Improved tests of genuine entanglement for multiqudits

We give an improved criterion of genuine multipartite entanglement for an important class of multipartite quantum states using generalized Bloch representations of the density matrices. The practical criterion is designed based on the Weyl operators and can be used for detecting genuine multipartite entanglement in higher dimensional systems. The test is shown to be significantly stronger than some of the most recent criteria.

Introduction. -Quantum entanglement is an important phenomenon in quantum systems responsible for quantum superiority. Many applications of quantum entanglement have been found, for instance, in entanglement swapping [1], quantum cryptography [2] and quantum secure communication [3].
The genuine multipartite entanglement (GME) is perhaps one of the most significant quantum phenomenon [4,5] and the study of measuring GME has been a nontrivial problem in quantum information [7][8][9][10][11][12][13][14][15][16][17]. Many criteria were found for tripartite states: separability of bipartite quantum systems via Bloch representation [18], sufficient tests [19] in the vicinity of the GHZ state, the W states and the PPT entangled states,sufficient conditions for three-particle entanglement [20],classification of mixed three-qubit state [21]. Similar criteria were found using local sum uncertainty relations [22] as well as for genuine tripartite entanglement based on partial transposition and realignment of density matrices [23]. Criterion for tripartite entanglement was also studied [24] in terms of quantum Fisher information. Genuine multipartite entanglement state was discussed in an electron-positron system in [17]. In [25], the authors detected GME based on local harnessing or dephasing quantum non-Markovian operations. Tests based on norms of correlation tensors were given in [26,27]. For tripartite and four-partite states, separability tests were given by using the Bloch representation [28], by the matrix method [29], by the upper bound of the Bloch vectors [30],and by using witness operators [31]. Detection of genuine tripartite entanglement was given by multiple sequential observers [32],and by local marginals [33]. For higher dimensional quantum system, the separable criteria and k-separable criteria for general n-partite quantum states were also presented in [34,35,41].
Block representation has been used in detecting bipartite entanglement widely, see for example [18,[36][37][38][39][40]. In this paper, we will use the generalized Bloch representation and the Weyl operators to study the genuine entanglement of multipartite quantum systems. The Weyl representation uses a uniformed generators of the Lie algebra su(d) to generate the principal basis, thus it is easier to treat higher dimensional cases in practical computation. The tests are presented by using some special matrices from the density matrices, and we derive a better mathematical bound for the norm. Our new test has improved some of the recently known tests obtained by similar or different consideration.
The layout of the paper is as follows. In section 2, we quickly review some basic notions concerning the Weyl operators as generalized Pauli spin matrices and discuss their main properties. We then construct certain tensors based on the density matrices of the quantum system to estimate its bound under various situations, which naturally lead to new criterion for the GME. In section 3, we discuss how to generalize our results to higher dimensional cases, and the conclusion is given in section 4. Our test is found to be significantly stronger than some of the recently available criteria when the number of particles is large.
Genuine tripartite entanglement. -We start by considering the GME for tripartite states. Let E ij be the p-1 unit matrices of size d, where (E ij ) kl = δ ki δ jl . Let ω be a fixed d-th primitive root of unity, then the principal basis matrices or the Weyl operators are defined by summed over Z d = Z/dZ. The Weyl operators obey the rule [42,43]: For a column vector X let X = √ X † · X be the norm, and we also use it for the norm of a square matrix viewed as a prolonged column vector.
Any state on H d can be written uniquely as summed over nonzero pairs of modulo d integers, where A is the (d 2 − 1)-vector of nonzero Weyl operators, T is the column vector of size For any state ρ ∈ H d1 1 ⊗ H d2 2 , ρ has the following Blochlike expression based on the Weyl operators: where T (1) , T (2) , T (12) are the matrices with entries u i2j2 ) † ), and A ds = (A isjs ) is the column vector of Weyl operators on H ds (s = 1, 2). Setting we have the following lemma.
Proof Suppose the tripartite quantum state is separable under the bipartition i|jk, then where the factor states are Therefore where t stands for transpose. Lemma 1 and 2 imply that where we have used the norm property and |α β| tr = |α |β for vectors |α and |β . A mixed state is called genuine multipartite entangled (GME) if it cannot be expressed as a convex combination of biseparable states. In this note we exclusively consider GME of symmetrically coherent quantum systems, which have the property that if the quantum state is biseparable in one bipartition i|jk, then the system will also be biseparable in other bipartitions and moreover when the quantum state is a convex sum of biseparable quantum states ρ = i p i ρ i , then the summands ρ i will obey N (ρ i ) ≤ N (ρ) for any bipartitioned N . This coherent condition is satisfied by physically important quantum states, while the state displays visible invariance under permutation symmetry in the Hilbert space.
For a symmetrically coherent tripartite state ρ, let T (ρ) = min{ N 1|23 tr , N 2|13 tr , N 3|12 tr } and define where the maximum is taken over all permutations (ijk) of (123). We stress that the maximum is needed to generate a lower bound, as a general state can be entangled in an unexpected bipartition.
We have the following test of GME.
Proof Suppose ρ is a biseparable mixed state, then ρ = ρ 1 + ρ 2 + ρ 3 , where ρ i are respectively separable states k with i p i + j r j + k s k = 1. As our quantum state is symmetric, using Theorem 1 and the coherent property To see how our test fares, we consider the following example.
Consider the tensor space ⊗ n i=1 H di . Let A (i) = (A factors. Then any quantum state ρ over ⊗ n i=1 H di can be uniquely written as where d = d 1 · · · d n , T (i) is the vector with components t Let A j be the sum of T i1···ij 2 , i.e.
Remark 2. Note that the first term of the bound is exactly that of [29]. For the case D ≥ d 2 , the second term sharper than that of [29] in this case. Actually in general, our bound is almost always significantly stronger than that of [29]. For instance when d i = d, the second term in the bound is less than − n (n−1)(n−2) (d n−2 − 1), which means that the new bound is significantly sharper.
Conclusions. -Using the generalized Bloch representations of density matrices via Weyl operators, we have come up with several general tests to determine genuine entanglement for multipartite quantum systems. Our approach starts with some finer upper bounds for the norms of the correlation tensors, which then lead to then new entanglement criteria for genuine tripartite entangled quantum states. The key technical point is based on certain matrices compiled from the subtensors of the correlation tensor of the density matrices. The results are then generalized to higher dimensional multipartite symmetrically coherent quantum systems to detect genuine entanglement in arbitrary dimensional quantum states. Compared with previously available criteria, our new results detect more situations, which are explained in details with several examples. When the number of the particles is large, our criteria is found to be significantly stronger to detect GME.