1. Introduction
Originally introduced by Schr
dinger [
1] the Einstein-Podolsky-Rosen (EPR) steering for bipartite systems was considered as a ’spooky action at distance’ [
2] in the sense that one party can steer another distant party’s state instantly. The concept of EPR steering was proposed by Wiseman, Jones, and Doherty in 2007 [
3]. Since then the EPR steering has been systematically studied. Many different methods were proposed to detect and quantify the steerability of bipartite quantum states [
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15], together with many applications in quantum information processing tasks including one-sided device-independent quantum key distribution, random generation and one-sided device-independent quantum self-testing of pure quantum states, subchannel discrimination, quantum communication et al. [
16,
17,
18,
19,
20,
21,
22,
23,
24,
25].
The EPR steering lies between quantum nonlocality and quantum entanglement. A bipartite state is quantum nonlocal if it does not admit a local hidden variable model [
26], while it is EPR steerable if it does not admit a hidden state model [
3].
Bipartite steering is defined as follows. Alice and Bob share a quantum state
. Alice performs black-box measurements
A with outcomes
a, denoted by
(
and
, with
denoting the identity operator). The set of unnormalized conditional states
on Bob’s side is called an assemblage. Each element in this assemblage is given by
Alice can not steer Bob if
admits a local hidden state model (LHS), i.e.,
admits the decomposition
where
denotes classical random variable which occurs with probability
satisfying
is the probability given by the black-box measurement on Alice’s side,
are some local hidden states. Bob performs measurement
B with outcomes
b, denoted by
on the assemblage. The joint probability is
.
is said to be a steerable state from Alice to Bob if
does not admit a local hidden variable-local hidden state (LHV-LHS) model of the form,
Different from quantum entanglement and quantum nonlocality, EPR steering is asymmetric in general, which means that Alice can steer Bob but not vice versa for some bipartite quantum states
[
27]. The bipartite quantum nonlocality and EPR steering can be detected by detecting the EPR steering and quantum entanglement of some newly constructed quantum states, respectively [
28,
29,
30].
The multipartite steering is an important resource in quantum communication networks [
31,
32,
33] and in one-sided or two-sided device-independent entanglement detections [
34,
35]. Some ambiguities exist in the definition of multipartite steering. With respect to the typical spooky action at a distance [
31,
32,
33], and the semi-device independent entanglement verification scheme [
34,
35], two different approaches have been introduced to define the multipartite steering [
31,
34,
35]. One approach is to define genuine multipartite steering in terms of the steering under bi-partitions. A tripartite state
is defined to be genuine tripartite steerable if the state does not admit the mixtures of bi-partitions where in each partition (e.g.,
) the two-party state (e.g.,
) is allowed to be steerable. Linear inequalities have been derived to detect this kind of genuine multipartite steering [
31] and used in experimental demonstrations [
32,
33].
Another approach to defining tripartite steering and genuine tripartite steering is given as follows [
34,
35]. Let
be the joint probability that Alice, Bob and Charlie perform measurements
A,
B and
C with outcomes
b and
c, given by measurements operators
,
and
, respectively. A quantum state
is said to be tripartite steerable from Alice (untrusted party) to Bob and Charlie (trusted parties) if
does not admit a fully LHV-LHS model such that
where
and
are the distributions from the local hidden states
and
, see Equation (
13) in [
35] and Equation (
2) in [
36].
The genuine tripartite steering has been defined in [
34,
35,
36]. Alice measures her system so as to nonlocally influence the state of the other two parties. The ensemble of the unnormalized states is given by
If the ensemble prepared on Bob’s and Charlie’s sides cannot be reproduced by a biseparable state as Equation (
6),
with
then
is not genuine tripartite steerable from Alice to Bob and Charlie. Therefore, if
is genuine tripartite steerable from Alice to Bob and Charlie, then each member of the ensemble (
5) can not be expressed as [
34,
36],
with
and
The first term on the right-hand side of (
7) stands for that Alice cannot steer Bob and Charlie. Bob and Charlie share entanglement and a local hidden entangled state
. The other two terms imply that there is no entanglement between Bob and Charlie, and Alice can steer one of the two systems but not both: the second (third) term stands for that Alice can steer Bob (Charlie) but not Charlie (Bob).
A state is genuine tripartite steerable from Alice to Bob and Charlie if the joint probability
does not admit a hybrid LHV-LHS model [
35,
36],
where
is the distribution on Alice’s side from black-box measurements performed on a quantum state,
and
are the distributions from measurements on quantum states
and
can be reproduced by quantum state
shared by Bob and Charlie.
and
are distributions from a quantum state with untrusted
and trusted
and
. When
and
are probabilities from the local hidden states
and
, respectively, since
and
are the trusted parties. We always use
or
c and
or
to represent the distribution from measurements on two parties with one party trusted and the other two untrusted in this paper.
A quantum state
is said to be tripartite steerable from (untrusted) Alice and Bob to (trusted) Charlie if the joint probability
does not admit a fully LHV-LHS model such that
where
and
are the probabilities from the black-box measurements,
is the distribution from local hidden state
, see also the definition given in [
35,
36].
The genuine tripartite steering from Alice and Bob to Charlie has also been defined in [
34,
35,
36]. Alice and Bob measure their systems so as to nonlocally influence the state of Charlie’s. The ensemble prepared on Charlie’s side cannot be reproduced by a biseparable state as Equation (
6). Each member in the ensemble of unnormalized states can not be given by
with
and
. The first term on the right-hand side of (
10) stands for that Alice and Bob cannot jointly steer Charlie, and the second (third) term stands for that only Bob (Alice) can steer the state of Charlie. A state is genuine tripartite steerable from Alice and Bob to Charlie if the joint probability
does not admit a hybrid LHV-LHS model such that
where
and
are the distributions on Alice’s and Bob’s sides, respectively, arising from black-box measurements performed on a quantum state.
is the distribution produced from black-box measurements performed on a quantum state.
is the distribution from the state
.
and
are probabilities from a 2-qubit quantum state with untrusted
and
and trusted
. When
are probabilities from the local hidden states
and
, respectively, since
is the trusted party.
Entropic steering inequalities and semi-definite-program have been adopted to investigate the detection of multipartite steering [
34,
36,
37]. In the following, we construct new quantum states with respect to given three-qubit states and detect the tripartite steering and genuine tripartite steering analytically in terms of the tripartite entanglement and the genuine tripartite entanglement of the newly constructed quantum states. The entanglement of the newly constructed states can be detected by using the entanglement witness without full tomography of the states. By detecting the entanglement of the newly constructed states, the tripartite steering and genuine tripartite steering can be detected without using any steering inequalities. Since the “complexity cost” (the number of possible patterns of joint detection outcomes that can occur, see [
38]) for the least complex demonstration of entanglement is less than the “complexity cost” for the least complex demonstration of EPR steering [
29,
38], our scheme reduces the “complex cost” in experimental steering demonstration.
2. Main Results
A quantum state is fully separable if the joint probability
satisfies the condition,
Fully separable states are neither tripartite steerable states from Alice to Bob and Charlie nor from Alice and Bob to Charlie. From (
4) and (
9) a state which is not tripartitely steerable from Alice to Bob and Charlie is not tripartitely steerable from Alice and Bob to Charlie, i.e., tripartite steering from Alice and Bob to Charlie is stronger than that from Alice to Bob and Charlie.
A quantum state is bi-separable if the joint probability
satisfies the condition,
where
A bi-separable quantum state must not be a genuine tripartite steerable state from Alice to Bob and Charlie or from Alice and Bob to Charlie. From (
8) and (
11) a state which is not genuine tripartite steerable from Alice to Bob and Charlie is not genuine tripartite steerable from Alice and Bob to Charlie. As a result, given in [
34], the noisy GHZ state demonstrates the genuine tripartite steering from Alice to Bob and Charlie in a larger region compared to that from Alice and Bob to Charlie. For general tripartite quantum states, the genuine tripartite steering from Alice and Bob to Charlie is also stronger than that from Alice to Bob and Charlie.
Theorem 1. Let be a three-qubit quantum state and with and the identity matrix. We have
- (i)
If is genuine tripartite entangled, then is genuine tripartite steerable from Alice to Bob and Charlie for ;
- (ii)
If is tripartite entangled, then is tripartite steerable from Alice to Bob and Charlie for
The statements in Theorem 1 are equivalent to the following:
- (i’)
If is not genuine tripartite steerable from Alice to Bob and Charlie, then is bi-separable for ;
- (ii’)
If is not tripartite steerable from Alice to Bob and Charlie, then is fully separable for .
Proof of Theorem 1. We prove the theorem by proving its converse negative proposition: if is not a genuine tripartite steerable state from Alice to Bob and Charlie, then is a bi-separable state; if is not a tripartite steerable state from Alice to Bob and Charlie, then is a fully-separable state.
Firstly we give the (unnormalized) conditional quantum state on Alice’s side after Bob and Charlie perform measurements and on . Then the Bloch sphere representation of the conditional state can be expressed according to the joint probabilities. Lastly from the condition that is not genuine steering or steering from Alice to Bob and Charlie, we prove that is the convex combination of some qubit quantum states if satisfies certain conditions.
Step 1. From (
14) we have the (unnormalized) conditional state on Alice’s side when Bob and Charlie perform measurements
and
on
,
where
are Pauli matrices
and
, respectively.
Step 2.y and
are given by the joint probabilities,
with
and
the eigenvectors of
with respect to the eigenvalues 1 and
of
, respectively.
Step 3. (I). If
is not a genuine tripartite steerable state from Alice to Bob and Charlie, the joint probabilities admit a hybrid LHV-LHS model as follows,
where
and
are probabilities from qubit states
and
on Bob’s and Charlie’s sides, respectively.
Step 4. We now prove that the following conditional state
is the convex combination of qubit quantum states when
satisfies certain conditions,
where
Since and
when
and
are semi-definite positive matrices with trace one. They are quantum states when
. Therefore,
with
and
and
and
. From (
13)
is a bi-separable state. Namely, if
is genuine tripartite entangled, then
is genuine tripartite steerable from Alice to Bob and Charlie for
.
Step 3’. (II). If
is not tripartite steerable from Alice to Bob and Charlie, the joint probabilites admit LHV-LHS model,
and
Step 4’. Therefore,
is given by the convex combination of some qubit quantum states when
satisfies certain condition,
where
Since
for
when
is a semi-definite positive matrix when
Therefore,
is a quantum state when
Since
from (
12),
is fully separable. Hence, if
is tripartite entangled,
must be tripartite steerable from Alice to Bob and Charlie for
. □
Theorem 2. Let be a three-qubit state and where and is the identity matrix. We have
a) If is genuine tripartite entangled, then is genuine tripartite steerable from Alice and Bob to Charlie for ;
b) If is tripartite entangled, then is tripartite steerable from Alice to Bob and Charlie for .
The proof of Theorem 2 is given in
Appendix A. The statements in Theorem 2 are also equivalent to the following:
- (a’)
If is not genuine tripartite steerable from Alice and Bob to Charlie, then is bi-separable for ;
- (b’)
If is not tripartite steerable from Alice to Bob and Charlie, then is fully separable for .
Next, we illustrate our theorems with detailed examples.
Example 1. Consider , where . The defined in Theorem 1 is a matrix with entries , . The state is genuine entangled if [39], and is entangled if one of the following three inequalities is satisfied: or [40]. Therefore, from Theorem 1 we have that when this state is tripartite steerable and also genuine tripartite steerable from Alice to Bob and Charlie. Similarly, according to the entanglement of , from Theorem 2 we obtain that is tripartitely steerable from Alice and Bob to Charlie when . While in [36], genuine tripartite steering from Alice to Bob and Charlie is detected only when Example 2. Consider with . Similar to Example 1, by using the entanglement criteria given in [39,40] and Theorem 1, we have that is genuine tripartite entangled when , and thus is genuine tripartite steerable from Alice to Bob and Charlie. When is an entangled state, and is tripartite steerable from Alice to Bob and Charlie. Furthermore, from the entanglement of and Theorem 2, we have that is tripartitely steerable from Alice and Bob to Charlie when . While in [36] is proved to be tripartite steerable from Alice to Bob and Charlie when and genuine tripartite steerable from Alice to Bob and Charlie when In [35] is shown to be tripartite steerable from Alice to Bob and Charlie when and genuine steerable when . is tripartite steerable from Alice and Bob to Charlie when and genuine steerable when . In [37] is shown to be tripartite steering from Alice to Bob and Charlie when for two measurement settings, and for three measurement settings. is tripartite steering from Alice and Bob to Charlie when for two measurement settings, and for three measurement settings. Hence, in the case of detecting genuine tripartite steering from Alice to Bob and Charlie, our proposed method is stronger compared with the criteria given in [35,36,37], and in the case of tripartite steering from Alice to Bob and Charlie, our proposed method is stronger with respect to the criteria in [36,37]. The results are listed in Table 1. Next, instead of the criteria given in [
39,
40] we first present improved separability criteria. Consider a three-qubit state
. Let
be the entries of the matrix
. If the state
is bi-separable, we have
and
under the bipartition
;
and
under the bi-partition
;
and
under the bi-partition
. Hence for any pure bi-separable quantum state
, we have
. The above inequalities are also satisfied for bi-separable mixed states by the convex roof construction. Therefore, we have
Proposition 1. Let be any three-qubit state and the entries of the matrix . Then is genuine tripartite entangled if Example 3. Let us consider now with . Using the inequality (17), we have that the state defined in Theorem 1 is genuine tripartite entangled when whereas from the result given in [39,40], is genuine tripartite entangled when . Hence, from Theorem 1 when is genuine tripartite steerable form Alice to Bob and Charlie. Concerning the tripartite steerability, it has been shown in [41] that is tripartite entangled if is not a positive semi-definite matrix, where Γ is the transpose with respect to subsystems or . From this criterion we have that is tripartite entangled when , i.e., is tripartite steerable form Alice to Bob and Charlie for . Similarly from the given in Theorem 2 and the criteria given [41], we have that is tripartite steerable form Alice and Bob to Charlie when . While in [36], is proved to be tripartitely steerable from Alice to Bob and Charlie when and no genuine tripartite steerability is detected. In [37], is shown to be tripartite steering from Alice to Bob and Charlie when for two measurement settings, and for three measurement settings. is tripartite steering from Alice and Bob to Charlie when for two measurement settings, and for three measurement settings. Hence, in the case of detecting tripartite steering and genuine tripartite steering from Alice to Bob and Charlie, our proposed method is stronger with respect to the criteria in [35,36,37]. The results are listed in Table 2. One point to be stressed here is that, instead of the numerical results based on a semi-definite program in [
34], our results are derived analytically. For the GHZ state and W state mixed white noise, our criteria are powerful in detecting the genuine tripartite steering from Alice to Bob and Charlie. Nevertheless, the criteria can not detect any genuine tripartite steering from Alice and Bob to Charlie, which illustrates that the genuine multipartite steering from Alice and Bob to Charlie is a kind of stronger quantum correlation and some more powerful criteria are needed.