Entanglement criterion independent on observables for multipartite Gaussian states based on uncertainty principle

Quantum entanglement is one of important resources for quantum communication. Entanglement criteria help us detect entangled states. One of important criteria is the local uncertainty relation (LUR) entanglement criteria, which is studied extensively. However, all existent LUR criteria are dependent on the chosen observables. In the paper, applying the uncertainty principle, we improve the LUR criteria to obtain entanglement criteria for multipartite Gaussian states, which are independent on observalbes.

, i i is the pair of the position and momentum operators in the ith mode (party) and f is a computable function. The criteria in 21 are available for N-party and N-mode CV states (that is, there is only one mode in each party) and dependent on observables û and v. However, all of the above criteria are dependent on the choice of observables. In the present paper, improving the LUR criteria, we devote to building an entanglement criterion of Gaussian states, which is independent on observables. Furthermore, it is mentioned that the criterion in 21 is available for N-party and N-mode CV states (that is, there is only one mode in each party). The criterion in the present paper can be executed for N-party systems with arbitrary modes in each party (See Theorem 2.1 and Corollary 2.2).
Next let us introduce some definitions and notations about Gaussian states. The characteristic function χ of an arbitrary n-mode density operator ρ is defined as is the n-mode Weyl displacement operator 23 . Here, a î † and a î are the creation and annihilation operators in the ith mode satisfying the canonical commutation relation a a a a a a [ , ] and [ , ] [ , ] 0 and ∈  m n 2 denote the covariance matrix (CM) and the mean of ρ, respectively. γ fulfills the Robertson-Schrödinger uncertainty relation . Let p î and q i be the position operator and momentum operator on the ith mode, hen the CM of ρ can be calculated as follows: where m j = tr(ρR j ) is the jth coordinate of the displacement vector m. Note that entanglement of a Gaussian state is independent on its displacement, so we assume that all Gaussian states is with zero displacement in the paper. We also recall that a multipartite quantum state ρ on

Results
is the pair of the position and momentum operators in the jth mode of the ith party. Sometime we write the set of all quantum states on H. We have the following main result. Proof. See the method section.

If ρ is fully separable, then for two sets of arbitrary real numbers
The following example helps us understand how to obtain the matrix M , , Next we will design a optimization program for entanglement criteria of Gaussian states. Firstly, we have the following corollary from Theorem 2.1.

Corollary 2.2 There exists entanglement among H H H
is obtained by removing the sth row and the sth column of Γ α β ρ M , , for all s ∈ {1, 2, …, n}\{i 1 , i 2 , …, i l }. Applying the Corollary 2.2, we can detect entanglement of a multi-party Gaussian state by solving the following optimization problem.

 
We design the following steps to solve the OP problem. S1. We compute and collect leading principal minors |Γ k (i 1 , for each variable respectively; S3. We get stationary points of the equation set consist of β β on all stationary points. Finally we obtain the minimal value of |Γ k (i 1 , i 2 , …, i l )|. Now, we consider an example of the multi-mode pure symmetric Gaussian state is introduced in 24 . Arbitrary a 5-mode pure symmetric Gaussian state ρ s has the following covariance matrix: where a ≥ 1 and In 24 , entanglement of the above state in the case a = 1.2 is discussed 16 . Here we take a = 1.1. We first deal with the partition 1|2|3|4|5, that is, five modes and five parties. In order to determine when the state ρ s with the covariance matrix in Eq. (1) is entangled, it follows from Theorem 2.1 and Corollary 2.2 that we need to check when the following matrix Γ s , which is restructured by M J s i 2 γ = − ρ , is not positive for some real scalars α i and β i , i = 1, 2, 3, 4, 5 When a = 1.1, we calculate and obtain that the minimal values of 2 × 2 five leading principal minors of Γ s are all negative by Mathematica. So Γ s is not positive. It follows that the corresponding symmetric Gaussian state is entangled in the partition 1|2|3|4|5.
Furthermore, now if one want to ask whether or not there exists entanglement among the second, fourth and fifth mode. Then we only need to check positivity of the following Γ s : When a = 1.1, the matrix (3) is not positive, and so there exists entanglement between the second, fourth and fifth mode, similar to the discussion of the case of partition 1|2|3|4|5.

Discussion
The local uncertainty relation (LUR) criterion is one of important classes of entanglement criteria for the continuous variable system. It is dependent on chosen observables. Here, we improve LUR criteria and obtain observable-independent entanglement criteria for arbitrary multi-party and multi-mode Gaussian states. In particular, the criteria can be implemented by a by a minimum optimization computer program. It is also mentioned that one of the further open problems is to discuss the computational complexity of the optimization procedure of the OP problem.

Methods
Before the proof of Theorem 2.1, we need the following lemmas. The following lemma can be checked straightforwardly.

Lemma 2. Let
. If ρ is fully separable, then ( ) , and as we know that entanglement of Gaussian states is independent on the mean, we have Note that entanglement of a Gaussian state is independent on its first moment (i.e., the mean) of a Gaussian state, so we assume that 〈U〉 = 0 = 〈V〉. On the other hand,