Steering, Entanglement, nonlocality, and the Einstein-
Podolsky-Rosen Paradox
Author
Wiseman, HM, Jones, SJ, Doherty, AC
Published
2007
Journal Title
Physical Review Letters
DOI
https://doi.org/10.1103/PhysRevLett.98.140402
Copyright Statement
© 2007 American Physical Society. This is the author-manuscript version of this paper.
Reproduced in accordance with the copyright policy of the publisher. Please refer to the
journal's website for access to the definitive, published version.
Downloaded from
http://hdl.handle.net/10072/18284
Link to published version
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Griffith Research Online
https://research-repository.griffith.edu.au
Steering, Entanglement, Nonlocality, and the EPR Paradox
H. M. Wiseman,
1
S. J. Jones,
1
and A. C. Doherty
2
1
Centre for Quantum Computer Technology, Centre for Quantum Dynamics, Griffith University, Brisbane 4111 Australia
2
School of Physical Sciences, University of Queensland, Brisbane 4072 Australia
The concept of steering was introduced by Schr¨odinger in 1935 as a generalization of the EPR
paradox for arbitrary pure bipartite entangled states and arbitrary measurements by one party. Until
now, it has never been rigorously defined, so it has n ot been known (for example) what mixed states
are steerable (that is, can be used to exhibit steering). We provide an operational defi nition, from
which we prove (by considering Werner states and Isotropic states) that steerable states are a strict
subset of the entangled states, and a strict superset of the states that can exhibit Bell-nonlocality.
For arbitrary bipartite Gaussian states we derive a linear matrix inequality th at decides th e question
of steerability via Gaussian measurements, and we relate this to the original EPR paradox.
The nonlocality of entangled states, a key feature of
quantum mechanics (QM), was first pointed out in 1935
by Eins tein Podolsky and Rosen (EPR) [1]. They consid-
ered a general non-factorizable pure state of two systems,
held by two parties (say Bob and Alice):
|Ψi =
∞
X
n=1
c
n
|ψ
n
i|u
n
i =
∞
X
n=1
d
n
|ϕ
n
i|v
n
i, (1)
where {|u
n
i} and {|v
n
i} are two orthonormal bases for
Alice’s system. If Alice chose to measure in the {|u
n
i}
({|v
n
i}) basis, then she could instantaneously project
Bob’s system into one of the states |ψ
n
i (|ϕ
n
i). For EPR,
the fact that the ensemble of |ψ
n
is is different from the
ensemble of |ϕ
n
is was problematic because “ the two sys-
tems no longer interact, [so] no real change can take place
in [Bob’s] system in consequence of anything that may be
done to [Alice’s] system.” Thus, they thought (wrongly)
that this nonlocality must be an artefact of the incom-
pleteness of QM. This intuition was supp orted by their
famous example (the EPR “paradox”) involving position
and momentum, which could be trivially resolved by con-
sidering local hidden variables (LHVs) for q and p.
The EPR pa per provoked an interesting response from
Schr¨odinger [2], who introduced the terms entangled state
for states o f the form of Eq. (1) and steering for Alice’s
ability to affect Bob’s state through her choice o f mea-
surement basis. Schr¨odinger had invented the quantum
state ψ for atoms [3], and, unlike EPR, believed it gave a
complete and correct description fo r a localized, isolated
system. Thus, he r e jected their suggested explanation of
steering in terms of LHVs. However, like EPR, he could
not easily accept no nlocality, and so suggested that QM
was incorrect in its des c ription of delocalized entangled
systems [2]. That is, he thought (wrongly) that Bob’s
system has a definite state, even if it is unknown to him,
so that steering would never be seen expe rimentally. We
call this a local hidden state (LHS) model for Bob.
In this Letter we revisit Schr¨o dinger’s concept of steer-
ing (which has received increasing attention in recent
years [4, 5, 6, 7, 8]) from a quantum information per-
sp e c tive. That is, we define it according to a task.
First, let us define the more familiar concept of Bell-
nonlocality [9] as a task, in this case with three parties.
Alice and Bob c an prepar e a shared bipartite sta te, and
repeat this any number of times. Each time, they mea-
sure their respective parts. Except for the preparation
step, communication be tween them is forbidden. Their
task is to convince Charlie (with whom they can com-
municate) that the state they can prepa re is entangled.
Charlie accepts QM as correct, but trusts neither Al-
ice nor Bob. If the correlations can be explained by a
LHV model then Charlie will not be convinced that the
state is entangled; the results could have been fabricated
from shared classical randomness. Conversely, if the cor-
relations between the results they report cannot be so
explained, then the state must be entangled. Therefore
they will succeed in their task iff (if and only if) they can
demonstrate Bell-nonlocality.
The ana logous definition for steering uses a task with
two parties. Alice can prepa re a bipartite quantum state
and send one part to Bob, and repeat this any number
of times. Each time, they measure their respective parts,
and communicate classically. Alice’s task is to convince
Bob that the state she can prepare is e ntangled. Bob (like
Schr¨odinger) accepts that QM describes the results of the
measurements he makes. However Bob does not trust
Alice. If the cor relations between Bob’s measurement
results and the re sults Alice reports can be explained by
a LHS model for Bo b then Bob will no t be convinced
that the state is entangled; Alice c ould have drawn a
pure state at rando m from some ensemble and sent it to
Bob, and then chosen her r e sult based on her knowledge
of this LHS. Conversely, if the correlations cannot be so
explained then the state must be entangled. Therefore
Alice will succ e e d in her task iff she can create genuinely
different ensembles for Bob, by steering Bob’s state.
As EPR and Schr¨odinger noted, steering may be
demonstrated using any pure entangled state, and the
same is true of Bell-nonlocality [10]. But in the labora-
tory states ar e mixed. In a seminal paper, Werner [11]
asked the question: c an all entangled states b e used to
demonstrate Bell-nonlocality? As Wer ner showed [11],
the surprising answer is: no — a hint of the complexity
2
of bound entanglement [12] still being uncovered.
In this Letter, we address the following questions: Can
all entangled states be used to demonstrate steering?
Doe s a demonstration of steering also demonstrate Bell-
nonlocality? We prove that in both cases the answer
is again: no. Thus, steerability is a distinct nonlocal
property of some bipartite quantum states, different from
both Bell-nonlocality and nonseparability.
This Le tter is structured as follows. We begin by find-
ing the ma thema tical formulation of the above opera-
tional definition of s teering. From this it follows that
steerability is stronger than nonseparability, and weaker
than Bell-nonlocality. We then show, using two-qubit
Werner states and Isotropic states, that this is a strict
hierarchy. Lastly, we consider Gaussian states with Gaus-
sian measurements. We determine the condition under
which steering can be demonstrated, and relate this to
the Reid c riterion for the EPR “para dox” [13].
Concepts of Quantum Nonlocality. — Let the set of
all observables on the Hilbert space for Alice’s system be
denoted D
α
. We denote an element of D
α
by A, and the
set of eigenvalues {a} of A by λ(A). By P (a|A; W) we
mean the probability that Alice will obtain the result a
when she measures A on a system with state matrix W .
We denote the measurements that Alice is able to per-
form by the set M
α
⊆ D
α
. (Note that, following Werner
[11], we are restricting to projective measurements.) The
corresponding notations for Bob, and for Alice and Bob
jointly, are obvious. Thus, for example,
P (a, b|A, B; W ) = Tr[(Π
A
a
⊗ Π
B
b
)W ], (2)
where Π
A
a
is the projector satisfying AΠ
A
a
= aΠ
A
a
.
The strongest sort of nonlocality in QM is Bell-non-
locality [9]. This is ex hibited in an experiment on state W
iff the co rrelations between a and b cannot be ex plained
by a LHV model. That is, if it is not the case that for all
a ∈ λ(A), b ∈ λ(B), for all A ∈ M
α
, B ∈ M
β
, we have
P (a, b|A, B; W ) =
X
ξ
℘(a|A, ξ)℘(b|B, ξ)℘
ξ
. (3)
Here, and below, ℘(a|A, ξ), ℘(b|B, ξ) and ℘
ξ
denote some
(po sitive, normalized) probability distributions, involv-
ing the LHV ξ. We say that a state is Bell-nonlocal iff
there exists a measurement set M
α
× M
β
that allows
Bell-nonlocality to be demonstrated.
A strictly weaker [11] concept is that of nonse parabil-
ity or entanglement. A nonse parable state is one tha t
cannot be written as W =
P
ξ
σ
ξ
⊗ ρ
ξ
℘
ξ
. Here, and
below, σ
ξ
∈ D
α
and ρ
ξ
∈ D
β
are some (positive, nor-
malized) quantum s tates. We can also give an op e ra-
tional definition, by allowing Alice and Bob the ability
to measure a quorum of local observables, so that they
can reconstruct the state W by to mography [14]. Thus a
state W is nonsepar able iff it is not the ca se that for all
a ∈ λ(A), b ∈ λ(B), for all A ∈ D
α
, B ∈ D
β
, we have
P (a, b|A, B; W ) =
X
ξ
P (a|A; σ
ξ
)P (b|B; ρ
ξ
)℘
ξ
. (4)
Bell-nonlocality a nd nonseparability a re both concepts
that are symmetric b e tween Alice and Bob. However
steering, Schr¨odinger’s term for the EPR effect, is in-
herently asymmetric. It is about whether Alice, by her
choice of measurement A, can collapse Bob’s system
into different types o f states in the different ens e mbles
E
A
≡
˜ρ
A
a
: a ∈ λ(A)
. Here ˜ρ
A
a
≡ Tr
α
[W (Π
A
a
⊗ I)] ∈
D
β
, where the tilde denotes that this state is unnor-
malized (its norm is the probability of its realization).
Of course Alice cannot affect Bob’s unconditioned state
ρ = Tr
α
[W ] =
P
a
˜ρ
A
a
— that would allow super-luminal
signalling. Nevertheless, a s Schr¨odinger said in 1935 [2],
“It is rather discomforting that the theory should allow
a system to be steere d . . . into one or the other type of
state at the experimenter’s mercy in spite of [her] having
no ac c e ss to it.” As stated earlier, he was “not satisfied
about there being enough e xperimental evidence for [it].”
The “experimental evidence” required by Schr¨odinge r
is precisely that required for Alice to succeed in the
“steering tas k” defined in the introduction. The experi-
ment can be rep e ated at will, and we assume Bob’s mea-
surements enable him to do state tomography. Prior to
all experiments, Bob demands that Alice announce the
possible ensembles
E
A
: A ∈ M
α
she can steer Bob’s
state into. In any given run (after he has received his
state), Bob should randomly pick an ensemble E
A
, and
ask Alice to prepare it. Alice should then do so, by mea-
suring A on her system, and announce to Bob the partic-
ular member ρ
A
a
she has prepared. Over many r uns , Bob
can verify that each state announced is indeed produced,
and with the correct frequency Tr[˜ρ
A
a
].
If Bob’s system did have a pre-existing LHS ρ
ξ
, then
Alice could attempt to fool Bob, using her knowledge
of ξ. This state would be drawn at random from some
prior ensemble of LHSs F = {℘
ξ
ρ
ξ
} with ρ =
P
ξ
℘
ξ
ρ
ξ
.
Alice would then have to announce a LHS ˜ρ
A
a
according
to so me stochastic map from ξ to a. If, for all A ∈ M
α
,
and for all a ∈ λ(A), there exists a ℘(a|A, ξ) such that
˜ρ
A
a
=
X
ξ
℘(a|A, ξ)ρ
ξ
℘
ξ
(5)
then Alice would have failed to convince Bob that she
can s teer his system. Conversely, if Bob cannot find any
ensemble F and map ℘(a|A, ξ) satisfying Eq. (5) then
Bob must admit that Alice can steer his system.
We can recast this definition as a ‘hybrid’ of Eqs. (3 )
and (4): Alice’s measurement strategy M
α
on state W
exhibits steering iff it is not the case that for all a ∈
λ(A), b ∈ λ(B), for all A ∈ M
α
, B ∈ D
β
, we have
P (a, b|A, B; W ) =
X
ξ
℘(a|A, ξ)P (b|B; ρ
ξ
)℘
ξ
. (6)
3
FIG. 1: (Color on-line.) Boundaries between classes of entan-
gled states for Werner (a) and Isotropic (b) states W
η
d
. The
bottom (blue) line is η
ent
(states are entangled iff η > η
ent
).
The next (red) line is η
steer
, defined analogously for steering.
The top (green) line with down-arrows is an upper bound on
η
Bell
, defined analogously for Bell-nonlocality. The up-arrows
are lower bounds on η
Bell
for d = 2. The three classes are thus
distinct. Dots join values at finite d with those at d = ∞.
Iff there exists a measurement strategy M
α
that exhibits
steering, we say that the state W is steerable (by Alice).
Clearly steer ability is stronger than nonseparability,
but Bell-nonlocality is stro nger than steerability. At least
one o f these re lations must be “strictly stronger”, because
of Werner’s result [11]. In the following sections we prove
that both are “strictly stronger”; se e Fig. 1.
Conditions for Steerability. — Below we derive nec-
essary and sufficient conditions for steerability for three
families of states. Crucial to the derivations is the con-
cept of an optimal en s emble F
⋆
= {ρ
⋆
ξ
℘
⋆
ξ
}. This is an
ensemble such tha t if it cannot satisfy Eq. (5) then no
ensemble can satisfy it. In finding an optimal ensemble
F
⋆
we us e the symmetries of W and M
α
:
Lemma 1 Consider a group G with a unitary represen-
tation U
αβ
(g) = U
α
(g) ⊗ U
β
(g) on the Hilbert space for
Alice and Bob. Say that ∀A ∈ M
α
, ∀a ∈ λ(A), ∀g ∈ G,
we have U
†
α
(g)AU
α
(g) ∈ M
α
and
˜ρ
U
†
α
(g)AU
α
(g)
a
= U
β
(g)˜ρ
A
a
U
†
β
(g). (7)
Then there exists a G-covariant optimal ensemble: ∀g ∈
G, {ρ
⋆
ξ
℘
⋆
ξ
} = {U
β
(g)ρ
⋆
ξ
U
†
β
(g)℘
⋆
ξ
}.
Proof. Say there exists an ensemble F = {ρ
ξ
℘
ξ
} sat-
isfying Eq. (5). Then under the conditions of lemma 1,
the G-covariant ensemble F
⋆
= { ρ
⋆
(g,ξ)
℘
ξ
dµ
G
(g)}, with
ρ
⋆
(g,ξ)
= U
β
(g)ρ
ξ
U
†
β
(g), satisfies Eq. (5) with the choice
℘
⋆
(a|A, (g, ξ)) = ℘(a|U
†
α
(g)AU
α
(g), ξ).
(i) Werner States. — This family of states in C
d
⊗C
d
was intro duced in Ref. [11]. We parametrize it by η ∈ R
such that W
η
d
is linear in η, W
η
d
is a product state for
η = 0, and the larges t permissible value for η is 1:
W
η
d
=
d − 1 + η
d − 1
I
d
2
−
η
d − 1
V
d
. (8)
Here I is the identity and V the “flip” operator (Vϕ⊗ψ ≡
ψ ⊗ ϕ). Werner states ar e nonseparable iff η > η
ent
=
1/(d+ 1) [11]. For d = 2, the Werner states violate a Bell
inequality if η > 1/
√
2 [15]. This places an upper bound
on η
Bell
, defined by W
η
d
being Bell-nonlocal iff η > η
Bell
.
For d > 2 only the trivial upper bound of 1 is known.
However, Werner found a lower bound on η
Bell
of 1 −1/d
[11], which is strictly greater that η
ent
.
We now show that Werner’s lower bound is in fact
equal to η
steer
, defined by W
η
d
being steerable iff η >
η
steer
. We allow Alice all possible measurement strate-
gies: M
α
= D
α
, and without loss of generality ta ke the
projectors to be r ank-one: Π
A
a
= |aiha|. For Werner
states, the conditions of lemma 1 are then satisfied for the
d-dimensional unitary group U(d). Specifically, g → U ,
and U
αβ
(g) → U ⊗ U [11]. Again without loss of gen-
erality we can take the optimal ensemble to cons ist of
pure states, in which case there is a unique covariant op-
timal ensemble, F
⋆
= {|ψihψ|dµ
H
(ψ)}, where dµ
H
(ψ)
is the Haar measure over U(d). Werner used the same
construction; his LHVs for Bob were in fact these LHSs.
Now we determine when E q. (5) can be satisfied by
this F
⋆
. Using ˜ρ
A
a
= ha|W
η
d
|ai it is s imple to show that,
for any A ∈ D
α
and a ∈ λ(A),
ha| ˜ρ
A
a
|ai = (1 − η)/d
2
. (9)
Using the methods of Werner’s proof we show that for
any positive no rmalized distribution ℘(a|A, ψ),
ha|
Z
dµ
H
(ψ) |ψihψ|℘(a|A, ψ) |ai ≥ 1/d
3
. (10)
The upper bound is attained for the choice [11]
℘(a|A, ψ) =
n
1 if |hψ|ai| ≤ |hψ|a
′
i|, ∀a
′
6= a
0 otherwise.
(11)
Comparing this with Eq. (9) we see that steering can be
demonstrated if (1 − η)/d
2
< 1/d
3
. More over, it is easy
to verify that when this inequality is saturated, Eq. (11)
satisfies Eq. (5). Thus η
steer
= 1 − 1/d.
Recently a new lower bound for η
Bell
was found for d =
2 [16], greater than η
steer
, as shown in Fig. 1. This proves
that steerability is strictly weaker than Bell-nonlocality
as well as being strictly stronger than non-separability.
(ii) Isotropic States. — This family, introduced in [17],
can be parametrized identically to the Werner states:
W
η
d
= (1 − η)I/d
2
+ ηP
+
, (12)
where P
+
= |ψ
+
ihψ
+
|, where |ψ
+
i =
P
d
i=1
|ii|ii/
√
d.
For d = 2 the Isotropic states are identical to Wer ner
states. Isotropic states are nonseparable iff η > η
ent
=
1/(d + 1) [17]. A no n-trivial upper bound on η
Bell
for all
d is known [18]; Ref. [16] gives a lower b ound for d = 2.
To determine η
steer
for isotropic states, we fo llow the
same method as for Werner states, except that this time
U
αβ
= U
∗
⊗ U [1 7]. Instead of Eq. (9) we obtain
ha| ˜ρ
A
a
|ai = η/d + (1 − η)/d
2
, (13)
4
and instea d of Eq. (10), we show that
Z
dµ
H
(ψ) |ha|ψi|
2
℘(a|A, ψ) ≤ H
d
/d
2
, (14)
where H
d
=
P
d
n=1
(1/n) is the Harmonic series. The
upper bound is attained for the choice
℘(a|A, ψ) =
n
1 if |hψ|ai| ≥ |hψ|a
′
i|, ∀a
′
6= a
0 otherwise.
(15)
Comparing this result with Eq. (13), we see that isotropic
states are steerable if η > (H
d
− 1) /(d − 1). Moreover,
it is easy to verify that when this inequality is saturated,
Eq. (15) satisfies Eq. (5). Thus η
steer
= (H
d
− 1) /(d−1).
(iii) Gaussian States. — Finally we investigate a gen-
eral (multimode) bipartite Gaussian state W [19]. Such
a state may be defined by its covariance matrix (CM)
V
αβ
. In (Alice, Bob) block form it appears as
CM[W ] = V
αβ
=
V
α
C
C
⊤
V
β
. (16)
This represents a valid state iff V
αβ
+ iΣ
αβ
≥ 0 [19].
This is a linear matrix inequality (LMI), in which Σ
αβ
=
Σ
α
⊕ Σ
β
is a symplectic matrix proportional to ~.
Rather than addressing steerability in general, we con-
sider the case where Alice can only make Gaussian mea-
surements, denoted by G
α
. A measurement A ∈ G
α
is descr ibed by a Gaussian p ositive operator with a CM
T
A
satisfying T
A
+iΣ
α
≥ 0 [19]. When Alice makes such
a measurement, Bob’s conditioned state ρ
A
a
is Gaussian
with a CM V
A
β
= V
β
− C(T
A
+ V
α
)
−1
C
⊤
[20].
Theorem 2 The Gaussian state W defined in Eq. (16)
is no t steerable by Alice’s Gaussian measurements iff
V
αβ
+ 0
α
⊕ iΣ
β
≥ 0. (17)
Proof. The proof has two parts. First, suppose Eq. (17)
is true. Then using matrix inversion formulas [20], one
sees that the matrix U ≡ V
β
− CV
−1
α
C
⊤
satisfies
U + iΣ
β
≥ 0 and ∀A ∈ G
α
, V
A
β
− U ≥ 0. (18)
The first LMI allows us to define an ensemble F
U
=
{ρ
U
ξ
℘
U
ξ
} of Gaussian states with CM[ρ
U
ξ
] = U, distin-
guished by their mean vectors (ξ). The second LMI im-
plies that ∀A, ρ
A
a
is a Gaussian mixture (over ξ) of such
states. Therefore W is not steerable by Alice.
Now suppose W is not steerable. Then there is some
ensemble F sa tisfying Eq. (5). From the fact that V
A
β
is independent of a, one sees that U =
P
ξ
℘
ξ
× CM[ρ
ξ
]
satisfies Eq. (1 8). But unless (17) is true, one sees that no
such U satsifying Eq. (18) exists (again us ing standard
matrix analysis [20]). Therefore (17) must be true.
For the simplest ca se where Alice and Bob each have
o
ne mode with correlated positions q and momenta p,
Reid [13] has argued the EPR “paradox” is demonstrated
iff the product of the conditional variances V (q
β
|q
α
) a nd
V (p
β
|p
α
) viola tes the uncertainty principle. It is eas y to
verify that this occurs under precisely the same condi-
tions a s when Eq. (17) is v iolated. This confirms tha t
the EPR “paradox” is merely a particular case of steer-
ing. As is well known [21], the Reid conditions are strictly
stronger than the conditions for nonsepa rability.
We conclude with a brief listing of open questions.
First, are there asymmetric states that are steerable by
Alice but not by Bob? Second, Bell-no nlocality is nec-
essary and sufficient for certain tasks [22], and likewise
nonseparability [23]. Is there a task (beyond the defining
one) for which steerability is similarly useful? Third, do
there exist steering analogs o f Be ll-operators and entan-
glement witnesses? Finally, we note that we expect many
applications of the concept of steering in quantum mea-
surement theor y and experimental quantum information.
This work was supported by the ARC and the State of
Queensland. We thank Rob Spekkens, Volkher Scholz,
Antonio Acin, and Michael Hall for useful discussions.
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