General spin systems without genuinely multipartite nonlocality

Abstract

There are multipartite entangled states in many-body systems which may be potential resources in various quantum applications. In this paper, based on clustering theorems, we show that both the gapped ground state and thermal state at an upper-bounded inverse temperature have no genuine multipartite nonlocality when disjoint regions are far away from each other. The present n-particle system shows only biseparable quantum correlations when the propagation relations show exponential decay. Similar result holds for spin systems with product states as initial states. These results show interesting features of quantum many-body systems with exponential decay of correlations.

Graphic abstract

Appendices

Appendix A: Proof of Fact 1

Since $\rho$ is a state of exponentially clustering of correlations, from Lemma 1 we have

$$\begin{aligned} |\langle E_{i}E_{j}E_{k}\rangle -\langle E_{i}\rangle \langle E_{j} E_{k}\rangle |\le & {} ce^{-\kappa \tau '}|X'| \nonumber \\\le & {} ce^{-\kappa \tau }|X| \end{aligned}$$

(A1)

where $i,j,k\in \{1,\ldots ,n\}$, $|X'|=\max \{|E_{i}|,|E_{j}|,|E_{k}|\}$ and $\tau '=\min \{\tau _{ij},\tau _{ik},\tau _{jk}\}$. With this result, we show Fact 1 by induction.

The case for $k=3$ is followed from Eq. (A1). For $k=m<n$, assume that we have

$$\begin{aligned}&|\langle E_{i_{1}}\cdots E_{i_{m}}\rangle -\langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-2}}\rangle \langle E_{i_{m-1}}E_{i_{m}}\rangle | \nonumber \\&\qquad \le (m-2)ce^{-\kappa \tau } |X| \end{aligned}$$

(A2)

Now, consider the case of $k=m+1$. In fact, we get that

$$\begin{aligned} \langle E_{i_{1}}\cdots E_{i_{m+1}}\rangle\le & {} \langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-2}}\rangle \langle E_{i_{m-1}}E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&+(m-2)ce^{-\kappa \tau } |X| \nonumber \\\le & {} \langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-2}}\rangle (\langle E_{i_{m-1}}\rangle \langle E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&+ce^{-\kappa \tau }) |X| +(m-2)ce^{-\kappa \tau } |X| \nonumber \\\le & {} \langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-1}}\rangle \langle E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&+(m-1)ce^{-\kappa \tau } |X| \end{aligned}$$

(A3)

and

$$\begin{aligned} \langle E_{i_{1}}\cdots E_{i_{m+1}}\rangle\ge & {} \langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-2}}\rangle \langle E_{i_{m-1}}E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&-(m-2)ce^{-\kappa \tau } |X| \nonumber \\\ge & {} \langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-2}}\rangle (\langle E_{i_{m-1}}\rangle \langle E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&-ce^{-\kappa \tau }) |X| -(m-2)ce^{-\kappa \tau } |X| \nonumber \\\ge & {} \langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-1}}\rangle \langle E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&-(m-1)ce^{-\kappa \tau } |X| \end{aligned}$$

(A4)

Combining the inequalities (A3) and (A4), we get that

$$\begin{aligned} (1-m)|X|ce^{-\kappa \tau }\le & {} \langle E_{i_{1}}\cdot \cdot \cdot E_{i_{m+1}}\rangle \nonumber \\&-\langle E_{i_{1}}\rangle \cdot \cdot \cdot \langle E_{i_{m-1}}\rangle \langle E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\\le & {} (m-1)ce^{-\kappa \tau } |X| \end{aligned}$$

(A5)

Therefore, it implies that for $k=s$ ($s\ge 3$) we have

$$\begin{aligned}&|\langle E_{i_{1}}\cdot \cdot \cdot E_{i_{s}}\rangle -\langle E_{i_{1}}\rangle \cdot \cdot \cdot \langle E_{i_{s-2}}\rangle \langle E_{i_{s-1}}E_{i_{s}}\rangle | \nonumber \\&\qquad \le (s-2)ce^{-\kappa \tau } |X| \end{aligned}$$

(A6)

Appendix B: The proof of Fact 5

Similar to Fact 1, from Lemma 3 we have

$$\begin{aligned}&|\langle E_{i}E_{j}E_{k}\rangle -\langle E_{i}\rangle \langle E_{j} E_{k}\rangle | \nonumber \\&\qquad \le ce^{-\kappa \tau }(e^{\kappa v t}-1) |X_{i}|\,|X_{j}|\,|X_{k}| \end{aligned}$$

(B1)

where $i,j,k\in \{1,\ldots ,n\}$. With this result, we show this fact by induction.

The case of $k=3$ is followed from Eq. (B1). Assume that for $k=m<n$ we have

$$\begin{aligned}&|\langle E_{i_{1}}\cdots E_{i_{m}}\rangle -\langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-2}}\rangle \langle E_{i_{m-1}}E_{i_{m}}\rangle | \nonumber \\&\qquad \le ce^{-\kappa \tau }(e^{\kappa v t}-1)\Sigma ^{m-2}_{i=1}\prod _{j=i}^m|X_{j}| \end{aligned}$$

(B2)

Define $\hat{\alpha }=\Sigma ^{m-2}_{i=1}\prod _{j=i}^{m+1}|X_{j}|$. For $k=m+1$, we get that

$$\begin{aligned} \langle E_{i_{1}}\cdots E_{i_{m+1}}\rangle\le & {} \langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-2}}\rangle \langle E_{i_{m-1}}E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&+\hat{\alpha }ce^{-\kappa \tau }(e^{\kappa v t}-1) \nonumber \\\le & {} \langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-2}}\rangle (\langle E_{i_{m-1}}\rangle \langle E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&+ce^{-\kappa \tau }(e^{\kappa v t}-1) |X_{i_{m-1}}|\,|X_{i_{m}}|\,|X_{i_{m+1}}|) \nonumber \\&+\hat{\alpha }ce^{-\kappa \tau }(e^{\kappa v t}-1) \nonumber \\\le & {} \langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-1}}\rangle \langle E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&+ce^{-\kappa \tau }(e^{\kappa v t}-1)\Sigma ^{m-1}_{i=1}\prod _{j=i}^{m+1}|X_{j}| \end{aligned}$$

(B3)

and

$$\begin{aligned} \nonumber \langle E_{i_{1}}\cdots {}E_{i_{m+1}}\rangle\ge & {} \langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-2}}\rangle \langle E_{i_{m-1}}E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&-\hat{\alpha }ce^{-\kappa \tau }(e^{\kappa v t}-1) \nonumber \\\ge & {} \langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-2}}\rangle (\langle E_{i_{m-1}}\rangle \langle E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&-ce^{-\kappa \tau }(e^{\kappa v t}-1) |X_{i_{m-1}}|\,|X_{i_{m}}|\,|X_{i_{m+1}}|) \nonumber \\&-\hat{\alpha }ce^{-\kappa \tau }(e^{\kappa v t}-1) \nonumber \\\ge & {} \langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-1}}\rangle \langle E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&-ce^{-\kappa \tau }(e^{\kappa v t}-1)\Sigma ^{m-1}_{i=1}\prod _{j=i}^{m+1}|X_{j}| \end{aligned}$$

(B4)

From the inequalities (B3) and (B4), we get that

$$\begin{aligned}&-ce^{-\kappa \tau }(e^{\kappa v t}-1)\Sigma ^{m-1}_{i=1}\prod _{j=i}^{m+1}|X_{j}| \nonumber \\&\qquad \le \langle E_{i_{1}}\cdots {} E_{i_{m+1}}\rangle -\langle E_{i_{1}}\rangle \cdots \langle E_{i_{m-1}}\rangle \langle E_{i_{m}}E_{i_{m+1}}\rangle \nonumber \\&\qquad \le ce^{-\kappa \tau }(e^{\kappa v t}-1)\Sigma ^{m-1}_{i=1}\prod _{j=i}^{m+1}|X_{j}| \end{aligned}$$

(B5)

So, from the inequality (B5), for any $k=s\ge 3$ it follows that

$$\begin{aligned} \nonumber&|\langle E_{i_{1}}\cdots {} E_{i_{s}}\rangle -\langle E_{i_{1}}\rangle \cdots \langle E_{i_{s-2}}\rangle \langle E_{i_{s-1}}E_{i_{s}}\rangle | \nonumber \\&\qquad \le ce^{-\kappa \tau }(e^{\kappa v t}-1)\Sigma ^{s-2}_{i=1}(|X_{i}|\,|X_{i+1}|\cdot \cdot \cdot |X_{s}|)\nonumber \\&\qquad \le \alpha ce^{-\kappa \tau }(e^{\kappa v t}-1) \end{aligned}$$

(B6)