Entanglement criteria and full separability of multi-qubit quantum states

Abstract

We introduce an entanglement criterion to exclude full separability of
quantum states. We present numerical evidence that the criterion is necessary
and sufficient for the class of GHZ diagonal three-qubit states and estimate
the volume of bound entangled states within this class. Finally, we extend our
approach to bound entangled states which are not GHZ diagonal.

Full text

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arXiv:1009.3782v2 [quant-ph] 1 Jan 2011
Entanglement criteria and full separability of
multi-qubit quantum states
Otfried G¨ uhne
Institut f¨ ur Quantenoptik and Quanteninformation,¨Osterreichische Akademie der
Wissenschaften, Technikerstr. 21A, 6020 Innsbruck, Austria
Institut f¨ ur Theoretische Physik, Universit¨ at Innsbruck, Technikerstr. 25, 6020 Innsbruck,
Austria
Fachbereich Physik, Universit¨ at Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany
Abstract
We introduce an entanglement criterion to exclude full separability of quantum
states. We present numerical evidence that the criterion is necessary and sufficient
for the class of GHZ diagonal three-qubit states and estimate the volume of bound
entangled states within this class. Finally, we extend our approach to bound
entangled states which are not GHZ diagonal.
1. Introduction
Entanglement is believed to be an important resource in quantum information
processing and consequently many works are devoted to its characterization [1,
2]. This characterization becomes complicated, if more than two particles are
entangled, since then different classes of entanglement exist. Some methods to
distinguish between the different classes have been presented in the literature [3–
17]. However, no general solution of the problem is known, not even for specific
families of states.
In this paper we present a criterion for the verification of multi-qubit entan-
glement. The criterion is formulated as a set of simple inequalities for the matrix
elements of the state. Our approach is inspired by some existing criteria [12], and
also by some recent works on entanglement in the family of graph states [16, 17].
We consider then density matrices which are diagonal in terms of Greenberger-
Horne-Zeilinger (GHZ) states and investigate the optimality of our criterion. For
that, we derive methods to prove that a given state is separable and it turns out
that all GHZ diagonal states under scrutiny are either detected by our criterion
or proven to be separable. This allows to estimate the volume of so-called bound
entangled states in this class of states. Bound entanglement is a weak form of
Preprint submitted to ElsevierJanuary 4, 2011

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entanglement, where no pure state entanglement can be distilled from; and this
phenomenon is central to many open problems in quantum information theory
[1]. Finally, we discuss with the help of an example how our ideas can be used to
characterize bound entanglement close to the three-qubit W state.
2. Definitions and statement of the problem
Before explaining our separability criteria, we introduce the notation and give
some examples of existing separability criteria and interesting quantum states.
2.1. Separability and entanglement
We consider an N-particle system with Hilbert space Htot= H⊗N. Any ma-
trix ̺ acting on Htotwhich is hermitian (̺ = ̺†), has no negative eigenvalues
(̺ ≥ 0) and is normalized [Tr(̺) = 1] is a valid density matrix of some quantum
state. By definition, a state is fully separable, if it can be written as a convex
combination of product states,
̺ =
?
k
pk|ak??ak| ⊗ |bk??bk| ⊗ ... ⊗ |zk??zk|, (1)
where the pkare non-negative (pk≥ 0) and normalized (?
words, they form a probability distribution. If a state cannot be written as in
Eq. (1) it is entangled in some sense. In general, it is not easy to check whether
a given quantum state is fully separable or not, see Refs. [3–13] for some existing
sufficient criteria for entanglement and Refs. [18, 19] for numerical tests to prove
separability. Especially if a state is only weakly entangled, proving entanglement
is not straightforward.
In this paper we will derive criteria, which allow to prove that a state is not
fully separable and hence contains some entanglement. It should be stressed
that there are more refined notions of entanglement for multiparticle systems
(e.g., genuine multipartite entanglement) which can be more relevant for special
situations (e.g., experiments). We will, however, concentrate on full separability
as the basic definition, for a discussion of recent results on the other classifications
see the reviews in Refs. [1, 2]. Moreover, we will focus our discussion on the case
of three qubits, but our results can directly be generalized to more particles.
kpk= 1); in other
2.2. A criterion for full separability in terms of the matrix elements
Let us explain a separability criterion for full separability derived in Ref. [12],
which we will generalize later. For three qubits, consider the 8×8-density matrix
with entries ̺i,j. Here and in the following, we always order the basis vectors in the
canonical way, {|000?,|001?,|010?,...,|111?}. Then, if the state is fully separable,
the entries fulfill
|̺1,8| ≤
2
6√̺2,2̺3,3̺4,4̺5,5̺6,6̺7,7. (2)

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The idea of the proof is as follows: It is easy to check that for a pure fully
separable state equality holds in Eq. (2). Then, |̺1,8| is convex in ̺ [a function
f(̺) is convex if f[p̺1+(1−p)̺2] ≤ pf(̺1)+(1−p)f(̺2)], while the right-hand
side of Eq. (2) is concave [i.e., f[p̺1+ (1 − p)̺2] ≥ pf(̺1) + (1 − p)f(̺2)]. This
implies the bound for mixed fully separable states.
The right-hand side of the inequality (2) may be replaced by other expres-
sions, for instance, |̺1,8| ≤
this criterion is able to detect the entanglement in a family of states which are
separable with respect to any bipartition, but not fully separable [12]. This is
then a bound entangled state and the criterion improves existing criteria for this
family significantly.
In the following, we will extend the criterion in Eq. (2) by considering more
than one off-diagonal element on the left-hand side. We will also discuss the
optimal choice of the right-hand side.
4√̺2,2̺3,3̺5,5̺8,8 holds for separable states. Then,
2.3. A bound entangled state
Let us now introduce a bound entangled state which motivates our approach.
Consider the family of three-qubit density matrices introduced by A. Kay [16],
̺AK(ˆ α) =
1
8 + 8ˆ α












4 + ˆ α
0
0
0
0
0
0
2
00
0
0
0
0
0
0
0
2
0
0
0
2
0
0
0
0
2
0
0
0
0
0
0
ˆ α
0
0
0
0
2
0
ˆ α
0
0
−2
0
0
−2
0
0
ˆ α
0
0
ˆ α
2
0
0
0
ˆ α
0
0
0
ˆ α
04 + ˆ α












.(3)
This matrix is a valid quantum state for ˆ α ≥ 2 and it is separable for ˆ α ≥
2√2. This has been proved in Ref. [16] by writing down an explicit separable
decomposition as in Eq. (1). Furthermore, using the algorithm outlined in Ref. [5]
it was shown numerically that the state is entangled for 2 ≤ ˆ α ≤ 2.828. Note that
for any ˆ α ≥ 2 the state has a positive partial transpose (PPT) for any bipartition.
This does not only imply that no entanglement can be distilled from it1; moreover,
since the state is diagonal in the GHZ basis (see also below), it also implies that
1At this point, one should mention that undistillable entanglement can arise in multiparticle
systems in a simple way: If ̺ is entangled with respect to one bipartition, but PPT with respect
to another bipartition, this implies already that ̺ is entangled, but multipartite undistillable.
In this paper, however, we use a more restrictive definition: we consider bound entangled
states, that are separable for any bipartition, but not fully separable. This means that they are
undistillable, even if arbitrary parties join.
3

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the state is separable for all bipartitions. Namely, from the fact that a two-
qubit Bell diagonal state with a positive partial transpose is separable, one can
conclude that a GHZ diagonal state that has a positive partial transposition is
also separable for that partition [20].
The state in Eq. (3) is an example of a GHZ diagonal state. These are of the
form
8
?
k=1
where the GHZ state basis consists of the eight vectors |GHZk? = |0x2x3? ±
|1¯ x2¯ x3? where xi, ¯ xi∈ {0,1} and xi?= ¯ xi. Alternatively, the GHZ diagonal states
can be written as
̺ =1
8
̺ =
pk|GHZk??GHZk|, (4)
?
?+λ2ZZ
?+λ3Z
?Z+λ4
?ZZ+λ5XXX +λ6Y Y X+λ7Y XY +λ8XY Y?,
(5)
where X,Y, and Z denote the the Pauli matrices and tensor product symbols have
been omitted. The observables occurring in Eq. (5) are the so-called stabilizing
observables of the GHZ states, see Ref. [21] for a further discussion.
GHZ diagonal states have been intensively discussed in the literature before
[3, 12, 20, 22] and are interesting from several perspectives: They have a simple
structure, since only the diagonal and the anti-diagonal elements of the matrix
can be nonzero, and they occur naturally in certain types of decoherence processes
[23]. Furthermore, any state can be transformed to a GHZ diagonal state (without
changing the fidelities of the GHZ states) by local operations [3, 21]. This means
that if the remaining GHZ diagonal state is entangled, the initial state must have
been entangled, too. On the other hand, if one has a GHZ diagonal state ̺1and
a second separable state ̺2which is mapped to ̺1by these local operations, then
̺1must be separable, too.
The state ̺AKis not detected by the criterion in Eq. (2) or variants thereof.
The reason lies in the fact that Eq. (2) considers only single offdiagonal elements,
and neglects the (phase) relations between them.
this paper to develop an improvement of Eq. (2) which takes into account all
offdiagonal elements at the same time. This will finally prove analytically that
the state ̺AKis entangled if 2 ≤ ˆ α < 2√2.
It is the main purpose of
3. The separability criterion
In this section we will now formulate the separability criterion. We will restrict
our attention to three qubits, but it should be stressed that our approach can
straightforwardly be generalized to an arbitrary number of qubits.
To start, consider a pure product state,
|φ? = (c1|0? + s1|1?) ⊗ (c2|0? + s2|1?) ⊗ (c3|0? + s3|1?).
4
(6)

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Here, the complex coefficients ciand sifulfill the normalization |ci|2+ |si|2= 1.
Let us consider the corresponding 8×8 density matrix ̺ = |φ??φ|. The offdiagonal
elements can be written as
̺1,8= c1c2c3s∗
̺2,7= c1c2s3s∗
̺3,6= c1s2c3s∗
̺5,4= s1c2c3c∗
1s∗
1s∗
1c∗
1s∗
2s∗
2c∗
2s∗
2s∗
3= κei(a+b+c),
3= κeia,
3= κeib,
3= κeic. (7)
Here, we used the notations a = φ1+ φ2− φ3, b = φ1− φ2+ φ3, and c =
−φ1+ φ2+ φ3, where the phases are defined via cks∗
we set κ = |c1s1||c2s2||c3s3|. Note that the other offdiagonal elements follow from
these and the fact that ̺ is hermitian.
From Eq. (7) one can conclude two things. First, for a product states the
absolute value of all the offdiagonal elements is the same. Second, the four phases
are not arbitrary; they depend only on three parameters.
To proceed, consider a linear functional like
k= |cksk|eiφk. Furthermore,
L(̺,?X) = ℜ(X1̺1,8+ X2̺2,7+ X3̺3,6+ X4̺5,4),
where?X = (X1,X2,X3,X4) is a vector of complex coefficients and ℜ(...) denotes
the real part. The functional L is compatible with convex combinations of the
quantum state, i.e., one has L[p̺1+ (1 − p)̺2] = pL(̺1) + (1 − p)L(̺2). For a
pure separable state, L is given by
κF(?X) with
F(?X)
−ℑ(X2)sin(a) + ℜ(X3)cos(b) − ℑ(X3)sin(b)
+ℜ(X4)cos(c) − ℑ(X4)sin(c).
(8)
L(|φ?)=
=ℜ(X1)cos(a + b + c) − ℑ(X1)sin(a + b + c) + ℜ(X2)cos(a)
(9)
In order to obtain an extension of the separability condition in Eq. (2) we need
two more facts. First, for given coefficients?X one may compute the maximum
given by
C(?X) = sup
a,b,c|F(?X)|(10)
For given values of?X this can usually be computed analytically (see also below)
or with a simple numerical optimization.
Second, we have to characterize κ for product states. For them we have
κ=
4√̺1,1̺4,4̺6,6̺7,7=
√̺k,k̺9−k,9−kfor all 1 ≤ k ≤ 4.
5
4√̺2,2̺3,3̺5,5̺8,8
=
(11)