Detection of $k$-partite entanglement and $k$-nonseparability of multipartite quantum states

Quantum entanglement is an important physical resource in quantum computation and quantum information processing and is also a key feature that distinguishes quantum theory from classical theory. So, it is very valuable for studying the characterization and detection of entanglement for general multipartite systems.

There are two different ways to characterize the entanglement of multipartite systems RPMK09 ; Guhne2009 , one is according to the question ”How many partitions are separable?”, the other is ”How many particles are entangled”. The former is described by $k$-separability, the latter by $k$-partite entanglement. $k$-separability provides a fine graduation of states according to their degrees of separability, while $k$-partite entanglement provides the hierarchic classification of states according to their degrees of entanglement. Detecting and characterizing $k$-partite entanglement provide a refined insight in entanglement dynamics.

For $N$-partite quantum systems, the $k$-partite entanglement and $k$-nonseparability are two different concepts of multipartite entanglement except that $2$-partite entanglement is equivalent to $N$-nonseparability and genuine $N$-partite entanglement is equivalent to $2$-nonseparability. Both $k$-nonseparability and $k$-partite entanglement can be used to characterize multipartite entanglement. There are some approaches to identify genuine $N$-partite entanglement such as based on spin-squeezing inequality VitaglianoHyllus11 and elements of density matrix GuhneSeevinck10 ; HuberMintert10 ; GanHong10 ; GanHong11 ; WuKampermann12 ; ChenMa12 , Bell-type inequalities Seevinck02 , semidefinite program and state extensions Doherty02 ; Doherty04 ; Doherty05 , covariance matrices Gittsovich05 , etc.

The $k$-partite entanglement of $N$-partite quantum systems has been studied by using different tools and some progress has been made. Some of the complete set of generalized spin squeezing inequalities employed to detect $k$-particle entanglement and bound entanglement VitaglianoHyllus11 . A hierarchic classification of all states from $2$-partite entanglement to $N$-partite entanglement based on Wigner-Yanase skew information was presented ChenZQ2011 . Entanglement criteria in terms of the quantum Fisher information Hyllus2012 ; Goth2012 were applied to detect several classes of $k$-partite entanglement. Further approach had led to $k$-particle entanglement criteria which was developed by a comparison of the Fisher information and the sum of variances of all local operators Gessner16 .

In recent years, $k$-nonseparability of $N$-partite quantum states has attracted more and more attention and extensively explored in various ways GaoYan14 ; GaoHong13 ; HongGaoYan12 ; ABMarcus10 ; Ananth15 ; HuberLlobet13 ; HongLuoSong15 ; HongLuo16 ; KlocklHuber15 . Some practical convenient inequalities only involving elements of density matrix for detecting $k$-nonseparability were constructed in GaoYan14 ; GaoHong13 ; HongGaoYan12 ; ABMarcus10 ; Ananth15 , which can be used to distinguish $N$-1 different classes of any high dimensional $N$-partite $k$-inseparable states. In HuberLlobet13 a method that derived from some inequalities in the form of entropy vector was developed to test $k$-nonseparability, whose essence is also elements of density matrix. $k$-nonseparable states can also be checked by quantum Fisher information and local uncertainty relations HongLuoSong15 ; HongLuo16 . From the Bloch vector decomposition of a density matrix, the tensor elements can reflect many important characteristics of multipartite quantum states. The correlation tensor norms used for determining $k$-nonseparability in multipartite states KlocklHuber15 .

Much effort has been put into the detection and characterization of $k$-nonseparability and $k$-partite entanglement in multipartite systems, but it is far from perfect because of the extremely complex structure of entanglement of multipartite quantum states RPMK09 ; Guhne2009 . This paper is devoted to giving powerful inequalities that provide, upon violation, experimentally accessible sufficient conditions for $k$-nonseparability and $k$-partite entanglement in $N$-partite systems. Compared with previous works, the resulting criteria are stronger. The inequalities for $k$-nonseparability detection given in Ananth15 can be seen as special case of our results.

In this paper, we will further study the structure of entanglement of multipartite quantum states and present some inequalities to detect $k$-partite entanglement and $k$-nonseparability of multipartite quantum states. In Sec. II, some concepts and symbols are introduced. In Sec. III and Sec. IV, we derive a full hierarchy of $k$-partite entanglement criteria and $k$-nonseparability criteria based on linear local operators to characterize multipartite entanglement. Finally, we show their application ability and point out that they can be adapted in present-day experiments.

The $k$-partite entanglement and $k$-nonseparability are two different concepts involving the partitions of subsystem in $N$-partite quantum system.

An $N$-partite quantum pure state $|\phi\rangle$ is said to be $k$-producible if it can be written as a tensor product $|\phi\rangle=\bigotimes\limits_{i=1}^{n}|\phi_{A_{i}}\rangle$ such that each $|\phi_{A_{i}}\rangle$ is a state of at most $k$ particles Guhne2009 . A pure state contains $k$-partite entanglement if it is not ($k$-1)-producible. For an $N$-partite mixed state $\rho$, if it can be written as a convex combination of $k$-producible pure states, then it is called $k$-producible. The individual pure states composing a $k$-producible mixed state may be $k$-producible under different partitions. If $\rho$ is not $k$-producible, we say that $\rho$ contains $(k+1)$-partite entanglement. Here $1\leq k\leq N-1$.

An $N$-partite quantum pure state $|\phi\rangle$ is called $k$-separable if there is a splitting of $N$ particles into $k$ partitions $A_{1},A_{2},\cdots,A_{k}$ such that $|\phi\rangle=\bigotimes\limits_{i=1}^{k}|\phi_{A_{i}}\rangle$ Guhne2009 . For an $N$-partite mixed state $\rho$, if it can be written as a convex combination of $k$-separable pure states, then it is called $k$-separable. If $\rho$ is not $k$-separable, we say that $\rho$ is $k$-nonseparable. Here $2\leq k\leq N$.

In particular, the $1$-producible (or $N$-separable) states are just fully separable states, and the quantum states being not $(N-1)$-producible (or 2-separable) are genuinely entangled states.

For the following statement, let’s introduce some symbols. Let $P$ be the global permutation operator performing simultaneous permutations on all subsystems in $(\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\cdots\otimes\mathcal{H}_{N})^{\otimes 2}$, while $P_{\alpha}$ be local permutation operators permuting the two copies of all subsystems contained in the set $\alpha$, that is,

Now we state our main results in the detection of $k$-partite entanglement.

Theorem 1. If $\rho$ is a $k$-producible quantum state of $N$-partite quantum system $\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\cdots\otimes\mathcal{H}_{N}$, where dim $\mathcal{H}_{i}=d_{i},i=1,2,\cdots,N$, then

where $r=\frac{N}{k}$ for $k|N$, $r=[\frac{N}{k}]+1$ for $k\nmid N$, $|\Phi\rangle=|\phi_{1}\rangle|\phi_{2}\rangle$ with $|\phi_{1}\rangle=\bigotimes\limits_{i=1}^{N}|x_{i}\rangle$ and $|\phi_{2}\rangle=\bigotimes\limits_{i=1}^{N}|y_{i}\rangle$ being any fully separable states of $N$-partite quantum system, and $\{\alpha\}$ consists of all possible nonempty proper subsets of $\{1,2,\cdots,N\}$. That is, an $N$-partite state $\rho$ does not satisfy inequality (1), then $\rho$ contains $k+1$-partite entanglement.

Proof. Let us first prove that inequality (1) holds for any $k$-producible pure state. Suppose that pure state $\rho=|\varphi\rangle\langle\varphi|$ with $|\varphi\rangle=\bigotimes\limits_{i=1}^{n}|\varphi_{A_{i}}\rangle$ is $k$-producible, where every vector $|\varphi_{A_{i}}\rangle$ involves at most $k$ parties. Then we have

Let $\alpha_{j_{1},\cdots,j_{m}}=A_{j_{1}}\cup A_{j_{2}}\cup\cdots\cup A_{j_{m}}$, $|\varphi_{\alpha_{j_{1},\cdots,j_{m}}}\rangle=\bigotimes\limits_{t=1}^{m}|\varphi_{A_{j_{t}}}\rangle$, and $|\varphi_{\overline{\alpha_{j_{1},\cdots,j_{m}}}}\rangle=\bigotimes\limits_{t=m+1}^{n}|\varphi_{A_{j_{t}}}\rangle$, we derive

Combining (2) and (3) gives that

which implies that inequality (1) holds for any $k$-producible pure state.

Next, we prove inequality (1) also holds for any $k$-producible mixed state. Suppose that $\rho=\sum\limits_{i}p_{i}\rho_{i}=\sum\limits_{i}p_{i}|\varphi_{i}\rangle\langle\varphi_{i}|$ is $k$-producible mixed state, where $\rho_{i}=|\varphi_{i}\rangle\langle\varphi_{i}|$ is $k$-producible. Then by triangle inequality and the Cauchy-Schwarz inequality, one has

The proof is complete.

Theorem 2. Suppose that $\rho$ is an $N$-partite density matrix acting on Hilbert space $\mathcal{H}^{\otimes N}$ with dim $\mathcal{H}=d$. Let $|\Psi^{st}_{ij}\rangle=|\psi^{s}_{i}\rangle|\psi^{t}_{j}\rangle$, where $|\psi^{s}_{i}\rangle=|x_{1}\cdots x_{i-1}sx_{i+1}\cdots x_{N}\rangle$ and $|\psi^{t}_{j}\rangle=|x_{1}\cdots x_{j-1}tx_{j+1}\cdots x_{N}\rangle$ are fully separable states of $\mathcal{H}^{\otimes N}$. If $\rho$ is a $k$-producible, then

for $2\leq k\leq N-1$, and

for $k=1$. Here $\{|\omega_{1}\rangle,\cdots,|\omega_{T}\rangle\}\subseteq\mathcal{H}$ is a quantum state set.

Of course, if an $N$-partite state $\rho$ does not satisfy the above inequality (4) (respectively, inequality (5)), then $\rho$ contains $k+1$-partite entanglement ($2\leq k\leq N-1$) (respectively, 2-partite entanglement).

It should be pointed that there are $\frac{1}{2}n(n-1)$ inequalities in eq.(5)), and violation of any one of them implies 2-partite entanglement.

Proof. To establish the validity of ineq. (4) and ineq. (5) for all $k$-producible states $\rho$, let us first verify that this is true for any $k$-producible pure state $\rho$.

Suppose that pure state $\rho=|\varphi\rangle\langle\varphi|$ is $k$-producible under partition $\{A_{1},A_{2},\cdots,A_{n}\}$, and $|\varphi\rangle=\bigotimes\limits_{l=1}^{n}|\varphi_{A_{l}}\rangle$. Then we have

in case of $i,j$ in same part, and

in case of $i,j$ in different parts ($i\in A_{l},j\in A_{l^{\prime}}$ with $l\neq l^{\prime}$) where $|\psi^{st}_{ij}\rangle:=|x_{1}\cdots x_{i-1}sx_{i+1}\cdots x_{j-1}tx_{j+1}\cdots x_{N}\rangle$. Moreover, (7) indicates that inequality (5) hols with equality for any 1-producible pure states.

Combining (6) and (7) ptoduces

Hence, inequality (4) holds for any $k$-producible pure state.

Now, let us prove (4) also holds for any $k$-producible mixed state. Suppose that $\rho=\sum\limits_{m}p_{m}\rho_{m}=\sum\limits_{m}p_{m}|\varphi_{m}\rangle\langle\varphi_{m}|$ is $k$-producible mixed state, where $\rho_{m}=|\varphi_{m}\rangle\langle\varphi_{m}|$ is $k$-producible. Then by triangle inequality and the Cauchy-Schwarz inequality, one has

as desired.

Next, we prove that (5) also holds for any 1-producible mixed state. Suppose that $\rho=\sum\limits_{m}p_{m}\rho_{m}=\sum\limits_{m}p_{m}|\varphi_{m}\rangle\langle\varphi_{m}|$ is $1$-producible mixed state, where $\rho_{m}=|\varphi_{m}\rangle\langle\varphi_{m}|$ is $1$-producible. Then by (7), triangle inequality and the Cauchy-Schwarz inequality, one has

Hence, inequality (5) holds for any 1-producible states. This complete the proof.

Here we present our main results in the detection $k$-nonseparability.

Theorem 3. If $\rho$ is a $k$-separable $N$-partite quantum state acting on Hilbert space $\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\cdots\otimes\mathcal{H}_{N}$, where dim $\mathcal{H}_{i}=d_{i},i=1,2,\cdots,N$, then

Here $|\Phi\rangle=|\phi_{1}\rangle|\phi_{2}\rangle$ with $|\phi_{1}\rangle=\bigotimes\limits_{j=1}^{N}|x_{j}\rangle$ and $|\phi_{2}\rangle=\bigotimes\limits_{j=1}^{N}|y_{j}\rangle$ being any fully separable states of $N$-partite quantum system, and $\{\alpha\}$ consists of all possible nonempty proper subsets of $\{1,2,\cdots,N\}$.

Of course, $\rho$ is a $k$-nonseparable $N$-partite state if it violates the above inequality (9).

The proof is similar to that of Theorem 1.

Criterion 1 of Ref. Ananth15 is the special case of Theorem 3 when $|\Phi\rangle=|00\cdots 0\rangle|(d_{1}-1)(d_{2}-1)\cdots(d_{N}-1)\rangle$.

Theorem 4. If $\rho$ is a $k$-separable $N$-partite quantum state acting on Hilbert space $\mathcal{H}^{\otimes N}$, where dim $\mathcal{H}=d$, then

for $2\leq k\leq N-1$. Inequality (5) holds for any $N$-separable states. Here $\{|\omega_{1}\rangle,\cdots,|\omega_{T}\rangle\}\subseteq\mathcal{H}$, $|\Psi^{st}_{ij}\rangle=|\psi^{s}_{i}\rangle|\psi^{t}_{j}\rangle$ with $|\psi^{s}_{i}\rangle=|x_{1}\cdots x_{i-1}sx_{i+1}\cdots x_{N}\rangle$ and $|\psi^{t}_{j}\rangle=|x_{1}\cdots x_{j-1}tx_{j+1}\cdots x_{N}\rangle$.

If an $N$-partite state $\rho$ does not satisfy the above inequality (10), then $\rho$ is not k-separable (k-nonseparable).

The proof is similar to that of Theorem 2.

We need to emphasize that inequality (5) holds for any $N$-separable states because $N$-separable states are the same as 1-producible states in $N$-partite quantum system.

Criterion 2 of Ref. Ananth15 is the special case of Theorem 4 when $|\psi\rangle=|00\cdots 0\rangle$ and $\{\omega_{1},\cdots,\omega_{T}\}=\{1,2,\cdots,d-1\}$.

In this section, we illustrate our main results with some explicit examples. We indeed detects $k$-partite entangled states and $k$-nonseparable mixed multipartite states which beyond all previously studied criteria.

For convenience of comparison, we first list the criteria which can identify $k$-partite entanglement and $k$-separability of some quantum states.

For an $N$-qubit quantum state $\rho$, let quantum Fisher information $F(\rho,H)=\sum\limits_{l,l^{\prime}}\dfrac{2(\lambda_{l}-\lambda_{l^{\prime}})^{2}}{(\lambda_{l}+\lambda_{l^{\prime}})}|\langle l|H|l^{\prime}\rangle|^{2}$, where $H=\frac{1}{2}\sum\limits_{i=1}^{N}\sigma_{z}^{(i)}$ with $\sigma_{z}^{(i)}=|0\rangle\langle 0|-|1\rangle\langle 1|$ acting on the $i$-th qubit.

(I) If

then $\rho$ is not $k$-producible and contains $(k+1)$-partite entanglement Hyllus2012 . Here $s=\lfloor\frac{N}{k}\rfloor$ is the largest integer smaller than or equal to $\frac{N}{k}$.

(II) If

then $\rho$ is $k$-nonseparable HongLuoSong15 .

Suppose that $\rho$ is an $N$ partite state acting on Hilbert space $\mathcal{H}^{\otimes N}$ with dim$\mathcal{H}=d$. Let $\{g_{m}\}_{m=1}^{d^{2}-1}$ be the SU($d$) generators and $G_{m}=\sum\limits_{i=1}^{N}g_{m}^{(i)}$, where $g_{m}^{(i)}$ means that $g_{m}$ acts on the $i$-th subsystem.

(III) If an $N$ partite state $\rho$ acting on Hilbert space $\mathcal{H}^{\otimes N}$ satisfies

then $\rho$ is not $2$-producible and contains 3-partite entanglement VitaglianoHyllus11 . Here $V(\rho,G_{m})=\textrm{tr}[\rho(G_{m})^{2}]-[\textrm{tr}\rho(G_{m})]^{2}$.

(IV) If $\rho$ is a $k$-separable $N$-partite quantum state acting on Hilbert space $\mathcal{H}^{\otimes N}$, where dim $\mathcal{H}=d$. Then Ananth15

Example 1. Consider the $10$-qubit mixed states,

where $|G\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes 10}+|1\rangle^{\otimes 10}),|\widetilde{G}\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes 10}-i|1\rangle^{\otimes 10}).$

We choose $|\Phi\rangle=|0\rangle^{\otimes 10}|1\rangle^{\otimes 10}$ for our Theorem 1 and Theorem 3.

The parameter ranges of 10-qubit mixed states $\rho(p,q)$ containing $(k+1)$-partite entanglement (with $k$ =3 and $k$ = 4) detected by Theorem 1 and (I) are illustrated in Fig. 1. We can see that the quantum states containing 4-partite entanglement in the area enclosed by red line $a$, the $p$ axis, blue line $a^{\prime}$ and $q$ axis are detected only by Theorem 1. Similarly, we find that the quantum states containing 5-partite entanglement in the area enclosed by orange line $b$, the $p$ axis, green line $b^{\prime}$ and $q$ axis are detected only by Theorem 1.

For 10-qubit mixed states $\rho(p,q)$, the parameter ranges for $k$-nonseparability (with $k$ =3 and $k$ = 4) detected by Theorem 3 and (II) are illustrated in Fig. 2. We can see that these $3$-nonseparable quantum states in the area enclosed by purple line $c$, $p$ axis, cyan line $c^{\prime\prime}$, line $q=1-p$, cyan line $c^{\prime}$ and $q$ axis are detected only by Theorem 3. Similarly, we find that these $4$-nonseparable quantum states in the area enclosed by magenta line $d$, $p$ axis, brown line $d^{\prime\prime}$, line $q=1-p$, brown line $d^{\prime}$ and $q$ axis are detected only by Theorem 3.

Example 2. Let us consider the family of 4-qutrit quantum states,

where $|W\rangle=\frac{1}{2\sqrt{2}}(\sum\limits_{i=1}^{2}|i000\rangle+|0i00\rangle+|00i0\rangle+|000i\rangle),$ and $\sigma|0\rangle=|1\rangle$, $\sigma|1\rangle=|2\rangle$, $\sigma|2\rangle=|0\rangle$.