# Multipartite steering inequalities based on entropic uncertainty relations

## Abstract

We investigate quantum steering for multipartite systems by using entropic uncertainty relations. We introduce entropic steering inequalities whose violation certifies the presence of different classes of multipartite steering. These inequalities witness both steerable states and genuine multipartite steerable states. Furthermore, we study their detection power for several classes of states of a three-qubit system.

Quantum steering is a type of quantum correlation, owned by some entangled
states of composite systems. It enables one subsystem to influence
the state of the others, with which it shares the entangled state,
by applying local measurements. The concept of quantum steering, for
bipartite systems, was introduced in the early days of quantum mechanics
by Schrodinger (1), who recognized that this class of states
allow one part “to steer” the state of the other into an eigenstate
of an arbitrary observable, and hence they express the “spooky action
at distance” discussed in (2). Nowadays we are aware that
three types of quantum entanglement exist: Bell nonlocality, steerability
and nonseparability. Bell nonlocality correlations are the strongest
ones and are owned by global states that violate some Bell inequalities
(3), which are related to the non existence of local hidden
variable (LHV) models. Then we have quantum steering, which was formalized
in 2007 by Wiseman *et.al* (4) as the incompatibility
of quantum mechanics predictions with a local hidden state (LHS) model,
where the parties have pre-determined states. Formally, given a bipartite
system owned by Alice and Bob that share a state , we
say that the correlations demonstrate quantum steering if the joint
measurement probabilities cannot be expressed as:

(1) |

where and are respectively the outcomes of the measurements
of Alice’s observable and Bob’s observable . In the
above equation represents the probability
of obtained from a quantum pre-determined state
that depends only on which occurs with probability
and not on . Instead, the conditional probability of
on an arbitrary state, which may depend on , will be indicated
as Conversely if for any choice of
measurements equation (1) holds, then the
state is called nonsteerable, in the sense that it admits a LHS model.

At the bottom of the hierarchy there is entanglement (5); (6),
which can be defined as the existence of states of composite systems
that cannot be given as a convex combination of states of the individual
subsystems, namely separable states. Interestingly, these three notions,
which can be only found in nonseparable states, coincide for pure
states.

All of these types of correlations have been generalized to multipartite
systems. However for steerability there exist different approaches
(8); (7); (9) that go beyond the bipartite
scenario. Here we consider the one discussed in (7),
which also allows one to discuss the notion of post-quantum steering
(10), which does not exist for bipartite systems.

In this paper we introduce a number of entropic inequalities whose
violation certifies multipartite steering. Steerability is an asymmetric
concept, i.e. one part steers the others. In multipartite systems
there exist several different steering scenarios, depending on how
many subsystems steer the others. For example in a tripartite system
we can have one subsystem that tries to steer the other two, a scenario
that we indicate as* one-to-two steering*, or two subsystem that
might steer the other one, which we refer as* two-to-one steering*.
As in the case of entanglement we have different levels of multipartite
steerability (7).

In the *one-to-two steering* scenario we say that the correlations
demonstrate multipartite steering (7) if the joint
measurement probabilities cannot be expressed as:

(2) |

where , and are the outcomes the observables and of Alice, Bob and Charlie respectively. In (2) Bob and Charlie’s quantum state is pre-determined, for a given their state is . A state is said instead to demonstrate genuine multipartite steering (7) if the joint measurement probabilities cannot be written as:

(3) | ||||

where and
satisfy:
In (3) there are three terms: in the first there
is no steering between Alice, Bob and Charlie, in the second Alice
can steer Charlie but not Bob, namely only Bob’s state is pre-determined
for a given , conversely in the third Alice can steer Bob
but not Charlie, which means that only Charlie’s state is pre-determined
for a given

In the *two-to-one steering* scenario we say that the correlations
demonstrate multipartite steering (7) if the joint
measurement probabilities cannot satisfy:

(4) |

Conversely if the above holds the state is nonsteerable from Alice and Bob to Charlie, indeed Charlie’s state is pre-determined by the value of In this scenario a state is said to be GMS (7) if the joint measurement probabilities cannot be expressed as:

(5) |

where . In (5) the first term shows that only Bob can steer Charlie, in the second only Alice and in the third Alice and Bob cannot jointly steer Charlie. However Alice and Bob can share entanglement.

As any type of quantum correlations, one of the problems connected with quantum steering is its detection. Several methods have been introduced in the last years for detecting steering in bipartite systems, for example (11); (12); (13); (14); (15). Here we are interested in entropic steering criteria such as the one defined in (14); (15); (16); (17). In (14) it was derived that a nonsteerable state satisfies:

(6) |

where is the conditional entropy of given and is the conditional entropy of computed on , that does not depend on Alice’s measurements. Thus any violation of (6) demonstrates steering from Alice to Bob. In (15) the inequality (6) was generalized to state-independent entropic uncertainty relations (EUR). As an example, given any two of Alice’s observables and and two of Bob’s observables and , for any nonsteerable state the following holds:

(7) |

where
with and
the eigenstates of and respectively. Eq. (7)
is a generalization of Maaseen and Uffink’s EUR (18) to nonsteerable
states, which can be violated only by steerable states from Alice
to Bob. Starting from (6) other inequalities of
the form (7) can be derived simply by considering
different EUR from the ones of (18), for example the ones derived
in (28); (24); (29); (22); (25); (23); (26); (20); (27); (19); (21).

In this paper we derive the following results:

we first show that (6) and (7)
can be generalized to tripartite steering, obtaining different sufficient
conditions for both steerable and GMS states. In the case of*
one-to-two steering *scenario where Alice, whose measurements are
uncharacterized, might steer Bob and Charlie’s state, we show that
any nonsteerable state satisfies the following set of entropic uncertainty
relations:

(8) |

where
with labeling the subsystem considered and
and being the eigenstates
of and respectively. Here is given by
for any observables.

We also show that

(9) |

holds for all nonsteerable states, where and indicates
the opposite of , i.e. if and
when .

The last inequality for nonsteerable states involves
the following quantity:

(10) |

We prove that for a nonsteerable state the following holds:

(11) |

where

For any non-GMS states we prove that the following
inequality is satisfied:

(12) |

Finally, we give the following state-dependent entropic uncertainty relation valid for all non-GMS states:

(13) |

where represents the Von Neumann conditional
entropy between the bipartition when the variable
occurs, while is the Von Neumann conditional
between Bob and Alice when occurs. Note that these quantities
can be negative (36) for entangled states, moreover their
lowest values and are reached
by maximally entangled states. We note that the inequality (13)
is not useful in the context of multipartite steering detection, if
one wants to understand the steering property of an unknown state,
since it requires the knowledge of the LHS model (3).
Conversely, the multipartite steering criteria (8-12)
can be exploited in the task of discovering the steering property
of unknown quantum states. In order to compare the power in detecting
steerability of the criteria in Section V we consider the steerability
robustness of the standard and states under white noise
and we show that criterion (9) detects more multipartite
steerable states than the others.

The results (8-12) are also extended,
with the same techniques, to the *two-to-one* steering scenario.

The paper is organized as follows: in section I we review bipartite
quantum steering by following the approach of (30). Here
we also report the derivations of (6) and (7).
In section II we review the definition of multipartite steering, which
it was introduced in (7). In section III we focus
on the *one-to-two steering* scenario and we derive the steering
inequalities (8,12). In section IV the
results for the *two-to-one steering* are discussed. Finally
in section V some steering states are considered in order to study
the detection power of these inequalities.

## I Bipartite quantum steering

### i.1 Definition and LHS model

Bipartite quantum steering (4) can be seen as the ability to nonlocally influence the set of possible quantum states of a given system through the measurements of another system sufficiently entangled with the first one. In the steering scenario Alice and Bob share a quantum state and Alice performs a measurement whose outcome occurs with probability . As a consequence of Alice’s measurements, Bob’s state is transformed into the state with probability . Here we do not require any characterization of Alice’s measurements, namely we only say that she performs an arbitrary measurement, and we suppose that Bob has full access to the conditional state and on his measurements. Namely, the information available to Bob is the collection of the post-measured states and their respective probabilities , which can be described with the following ensemble of unnormalized states:

(14) |

Each member of (14) is given by:

(15) |

where and are Alice’s POVM elements. The ensemble (14) represents the set of possible quantum states that can be nonlocally influenced when steering correlations are owned by . Therefore the LHS model formally represents the minimal requirement on (14) in order to avoid this nonlocal influence, then steering is defined as the possibility of remotely generating ensembles that could not be produced by a LHS model. This model can be thought in the following way: a source sends, according to a probability distribution , a classical message to Alice, her probability of obtaining depends now on : . To each there corresponds a pre-determined state of Bob , which is sent to Bob with the same probability Bob’s ensemble (14), that now does not depend on Alice’s measurements, is given by:

(16) |

The definition of steering is as follows: an ensemble (14) is said to demonstrate bipartite steering if it does not admit a decomposition of the form (16). Moreover a quantum state is said to be steerable from Alice to Bob if there exists a measurement in Alice’s part that produces an ensemble that demonstrates steering. This is an asymmetric concept that also implies entanglement. Indeed suppose that is separable, namely we have . After Alice has performed a measurement, Bob’s ensemble becomes:

(17) | ||||

which is of the form (16). Since it implies nonseparability, steering detection can be seen as an entanglement detection task where one part, the one that steers, performs arbitrary measurements and its system remains completely uncharacterized, namely we do not assume anything on it, not even its dimension. The existence of a LHS model can be written also in terms of joint probabilities of measurements, namely by the condition (1). Indeed we have:

(18) | ||||

### i.2 Entropic uncertainty steering inequalities

Here we review the techniques used in (14) and (15) to derive (6) and (7). Suppose that a state admits a LHS model, then (1) holds. Note first that:

(19) |

with

(20) |

where the last equality holds since the state admits a LHS model. Given we consider the relative entropy between and which is always nonnegative. Namely we have:

(21) |

The above can be written as a sum of two terms. The first is given by:

(22) | |||

The second, by using (20), can be expressed as:

(23) |

Therefore (21) implies:

(24) |

which leads to (6) by averaging over , that provides a sufficient condition to detect steering states, indeed any violation of it implies the presence of bipartite quantum steering. If we now consider a sum as , we find:

(25) |

In the right-hand side of (25) depends on , namely the two entropies are computed over the state . However for any state Maaseen and Uffink’s EUR (18) holds, namely we have: , which together with , implies (7):

(26) |

Since the above must be valid for any nonsteerable state, any violation of it indicates the presence of a steerable state.

## Ii Multipartite quantum steering

In this section we start reviewing the concept of quantum steering
for multipartite systems. We focus on the tripartite case, where there
are two possible scenarios, following the approach given in (30); (7).
In the first case, which can be named *one-to-two steering scenario,*
Alice measures her system and wants to nonlocally influence the state
of the other two. The available information is encoded in the following
ensemble of unnormalized states:

(27) |

where is a POVM of Alice’s
measurements. The second possibility, the *two-to-one steering
scenario, *consists in two parties, say Alice and Bob that, by measuring
their systems, want to influence the states of the third party. In
this case, the post-measured ensemble of states is given by:

(28) |

where , are POVMs of Alice and Bob’s respectively. Multipartite steering scenario therefore consists of all the asymmetric scenarios, where some subset of the parties have full control on their subsystems, and they want to steer the state of the remaining subsets. Just like entanglement, which has a much richer structure in the multipartite case in than the bipartite one since different notions of separability can be introduced, also steerability have different levels for multipartite systems. In the case of a tripartite system we have two notions: multipartite steering and the genuine multipartite steering, which refers to the impossibility to explain the correlations between measurement outcomes in terms of different LHS models.

## Iii one-to-two steering scenario

### iii.1 LHS models

Let us first focus on the *one-to-two steering scenario.* If
Alice cannot nonlocally influence Bob and Charlie the ensemble (27)
becomes:

(29) |

In the above there is no steering from Alice to Bob and Charlie and each member of the ensemble is prepared in a separable state of Bob and Charlie. Note that the above can be thought as a multipartite LHS model where, with probabilities Alice receives and outputs with probability , while Bob and Charlie’s states are pre-determined by the value of Any tripartite state that can produce an ensemble that cannot be written as (29) is said to be multipartite steering. An example is provided by , where . Indeed Alice can prepare an ensemble that cannot be written as (29). This LHS model can be expressed in terms of joint probabilities as (2), indeed:

(30) |

which implies (2) by using (29).
Note that a slightly different definition of this form of multipartite
steering exists (7). Indeed, one could require
that entanglement between Bob and Charlie is present. As a consequence,
Bob and Charlie’s pre-determined state would be ,
instead of . However,
here we consider only the case where there is no entanglement between
Bob and Charlie, since we are interested in detecting the possible
simplest form of these quantum correlations.

If the state is non-GMS then the ensemble (27)
can be expressed as:

(31) |

where and . Each member of this ensemble can be expressed as a sum of three terms. In the first one there is no steering between Alice and Bob-Charlie. In the other two, which are made of separable states only of Bob and Charlie, Alice can steer one of the two subsystems but not the other. This can be thought in terms of a hybrid-LHS model in the following way. The hidden variable discriminates between different situations: in the first the global state of Bob and Charlie is pre-determined, that is , and this state can be entangled; in the other two determines the state of just one subsystem, the other is not pre-determined. The previous example can now lead to an ensemble of the form (31). Any tripartite states that cannot produce an ensemble such (31) is said to be genuine multipartite steering. Conversely if (31) can be produced, the state is non-GMS. By using (30) and (31) we can express this hybrid-LHS model in terms of joint probabilities:

(32) | ||||

which is exactly the requirement (3).

### iii.2 Entropic uncertainty multipartite steering inequalities

In this section we derive entropic steering inequalities for a multipartite system, with the aim to discriminate also the different notions of multipartite steering. We start by considering a nonsteerable state and show that it must imply some inequalities, then we use them to formulate sufficient conditions for multipartite steering detection. If the state is nonsteerable it satisfies:

(33) |

with

(34) |

where the last equality holds since (2) holds. As in the bipartite case, we now consider the relative entropy between and , which has to verify:

(35) |

The above, with (33) and (34), implies:

(36) |

and by averaging over we arrive at:

(37) |

Since Bob and Charlie share a separable state, for we also have:

(38) |

Now by considering the relative entropy between and for , we can derive in the same way:

(39) |

The entropic uncertainty relations (8), namely , can be derived simply by noting that the following holds for any state:

(40) |

with , since From the above
we can see that if we considered EUR different from the ones of (18),
we could find other EUR for nonsteerable states. These relations can
be used to define sufficient conditions for multipartite steering.
Indeed any violation indicates its presence.

Let us consider complementary observables, namely observables whose
eigenbasis are mutually unbiased ^{1}

(41) |

(42) |

for ; being the dimension of the system and

We want now to extend these results to the case of GMS states. Suppose
now that the state of the system is non-GMS, then we have:

(43) |

Since (32) holds, each term can be written as follows:

(44) |

(45) |

(46) |

Equation 43 can be also expressed as where is a classical variable such that We consider now the relative entropy between and which must be nonnegative:

(47) |

The above quantity is a sum of two terms. The first one is:

(48) |

that is since (43) holds. The second one is:

(49) |

which, by using the decomposition of given by (44), (45) and (46), can be written as a sum of three terms:

(50) |

(51) |

(52) |

After reordering the terms and averaging over , that for example implies and similar relations, we arrive at:

(53) | ||||

where Since and for any and we finally arrive at:

(54) | ||||

Then by using (40) we can derive the following state-dependent entropic uncertainty relations:

(55) | ||||

Note that in general hence the above implies:

(56) |

which for complementary observables becomes:

(57) |

where .

Now we focus on the conditional entropies
and . Since
the three terms (44), (45) and (46) become:

(58) |

(59) |

(60) |

From the above relations we derive, with the same arguments that we have used in the previous case,

(61) | ||||

The same holds for Charlie:

(62) | ||||

Any violation of one of the above EUR indicates that the state is GMS. In order to define stronger steering criteria we can also look at conditional entropies of the form and where measurements on parts different from Alice are performed. As an example we consider therefore we are interested in the probability:

(63) |

Since the state is non-GMS, the terms in the above equation can be written as follows:

(64) |

(65) |

(66) |

From the above we can derive in the usual way that:

(67) |

The same holds for Bob:

(68) |

In terms of entropic uncertainty relations we have:

(69) |

(70) |

these two equations are equivalent to eqs. (9). Moreover, when combined with (55) they imply eq. (12), namely:

(71) |

where
and

A nonsteerable state satisfies equation (11) instead, namely:

(72) |

Indeed for a nonsteerable state we have shown that . Then in this case we also have: