Detection of Network and Genuine Network Quantum Steering

Strong quantum correlations like quantum nonlocality in quantum networks can be established among distant parties sharing physical resources emitted by independent sources. The quantum network is important in quantum information processing such as quantum cryptography, quantum communication and distributed quantum computation. Compared with the case of the parties sharing quantum entanglement from a single common source, it is much more difficult to characterize the correlations in quantum networks because of the non-convexity of the local network spaces. In quantum network scenarios, the parties hold the physical systems from different sources which are assumed to be independent with each other. Entanglement swapping is the simplest network scenario, where Alice, Bob and Charlie share two independent sources. When Bob performs the joint measurements on his two parties, Alice and Charlie can share the entanglement [1]. In the bilocality scenario Alice and Charlie admit two independent hidden variables. It is then generalized to the $n$-locality scenarios.

Many efficient methods have been proposed to detect the quantum nonlocality in quantum networks, such as the explicit parametrization of network local models [2, 3], hierarchies of relaxations of the sets of compatible correlations [4], inflation technique [6, 5], network Bell inequalities [8, 7, 13, 9, 10, 11] and numerical approaches [15, 14].

Motivated by the quantum network nonlocality, the network quantum steering has been proposed in [16], in which the intermediate parties are untrusted while the endpoints are trusted. Under the measurement performed on the intermediate parties, quantum network steering emerges when the sub-normalized state is entangled. Inequalities have been constructed to detect the network steering in the star network scenario with the central party trusted and all the edge parties untrusted [17]. But

In this paper, we introduce the network steering and the genuine network steering from the central parties to the edge parties in star networks, from the aspect of the probability theory. Linear and nonlinear inequalities are constructed to verify the network steering and the genuine network steering for both linear and star networks when the central party performs the fixed measurement. These inequalities can detect more network steering than those of the network n-nonlocality. We demonstrate that the biseparable sub-normalized state is genuine network steerable by example.

Consider the bipartite EPR-steering scenario [18, 19]. Alice and Bob share a bipartite quantum state $\rho_{AB}$. Alice performs a set of black-box measurements $\mathbf{x}$ on $\rho_{AB}$ with outcomes $\mathbf{a},$ denoted by $M_{\mathbf{x}}^{\mathbf{a}}$. The set of sub-normalized states $\{\tau_{\mathbf{x}}^{\mathbf{a}}\}_{\mathbf{x},\mathbf{a}}$ on Bob’s side is called an assemblage. Each element in the assemblage is given by $\tau_{\mathbf{x}}^{\mathbf{a}}={\rm{Tr}}_{A}[(M_{\mathbf{x}}^{\mathbf{a}}\otimes\rm{I}_{2})\rho_{AB}],$ where $\rm{I}_{2}$ is the identity matrix. Alice can not steer Bob if $\tau_{\mathbf{x}}^{\mathbf{a}}$ admits a local hidden state (LHS) model as follows,

$\displaystyle\begin{aligned} \tau_{\mathbf{x}}^{\mathbf{a}}=\sum\limits_{\lambda}p(\lambda)p(\mathbf{a}|\mathbf{x},\lambda)\rho_{\lambda},\end{aligned}$

(1)

where $\lambda$ denotes the classical random variable distributed according to $p(\lambda)$ satisfying $\sum\limits_{\lambda}p(\lambda)=1,$ $\rho_{\lambda}$ is the hidden local state of Bob and $p(\mathbf{a}|\mathbf{x},\lambda)$ is the local response function of Alice. If there are measurements such that $\tau_{\mathbf{x}}^{\mathbf{a}}$ does not admit an LHS model, $\rho_{AB}$ is said to be steerable from Alice to Bob.

The EPR-steering of $\rho_{AB}$ from Alice to Bob can also be described by the joint probability. Bob performs measurements $\mathbf{y}$ on the assemblage $\{\tau_{\mathbf{x}}^{\mathbf{a}}\}_{\mathbf{x},\mathbf{a}}$ with outcomes $\mathbf{b},$ denoted by $M_{\mathbf{y}}^{\mathbf{b}}$. The joint probabilities are $p(\mathbf{a},\textsl{b}|\textsl{x},\textsl{y})={\rm{Tr}}[M_{\textsl{y}}^{\textsl{b}}\sigma_{\textsl{x}}^{\textsl{a}}]$. $\rho_{AB}$ is steerable from Alice to Bob if there exist measurements $M_{\textsl{x}}^{\textsl{a}}$ and $M_{\textsl{y}}^{\textsl{b}}$ such that the joint probability does dot admit the following local hidden variable-local hidden state (LHV-LHS) model,

$\displaystyle\begin{aligned} p(\textsl{a},\textsl{b}|\textsl{x},\textsl{y})=\sum\limits_{\lambda}p(\lambda)p(\textsl{a}|\textsl{x},\lambda)p_{Q}(\textsl{b}|\textsl{y},\rho_{\lambda}).\end{aligned}$

(2)

Consider the EPR-steering scenario of tripartite state $\rho_{ABC}$ shared by Alice, Bob and Charlie [20, 21, 23, 22]. Alice, Bob and Charlie perform measurements $\textsl{x},$ y and z with outcomes $\textsl{a},$ b and $\textsl{c},$ denoted by $M_{\textsl{x}}^{\textsl{a}},$ $M_{\textsl{y}}^{\textsl{b}}$ and $M_{\textsl{z}}^{\textsl{c}}$, respectively. The joint probability is given by $p(\textsl{a},\textsl{b},\textsl{c}|\textsl{x},\textsl{y},\textsl{z})=\rm{Tr}[\rho_{ABC}(M_{\textsl{x}}^{\textsl{a}}\otimes M_{\textsl{y}}^{\textsl{b}}\otimes M_{\textsl{z}}^{\textsl{c}})]$. $\rho_{ABC}$ is said to be tripartite steerable from Alice and Bob to Charlie if there are measurements $M_{\textsl{x}}^{\textsl{a}},$ $M_{\textsl{y}}^{\textsl{b}}$ and $M_{\textsl{z}}^{\textsl{c}}$ such that the joint probability $p(\textsl{a},\textsl{b},\textsl{c}|\textsl{x},\textsl{y},\textsl{z})$ does not satisfy the fully local hidden variable and local hidden state model as follows,

$\displaystyle\begin{aligned} &p(\textsl{a},\textsl{b},\textsl{c}|\textsl{x},\textsl{y},\textsl{z})\\ =&\sum\limits_{\lambda}p(\lambda)p(\textsl{a}|\textsl{x},\lambda)p(\textsl{b}|\textsl{y},\lambda)p_{Q}(\textsl{c}|\textsl{z},\rho^{C}_{\lambda}).\end{aligned}$

(3)

The state $\rho_{ABC}$ is tripartite steerable from Alice to Bob and Charlie if there exist measurements $M_{\textsl{x}}^{\textsl{a}},$ $M_{\textsl{y}}^{\textsl{b}}$ and $M_{\textsl{z}}^{\textsl{c}}$ such that the joint probability $p(\textsl{a},\textsl{b},\textsl{c}|\textsl{x},\textsl{y},\textsl{z})$ does not satisfy the local hidden variable and fully local hidden state model as follows,

$\displaystyle\begin{aligned} &p(\textsl{a},\textsl{b},\textsl{c}|\textsl{x},\textsl{y},\textsl{z})\\ =&\sum\limits_{\lambda}p(\lambda)p(\textsl{a}|\textsl{x},\lambda)p_{Q}(\textsl{b}|\textsl{y},\rho^{B}_{\lambda})p_{Q}(\textsl{c}|\textsl{z},\rho^{C}_{\lambda}).\end{aligned}$

(4)

where $\rho_{\lambda}^{B}$ and $\rho_{\lambda}^{C}$ are the local hidden states of Bob and Charlie, respectively.

The other tripartite steering of $\rho_{ABC}$ from Alice to Bob and Charlie is defined as in [23], for which there exist measurements $M_{\textsl{x}}^{\textsl{a}},$ $M_{\textsl{y}}^{\textsl{b}}$ and $M_{\textsl{z}}^{\textsl{c}}$ such that the joint probability $p(\textsl{a},\textsl{b},\textsl{c}|\textsl{x},\textsl{y},\textsl{z})$ does not satisfy the local hidden variable and bipartite local hidden state model as follows,

$\displaystyle\begin{aligned} &p(\textsl{a},\textsl{b},\textsl{c}|\textsl{x},\textsl{y},\textsl{z})\\ =&\sum\limits_{\lambda}p(\lambda)p(\textsl{a}|\textsl{x},\lambda)p_{Q}(\textsl{b},\textsl{c}|\textsl{y},\textsl{z},\rho^{BC}_{\lambda}),\end{aligned}$

(5)

where $\rho_{\lambda}^{BC}$ is the local hidden state of Bob and Charlie. In both scenarios given by Eq.(4) and Eq.(5), a source sends a classical message $\lambda$ to Alice with the probability $p(\lambda)$. The difference between (4) and (5) is that the corresponding local quantum states sent to Bob and Charlie are separable and entangled respectively.

Consider the networks that the $n$ parties are arranged in the star networks. For the simplest scenario which has three parties and two sources, the first two parties Alice and Bob₁ share the state $\rho_{AB_{1}}$ and the last two parties Bob₂ and Charlie share the state $\rho_{B_{2}C}.$ The intermediate party Bob, Bob₁ and Bob₂, performs the fixed measurement y (without the input) with outcomes b,denoted as $M_{\textsl{y}}^{\textsl{b}}$, The sub-normalized state under the measurement is

$$\sigma_{\textsl{b}}^{AC}={\rm{Tr}}_{B}[(\rm{I}_{\it{A}}\otimes M^{\textsl{b}}_{\textsl{y}}\otimes\rm{I}_{\textit{C}})(\rho_{\textit{AB}_{1}}\otimes\rho_{\textit{B}_{2}\textit{C}})].$$

(6)

$\sigma_{b}^{AC}$ admits a network local hidden state model (NLHS) if $\sigma_{b}^{AC}$ satisfies the following condition,

$\displaystyle\begin{aligned} \sigma_{\textsl{b}}^{AC}=\sum\limits_{\lambda_{1},\lambda_{2}}p(\lambda_{1})p(\lambda_{2})p(\textsl{b}|\lambda_{1},\lambda_{2})\rho_{\lambda_{1}}^{A}\otimes\rho_{\lambda_{2}}^{C}.\end{aligned}$

(7)

The network state $\rho_{ABC}=\rho_{AB_{1}}\otimes\rho_{B_{2}C}$ demonstrates the network steering from the central party to the endpoint parties if there exists a fixed measurement $M_{\textsl{y}}^{\textsl{b}}$ such that $\sigma_{\textsl{b}}^{AC}$ does not admit NLHS [16].

The entanglement of $\sigma_{\textsl{b}}^{AC}$ can rule out the NLHS model from Bob to Alice and Charlie, but entanglement detection is a difficult problem, especially for high dimensional quantum states. In [16], the authors only investigated the network steering scenarios with respect to some special states such as separable and unsteerable ones. To investigate the network steering for general quantum states, we define the network steering from the aspect of joint probabilities. We derive inequalities to detect if the sub-normalized states under the measurement performed by the central party admit the NLHS model, thus detecting the network steering.

Bob performs a fixed measurement y with outcome b. The endpoint parties Alice and Charlie perform the measurements x and z on the sub-normalized state $\sigma_{\textsl{b}}^{AC}$ with outcomes a and c. The joint probability $p(\textsl{a},\textsl{b},\textsl{c}|\textsl{x},\textsl{z})={\rm{Tr}}[(M_{\textsl{x}}^{\textsl{a}}\otimes M_{\textsl{y}}^{\textsl{b}})\sigma_{\textsl{b}}^{\textit{AC}}]$ admits a network local hidden variable and local hidden state model (NLHV-LHS) from the central party Bob to the endpoint parties Alice and Charlie if the joint probability satisfies the following condition,

$\displaystyle\begin{aligned} &p(\textsl{a},\textsl{b},\textsl{c}|\textsl{x},\textsl{z})\\ =&\sum\limits_{\lambda_{1},\lambda_{2}}p(\lambda_{1})p(\lambda_{2})p_{Q}(\textsl{a}|\textsl{x},\rho_{\lambda_{1}}^{A})\\ &\times p(\textsl{b}|\lambda_{1},\lambda_{2})p_{Q}(\textsl{c}|\textsl{z},\rho_{\lambda_{2}}^{C}),\end{aligned}$

(8)

where $p_{Q}(\textsl{a}|\textsl{x},\rho_{\lambda_{1}}^{A})$ ($p_{Q}(\textsl{c}|\textsl{z},\rho_{\lambda_{2}}^{C})$) is the probability generated from Alice’s (Charlie’s) system, $p(\textsl{b}|\lambda_{1},\lambda_{2})$ is the probability from Bob’s system $B_{1}B_{2}$.

Consider the network composed of three parties Alice, Bob and Charlie and two resources $\lambda_{1}$ and $\lambda_{2}$. Alice and Charlie perform the mutually unbiased measurements $x^{i}$ with outcomes $a$ and $z^{i}$ with outcomes $c$ $(i=1,2,\,a,c=0,1)$ and Bob performs the fixed measurement $y=\{G_{00},G_{01},G_{10},G_{11}\}$ with four possible outcomes $b\equiv\{b_{1},b_{2}\}\in\{0,1\}.$ Denote $y_{1}=G_{00}-G_{11}-(G_{01}-G_{10}),$ $y_{2}=G_{00}-G_{11}+G_{01}-G_{10}$ and $y_{3}=G_{00}+G_{11}-G_{01}-G_{10}$. Then

$\displaystyle\begin{aligned} &\langle x^{i}\otimes y^{j}\otimes z^{k}\rangle=\sum\limits_{a,b,c}(-1)^{a+t\cdot b+c}p(a,b,c|x^{i},z^{k}),\\ \end{aligned}$

with $\{b_{1},b_{2}\}$ the string of 2 bits representing $b=0,1,2$ and $3,$ $t$ the string of 2 bits representing $j=1,2,3.$ We have the following Theorem, see proof in Appendix.

Theorem 1. For line network, the probabilities that admit the NLHS-LHV model from Bob to Alice and Charlie satisfy the following inequalities,

$\displaystyle\sum_{i=1}^{3}|\langle x^{i}\otimes y^{i}\otimes z^{i}\rangle|\leq 1.$

(9)

For the case of star networks, let us consider a star network composed of a central party $B=B_{1}\cdots B_{n}$ and $n$ edge parties $A_{i}$ shared by Bob and Alice_i, $i=1,\cdots,n$. The central party is separately connected to the $n$ edge parties. The edge parties perform the measurements $\bigotimes\limits_{k=1}^{n}M_{\textsl{x}_{k}}^{\textsl{a}_{k}}$ and the central party performs the fixed measurement $M_{\textsl{y}}^{\textsl{b}}$. The quantum state $\rho_{AB}=\bigotimes\limits_{k=1}^{n}\rho_{A_{k}B_{k}}$ admits an NLHV-LHS model from the central party to the edge parties if the joint probability satisfies

$\displaystyle\begin{aligned} &p(\textsl{a}_{1},\cdots,\textsl{a}_{n},\textsl{b}|\textsl{x}_{1},\cdots,\textsl{x}_{n})=\rm{Tr}[(\bigotimes\limits_{k=1}^{n}M_{\textsl{x}_{\it{k}}}^{\textsl{a}_{\it{k}}}\otimes M_{\textsl{y}}^{\textsl{b}})\bigotimes\limits_{k=1}^{n}\rho_{\it{A}_{\it{k}}\it{B}_{\it{k}}}]\\ &=\sum\limits_{\lambda_{k}}p(\lambda_{1})\cdots p(\lambda_{n})p_{Q}(\textsl{a}_{1}|\textsl{x}_{1},\rho_{\lambda_{1}}^{A_{1}})...p_{Q}(\textsl{a}_{\textsl{n}}|\textsl{x}_{n},\rho_{\lambda_{n}}^{A_{n}})\\ &\times p(\textsl{b}|\lambda_{1},\cdots,\lambda_{n})\end{aligned}$

with $\textsl{b}=\{\textsl{b}_{1}\cdots\textsl{b}_{n}\}$ and $p_{Q}(\textsl{a}_{k}|\textsl{x}_{k},\lambda_{k})=\rm{Tr}[M_{\textsl{x}_{\it{k}}}^{\textsl{a}_{\it{k}}}\rho_{\lambda_{k}}^{\it{A}_{\it{k}}}].$

Let the edge parties perform the mutually unbiased measurements $x_{k}^{i_{k}}(k=1,\cdots,n)$ and the fixed measurement performed by Bob is given by $\textsl{y}=\{G_{0\cdots 0},G_{0\cdots 01},\cdots,G_{1\cdots 1}\}$ with $2^{n}$ possible outcomes $\textsl{b}\equiv\{b_{1},\cdots,b_{n}\}\in\{0,1\}^{n}$. Set

$$y^{i_{1}\cdots i_{n}}=\left\{\begin{aligned} &\sum\limits_{t_{1}=0,t_{2}\cdots t_{n}}(-1)^{I\cdot T}(G_{t_{1}\cdots t_{n}}-G_{\bar{t}_{1}\cdots\bar{t}_{n}}),\\ &\hskip 142.26378pti_{1}\cdots i_{n}\in C\\ &\sum\limits_{t_{1}=0,t_{2}\cdots t_{n}}(-1)^{I_{0}\cdot T}(G_{t_{1}\cdots t_{n}}+G_{\bar{t}_{1}\cdots\bar{t}_{n}}),\\ &\hskip 142.26378pti_{1}\cdots i_{n}\in C^{\prime}\end{aligned}\right.$$

(10)

where $I=\{i_{1}-1,\cdots,i_{n}-1\},$ $T=\{t_{1},\cdots,t_{n}\}$, $``\cdot"$ represents the inner product, $I_{0}=\{i_{1},\cdots,i_{n}\},$ $y_{1}^{1}=y^{1\cdots 1},$ $y_{1}^{2}=y^{2\cdots 2}$ and $y_{1}^{3}=y^{3\cdots 3},$ $C=\{111\cdots 1,11\cdots 122,\cdots,221\cdots 1,\cdots 22221\cdots 1,\cdots\}$ (each string has either zero or even number of 2) and $C^{\prime}=\{330\cdots 0,00\cdots 033,\cdots,330\cdots 0,\cdots 33330\cdots 0,\cdots\}$ (each string has even number of 3). Then we have

$\displaystyle\begin{aligned} &\langle x^{i_{1}}_{1}\otimes x^{i_{2}}_{2}\otimes\cdots\otimes x^{i_{n}}_{n}\otimes y^{i_{1}\cdots i_{n}}\rangle\\ =&\sum\limits_{a_{k}^{i_{k}},\textsl{b}}(-1)^{\sum\limits_{k}a_{k}^{i_{k}}+\Re}p(a_{1}^{i_{1}},\cdots,a_{n}^{i_{n}},\textsl{b}|x_{1}^{i_{1}},\cdots,x_{n}^{i_{n}}),\\ \end{aligned}$

where $\Re=I\cdot\Delta+b_{1}$ when $i_{1}\cdots i_{n}\in C,$ and $\Re=I_{0}\cdot\Delta$ when $i_{1}\cdots i_{n}\in C^{\prime},$ $\Delta=\{b_{1},\cdots,b_{n}\}$ the string of 2 bits representing $b=0,1,\cdots,2^{n}-1$ when $b_{1}=0$ and $\Delta=\{\bar{b}_{1},\cdots,\bar{b}_{n}\}$ when $b_{1}=1.$

We have the following conclusions, see proof in Appendix.

Theorem 2. (a) If $\rho_{AB}=\bigotimes\limits_{k=1}^{n}\rho_{A_{k}B_{k}}$ admits NLHV-LHS from the central party to the edge parties when Bob performs the fixed measurement in the two measurement settings, we have
(1) When $n$ is an odd number,

$\displaystyle\begin{aligned} &\sum\limits_{i_{1},\cdots,i_{n}\in C}|\langle x^{i_{1}}_{1}\otimes x^{i_{2}}_{2}\otimes\cdots\otimes x^{i_{n}}_{n}\otimes y^{i_{1}\cdots i_{n}}\rangle|^{\frac{2}{n}}\\ &\leq 2^{n-2}\end{aligned}$

(11)

and

$\displaystyle\begin{aligned} &\sum\limits_{i_{1},\cdots,i_{n}\in C}|\langle x^{i_{1}}_{1}\otimes x^{i_{2}}_{2}\otimes\cdots\otimes x^{i_{n}}_{n}\otimes y^{i_{1}\cdots i_{n}}\rangle|^{\frac{1}{n}}\\ &\leq 2^{n-2}\sqrt{2},\end{aligned}$

(12)

(2) When $n$ is an even number,

$\displaystyle\begin{aligned} &|\langle x^{1}_{1}\otimes x^{1}_{2}\otimes\cdots\otimes x^{1}_{n}\otimes y_{1}^{1}\rangle|^{\frac{2}{n}}\\ +&|\langle x^{2}_{1}\otimes x^{2}_{2}\otimes\cdots\otimes x^{2}_{n}\otimes y_{1}^{2}\rangle|^{\frac{2}{n}}\leq 1.\end{aligned}$

(13)

and

$\displaystyle\begin{aligned} &|\langle x^{1}_{1}\otimes x^{1}_{2}\otimes\cdots\otimes x^{1}_{n}\otimes y_{1}^{1}\rangle|^{\frac{1}{n}}\\ +&|\langle x^{2}_{1}\otimes x^{2}_{2}\otimes\cdots\otimes x^{2}_{n}\otimes y_{1}^{2}\rangle|^{\frac{1}{n}}\leq\sqrt{2}.\end{aligned}$

(14)

(b) If $\rho_{AB}=\bigotimes\limits_{k=1}^{n}\rho_{A_{k}B_{k}}$ admits NLHV-LHS from the central party to the edge parties when Bob performs the fixed measurement in three measurement settings, we have
(1) When $n$ is an odd number,

$\displaystyle\begin{aligned} &\sum\limits_{i_{1},\cdots,i_{n}\in C\cup C^{\prime}}|\langle x^{i_{1}}_{1}\otimes x^{i_{2}}_{2}\otimes\cdots\otimes x^{i_{n}}_{n}\otimes y^{i_{1}\cdots i_{n}}\rangle|^{\frac{2}{n}}\\ &\leq 2^{n-2}+2^{n-2}-1\end{aligned}$

(15)

and

$\displaystyle\begin{aligned} &\sum\limits_{i_{1},\cdots,i_{n}\in C\cup C^{\prime}}|\langle x^{i_{1}}_{1}\otimes x^{i_{2}}_{2}\otimes\cdots\otimes x^{i_{n}}_{n}\otimes y^{i_{1}\cdots i_{n}}\rangle|^{\frac{1}{n}}\\ &\leq 2^{n-2}\sqrt{3}+2^{n-2}-1.\end{aligned}$

(16)

(2) When $n$ is an even number,

$\displaystyle\begin{aligned} &|\langle x^{1}_{1}\otimes x^{1}_{2}\otimes\cdots\otimes x^{1}_{n}\otimes y_{1}^{1}\rangle|^{\frac{2}{n}}\\ +&|\langle x^{2}_{1}\otimes x^{2}_{2}\otimes\cdots\otimes x^{2}_{n}\otimes y_{1}^{2}\rangle|^{\frac{2}{n}}\\ +&|\langle x^{3}_{1}\otimes x^{3}_{2}\otimes\cdots\otimes x^{3}_{n}\otimes y_{1}^{3}\rangle|^{\frac{2}{n}}\leq 1\end{aligned}$

(17)

and

$\displaystyle\begin{aligned} &|\langle x^{1}_{1}\otimes x^{1}_{2}\otimes\cdots\otimes x^{1}_{n}\otimes y_{1}^{1}\rangle|^{\frac{1}{n}}\\ +&|\langle x^{2}_{1}\otimes x^{2}_{2}\otimes\cdots\otimes x^{2}_{n}\otimes y_{1}^{2}\rangle|^{\frac{1}{n}}\\ +&|\langle x^{3}_{1}\otimes x^{3}_{2}\otimes\cdots\otimes x^{3}_{n}\otimes y_{1}^{3}\rangle|^{\frac{1}{n}}\leq\sqrt{3},\end{aligned}$

(18)

where $x_{k}^{i_{k}}$ $(i_{k}=1,2,3)$ are all mutually unbiased measurements for $k=1,\cdots,n,$ $x_{1}^{0}=x_{2}^{0}=\cdots x_{n}^{0}=\rm{I}_{2}$ are the identity matrices.

In the above studies we have concerned the sub-normalized state under one fixed measurement performed by the central party. When the central party performs four measurements $B_{i}$ $(i=1,\cdots,4)$, and the edge parties performs three settings of measurements, we have the following, see proof in Appendix.

Theorem 3. The star network state $\rho_{A_{1}\cdots A_{n}B}$ admits NLHS model from the central party to the edge parties if

$\displaystyle\begin{aligned} |J_{1}|^{\frac{1}{n}}+|J_{2}|^{\frac{1}{n}}+|J_{3}|^{\frac{1}{n}}+|J_{4}|^{\frac{1}{n}}\leq 4,\end{aligned}$

(19)

where

$\displaystyle\begin{aligned} &J_{1}=\langle\bigotimes\limits_{i=1}^{n}(x_{i}^{1}+x_{i}^{2}+x_{i}^{3})\otimes B_{1}\rangle,\\ &J_{2}=\langle\bigotimes\limits_{i=1}^{n}(x_{i}^{1}+x_{i}^{2}-x_{i}^{3})\otimes B_{2}\rangle,\\ &J_{3}=\langle\bigotimes\limits_{i=1}^{n}(x_{i}^{1}-x_{i}^{2}+x_{i}^{3})\otimes B_{3}\rangle,\\ &J_{4}=\langle\bigotimes\limits_{i=1}^{n}(-x_{i}^{1}+x_{i}^{2}+x_{i}^{3})\otimes B_{4}\rangle\\ \end{aligned}$

and $x_{i}^{j}$ $(i=1,\cdots,n,j=1,2,3)$ are all mutually unbiased measurements.

As an example let us consider the star network $\rho_{A_{1}\cdots A_{n}B}=\bigotimes\limits_{i=1}^{n}\rho_{A_{k}B_{k}}$, where $\rho_{A_{k}B_{k}}=\frac{1-p}{4}\rm{I}_{4}+p|\phi\rangle\langle\phi|$ with $|\phi\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle).$ Let $B_{1}=\bigotimes\limits_{i=1}^{n}(x_{i}^{1}+x_{i}^{2}+x_{i}^{3}),$ $B_{2}=\bigotimes\limits_{i=1}^{n}(x_{i}^{1}+x_{i}^{2}-x_{i}^{3}),$ $B_{3}=\bigotimes\limits_{i=1}^{n}(x_{i}^{1}-x_{i}^{2}+x_{i}^{3})$ and $B_{4}=-x_{i}^{1}+x_{i}^{2}+x_{i}^{3}.$ From the theorem 3, we have that $\rho_{A_{1}\cdots A_{n}B}$ demonstrates the network steering when $p>\frac{\sqrt{3}}{3}.$ However, it has been shown that the state demonstrates the network nonlocality for $p>\frac{\sqrt{3}}{2}$ when the edge parties perform three settings of measurements [12].

In the following examples, we set $x_{k}^{i_{k}}$ to be pauli matrices, $G_{t_{1}\cdots t_{n}}=|\psi^{+}_{t_{1}\cdots t_{n}}\rangle\langle\psi^{+}_{t_{1}\cdots t_{n}}|$ and $G_{\bar{t}_{1}\cdots\bar{t}_{n}}=|\psi^{-}_{t_{1}\cdots t_{n}}\rangle\langle\psi^{-}_{t_{1}\cdots t_{n}}|$, where

$$|\psi^{+}_{t_{1}\cdots t_{n}}\rangle=\frac{1}{\sqrt{2}}(|t_{1}\cdots t_{n}\rangle+|\bar{t}_{1}\cdots\bar{t}_{n}\rangle),$$

$$|\psi^{-}_{t_{1}\cdots t_{n}}\rangle=\frac{1}{\sqrt{2}}(|t_{1}\cdots t_{n}\rangle-|\bar{t}_{1}\cdots\bar{t}_{n}\rangle)$$

with $\bar{t}$ representing the bit flip of $t,$ $t_{1}=0$ and $t_{i}\in\{0,1\}$ for $i\geq 2$. For the special case of $n=2,$ we have $G_{00}=|\psi_{00}^{+}\rangle\langle\psi_{00}^{+}|,$ $G_{11}=|\psi_{00}^{-}\rangle\langle\psi_{00}^{-}|,$ $G_{01}=|\psi_{01}^{+}\rangle\langle\psi_{01}^{+}|$ and $G_{10}=|\psi_{01}^{-}\rangle\langle\psi_{01}^{-}|,$ where $|\psi_{00}^{\pm}\rangle=\frac{1}{\sqrt{2}}(|00\rangle\pm|11\rangle)$ and $|\psi_{01}^{\pm}\rangle=\frac{1}{\sqrt{2}}(|01\rangle\pm|10\rangle).$

As another example let us consider the star network $\rho_{A_{1}\cdots A_{n}B}=\rho_{A_{1}B_{1}}\otimes\cdots\otimes\rho_{A_{n}B_{n}}$ with $\rho_{A_{i}B_{i}}=\frac{1}{4}(\rm{I}_{4}+c_{1}\sigma_{1}\otimes\sigma_{1}+c_{2}\sigma_{2}\otimes\sigma_{2}+c_{3}\sigma_{3}\otimes\sigma_{3})$ and $B=B_{1}B_{2}\cdots B_{n}$. From (LABEL:2-set-neven-1), $\rho_{A_{1}\cdots A_{n}B}$ is steerable from the central party to the edge parties when $\max\{c_{1}^{2}+c_{2}^{2},c_{1}^{2}+c_{3}^{2},c_{2}^{2}+c_{3}^{1}\}>1$ in two measurement settings (which is the same as the result in [26]), and from (LABEL:3-set-neven-1) $\rho_{A_{1}\cdots A_{n}B}$ is steerable when $c_{1}^{2}+c_{2}^{2}+c_{3}^{2}>1$ in three measurement settings if $n$ is even. When $n$ is odd, from (LABEL:2-set-nodd-1) and (LABEL:2-set-nodd-2) $\rho_{A_{1}\cdots A_{n}B}$ is steerable from the central party to the edge parties when

$$\sum\limits_{i_{1}\cdots i_{n}\in C}\sqrt[n]{(c_{i_{1}}\cdots c_{i_{n}})^{2}}>2^{n-2}$$

or

$$\sum\limits_{i_{1}\cdots i_{n}\in C}\sqrt[n]{|c_{i_{1}}\cdots c_{i_{n}}|}>2^{n-2}\sqrt{2}$$

in two measurement settings. From (LABEL:3-set-nodd-1) and (LABEL:3-set-nodd-2), $\rho_{A_{1}\cdots A_{n}B}$ is steerable when

$$\sum\limits_{i_{1}\cdots i_{n}\in C\bigcup C^{\prime}}\sqrt[n]{(c_{i_{1}}\cdots c_{i_{n}})^{2}}>2^{n-1}-1$$

or

$$\sum\limits_{i_{1}\cdots i_{n}\in C\bigcup C^{\prime}}\sqrt[n]{|c_{i_{1}}\cdots c_{i_{n}}|}>2^{n-2}\sqrt{3}+2^{n-2}-1$$

in three measurement settings with $c_{0}=1$.

In particular, when $|c_{1}|=|c_{2}|=|c_{3}|=p,$ we have that $\rho_{A_{1}\cdots A_{n}B}$ is steerable for $p>\frac{1}{\sqrt{2}}$ in two measurement settings, and for $p>\frac{1}{\sqrt{3}}$ (even $n$) or $p>p_{0}$ (odd $n$) in three measurement settings, where $p_{0}$ is the solutions of $2^{n-1}p+C_{n}^{n-1}p^{\frac{n-1}{n}}+\cdots+C_{n}^{2}p^{\frac{2}{n}}=2^{n-2}\sqrt{3}+2^{n-2}-1$ ($p_{0}=0.589$ for $n=3$). Nevertheless, it has been shown in [12] that $\rho_{A_{1}\cdots A_{n}B}$ is not of n-locality when $p>\frac{\sqrt{3}}{2}$ in three measurement settings and when $p>0.741$ in four measurement settings.

The general two-qubit quantum state $\rho_{A_{i}B_{i}}=\frac{1}{4}(\rm{I}_{4}+\sum\limits_{j}(a_{j}\sigma_{j}\otimes\rm{I}_{2}+b_{j}\rm{I}_{2}\otimes\sigma_{j})+\sum\limits_{kl}c_{kl}\sigma_{k}\otimes\sigma_{l})$ is locally unitary equivalent to the state $\rho^{\prime}_{A_{i}B_{i}}=\frac{1}{4}(\rm{I}_{4}+\sum\limits_{j}(a^{\prime}_{j}\sigma^{\prime}_{j}\otimes\rm{I}_{2}+b^{\prime}_{j}\rm{I}_{2}\otimes\sigma^{\prime}_{j})+\sum\limits_{k}d_{k}\sigma^{\prime}_{k}\otimes\sigma^{\prime}_{k})$, where $C=[c_{kl}]=U.D.V^{*}$ is the singular value decomposition of $C$ and $D=\rm{Diag}\{d_{1},d_{2},d_{3}\}$. Then $\rho_{A_{1}\cdots A_{n}B}$ is steerable from the central party to the edge parties when $\max\{d^{2}_{1}+d^{2}_{2},d_{1}^{2}+d_{3}^{2},d_{2}^{2}+d_{3}^{2}\}>1$ in two measurement settings, and when $d_{1}^{2}+d_{2}^{2}+d_{3}^{2}=\rm{Tr}[CC^{T}]>1$ in three measurement settings for even $n$. For odd $n$ $\rho_{A_{1}\cdots A_{n}B}$ is steerable from the central party to the edge parties when

$$\sum\limits_{i_{1}\cdots i_{n}\in C}\sqrt[n]{|d_{i_{1}}\cdots d_{i_{n}}|^{2}}>2^{n-2}$$

or

$$\sum\limits_{i_{1}\cdots i_{n}\in C}\sqrt[n]{|d_{i_{1}}\cdots d_{i_{n}}|}>2^{n-2}\sqrt{2}$$

in two measurement settings, and when

$$\sum\limits_{i_{1}\cdots i_{n}\in C\bigcup C^{\prime}}\sqrt[n]{(d_{i_{1}}\cdots d_{i_{n}})^{2}}>2^{n-1}-1$$

or

$$\sum\limits_{i_{1}\cdots i_{n}\in C\bigcup C^{\prime}}\sqrt[n]{|d_{i_{1}}\cdots d_{i_{n}}|}>2^{n-2}\sqrt{3}+2^{n-2}-1$$

in three measurement settings.

The inequalities in Theorem 1 and Theorem 2 can be used to detect the network steering when the central party performs a single unknown measurement in the star network scenarios composed of bell-diagonal states and general two-qubit states. Compared with the violation of $n$-locality, the violations of our inequalities detect more network steering for the star network scenarios composed of Werner states.

Consider the star network with state $\rho_{\it{A_{1}A_{2}A_{3}B}}$, where Bob performs one fixed joint measurement $M_{\textsl{y}}^{\textsl{b}}$. Entanglement can be generated among the edge parties when the central party performs the joint measurement. Here we consider if the network assemblage $\{\sigma_{b}^{\it{A_{1}A_{2}A_{3}}}\}$ can be composed of the bi-separable local hidden states with $\sigma_{\textsl{b}}^{\it{A_{1}A_{2}A_{3}}}=\rm{Tr}_{B}[(\rm{I}_{8}\otimes M_{\textsl{y}}^{\textsl{b}})\rho_{\it{A_{1}A_{2}A_{3}B}}]$. The three sources send three classical messages $\lambda_{i}$ $(1\leq i\leq 3)$ to the central party $B$. One source generates a quantum state $\rho_{\lambda_{3}}^{\it{A_{3}}}$ ($\rho_{\lambda_{1}}^{\it{A_{1}}}$ or $\rho_{\lambda_{2}}^{\it{A_{2}}}$) with probability $p(\lambda_{3})$ ($p(\lambda_{1})$ or $p(\lambda_{2})$). The other two sources generate entangled states $\rho_{\lambda_{1}\lambda_{2}}^{\it{A_{1}A_{2}}},$ $\rho_{\lambda_{2}\lambda_{3}}^{\it{A_{2}A_{3}}}$ or $\rho_{\lambda_{1}\lambda_{3}}^{\it{A_{1}A_{3}}}$ with the probabilities $p(\lambda_{1},\lambda_{2}),$ $p(\lambda_{2},\lambda_{3})$ and $p(\lambda_{1},\lambda_{3})$ randomly, which are sent to the edge parties. The total probability that $\it{A_{1}A_{2}}$ ($\it{A_{2}A_{3}}$, $\it{A_{1}A_{3}}$) receive the entangled local hidden states is $q_{1}$ ($q_{2}$, $q_{3}$).

Then the sub-normalized state $\sigma_{\textsl{b}}^{\it{A_{1}A_{2}A_{3}}}$ admits bi-separable local hidden states (BLHS) if

$\displaystyle\begin{aligned} &\sigma_{\textsl{b}}^{\it{A_{1}A_{2}A_{3}}}\\ =&q_{1}\sum\limits_{\lambda_{1},\lambda_{2},\lambda_{3}}p(\lambda_{1},\lambda_{2})p(\lambda_{3})p(\textsl{b}|\lambda_{1},\lambda_{2},\lambda_{3})\rho_{\lambda_{1}\lambda_{2}}^{\it{A_{1}A_{2}}}\otimes\rho_{\lambda_{3}}^{\it{A_{3}}}\\ +&q_{2}\sum\limits_{\lambda_{1},\lambda_{2},\lambda_{3}}p(\lambda_{1})p(\lambda_{2},\lambda_{3})p(\textsl{b}|\lambda_{1},\lambda_{2},\lambda_{3})\rho_{\lambda_{2}\lambda_{3}}^{\it{A_{2}A_{3}}}\otimes\rho_{\lambda_{1}}^{\it{A_{1}}}\\ +&q_{3}\sum\limits_{\lambda_{1},\lambda_{2},\lambda_{3}}p(\lambda_{2})p(\lambda_{1},\lambda_{3})p(\textsl{b}|\lambda_{1},\lambda_{2},\lambda_{3})\rho_{\lambda_{1}\lambda_{3}}^{\it{A_{1}A_{3}}}\otimes\rho_{\lambda_{2}}^{\it{A_{2}}}\\ \end{aligned}$

with $\sum\limits_{i}q_{i}=1.$ The sketch of bi-separable local hidden states model of $\sigma_{\textsl{b}}^{\it{A_{1}A_{2}A_{3}}}$ is shown in Fig. (1). From the aspect of probabilities, we have that $\rho_{A_{1}A_{2}A_{3}B}$ admits network local hidden variable and bi-separable local hidden states (NLHV-BLHS) model if

$\displaystyle\begin{aligned} &p(\textsl{a}_{1},\textsl{a}_{2},\textsl{a}_{3},\textsl{b}|\textsl{x}_{1},\textsl{x}_{2},\textsl{x}_{3})\\ =&q_{1}\sum\limits_{\lambda_{1},\lambda_{2},\lambda_{3}}p(\textsl{b}|\lambda_{1},\lambda_{2},\lambda_{3})p(\lambda_{1},\lambda_{2})p(\lambda_{3})\\ &\times p_{Q}(\textsl{a}_{1},\textsl{a}_{2}|\textsl{x}_{1},\textsl{x}_{2},\rho_{\lambda_{1}\lambda_{2}}^{\it{A_{1}A_{2}}})p_{Q}(\textsl{a}_{3}|\textsl{x}_{3},\rho_{\lambda_{3}}^{\it{A_{3}}})\\ +&q_{2}\sum\limits_{\lambda_{1},\lambda_{2},\lambda_{3}}p(\textsl{b}|\lambda_{1},\lambda_{2},\lambda_{3})p(\lambda_{2},\lambda_{3})p(\lambda_{1})\\ &\times p_{Q}({\textsl{a}}_{2},\textsl{a}_{3}|\textsl{x}_{2},\textsl{x}_{3},\rho_{\lambda_{2}\lambda_{3}}^{\it{A_{2}A_{3}}})p_{Q}(\textsl{a}_{1}|\textsl{x}_{1},\rho_{\lambda_{1}}^{\it{A_{1}}})\\ +&q_{3}\sum\limits_{\lambda_{1},\lambda_{2},\lambda_{3}}p(\textsl{b}|\lambda_{1},\lambda_{2},\lambda_{3})p(\lambda_{1},\lambda_{3})p(\lambda_{2})\\ &\times p_{Q}(\textsl{a}_{1},\textsl{a}_{3}|\textsl{x}_{1},\textsl{x}_{3},\rho_{\lambda_{1}\lambda_{3}}^{\it{A_{1}A_{3}}})p_{Q}(\textsl{a}_{2}|\textsl{x}_{2},\rho_{\lambda_{2}}^{\it{A_{2}}}),\end{aligned}$

(20)

Generally we have star networks with state $\rho_{A_{1}\cdots A_{n}B}=\bigotimes\limits_{i=1}^{n}\rho_{A_{i}B_{i}}$. Bob performs one fixed measurement y defined in Theorem 2. It admits bi-separable local hidden states (BLHS) model if

$\displaystyle\begin{aligned} &\sigma_{\textsl{b}}^{A_{1}\cdots A_{n}}\\ =&\sum\limits_{\lambda_{1}\cdots\lambda_{n}}p(\textsl{b}|\lambda_{1},\cdots,\lambda_{n})\\ &\times\sum\limits_{s=1}^{\lfloor\frac{n}{2}\rfloor}\sum\limits_{t=1}^{C_{n}^{s}}q^{t}_{s}p(\Lambda^{t}_{s})\rho^{\Gamma^{t}_{s}}_{\Lambda^{t}_{s}}\otimes p(\tilde{\Lambda}^{t}_{s})\rho^{\tilde{\Gamma}^{t}_{s}}_{\tilde{\Lambda}^{t}_{s}}\end{aligned}$

(21)

where $\sum\limits_{s,t}q_{s}^{t}=1,$ and $\Lambda^{t}_{s}$ and $\tilde{\Lambda}^{t}_{s}$ are two disjoint subsets of the set $\{\lambda_{1},\cdots,\lambda_{n}\}$ with the number of the elements $s$ and $n-s$ respectively. The number of the sets $\Lambda_{s}^{t}$ is $C_{n}^{s}$ with $C_{n}^{s}$ being the combinations. Let $\Lambda^{t}_{s}=\{\lambda_{i}|i\in K^{t}_{s}\}$ and $\tilde{\Lambda}^{t}_{s}=\{\lambda_{i}|i\in\overline{K^{t}_{s}}\}$ with $K^{t}_{s}$ being the set obtained by selecting $s$ elements from $\{1,2,\cdots,n\}$ and $\overline{K^{t}_{s}}$ being the complement of $K^{t}_{s}.$ Then $\Gamma^{t}_{s}=\{\textit{A}_{i}|i\in K^{t}_{s}\}$ and $\tilde{\Gamma}^{t}_{s}=\{\textit{A}_{i}|i\in\overline{K^{t}_{s}}\}.$