Entanglement detection beyond the computable cross-norm or realignment criterion
Separability problem, to decide whether a given state is entangled or not, is a fundamental problem in quantum information theory. We propose a powerful and computationally simple separability criterion, which allows us to detect the entanglement of many bound entangled states. The criterion is strictly stronger than the criterion based on Bloch representations, the computable cross-norm or realignment criterion and its optimal nonlinear entanglement witnesses. Furthermore, this criterion can be generalized to an analogue of permutation separability criteria in the even-partite systems.
pacs:03.67.Mn, 03.65.Ta, 03.65.Ud
Entanglement is one of the most fascinating features of quantum theory that has puzzled generations of physicist. While initially the discussion was mainly driven by conceptual and philosophical consideration Bell (), in recent years the focus has shifted to mathematical aspects and practical applications. It was realized that entanglement is an essential resource in quantum information and acts an important role in many other physical phenomenon e.g. quantum phase transition QPT (). Therefore, the detection and quantification of entanglement became fundamental problems in quantum information theory werner (). However, entanglement is not yet fully understood and it is a challenging task and remains an open question to decide whether a given state is entangled or not despite a great deal of effort in the past years review1 (); Peres (); CCN1 (); CCN (); permutation1 (); permutation2 (); dV (); LUR1 (); witness1 (); nonlinear (); optimize (); Yu (); CM ().
Completely solving the separability problem is far away from us, this is in fact a NP-hard problem as proved in Gurvits (). Nevertheless, there are still many efficient conditions for entanglement such as the partial transposition criterion Peres (), the computable cross-norm or realignment (CCNR) criterion CCN1 (); CCN (), the permutation separability criteria permutation1 (); permutation2 (), the criterion based on Bloch representations dV (), local uncertainty relations LUR1 (), entanglement witnesses witness1 () and the covariance matrices (CM) approach CM (), etc.. The CCNR criterion is only a necessary condition for arbitrary dimensional systems. However, it can detect many bound entangled states where the partial transposition criterion fails. Recently, Gühne et al. proposed its nonlinear entanglement witness based on local uncertainty relations nonlinear (). The nonlinear entanglement witness is strictly stronger than the original criterion.
In this paper, we propose a practical criterion, based on which has similar properties as covariance matrices CM (). The criterion is strictly stronger than the dV criterion (i.e. the criterion based on Bloch representations), the CCNR criterion and its optimal nonlinear entanglement witnesses. Then we apply our criterion of separability to a bound entangled state with white noise. Finally, we generalize this criterion to multipartite entanglement and propose an analogue of permutation separability criteria in even-partite systems. It is worth noticing that our method proposed in this paper may be used to improve many other separability criteria.
Bipartite systems.– Before embarking on our criteria, it is worth introducing the CCNR criterion and its nonlinear witnesses. The CCNR criterion states that if is separable, the following inequality must be hold CCN (),
where stands for the trace norm (i.e. the sum of the singular values). The realignment operation , with scalar product in Hilbert Schmidt space of operators, and for a general operator it is given by linearity expanding in a product basis CCN1 (); CCN (); permutation1 (); Appendix (). Refs. nonlinear (); optimize () put forward its nonlinear witnesses and their optimal form, respectively,
where , are complete sets of local orthogonal bases Yu () for subsystems and respectively, and is defined as . For separable states and hold. Conversely, if any state violates one of the three inequalities, it is indeed entangled. Actually, can be expressed as , which will be proved in Proposition 1. In the following, we will propose a separability criterion. It is a slightly modified and therefore improved version of the optimal nonlinear witness Eq. (3).
Theorem 1. If a bipartite density matrix is separable, then the following inequality holds,
where and are reduced density matrices for subsystems and .
Proof.– A separable bipartite density matrix can be written as , and its reduced density matrices are , , where is a probability distribution and the , are pure states describing subsystems and , respectively.
It is easy to conclude that . We have reviewed the realignment operation in Appendix (). Therefore,
where we have used . The first inequality holds due to the convex property of the trace norm and the second one holds by applying the Cauchy-Schwarz inequality.
Obviously, Theorem 1 has a similar form of the CCNR criterion, using and instead of and 1, respectively. It suggests that using we can obtain some new separability criteria. In the following, it will be shown that Theorem 1 is strictly stronger than the CCNR criterion and its nonlinear witnesses (2), with an example and a strict proof.
Example 1.– Paweł Horodecki introduced a bound entangled state in Ref. bound (), and the density matrix is real and symmetric,
where . Let us consider a mixture of this state with white noise,
and show the curves , , with respect to the CCNR criterion, its optimal nonlinear witness, and Theorem 1 in Fig. 1.
In Ref. CCN (), it is found that the state still has entanglement when , , using the CCNR criterion. According to Theorem 1, one can obtain an upper bound , for which is still entangled. Furthermore, states which can be detected by the CCNR criterion or its nonlinear witnesses also violate Theorem 1 (see Fig. 1). Proposition 1 will provide a strict proof.
Proposition 1. Any state which can be detected by the CCNR criterion or its nonlinear witnesses (2) also violates Theorem 1.
Proof.– It is worth noticing that Eq. (3) is equivalent to , i.e., the sum of singular values of matrix is equal to the trace norm of . Note that . Therefore, , where . Moreover, Theorem 1 can be written as a nonlinear witness, .
Due to , it can be concluded that Theorem 1 is strictly stronger than the optimized nonlinear witness Eq. (3). Since Eq. (3) is not only a nonlinear form of Eq. (1) but also an optimal form of Eq. (2), it is strictly stronger than Eqs. (1) and (2) nonlinear (); optimize (). Thus, Proposition 1 holds.
Example 2.– Let us consider a noisy singlet state introduced in Ref. nonlinear (), , where and . Actually, the state is entangled for any nonlinear (). The CCNR criterion and its optimal nonlinear witness can detect the entanglement for all and , respectively. Using Theorem 1, one finds that the state still has entanglement when . One might expect Theorem 1 to be a necessary and sufficient condition for entanglement in two-qubit system. Unfortunately, this is not the case. However, considering the enhancement with local filtering operations, Theorem 1 becomes a necessary and sufficient condition for two qubits CM ().
Proposition 2. Theorem 1 is strictly stronger than the dV criterion.
Proof.– For bipartite systems, the Bloch representation can be written as , where and denote the generators of SU(M) and SU(N). The coefficients , , form the real matrix , and column vectors , , respectively. The dV criterion states that if a bipartite state is separable then must hold dV (). Notice that Theorem 1 is equivalent to . With the help of the triangle inequality of trace norm, the left-hand side (LHS) can be bounded as . For the right-hand side (RHS), we find . From , we can conclude that holds. Thus, any state which satisfies Theorem 1 must satisfy the dV criterion as well, i.e., Theorem 1 is strictly stronger than the dV criterion.
It was pointed out in Ref. Referee (), if one considers the enhancement of separability criteria with local filtering operations introduced in Ref. CM (), Theorem 1 reduces to the dV criterion Referee (). By constructive algorithms, states can be transformed (preserving entanglement or separability) to a filter normal form (FNF) or arbitrarily close to it CM (). For states in the FNF, holds. Therefore, for these states Theorem 1 is equivalent to the dV criterion. Hence, Theorem 1 is not expected to improve our entanglement detection capability if one is able to enhance other criteria with local filters. However, there are still some advantages Referee (). Firstly, separability criteria under filtering require numerical algorithms, which might pose problems as the dimensionality increases, while Theorem 1 is completely analytical. Secondly, the results rely on interesting, original and relatively simple ideas which might be used to improve other criteria. Finally, we will generalize Theorem 1 to the multipartite setting and derive a criterion (Theorem 2) for states with an even number of subsystems. To compare Theorem 1 with inequality (8) in CM () in the FNF, one can conclude that for they coincide, where () is the dimension of subsystem A (B). If is small, Theorem 1 is slightly better than inequality (8) in CM (), if is large, inequality (8) in CM () is drastically better than Theorem 1 CM ().
The transformation used in Theorem 1 can also be used to obtain a criterion which is similar to the partial transposition criterion.
Proposition 3. If a bipartite density matrix is separable, then the following inequality holds,
where stands for a partial transpose with respect to the subsystem .
Proof.– For a separable bipartite density matrix , it can be concluded that where we used the equalities horn2 () and . Notice that , and . Thus, the eigenvalues of can be written as , () and the singular values are , . Due to , it is obtained that Similarly, can be gotten. Thus, we have with the Cauchy-Schwarz inequality.
Actually, there is a tiny difference between Theorem 1 and Proposition 3. Right hand side of Eq. (8) is exactly two times as large as the one of Eq. (4). However, the coefficient 2 cannot be replaced with a smaller number. For example, when the separable state is substituted into inequality (8), the equal sign holds. It is one of the reasons that Proposition 3 is not as strong as Theorem 1. Consider Example 2, it can only detect entanglement for .
Multipartite systems.– Theorem 1 and Proposition 3 can be generalized to even-partite systems. A mixed state of an -partite system is defined to be separable if it can be represented in the form , where is a probability distribution and are pure states of subsystems. When is an even number, there are two different classes of bipartite partitions and introduced in Cai (). denotes that both sides of bipartite partition contain odd number of parties, and means even-number parties in each side note (). For instance, and note2 () when . An operator of their linear combination can be defined
where and . For and , and , respectively. In the following, we will present parallel criteria of permutation separability criteria based on . Recall the permutation operation introduced in permutation1 (), where acts on the th and th parties while leaves untouched the rest subsystems.
Theorem 2 (General criteria). If an -partite density matrix is separable ( is an even number), then the following inequalities
hold for separable states, where and is a rearrangement of .
Proof.- According to the proof of Theorem 1 and Proposition 3, it can be concluded that where we have used , and . Inequality (10) can also be proved with the same method.
has a certain meaning. Notice that all of the criteria presented in this paper can be viewed as parallel criteria of the CCNR, partial transposition and permutation separability criteria based on , and they are independent on the original criteria. It is considered that and the covariance matrix are of similar construction, where having the same singular values as can be viewed as evidence. Moreover, in multipartite systems seems to contain genuine entanglement information in the sense of explanation in Ref. Cai (). Therefore, the operator can be viewed as removing some local and separable information from . We make a conjecture that can also be used to obtain some other separable criteria.
In conclusion, we have presented a more powerful separability criterion, which is strictly stronger than the dV criterion, the CCNR criterion and its optimal nonlinear entanglement witnesses . The criterion is computationally simple and has been generalized in even-partite systems. It is worth noting that many other separability criteria may be improved with the method proposed in this paper. It is an interesting open question whether our criteria can be used to obtain lower bounds on the concurrence concurrence (); mintert (); chen (), since Ref. chen () has derived a lower bound of the concurrence based on the partial transposition and CCNR criteria.
We thank Heng Fan and Otfried Gühne for helpful discussions, and anonymous referees for pointing out that Theorem 1 reduces to the dV criterion for states in the FNF and other suggestions. This work was funded by the National Fundamental Research Program (Grant No. 2006CB921900), the National Natural Science Foundation of China (Grants No. 10674127 and No. 60621064), the Innovation Funds from the Chinese Academy of Sciences, Program for NCET.
Note added.– After resubmission of the revised manuscript, we became aware of a very recent preprint by Gittsovich et al. CM2 (), which has proved that Theorem 1 is a corollary of the CM criterion.
Appendix.– Here, We review the realignment operation introduced by Chen and Wu CCN (). For each matrix , the vector is defined as . Suppose is an block matrix with block size . The realignment is defined as . Therefore, a straightforward conclusion holds, CCN1 (); CCN (); permutation1 (), where holds.
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