Steering Witness and Steering Criterion of Gaussian States.

Quantum steering is an important quantum resource, which is intermediate between entanglement and Bell nonlocality. In this paper, we study steering witnesses for Gaussian states in continuous-variable systems. We give a definition of steering witnesses by covariance matrices of Gaussian states, and then obtain a steering criterion by steering witnesses to detect steerability of any (m+n)-mode Gaussian states. In addition, the conditions for two steering witnesses to be comparable and the optimality of steering witnesses are also discussed.

1. Introduction

Entanglement plays an important role in quantum information theory, which has been widely used in quantum information processing [1,2,3]. The detection of entanglement has attracted much attention in recent years (see [4,5,6,7,8,9,10,11,12,13,14,15]). Among these criteria, the entanglement witness (EW) criterion provides a sufficient and necessary condition for the separability of a bipartite quantum state ([5]). A self-adjoint operator W acting on a separable complex Hilbert space is an EW if W is not positive and holds for all separable states . It was shown that a bipartite state is entangled if and only if there exists at least one EW detecting it. Obviously, there does not exist an EW that can detect all entangled states. So, the concept of the optimal EW is proposed in [8] and some methods are given to check the optimality of EWs, for example, see [8,11,12,15].

In 1935, Einstein, Podolsky and Rosen (EPR) first discovered the anomalous phenomenon of quantum states in multipartite quantum systems, which is contrary to the classical mechanics ([16]). In order to capture the essence of the EPR paradox, the notion of EPR steering was first introduced by Schrdinger in [17]. EPR steering is a quantum correlation between entanglement and Bell nonlocality. Different from entanglement and nonlocality, this correlation is inherently asymmetry with respect to the observers.

In recent years, EPR steering has attracted many authors’ attention. It has been shown that EPR steering plays a fundamental role in various quantum protocols, secure communication, and other fields ([18,19,20]). Various EPR steering criteria have been derived. For example, Cavalcanti and James in [21] obtained the experimental criterion of EPR steering from entropy uncertainty relations. Ji et al. in [22] obtained steerability criteria by using covariance matrices of local observables, which are applicable for both finite- and infinite-dimensional quantum systems. Wittmann et al. in [23] gave EPR steering inequalities with three Pauli measurements; and then, as a generalization of the Pauli matrices, Marciniak et al. in [24] found EPR steering inequalities with mutually unbiased bases. For continuous-variable systems, the authors in [25] performed a systematic investigation of EPR steering for bipartite Gaussian states by pseudospin measurements. Kogias and Adesso [26] gave a measure of EPR steering for two-mode continuous variable states.

Inspired by EW, in this paper, we will try to consider quantum EPR steering witness for Gaussian states in continuous-variable systems. This paper is organized as follows. In Section 2, we recall the concepts of Gaussian states and quantum EPR steering, and some known EPR steering criteria for Gaussian states. Section 3 is devoted to giving a definition of steering witness for Gaussian states in terms of covariance matrices and then discussing properties of steering witness. Based on steering witnesses, a steering criterion for bipartite -mode Gaussian states are obtained. The conditions for two steering witnesses to be comparable are given and the optimality of steering witnesses are also obtained. Section 4 is a brief conclusion.

2. Definition and Criterion of Gaussian Quantum EPR Steering

In this section, we briefly introduce the notion of Gaussian quantum steering.

Gaussian states. Recall that a quantum system is associated with a separable complex Hilbert space H. A quantum state on H is a positive operator with trace 1. For arbitrary state in an n-mode continuous-variable system with state space H, its characteristic function is defined as where , is the Weyl displacement operator, . As usual, and () stand for, respectively, the position and momentum operators, where and are the creation and annihilation operators in the kth mode, satisfying the Canonical Commutation Relation (CCR) Particularly, is called a Gaussian state if is of the form where is called the mean or the displacement vector of and is the covariance matrix (CM) of defined by with ([27]). Here stands for the algebra of all matrices over the real field . So, any Gaussian state with CM and displacement vector will be represented as . Note that is real symmetric and satisfies the condition , where with for each k. Assume that is any -mode Gaussian state. Then its CM can be written as

Qauntum steering. Now let us recall the definition of steerability. A measurement assemblage is a collection of positive operators satisfying for each x. Such a collection represents one positive-operator-valued measurement (POVM), describing a general quantum measurement, for each x. In a (bipartite) steering scenario, one party performs measurements on a shared state , which steers the quantum state of the other particle. If Alice performs a set of measurements , then the collection of sub-normalized “steered states” of Bob are an assemblage with If every assemblage on Bob can be explained by a local hidden state (LHS) model, of the form where is a hidden variable, distributed according to , are “hidden states” of Bob, and are local “response functions” of Alice, then we say that it has LHS form, or does not demonstrate steering ([28]). If there exist measurements such that does not admit such an LHS decomposition, we say that the state is steerable from A to B. If for all measurements we can never demonstrate steering with a given state, we say it is unsteerable from A to B. Symmetrically, we can define the steerability of from B to A. Steering is a quantum correlation between entanglement and Bell nonlocality. However, unlike Bell nonlocality and nonseparability, which are symmetric between Alice and Bob, steering is inherently asymmetric.

Gaussian Positive Operator-Valued Measurement. An m-mode Gaussian Positive Operator-Valued Measurement (GPOVM) is defined as where is the m-mode Weyl displacement operator, , is a zero mean m-mode Gaussian state with CM , which is called the seed state of the GPOVM . So, we can denote a GPOVM with the seed CM by ([29,30]).

A criterion for unsteerability of Gaussian states. For arbitrary bipartite Gaussian states, the authors in [28] derived a linear matrix inequality that decides the question of steerability via GPOVMs.

[ where

By Theorem 1, under the restriction of GPOVMs, Equation (

3. Steering Witness for Gaussian States and Their Comparability

In this section, we will first give a definition of steering witness for Gaussian states, and then discuss some properties of steering witness.

Denote by the set of all real symmetric matrices. Note that a CM can describe a physical quantum state if and only if it satisfies the bona fide uncertainty principle relation . Let stand for the set of all CMs satisfying uncertainty principle relations in -mode continuous-variable systems, that is, For the convenience, write . Let We call the elements in unsteerable CMs from A to B as Theorem 1. It is easily checked that is a closed and convex set. The following result gives another property of .

Assume that

For any positive matrix , by Equation (3), it is obvious that . For any , as and we get . □

Next, write We call any element W in the steering witness from A to B in -mode bipartite continuous-variable systems with subsystems A and B, where .

The following theorem gives a criterion of detecting steerability of any -mode Gaussian states by steering witnesses.

(Steering witness criterion) Assume that

For the “only if” part, if holds for all , by Equation (5), . It follows from Theorem 1 that is unsteerable from A to B.

For the “if” part, on the contrary, suppose that is steerable from A to B. Then is steerable from A to B as Theorem 1, that is, with . Since the set is convex and closed, by the Hahn-Banach theorem, there exists some such that

We claim . Otherwise, assume . If is not positive, there is a negative eigenvalue of with the corresponding eigenvector . Take any , any and let . By Proposition 1, . Note that This means that, for sufficient large , we have , which yields a contradiction to Equation (6). Thus, is positive, and so . Further, we can conclude . In fact, by Williamson normal form Theorem, for any CM , there exists a symplectic matrix such that with . So However, this leads to a contradiction with .

Hence . By letting in Equation (6) yields which implies with , a contradiction. Therefore, is unsteerable from A to B, and thus is unsteerable from A to B. □

By Theorem 2, we see that, for any

In the rest part, we will discuss the properties of steering witnesses. Given a steering witness , denote the set of CMs detected by W by It is obvious that any two steering witnesses and have one of the following three relations:

(1) or ;

(2) ;

(3) and , .

For any two steering witnesses

Particularly, for a steering witness W, we say that W is optimal if there is no other steering witness finer than W.

The following result gives the relation of two comparable steering witnesses.

Suppose that

Assume that are two steering witnesses with and .

(i) Assume that , but . Take any and any positive number . Write Then and So for all .

On the other hand, note that as . Take any with . Then and so This implies for such x, a contradiction.

(ii) Assume that . Letting , then and By (i), we have , and so .

(iii) If , by taking and with , we have and . Write . It is obvious that and By (i), one gets , that is, . So Note that the last inequality is due to (ii). Thus, Equation (7) implies and hence

Finally, we will show In fact, for any , we have , and by (ii), . Thus, , and so □

In the following theorem, we give a necessary and sufficient condition for two steering witnesses to be comparable.

Suppose that

Assume that are two steering witnesses. If there exists some and some positive matrix with for all such that then, for any , we have It follows that . So . By Definition 1, one obtains .

Conversely, if , by taking and by Theorem 3, we have for all , that is, On the other hand, for any , by Theorem 3 (iii), one has , that is, Combining Equations (8) and (9) gives

Now, let with . Obviously, and holds for all by Equation (8).

Finally, if X is not positive, then there is a negative eigenvalue of X with the corresponding eigenvector . Take any , any and let . Obviously, . Note that Also note that by Equation (10). These yield a contradiction. So X is positive.

The proof of the theorem is finished. □

For the optimality of steering witnesses, we have

Suppose that

The “if” part is obvious by Theorem 4.

For the “only if” part, assume that there is some and some positive matrix satisfying for all such that is a steering witness. Then , where and with for all . By Theorem 4 again, . A contradiction. □

Finally, we discuss the question when different steering witnesses can detect some common steering CMs.

For any two steering witnesses

To prove the theorem, two lemmas are needed.

Suppose that

For two steering witnesses with , we have for all . By Theorem 3(ii), , and so . This means , that is, .

In addition, if , then and as . Thus This implies . So . □

Assume that

Assume, on the contrary, that and . Take for . Write Note that is a convex set. So and thus . Hence there exists some such that and . If , then , and for sufficiently small , we have A contradiction. Similarly, if , by considering for sufficiently small , one can also obtain a contradiction.

Therefore, or . The proof is completed. □

Take any two steering witnesses . If there exists some such that is not a steering witness, then , that is, .

For the “only if” part, assume that and for all . Then . Since , by Lemma 2, we have either or . When varies from 0 to 1 continuously, varies from to continuously. Denote . If , then there must be exist some with such that is not a steering witness, that is, . Otherwise, for all with , we have . Since and , for all , one has and for sufficiently small , a contradiction. Hence .

Similarly, one can show . So there exists some , such that is not a steering witness. The proof is finished. □

4. Conclusions

Quantum EPR steering is an important quantum resource. It is a fundamental and important question of how to detect steerability of quantum states. In this paper, we investigated steering witnesses of Gaussian states in continuous-variable systems. We give a definition of steering witnesses by covariance matrices of quantum states, and then present a steering witness criterion of any -mode Gaussian state to be unsteerable by the Hahn-Banach theorem. In addition, the conditions for any two steering witnesses to be comparable and the optimality of steering witnesses are also discussed. Our investigations may highlight further researches on steering witnesses.