Effects of measurement dependence on tilted CHSH Bell tests

Abstract

Device-independent (DI) randomness generations based on tilted CHSH Bell tests, different from other Bell tests, use a little entanglement but can produce quite a lot of certified randomness. In all these protocols, the violations of Bell inequalities which imply the generation of certified randomness rely on an assumption, i.e., measurement independence. Since ensuring measurement independence in a practical Bell test is difficult, it is crucial to explore effects of relaxing this assumption (called as measurement dependence). Naturally, a question, how measurement dependence affects tilted CHSH Bell tests, arise. In this paper, we answer this question. Concretely, we introduce measurement dependence with flexible lower bounds which are different from fixed lower bound and then qualify the effect of measurement dependence on tilted CHSH Bell tests with different input distributions. The results show that the effect of measurement dependence on tilted CHSH Bell tests with flexible lower bound of measurement dependence is more powerful than that of only considering fixed lower bound. The relevance of this work highlights the study of DI randomness amplification and other DI quantum information tasks.

Appendix

1.1 Appendix A: the general input distributions

We show the deduction of Theorem 1.

Proof

Based on the lower bound and upper bound of measurement dependence, we get

$$\begin{aligned} l\le p'\left( X_{j},Y_{k}|\lambda \right) \le h, \quad \forall {j, k,\lambda ,} \end{aligned}$$

(19)

where $j,k\in \{0,1\}$.

For an arbitrary tuple $(X_{j},Y_{k},\lambda )$, let

$$\begin{aligned} p\left( X_{j}, Y_{k}|\lambda \right) =\frac{p'\left( X_{j}, Y_{k}|\lambda \right) -l}{1-4l}. \end{aligned}$$

(20)

The constrains on $p'(X_{j}, Y_{k}|\lambda )$ can be converted to the following constrains:

$$\begin{aligned} 0\le & {} p\left( X_{j}, Y_{k}|\lambda \right) \le \frac{h-l}{1-4l}, \quad \forall j, k, \lambda , \end{aligned}$$

(21)

$$\begin{aligned} \sum _{j,k}p\left( X_{j}, Y_{k}|\lambda \right)= & {} \sum _{j,k} \frac{p'\left( X_{j}, Y_{k}|\lambda \right) -l}{1-4l}\nonumber \\= & {} \frac{\sum _{j,k}\left[ p'\left( X_{j}, Y_{k}|\lambda \right) -l\right] }{1-4l}\nonumber \\= & {} \frac{\sum _{j,k}p'\left( X_{j}, Y_{k}|\lambda \right) -4l}{1-4l}\nonumber \\= & {} 1, \end{aligned}$$

(22)

where the last equality holds according to $\sum _{j,k}p'(X_{j}, Y_{k}|\lambda )=1$.

So, we focus on the case that $l=0$.

Let $p_{A}(-1|X_{j},\lambda )=m_{j},$ $p_{B}(-1|Y_{k},\lambda )=n_{k},$ $p(-1,-1|X_{j},Y_{k},\lambda )=c_{j,k},$ we get

$$\begin{aligned} p\left( -1,1|X_{j},Y_{k},\lambda \right)= & {} m_{j}-c_{j,k},\nonumber \\ p\left( 1,-1|X_{j},Y_{k},\lambda \right)= & {} n_{k}-c_{j,k},\nonumber \\ p\left( 1,1|X_{j},Y_{k},\lambda \right)= & {} 1+c_{j,k}-m_{j}-n_{k}. \end{aligned}$$

(23)

Hence, $p(a, b|X_{j},Y_{k},\lambda )\in \{c_{j,k}, m_{j}-c_{j,k}, n_{k}-c_{j,k}, 1+c_{j,k}-m_{j}-n_{k}\}$.

As we know, $c_{j,k}$ satisfies

$$\begin{aligned} \max \{0,m_{j}+n_{k}-1\}\le c_{j,k}\le \min \{m_{j},n_{k}\}. \end{aligned}$$

(24)

where

$$\begin{aligned} \min \{x,y\}= & {} \frac{1}{2}\left[ x+y-|x-y|\right] ,\nonumber \\ \max \{x,y\}= & {} \frac{1}{2}\left[ x+y+|x-y|\right] . \end{aligned}$$

(25)

Based on the definition of $\langle X_{j}Y_{k}\rangle$, we get

$$\begin{aligned} \begin{aligned} \left\langle X_{j}Y_{k}\right\rangle&=\sum p\left( a=b|X_{j},Y_{k}\right) -\sum p\left( a\ne b|X_{j},Y_{k}\right) \\&=1+4c_{jk}-2\left( m_{j}+n_{k}\right) . \end{aligned} \end{aligned}$$

(26)

So, by using Eqs. (24), (25), (26), we have

$$\begin{aligned} 2\left| m_{j}+n_{k}-1\right| -1\le \left\langle X_{j}Y_{k}\right\rangle \le 1-2\left| m_{j}+n_{k}\right| . \end{aligned}$$

(27)

Thus, $\widetilde{I}_{\alpha }^{\beta }$ can be described by

$$\begin{aligned} \begin{aligned} \widetilde{I}_{\alpha }^{\beta }=\,&\beta \left\langle X_{0}\right\rangle +\alpha \left\langle X_{0}Y_{0}\right\rangle +\alpha \left\langle X_{0}Y_{1}\right\rangle +\,\left\langle X_{1}Y_{0}\right\rangle -\left\langle X_{1}Y_{1}\right\rangle \\ =\,&\sum _{\lambda }p(\lambda )\left[ \beta p\left( \lambda |X_{0}\right) \langle X_{0}\rangle _{\lambda }+\alpha p\left( \lambda |X_{0}Y_{0}\right) \langle X_{0}Y_{0}\rangle _{\lambda }+\alpha p\left( \lambda |X_{0}Y_{1}\right) \langle X_{0}Y_{1}\rangle _{\lambda }\right. \\&\left. +\,p\left( \lambda |X_{1}Y_{0}\right) \langle X_{1}Y_{0}\rangle _{\lambda } -p\left( \lambda |X_{1}Y_{1}\right) \langle X_{1}Y_{1}\rangle _{\lambda }\right] \\ \le \,&\sum _{\lambda }p(\lambda )\left[ \beta p\left( \lambda |X_{0}\right) \left( 1-2m_{0}\right) +(\alpha -1)p\left( \lambda |X_{0}Y_{0}\right) \left( 1-2|m_{0}-n_{0}|\right) \right. \\&+\,(\alpha -1)p\left( \lambda |X_{0}Y_{1}\right) \left( 1-2|m_{0}-n_{1}|\right) +p\left( \lambda |X_{0}Y_{0}\right) \left( 1-2|m_{0}-n_{0}|\right) \\&+\,p\left( \lambda |X_{0}Y_{1}\right) \left( 1-2|m_{0}-n_{1}|\right) +p\left( \lambda |X_{1}Y_{0}\right) \left( 1-2|m_{1}-n_{0}|\right) \\&\left. -\,p\left( \lambda |X_{1}Y_{1}\right) \left( 2|m_{1}+n_{1}-1|-1\right) \right] \\ =\,&\beta +2(\alpha -1)-2\sum _{\lambda }p(\lambda )\left[ \beta p\left( \lambda |X_{0}\right) m_{0}+(\alpha -1)p\left( \lambda |X_{0}Y_{0}\right) |m_{0}-n_{0}|\right. \\&\left. +\,(\alpha -1)p\left( \lambda |X_{0}Y_{1}\right) |m_{0}-n_{1}|\right] +4-2\sum _{\lambda }p(\lambda )\left[ p\left( \lambda |X_{0}Y_{0}\right) |m_{0}-n_{0}|\right. \\&\left. +\,p\left( \lambda |X_{0}Y_{1}\right) |m_{0}-n_{1}| +p\left( \lambda |X_{1}Y_{0}\right) |m_{1}-n_{0}| -p\left( \lambda |X_{1}Y_{1}\right) |m_{1}+n_{1}-1|\right] \\ \le \,&\beta +2\alpha +2-2\beta \min p\left( X_{0} |\lambda \right) m_{1}-4(\alpha -1)\min _{j,k\in \{0,1\}} p\left( X_{j}Y_{k}|\lambda \right) |n_{0}-n_{1}|\\&-\,8(2G-1)\min _{j,k\in \{0,1\}}p\left( X_{j}Y_{k}|\lambda \right) \\ \le \,&\beta +2\alpha +2-8(2G-1)\sum _{\lambda }p(\lambda ) \min _{j,k\in \{0,1\}}p\left( X_{j}Y_{k}|\lambda \right) ,\\ \end{aligned} \end{aligned}$$

(28)

when the conditions $m_{1}=0$ and $n_{0}=n_{1}$ can be satisfied, “=” holds.

In the following, based on results of Ref. [18], we get the values of $\min p(X_{j},Y_{k}|\lambda )$:

• (a) suppose that $h\ge \frac{1}{3}$, we always find that $\min p(X_{j},Y_{k}|\lambda )=0$.

• (b) if $\frac{1}{4}\le h \le \frac{1}{3},$ let $p(X_{j},Y_{k}|\lambda )=h$ except for $p(X_{j_{1}},Y_{k_{1}}|\lambda )$ where $(j_{1},k_{1})\ne (j,k)$, so, $\min p(X_{j},Y_{k}|\lambda )=p(X_{j_{1}},Y_{k_{1}}|\lambda )=1-3h$. Then, Eq. (28) can be given by

  $$\begin{aligned} \begin{aligned}&\widetilde{I}_{\alpha }^{\beta }={\left\{ \begin{array}{ll} \beta +2\alpha +2, &{}\quad {h\ge \frac{1}{3}},\\ \beta +2\alpha +2-8(2G-1)(1-3h),&{}\quad {\frac{1}{4}\le h\le \frac{1}{3}}. \end{array}\right. } \end{aligned} \end{aligned}$$

  (29)

Next, $I_{\alpha }^{\beta }$ can be reproduced as

$$\begin{aligned} \begin{aligned} I_{\alpha }^{\beta }=&\,\beta \langle X_{0}\rangle +\alpha \langle X_{0}Y_{0}\rangle +\alpha \langle X_{0}Y_{1}\rangle +\langle X_{1}Y_{0}\rangle -\langle X_{1}Y_{1}\rangle \\ =&\,\beta +2(\alpha -1)-2\sum _{\lambda }p(\lambda )\left[ \beta p'\left( \lambda |X_{0}\right) m_{0}+(\alpha -1)p'\left( \lambda |X_{0}Y_{0}\right) |m_{0}-n_{0}|\right. \\&\left. +\,(\alpha -1)p'\left( \lambda |X_{0}Y_{1}\right) |m_{0}-n_{1}|\right] +4-2\sum _{\lambda }p(\lambda )\left[ p'\left( \lambda |X_{0}Y_{0}\right) |m_{0}-n_{0}|\right. \\&\left. +\,p'\left( \lambda |X_{0}Y_{1}\right) |m_{0}-n_{1}|+p'\left( \lambda |X_{1}Y_{0}\right) |m_{1}-n_{0}|-p'\left( \lambda |X_{1}Y_{1}\right) |m_{1}+n_{1}-1|\right] \\ \le \,&\beta +2(\alpha -1)-2\beta \frac{\min p\left( X_{0}|\lambda \right) -2l}{1-2l}m_{1}-4(\alpha -1)\min _{(0,0),(0,1)}\frac{p'\left( X_{j}Y_{k} |\lambda \right) -l}{1-4l}\\&|n_{0}-n_{1}|+4-8(2G-1) \min _{j,k\in \{0,1\}} \frac{p'\left( X_{j}Y_{k}|\lambda \right) -l}{1-4l}\\ \le&\,\frac{1}{1-4l}\left[ \beta +2\alpha +2-2\beta \min \left( p(X_{0}|\lambda \right) m_{1}-l)-4(\alpha -1) \min _{j,k\in \{0,1\}}\left( p\left( X_{j}Y_{k}|\lambda \right) \right. \right. \\&\left. \left. -l\right) |n_{0}-n_{1}|-8(2G-1) \min _{j,k\in \{0,1\}} \left( p\left( X_{j}Y_{k}|\lambda \right) -l\right) \right] .\\ \end{aligned} \end{aligned}$$

(30)

Based on Eqs. (28), (30), we get the relation between $I_{\alpha }^{\beta }$ and $\widetilde{I}_{\alpha }^{\beta }$ in the following

$$\begin{aligned} I_{\alpha }^{\beta }=(1-4l)\widetilde{I}_{\alpha }^{\beta } +4\beta l+8(\alpha -1)l+8l. \end{aligned}$$

(31)

Thus, we give the following results:

(a)    when $\frac{h-l}{1-4l}\ge \frac{1}{3}$, that is, $3h+l\ge 1$, we get

$$\begin{aligned} \begin{aligned} I_{\alpha }^{\beta }&=(1-4l)\widetilde{I}_{\alpha }^{\beta }+4\beta l+8(\alpha -1)l+8l\\&=\beta +2\alpha +2-8l, \end{aligned} \end{aligned}$$

(32)

(a)    when $\frac{h-l}{1-4l}< \frac{1}{3}$, that is, $3h+l< 1$, we obtain

$$\begin{aligned} I_{\alpha }^{\beta }=\tilde{I}_{\alpha }^{\beta } =\beta +2\alpha +2-8(2G-1)(1-3h). \end{aligned}$$

(33)

All in all, with flexible lower bounds of measurement dependence, we get

$$\begin{aligned} \begin{aligned}&I_{\alpha }^{\beta }(G,l,h)={\left\{ \begin{array}{ll}\beta +2\alpha +2-8l, &{}\quad {3h+l\ge 1},\\ \beta +2\alpha +2-8(2G-1)(1-3h),&{}\quad {3h+l<1}. \end{array}\right. } \end{aligned} \end{aligned}$$

(34)

We complete proof of Theorem 1. $\square$

1.2 Appendix B: the factorizable input distributions

We show the deduction of Theorem 2.

Proof

The derivation process of $\widetilde{I}_{\alpha }^{\beta }$ is similar to the proof of Theorem 1. Firstly, we still discuss the case that fixes lower bound.

Based on Eq. (28), we only compute the bound of $\min p(X_{j},Y_{k}|\lambda )$ in the factorizable input distribution as follows.

• (a) Suppose that $h\ge \frac{1}{2}$, we always find that $l=\min p(X_{j},Y_{k}|\lambda )=0$.

• (b) If $\frac{1}{4}\le h \le \frac{1}{2},$ we have

  $$\begin{aligned} \begin{aligned} \min p\left( X_{j},Y_{k}|\lambda \right)&=\left( 1-\max p\left( X_{j}|\lambda \right) \right) \left( 1-\max p\left( Y_{k}|\lambda \right) \right) \\&=1-\max p\left( X_{j}|\lambda \right) -\max p\left( Y_{k}|\lambda \right) +h. \end{aligned} \end{aligned}$$

  (35)

  By computation, we get $\min p\left( X_{j},Y_{k}|\lambda \right) =\frac{1}{2}-h$.

Then, Eq. (28) in the factorizable input distributions can be given by

$$\begin{aligned} \begin{aligned}&\widetilde{I}_{\alpha }^{\beta }={\left\{ \begin{array}{ll} \beta +2\alpha +2, &{}\quad {h\ge \frac{1}{2}},\\ \beta +2\alpha +2-4(2G-1)(1-2h),&{}\quad {\frac{1}{4}\le h<\frac{1}{2}}. \end{array}\right. } \end{aligned} \end{aligned}$$

(36)

Similar to the method of Theorem 1, we consider the case of flexible lower bound. By combining Eq. (36) with Eq. (31) , we get

$$\begin{aligned} \begin{aligned}&I_{\alpha }^{\beta }={\left\{ \begin{array}{ll} \beta +2\alpha +2-8l, &{}\quad {h+l\ge \frac{1}{2}},\\ \beta +2\alpha +2-4(2G-1)(1-2h),&{}\quad { h+l<\frac{1}{2}}. \end{array}\right. } \end{aligned} \end{aligned}$$

(37)

We complete proof of Theorem 2. $\square$