Separability Criterion for Multipartite Pure States

TL;DR

This paper introduces new criteria for determining the separability of multipartite pure quantum states, including an efficient algorithm that does not require Schmidt decomposition, advancing the analysis of quantum entanglement.

Contribution

The paper presents novel separability criteria and an algorithm that simplifies the process without needing Schmidt decomposition, improving the analysis of multipartite quantum states.

Findings

Established a relationship between separability and Schmidt decomposability.

Derived a criterion that does not require Schmidt decomposition.

Provided an effective algorithm for separability judgment.

Abstract

In this paper, we give out some effective criterions which can be used to judge the separability of multipartite pure states. We obtain the relationship between separability and Schmidt decomposable of multipartite pure states in Theorem1. The first criterion derived from Theorem2 dose not need the Schmidt decomposition which is hard to find for multipartite states. Theorem3 is more profound which can be used to deduce Corollary1 which is one of the main results in [1]. Finally, we give out an algorithm which can be used to judge the separability of multipartite pure states effectively.

Equations

$$|\psi ⟩=|{\psi }^{A_1}⟩\otimes |{\psi }^{A_2}⟩\otimes \cdots \otimes |{\psi }^{A_n}⟩,$$

$$|\psi ⟩=\sum _{s=1}^D{\lambda }_s|e_s^{A_1}⟩|e_s^{A_2}⟩\cdots |e_s^{A_n}⟩,$$

$$\det \left(M_{|\psi ⟩}^{A_k}-E_k\right)=0,k=1,2,\cdots ,n.$$

$$M_{|\psi\rangle}^{A_{k}}=\left(\begin{array}[]{cccc}1&0&\cdots&0\\ 0&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&0\end{array}\right),\quad k=1,2,\cdots,n.$$

$$M_{|\phi\rangle}^{A_{k}}=\left(\begin{array}[]{ccccc}{\frac{1}{2}}&0&\cdots&0&0\\ 0&0&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&0&0\\ 0&0&\cdots&0&{\frac{1}{2}}\end{array}\right),\quad k=1,2,\cdots,n.$$

$$M_{k}=\left(\begin{array}[]{cccc}a_{11\cdots 111\cdots 111}&a_{11\cdots 121\cdots 111}&\cdots&a_{11\cdots 1d_{k}1\cdots 111}\\ a_{11\cdots 111\cdots 112}&a_{11\cdots 121\cdots 112}&\cdots&a_{11\cdots 1d_{k}1\cdots 112}\\ \vdots&\vdots&\ddots&\vdots\\ a_{11\cdots 111\cdots 11d_{n}}&a_{11\cdots 121\cdots 11d_{n}}&\cdots&a_{11\cdots 1d_{k}1\cdots 11d_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ a_{i_{1}i_{2}\cdots i_{k-1}1i_{k+1}\cdots i_{n}}&a_{i_{1}i_{2}\cdots i_{k-1}2i_{k+1}\cdots i_{n}}&\cdots&a_{i_{1}i_{2}\cdots i_{k-1}d_{k}i_{k+1}\cdots i_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ a_{d_{1}d_{2}\cdots d_{k-1}1d_{k+1}\cdots d_{n}}&a_{d_{1}d_{2}\cdots d_{k-1}2d_{k+1}\cdots d_{n}}&\cdots&a_{d_{1}d_{2}\cdots d_{k-1}d_{k}d_{k+1}\cdots d_{n}}\end{array}\right),\quad k=1,2,\cdots,n.$$

$$M_{|\psi ⟩}^{A_k}=M_k{M}_k^{†},k=1,2,\cdots ,n.$$

$$a_{i_1{i}_2\cdots i_{k-1}s{i}_{k+1}\cdots i_n}\cdot M_k^t=a_{i_1{i}_2\cdots i_{k-1}t{i}_{k+1}\cdots i_n}\cdot M_k^s,$$

$$(t=1,2,\cdots ,s-1,s+1,\cdots ,d_k;k=1,2,\cdots ,n).$$

$$M_{k}=\left(\begin{array}[]{cc}{\frac{\sqrt{2}}{2}}&0\\ 0&0\\ \vdots&\vdots\\ 0&0\\ 0&{\frac{\sqrt{2}}{2}}\end{array}\right),\quad k=1,2,\cdots,n.$$

Full text

The state is one of the fundamental concepts in quantum computation and information which can be classified into pure state and mixed state. In view of the purification NC2000 , we always pay attention to the pure states with related technique for quantum computation and quantum information. The pure states can be classified into pure separable states and pure entanglement states. Entanglement is a valuable physical resource for accomplishing many useful quantum computing and quantum information processing tasks NC2000 . For certain tasks such as superdense coding Ben1992 and quantum teleportation Ben1993 , it has been demonstrated that entanglement is an indispensable ingredient. For many other tasks entanglement is also used to enhance the efficiency Childs2000 ; Acin2001 ; DA2001 ; ZFJi2006 . The question of quantifying entanglement of multipartite quantum states is fundamental to the whole field of quantum information and in general to the physics of multicomponent quantum systems. Separability is a theoretical foot stone to define the measures of entanglement. As a result, the problem of separability, that is whether a quantum state is separable or entangled, is fundamental. Some authors have given some separability criterions, valid under certain conditions Peres1996 ; Wu2000 ; Vcoffman2001 ; JEisert2004 ; YuChSh2005 ; DafaLi2006 . In this letter, we will devote to the separability criterions for any multipartite pure state.

Let $|\psi\rangle$ be a pure state of a composite system AB possessed by Alice and Bob, then we know that $|\psi\rangle$ has Schmidt decomposition $|\psi\rangle=\sum_{i}{\lambda_{i}|i^{A}\rangle|i^{B}\rangle}$ NC2000 . It is also known that a state of a bipartite system is separable if and only if it has Schmidt number 1 NC2000 . So we can judge a bipartite pure state to be separable or not by computing its Schmidt number. Unfortunately, the Schmidt decomposition does not always exist for any multi($n$)-partite pure state when $n>2$. In order to extend the forward criterion to any multipartite pure state, some pioneers have paid their attentions to the conditions of the occurrence of Schmidt decomposition for multipartite pure states. Peres Peres1995 presented a necessary and sufficient condition for the occurrence of Schmidt decomposition for a tripartite pure state and Peres1996 showed that the positivity of the partial transpose of a density matrix is a necessary condition for separability. Unfortunately, this criterion is only necessary for a pure state to be separable, but not sufficient. Thapliyal Thap1999 showed that a multipartite pure state is Schmidt decomposable if and only if the deduced density matrices obtained by tracing out any party are separable. We should note that the separable states in Thapliyal’s criterion contain the pure separable states and the mixed separable states. Making use of the Schmidt decomposition we will get the first criterion.

Let $|\psi\rangle$ be a pure state in a $d$ dimensional $n$-partite quantum system $H$, which is composed by the subsystems $H^{A_{1}},H^{A_{2}},\cdots,H^{A_{n}}$. So we have $H=H^{A_{1}}\otimes H^{A_{2}}\otimes\cdots\otimes H^{A_{n}},d=\prod_{k=1}^{n}{d_{k}}$ and denote $D=\max{\{d_{1},d_{2},\cdots,d_{n}\}}$ where $d_{k}$ is the dimension of $k$-th subsystem $H^{A_{k}}$ for $k=1,2,\cdots,n$. Because $|\psi\rangle$ is a pure state in $H$, we have $|\psi\rangle=\sum\limits_{i_{1}=1}^{d_{1}}{\sum\limits_{i_{2}=1}^{d_{2}}{\cdots\sum\limits_{i_{n}=1}^{d_{n}}{a_{i_{1}i_{2}\cdots i_{n}}|e_{i_{1}}^{A_{1}}\rangle|e_{i_{2}}^{A_{2}}\rangle\cdots|e_{i_{n}}^{A_{n}}\rangle}}}$, where $\{|e_{i_{k}}^{A_{k}}\rangle\}_{i_{k}=1}^{d_{k}}$ are the orthonormal basis of subsystems $H^{A_{k}}(k=1,2,\cdots,n)$ and $a_{i_{1}i_{2}\cdots i_{n}}\in\mathbb{C}$ are the amplitudes with $\sum\limits_{i_{1}=1}^{d_{1}}{\sum\limits_{i_{2}=1}^{d_{2}}{\cdots\sum\limits_{i_{n}=1}^{d_{n}}{|a_{i_{1}i_{2}\cdots i_{n}}|^{2}}}}=1$. By definition, a $n$-partite pure state $|\psi\rangle$ is separable if and only if there exits $n$ pure states $|\psi^{A_{k}}\rangle(k=1,2,\cdots,n)$ respectively belonging to subsystems $H^{A_{k}}(k=1,2,\cdots,n)$ and $|\psi\rangle=|\psi^{A_{1}}\rangle\otimes|\psi^{A_{2}}\rangle\otimes\cdots\otimes|\psi^{A_{n}}\rangle$. Generalizing the separability condition for a bipartite pure state we have the following theorem.

[TABLE]

where $|\psi^{A_{k}}\rangle$ is a pure state in $H^{A_{k}}$ for $k=1,2,\cdots,n$ respectively. Let $|e_{1}^{A_{k}}\rangle=|\psi^{A_{k}}\rangle(k=1,2,\cdots,n)$ and choose $\{|e_{i_{k}}^{A_{k}}\rangle\}_{i_{k}=2}^{d_{k}}$ to make $\{|e_{i_{k}}^{A_{k}}\rangle\}_{i_{k}=1}^{d_{k}}$ to be the orthonormal basis of subsystems $H^{A_{k}}$ for $k=1,2,\cdots,n$. Taking into account Eq. (1), if we choose $\lambda_{1}=1,\lambda_{2}=\lambda_{3}=\cdots=\lambda_{D}=0$ and $|e_{i_{k}}^{A_{k}}\rangle=0(i_{k}=d_{k}+1,d_{k}+2,\cdots,D)$, then we have

[TABLE]

which is actual the Schmidt decomposition and we know that the Schmidt number of $|\psi\rangle$ is 1.

On the other hand, if a $n$-partite pure state $|\psi\rangle$ is Schmidt decomposable and has Schmidt number 1. Supposing that $\lambda_{1}\neq 0$, then we have $|\psi\rangle=\sum\limits_{s}{\lambda_{s}|e_{s}^{A_{1}}\rangle|e_{s}^{A_{2}}\rangle\cdots|e_{s}^{A_{n}}\rangle}=|e_{1}^{A_{1}}\rangle|e_{1}^{A_{2}}\rangle\cdots|e_{1}^{A_{n}}\rangle$, where $|e_{1}^{A_{k}}\rangle$ is a pure state in $H^{A_{k}}$ for $k=1,2,\cdots,n$. ∎

Taking into account Theorem 1, we can judge the separability of any $n$-partite pure state by finding its Schmidt decomposition. In order to obtain the Schmidt decomposition of a $n$-partite pure state $|\psi\rangle$, we need to compute (1) the density operator $\rho_{|\psi\rangle}$, (2) the reduced density operators $\rho_{|\psi\rangle}^{A_{k}}(k=1,2,\cdots,n)$ and (3) the eigenvalues $\lambda_{i_{k}}$ of $\rho_{|\psi\rangle}^{A_{k}}(k=1,2,\cdots,n)$. However it is hard to compute all the eigenvalues of high dimensional density operators exactly. Theorem 1 does not give us an effective criterion. In the following, we will introduce some effective separability criterions.

Any quantum state can be represent by a density operator. For a $n$-partite pure state $|\psi\rangle$ in $H$, its density operator is $\rho_{|\psi\rangle}={|\psi\rangle\langle\psi|}$. Taking the partial trace operation on each $k$-th subsystem $H^{A_{k}}$, we get reduced density operators $\rho_{|\psi\rangle}^{A_{k}}=\mathop{\rm tr}\nolimits_{H^{A_{k}}}{\left(\rho_{|\psi\rangle}\right)}$ (with $O\left(\prod\limits_{{s=1}\atop{s\neq k}}^{n}{d_{s}^{2}d_{k}}\right)$ multiplication operations and $O\left(\sum_{t=1}^{n}{d_{t}}\right)$ plus operations in the worst case) for $k=1,2,\cdots,n$.

In order to judge the separability of a $n$-partite pure state there is no need to find the Schmidt decomposition since we have the following theorem.

[TABLE]

where $M_{|\psi\rangle}^{A_{k}}$ are the matrices related to the reduced density operators $\rho_{|\psi\rangle}^{A_{k}}$ for $k=1,2,\cdots,n$. $E_{k}$ are the identity matrices and have the same dimension with $M_{|\psi\rangle}^{A_{k}}$ for $k=1,2,\cdots,n$.

By using the characterizations of density operators (trace and positivity conditions) NC2000 , we can easily prove the following lemma.

Then we can give out the proof of Theorem 2 by using Lemma 1.

If the $n$-partite pure state $|\psi\rangle$ is Schmidt decomposable and has Schmidt number 1, then we have $|\psi\rangle=|e_{1}^{A_{1}}\rangle|e_{1}^{A_{2}}\rangle\cdots|e_{1}^{A_{n}}\rangle$. We can calculate the density operator $\rho_{|\psi\rangle}={|\psi\rangle\langle\psi|}=|e_{1}^{A_{1}}\rangle|e_{1}^{A_{2}}\rangle\cdots|e_{1}^{A_{n}}\rangle\langle e_{1}^{A_{1}}|\langle e_{1}^{A_{2}}|\cdots\langle e_{1}^{A_{n}}|$ and reduced density operators

$\rho_{|\psi\rangle}^{A_{k}}=\mathop{\rm tr}\nolimits_{H^{A_{k}}}{\left(\rho_{|\psi\rangle}\right)}=|e_{1}^{A_{1}}\rangle|e_{1}^{A_{2}}\rangle\cdots|e_{1}^{A_{k-1}}\rangle|e_{1}^{A_{k+1}}\rangle\cdots|e_{1}^{A_{n}}\rangle\langle e_{1}^{A_{1}}|\langle e_{1}^{A_{2}}|\cdots\langle e_{1}^{A_{k-1}}|\langle e_{1}^{A_{k+1}}|\cdots\langle e_{1}^{A_{n}}|,(k=1,2,\cdots,n)$.

So we have

[TABLE]

This means that $M_{|\psi\rangle}^{A_{k}}(k=1,2,\cdots,n)$ only have 1 to be their nonzero eigenvalues. So Eq. (3) is obtained.

On the other hand, all the matrices $M_{|\psi\rangle}^{A_{k}}(k=1,2,\cdots,n)$ only have 1 to be their nonzero eigenvalues. Denote $|f_{1}^{A_{k}}\rangle$ to be the eigenvectors of $M_{|\psi\rangle}^{A_{k}}$ corresponding to the eigenvalues 1 for $k=1,2,\cdots,n$ respectively. Then we have $|\psi\rangle=|f_{1}^{A_{1}}\rangle|f_{1}^{A_{2}}\rangle\cdots|f_{1}^{A_{n}}\rangle$ which is the Schmidt decomposition of $|\psi\rangle$ with Schmidt number 1. So $|\psi\rangle$ is separable. ∎

Theorem 2 can be used to judge the separability of any n-partite pure state. For example, $n$-cat state (in honor of Schr$\ddot{o}$dinger’s cat ) is $|\psi\rangle={\frac{\sqrt{2}}{2}}\left(|0^{\otimes n}\rangle+|1^{\otimes n}\rangle\right)$ with the Einstein-Podolsky-Rosen-Bohm pair ${\frac{\sqrt{2}}{2}}\left(|00\rangle+|11\rangle\right)$ when $n=2$ and the Greenberger-Hone-Zeilinger-Mermin state ${\frac{\sqrt{2}}{2}}\left(|000\rangle+|111\rangle\right)$ when $n=3$. We have the density operator $\rho_{|\psi\rangle}={\frac{1}{2}}\left({|0^{\otimes n}\rangle\langle 0^{\otimes n}|}+|0^{\otimes n}\rangle\langle 1^{\otimes n}|+|1^{\otimes n}\rangle\langle 0^{\otimes n}|+{|1^{\otimes n}\rangle\langle 1^{\otimes n}|}\right)$ and reduced operators $\rho_{|\psi\rangle}^{A_{k}}={\frac{1}{2}}\left({|0^{\otimes(n-1)}\rangle\langle 0^{\otimes(n-1)}|}+{|1^{\otimes(n-1)}\rangle\langle 1^{\otimes(n-1)}|}\right)$ which means that

[TABLE]

Then we have $\det{\left(M_{|\psi\rangle}^{A_{k}}-E_{k}\right)}={\frac{1}{4}}\neq 0(k=1,2,\cdots,n)$ which means that the $n$-cat state is an entanglement pure state by Theorem 2.

Theorem 2 give us an separability criterion for any $n$-partite pure state. The total number of times the criterion has to run, in the worst case, is $O(nd^{3})$ with the most operations being used for computing the determinants.

For a $n$-partite pure state, we have $|\psi\rangle=\sum\limits_{i_{1}=1}^{d_{1}}{\sum\limits_{i_{2}=1}^{d_{2}}{\cdots\sum\limits_{i_{n}=1}^{d_{n}}{a_{i_{1}i_{2}\cdots i_{n}}|e_{i_{1}}^{A_{1}}\rangle|e_{i_{2}}^{A_{2}}\rangle\cdots|e_{i_{n}}^{A_{n}}\rangle}}}$. Let $M_{k}$ are $\left(\prod\limits_{{s=1}\atop{s\neq k}}^{n}{d_{s}}\right)\times d_{k}$ matrices of the amplitudes $a_{i_{1}i_{2}\cdots i_{n}}(i_{s}=1,2,\cdots,d_{s})$ of the form

[TABLE]

We have the following lemma without proof.

[TABLE]

where $M_{k}(k=1,2,\cdots,n)$ are given by Eq. (4) and $M_{k}^{\dagger}$ are the Hermitian of $M_{k}(k=1,2,\cdots,n)$.

To avoid the Schmidt decomposition, by using Lemma 1 and Lemma 2 we also obtain the following theorem.

Using Theorem 3, we can easily prove Corollary 1 which is the main result of Dafa Li in DafaLi2006 . In fact, Theorem 3 and Corollary 1 are equivalent, but this does not mean they have the same efficiency. Corollary 1 can give us another criterion with $O(nd^{2})$ times to be used in the worst case. By using Theorem 3, we obtain a more effective criterion.

[TABLE]

where $a_{i_{1}i_{2}\cdots i_{k-1}si_{k+1}\cdots i_{n}}$ are any fixed nonzero elements in matrices $M_{k}$ for $k=1,2,\cdots,n$ respectively and $M_{k}^{t}$ is the $t$-th $(t=1,2,\cdots,d_{k})$ column of the matrix $M_{k}$ for $k=1,2,\cdots,n$.

If the rank of matrices $M_{k}(k=1,2,\cdots,n)$ are equal to 1, then any two columns $M_{k}^{s},M_{k}^{t}(s,t=1,2,\cdots,d_{k})$ of $M_{k}(k=1,2,\cdots,n)$ are linearly dependent. This will imply Eq. (2).

On the other hand, Eq. (2) implies that any two columns of $M_{k}(k=1,2,\cdots,n)$ are linearly dependent. Since $M_{k}(k=1,2,\cdots,n)$ are not zero matrices, We have the rank of matrices $M_{k}(k=1,2,\cdots,n)$ are equal to 1. ∎

For example, considering the $n$-cat state $|\psi\rangle={\frac{\sqrt{2}}{2}}\left(|0^{\otimes n}\rangle+|1^{\otimes n}\rangle\right)$, we have

[TABLE]

We can see ${\frac{\sqrt{2}}{2}}\cdot M_{k}^{2}\neq 0\cdot M_{k}^{1}$, where $M_{k}^{1},M_{k}^{2}$ are the first and second columns of $M_{k}(k=1,2,\cdots,n)$. So $n$-cat state is an entanglement state as we know.

Using Corollary 2, we have an effective algorithm which can be used to judge the separability of a $n$-partite pure state. The algorithm contains the following 3 steps. Step1: Construct matrices $M_{k}(k=1,2,\cdots,n)$ from the $n$-partite pure state $|\psi\rangle$ by Eq. 4; Step2: Deleting the zero columns and the zero rows of the matrices $M_{k}(k=1,2,\cdots,n)$. The result matrices are also denoted by $M_{k}(k=1,2,\cdots,n)$. Step 3: If Eq. (2) does not true for some $t(t=1,2,\cdots,s-1,s+1,\cdots,d_{k};k=1,2,\cdots,n)$, then stop and we get $|\psi\rangle$ is an entanglement pure state, otherwise, $|\psi\rangle$ is a separable pure state. The total number of times the algorithm has to run, in the worst case, is $O(nd)$.

In this letter, we have given some necessary and sufficient conditions for the separability of a multipartite pure state. Using these conditions we get some criterions which can be used to judge the separability of a multipartite pure state. Finally, an effective algorithm deduced from Corollary 2 has been provided.