac{1}{4^2}(\hat{I}+ 0.65\cdot \sigma _1\otimes \sigma _1- 0.12\cdot \sigma _2\otimes \sigma _2 + c_{3} \cdot \sigma _3\otimes \sigma _3 )$. The red solid line, blue solid line, black solid line, and magenta solid line represent the values of observational entropy with single coarse-graining ($S_{O(\mathcal {C}_{1})}(\rho )$), the values of observational entropy with two coarse-grainings ($S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho )$), the values of local observational entropy ($S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho )$), the values of local observational entropy ($S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\tilde{\rho })$), respectively. The black dotted line represents the values of observational entropy with single coarse-graining ($S_{O(\mathcal {C}_{2})}(\rho )$ or $S_{O(\mathcal {C}_{2})}(\tilde{\rho })$), which is equal to the maximal entropy of $\rho$ and $\tilde{\rho }$. The red and blue diamonds represent the intersection of local and other observational entropies (Color figure online)

Fig. 4

This graph shows the difference between observational entropy and von Neumann entropy for state $\rho$, where $\rho =\frac{1}{4^2}(\hat{I}+ 0.03\cdot \sigma _1\otimes \sigma _1- 0.5\cdot \sigma _2\otimes \sigma _2 + c_{3} \cdot \sigma _3\otimes \sigma _3 )$. The red solid line, blue solid line, and black solid line represent the difference between observational entropy with single coarse-graining and von Neumann entropy ($S_{O(\mathcal {C}_{1})}(\rho )-S_{VN}(\rho )$), the difference between observational entropy with multiple coarse-grainings and von Neumann entropy ($S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho )-S_{VN}(\rho )$), and the difference between local observational entropy and von Neumann entropy ($S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho )-S_{VN}(\rho )$), respectively. For $c_{1}=0.03$ and $c_{2}=-0.5$, we have $1-c_{1}+c_{2}+c_{3}=0.47+c_{3}$ and $1+c_{1}+c_{2}-c_{3}=0.53-c_{3}$ (40). The red dotted line indicates that the value of von Neumann entropy is meaningless when $c_{3}=-0.47$ and $c_{3}=0.53$, respectively. Point (0.2489, 0.23766) is the intersection of [$S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho )-S_{VN}(\rho )$] and [$S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho )-S_{VN}(\rho )$]. Point (0.3321, 0.28898) is the intersection of [$S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho )-S_{VN}(\rho )$] and [$S_{O(\mathcal {C}_{1})}(\rho )-S_{VN}(\rho )$]. Point $(-0.04835, 0.16158)$ is the minimum point of [$S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho )-S_{VN}(\rho )$] (Color figure online)

From Fig. 3, we can obtain the following conclusions. Observational entropy is nonincreasing with each added coarse-graining ($S_{O(\mathcal {C}_{1})}(\rho )\ge S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho )$) (Lemma 5). Observational entropies are less than maximal entropy (Lemma 3). Moreover, observational entropy with single coarse-graining (or total observational entropy, $S_{O(\mathcal {C}_{1})}(\rho )$) is not always larger than local observational entropy ($S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho )$). Since $\rho$ and $\tilde{\rho }$ share one $c_{3}$, their observational entropy with single or multiple coarse-graining is the same, but local observational entropies are different.

From Fig. 4, we can obtain the following conclusions. Observational entropy is not less than von Neumann entropy (Lemma 3). Second, the intersection of [$S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho )-S_{VN}(\rho )$] and [$S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho )-S_{VN}(\rho )$] shows that $S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho )$ and $S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho )$ are equal for $c_{3} =0.2489$ (as shown in Fig. 3, the blue diamonds). Moreover, if $-0.47 < c_{3} \le 0.2489$, we have $S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho ) \le S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho )$. Otherwise, if $0.2489 \le c_{3} < 0.53$, we have $S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho ) \ge S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho )$. The intersection of [$S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho )-S_{VN}(\rho )$] and [$S_{O(\mathcal {C}_{1})}(\rho )-S_{VN}(\rho )$] shows that $S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho )$ and $S_{O(\mathcal {C}_{1})}(\rho )$ are equal for $c_{3} =0.3321$ (as shown in Fig. 3, the blue diamonds). Moreover, if $-0.47 < c_{3} \le 0.3321$, we have $S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho ) \le S_{O(\mathcal {C}_{1})}(\rho )$. Otherwise, if $0.3321 \le c_{3} < 0.53$, we have $S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho ) \ge S_{O(\mathcal {C}_{1})}(\rho )$. What is more, $[S_{O(\mathcal {C}_{1})}(\rho )-S_{VN}(\rho )] \ge [S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho )-S_{VN}(\rho )]$ show that $S_{O(\mathcal {C}_{1})}(\rho ) \ge S_{O(\mathcal {C}_{1}, \mathcal {C}_{2})}(\rho )$, which reveal the fact that observational entropy is nonincreasing with each added coarse-graining. On the other hand, The result $[S_{O(\mathcal {C}_{A}\otimes \mathcal {C}_{B})}(\rho )-S_{VN}(\rho )]\ge 0.16158$ shows that quantum correlation entropy is non-negative [1].

If we add the number of subsystems, e.g., $\hat{\rho }=\frac{1}{4^3}(\hat{I}+\sum _{j=1}^3 c_j \sigma _j\otimes \sigma _j\otimes \sigma _j)$. We can verify that

$$\begin{aligned} S_{VN}(\hat{\rho })= & {} -\frac{1+\sqrt{c^2_1+c^2_2+c^2_3}}{2}\log _2 (1+\sqrt{c^2_1+c^2_2+c^2_3})\\&-\frac{1-\sqrt{c^2_1+c^2_2+c^2_3}}{2}\log _2 (1-\sqrt{c^2_1+c^2_2+c^2_3})+6. \end{aligned}$$

By selecting coarse-graining as $\mathcal {C}_{t}=\{\hat{P}^{t}_{x}: \hat{P}^{t}_{x}=\sum _{i}^{l}|i \rangle \langle i|\otimes \sum _{j}^{m}|j \rangle \langle j|\otimes \sum _{i^{'}}^{n}|i^{'} \rangle \langle i^{'}|, \sum _{x}\hat{P}^{t}_{x}=\hat{I}_{64}, i\le l\le 3, j\le m\le 3, i^{'}\le n\le 3, t\in N^{+} \}$, where $|i \rangle \langle i|$ $(i=0, 1, 2, 3)$, $|j \rangle \langle j|$ $(j=0, 1, 2, 3)$ and $|i^{'} \rangle \langle i^{'}|$ $(i^{'}=0, 1, 2, 3)$ are standard orthogonal basis of 4-dimensional Hilbert space, and $\hat{I}_{64}$ stands for the corresponding identity operator of $\mathcal {H}$. We can calculate observational entropy as $S_{O(\mathcal {C}_{t})}(\hat{\rho })=g(c_3)+6$ or $S_{O(\mathcal {C}_{t})}(\hat{\rho })=6$, where $g(c_3)$ is a function of $c_3$ on $[-1,1]$ and $g(0)=0$.

For example, we choose the coarse-graining $\mathcal {C}_{3}=\{\hat{P}^{3}_{0}, \hat{P}^{3}_{1}, \hat{P}^{3}_{2}, \hat{P}^{3}_{3}\}$ as follows.

$$\begin{aligned} \begin{aligned} \hat{P}^{3}_{0}&=|0 \rangle \langle 0|\otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|+ |2 \rangle \langle 2|)\otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|+ |3 \rangle \langle 3|),\\ \hat{P}^{3}_{1}&=(|1 \rangle \langle 1|+ |2 \rangle \langle 2|+ |3 \rangle \langle 3|)\otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|+ |2 \rangle \langle 2|)\otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|+ |3 \rangle \langle 3|),\\ \hat{P}^{3}_{2}&=\hat{I}\otimes |3 \rangle \langle 3|\otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|+ |3 \rangle \langle 3|)\\ \hat{P}^{3}_{3}&=\hat{I}\otimes \hat{I}\otimes |2 \rangle \langle 2|. \end{aligned} \end{aligned}$$

We perform the coarse-graining measurement on $\hat{\rho }$ with probabilities $p_{3x}=tr(\hat{P}^{3}_{x}\hat{\rho }\hat{P}^{3}_{x})$ and volumes $V_{3x}=tr(\hat{P}^{3}_{x})$, $x=0, 1, 2, 3$. We can verify that

$$\begin{aligned} p_{30}= & {} \frac{9-c_{3}}{64},~~V_{30}=9,~~ and~~ p_{31}=\frac{27+c_{3}}{64},~~V_{31}=27,~~ and~~\\ p_{32}= & {} \frac{3}{16},~~V_{32}=12~~ and,~~ p_{33}=\frac{1}{4},~~V_{33}=16. \end{aligned}$$

According to the definition of observational entropy, we have

$$\begin{aligned} \begin{aligned} S_{O(\mathcal {C}_{3})}(\rho )&=-\sum ^{3}_{x=0}p_{3x}\log _{2}\frac{p_{3x}}{V_{3x}}\\&=-\frac{9-c_{3}}{64}\log _{2} (9-c_{3})-\frac{27+c_{3}}{64}\log _{2} (27+c_{3})+\frac{99+c_{3}}{64}\log _{2} 3+6. \end{aligned} \end{aligned}$$

Since $|c_{3}|\le 1$, we have $S_{O(\mathcal {C}_{3})}(\hat{\rho })\le 6$ and $S_{O(\mathcal {C}_{3})}(\hat{\rho })= 6$ if $c_{3}=0$. On the other hand, we choose another coarse-graining $\mathcal {C}_{4}=\{\hat{P}^{4}_{0}, \hat{P}^{4}_{1}, \hat{P}^{4}_{2}, \hat{P}^{4}_{3}\}$ as follows.

$$\begin{aligned} \begin{aligned} \hat{P}^{4}_{0}&=(|0 \rangle \langle 0|+ |1 \rangle \langle 1|)\otimes (|2 \rangle \langle 2|+ |3 \rangle \langle 3|)\otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|),\\ \hat{P}^{4}_{1}&=(|0 \rangle \langle 0|+ |1 \rangle \langle 1|)\otimes (|2 \rangle \langle 2|+ |3 \rangle \langle 3|)\otimes (|2 \rangle \langle 2|+ |3 \rangle \langle 3|),\\ \hat{P}^{4}_{2}&=(|0 \rangle \langle 0|+ |1 \rangle \langle 1|)\otimes (|0 \rangle \langle 0|+ |1 \rangle \langle 1|) \otimes \hat{I}.\\ \hat{P}^{4}_{3}&=(|2 \rangle \langle 2|+ |3 \rangle \langle 3|)\otimes \hat{I} \otimes \hat{I}. \end{aligned} \end{aligned}$$

We perform the coarse-graining measurement on $\hat{\rho }$ with probabilities $p_{4x}=tr(\hat{P}^{4}_{x}\hat{\rho }\hat{P}^{4}_{x})$ and volumes $V_{4x}=tr(\hat{P}^{4}_{x})$, $x=0, 1, 2, 3$. We can verify that

$$\begin{aligned} p_{40}= & {} \frac{1}{8},~~V_{40}=8,~~ and~~ p_{41}=\frac{1}{8},~~V_{41}=8,~~ and~~\\ p_{42}= & {} \frac{1}{4},~~V_{42}=16,~~ and~~ p_{43}=\frac{1}{2},~~V_{43}=32. \end{aligned}$$

According to the definition of observational entropy, we have

$$\begin{aligned} \begin{aligned} S_{O(\mathcal {C}_{4})}(\hat{\rho })&=-\sum ^{3}_{x=0}p_{4x}\log _{2}\frac{p_{4x}}{V_{4x}}\\&=-\frac{1}{8}\log _{2} \frac{1}{64} -\frac{1}{8}\log _{2} \frac{1}{64} -\frac{1}{4}\log _{2} \frac{1}{64}-\frac{1}{2}\log _{2} \frac{1}{64}=6. \end{aligned} \end{aligned}$$

The above results show that if $p_{4x}$ is independent of $c_3$, the value of observational entropy is equal to the maximal entropy. From Lemma 3, we have $S_{O(\mathcal {C}_{3})}(\hat{\rho }) \le S_{O(\mathcal {C}_{4})}(\hat{\rho })$. In addition, for a family X-states as $\hat{\rho }$, we can also calculate multiple observational entropy and local observational entropy, which can verify the relevant properties of observational entropy, and make graphs like Figs. 3 and 4 to explain these conclusions.