Extended reduction criterion and lattice states

Extended reduction criterion and lattice statesFabio Benatti, Roberto Floreanini, and Alexandra M. Liguori Citation: Journal of Mathematical Physics 48, 052103 (2007); doi: 10.1063/1.2734867 View online: http://dx.doi.org/10.1063/1.2734867 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/48/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On the reduction criterion for random quantum states J. Math. Phys. 55, 112203 (2014); 10.1063/1.4901548 Classification of bi-qutrit positive partial transpose entangled edge states by their ranks J. Math. Phys. 53, 052201 (2012); 10.1063/1.4712302 On states of perfect correlation J. Math. Phys. 49, 112101 (2008); 10.1063/1.3012383 A combinatorial approach for studying local operations and classical communication transformations ofmultipartite states J. Math. Phys. 46, 122105 (2005); 10.1063/1.2142840 On extremal quantum states of composite systems with fixed marginals J. Math. Phys. 45, 4035 (2004); 10.1063/1.1776642

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Extended reduction criterion and lattice statesFabio Benattia�

Dipartimento di Fisica Teorica, Università di Trieste, Strada Costiera 11, 34014 Trieste,Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34100 Trieste,Italy

Roberto FloreaniniIstituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34100 Trieste, Italy

Alexandra M. LiguoriDipartimento di Fisica Teorica, Università di Trieste, Strada Costiera 11, 34014 Trieste,Italy

�Received 29 December 2006; accepted 30 March 2007; published online 15 May 2007�

We study a particular class of states of a bipartite system consisting of two four-level parties. By means of an adapted extended reduction criterion we associatetheir entanglement properties to the geometric patterns of a planar lattice consistingof 16 points. © 2007 American Institute of Physics. �DOI: 10.1063/1.2734867�

I. INTRODUCTION

In recent years, due to the rapid growth of quantum information, communication, and com-putation, the necessity of identifying, quantifying and classifying entangled states as a physicalresource has given birth to quantum entanglement theory �see the review from Ref. 1 and refer-ences therein�. Though the focus is now shifting to multipartite entanglement, still generic quan-tum correlations in finite dimensional bipartite systems are not completely understood.

From a mathematical point of view, the lack of exhaustive criteria capable of witnessing thepresence of bipartite entanglement is nothing else but the absence of a complete characterizationof positive, but not completely positive maps.2–4 Looking for new classes of entangled states5,6

and for dedicated entanglement witnesses7 may thus improve the comprehension of the algebraicstructure behind physical phenomena such as separability and free and bound entanglements.1

Recently, an extension of the so-called reduction map8 has been proposed9 that is tailor madefor revealing the entanglement of states with a specific structure with respect to angular momen-tum. Subsequently, this map was shown10 to be a particular instance of a larger class of indecom-posable positive maps.2–4 In the following, we shall adapt the extended reduction criterion to thestudy of the class of bipartite states of two four-level parties which were introduced in Ref. 11 andfurther studied in Ref. 12. For reasons that will soon become obvious, these 16�16 densitymatrices have been called lattice states.

These states are characterized by subsets of a discrete planar lattice with 16 elements and canbe grouped together into equivalence classes characterized by specific geometric patterns. Bymeans of the extended reduction map, we shall associate most of them with specific properties ofthe corresponding lattice states like that of being separable, Nonpositive under Partial Transposi-tion �NPT� entangled, or Positive under Partial Transposition �PPT� entangled. Because of this,besides enriching the phenomenology of entangled states, the class of lattice states may also turnout to be a useful arena for the approach based on the discrete Wigner functions.13–16

The plan of the paper is as follows: we shall first introduce the lattice states and their presently

a�Electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS 48, 052103 �2007�

48, 052103-10022-2488/2007/48�5�/052103/13/$23.00 © 2007 American Institute of Physics


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http://dx.doi.org/10.1063/1.2734867

http://dx.doi.org/10.1063/1.2734867

known properties; then, we shall improve their classification by using the extended reductioncriterion; finally, we shall completely characterize some subclasses of them and discuss whichones need stronger entanglement witnesses than those available so far.

II. LATTICE STATES

The construction of lattice states is as follows. Let ��, 0��3, be the Pauli matricessupplemented with �0= I2, the 2�2 identity matrix, and consider the totally symmetric state,

��+4� =

1

2�i=1

4

�ii� � C16,

where ��i�i=14 is a fixed orthonormal basis in C4. Denoting the tensor products of pairs of Pauli

matrices by

�� ª �� , �1�

the action of I4 � ��, I4= I2 � I2, on ��+4� yields an orthonormal basis

�� ª �I4 � ��+4� � C16,

with corresponding orthogonal projectors

P�� ª �� = �I4 � ��+4��+

4��I4 � �� . �2�

Let L16 denote the finite square lattice consisting of 16 points labeled by pairs �� ,��,

L16 ª ��,��:�,� = 0,1,2,3 .

We shall also denote by C� and R� the columns and rows of the lattice L16,

C� = ��,�� L16:� = 0,1,2,3, R� = ��,�� L16:� = 0,1,2,3 .

To every subset I�L16 of cardinality NI, we associate a mixed state equidistributed over theorthogonal projectors P�� labeled by pairs in I,

�I =1

NI�

��,��I

P��.

We shall refer to the �I as lattice states:1 they are a particular subclass of the mixed states

�� = ��,��L16

��P��, �� 0, ��,��L16

�� = 1 �3�

that commute with the projectors P��.It turns out that properties of these states like that of being PPT or bound entangled are related

to certain specific geometrical patterns of the subsets that label them. Indeed, the following resultshave been proved �see Refs. 11 and 12�:

Proposition 1:

1. A nececessary and sufficient condition for a lattice state �I to be PPT is that for every�� ,��L16, the number of points on C� and R� different from �� ,�� and beloging to I benot greater than NI /2. In terms of the characteristic functions I�� ,��=1 if �� ,�� I, =0otherwise, a lattice state �I is PPT if and only if for all �� ,��L16,

1Note that the orthogonal projections P�� in Eq. �2� result from the action of a completely positive map on the totallysymmetric projector onto ��+

4�, while the lattice states are equally weighted combinations of the latter. As such they mayin principle be experimentally implemented by local rotations, provided one is able to prepare ��+

4�.

052103-2 Benatti, Floreanini, and Liguori J. Math. Phys. 48, 052103 �2007�


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�=0,��

3

I��,� + ��=0,��

3

I��,�� NI

2. �4�

2. A sufficient condition for a PPT lattice state �I to be entangled is that there exists at least apair �� ,��L16 such that only one point on C� and R� and different from �� ,�� belongs toI. Equivalently, �I is entangled if a pair �� ,��L16 exists such that

�=0,��

3

I��,� + ��=0,��

3

I��,�� = 1, ��,�� I . �5�

Remarks 1:

1. A necessary and sufficient condition for PPTness can be worked out for the more generalstates in Eq. �3�; it reads

�=0,��

3

�� + ��=0,��

3

�� 1

2. �6�

2. From Ref. 12 all states �I with NI=1,2 ,3 ,5 ,7 are NPT. Therefore in order for �I to be PPT,NI must be equal to 4, 6, 8, or 9 or larger.

3. Since the maximum number of points on C��R� different from �� ,�� is at most 7, all stateswith NI=14,15,16 are PPT. Those with NI=15,16 are also separable since NI=16 meansthat �I is the maximally mixed state, while, for NI=15, �I is easily seen to be a separableisotropic state.8,11

4. If the condition in Eq. �5� is satisfied for one pair �� ,��L16, then NI=16−6=10 at themost. In the following, we shall use an entanglement witness which will allow us to add toI the site �� ,�� from a pair �C� ,R�� satisfying Eq. �5�. This will open the possibility ofchecking the entanglement of lattice states with NI=11.

5. From general arguments �see Ref. 17� we know all PPT lattice states �I with NI=4 areseparable.

According to the previous proposition, all �I not fulfilling Eq. �4� are NPT and thus entangled,while only some of those which are PPT are recognized as entangled by Eq. �5�.

As an example of how the geometric picture works, consider the following lattice subsets,where the crosses mark those sites which contribute to �I:

NI = 8

3

2 � � �

1 � �

0 � � �

0 1 2 3

, NI = 10

3 � � �

2 � �

1 � �

0 � � �

0 1 2 3

,

NI = 8

3 � � �

2 � �

1 � � �

0

0 1 2 3

, NI = 10

3 � � �

2 � �

1 � �

0 � � �

0 1 2 3

,

052103-3 Extended reduction criterion and lattice states J. Math. Phys. 48, 052103 �2007�


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NI = 10

3 � � �

2 � � � �

1 � � �

0

0 1 2 3

, NI = 11

3 � � �

2 � � � �

1 � � �

0 �

0 1 2 3

.

All subsets above identify PPT lattice states according to Eq. �4�, but, according to Eq. �5�,only those in the left column correspond to �bound� entangled states with certainty.

In the following section, we will develop a criterion strictly stronger than Eq. �5�: it willessentially amount to the removal of the request �� ,�� I in Eq. �5�. This allows us to enrich theclass of bound entangled lattice states, proving that also the states on the right hand side areentangled. On the other hand, in Sec. III B, we will further show that some PPT �I which are notrecognized as entangled by the stronger criterion are indeed separable.

III. LATTICE STATES AND THE EXTENDED REDUCTION CRITERION

Starting from the proof of indecomposability given in Ref. 9 for SO�3�-invariant states andadapting the argument of Ref. 10 we extend the reduction criterion8 to the lattice states.

We define an antiunitary map V by its action on a generic 4�4 matrix B�M4�C�: V�B�=VBTV†, where T denotes transposition with respect to a fixed orthonormal basis and V :C4�C4

is unitary and antisymmetric, VV†=V†V= I4, VT=−V.It follows that, for all ��C4, by expanding with respect to the chosen basis,

�*�V�� = �j,k=1

4

� jVjk�k = �*�VT�� = − �*�V�� .

Thus, V�� is orthogonal to �� so that ��− V�� is a projector for all ��C4,whence the linear map on M4�C�,

�V�B� = �Tr�B��I4 − B − V�B� , �7�

preserves positivity.Remark 2: Since we shall later exhibit PPT lattice states such that �id � �V��I� is not

positive definite, it turns out that �V is not decomposable.9,10

Notice that the request of antisymmetry and the fact that T��2�=−�2 give the followingexpansion of V along the tensor products ��:

V = ��2

v�2��2 + ��2

v2��2�, �8�

where v�2 and v2� are complex coefficients satisfying unitarity constraints.In order to study the entanglement witnessing power of �V for the lattice states, we shall use

that V is unitarily equivalent to the transposition and elaborate on the partial action of the latteron the projections P�� in Eq. �2�. In passing, this will provide a proof of Eq. �4� alternative to thatin Ref. 11

Before proceeding, we make the following useful observations concerning the algebraic rela-tions among the Pauli matrices and the lattice structure.

The products of two Pauli matrices define four 4�4 Hermitian, unitary matrices ��,

�� = ��=0

3

�� , ��

� =1

2Tr�� . �9�

From the explicit expression of the matrix elements and the cyclicity of the trace operation itfollows that



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�� = ��

� = �� , ��

� �* = �� = ��

� . �10�

The matrices �� are thus Hermitian and multiplication by �� on the left of both sides of Eq. �9�shows that they are also unitary. Explicitly, �0= I4 while

�1 =�0 1 0 0

1 0 0 0

0 0 0 i

0 0 − i 0�, �2 =�

0 0 1 0

0 0 0 − i

1 0 0 0

0 i 0 0�, �3 =�

0 0 0 1

0 0 i 0

0 − i 0 0

1 0 0 0� . �11�

Further, given 0�� ,��3 the Pauli matrix satisfying Eq. �9� is unique, thus any fixed �determines a map i� : �0,1 ,2 ,3�� i�� 0,1 ,2 ,3 defined by

�� = ��=0

3

�� = ��i��

� �i��. �12�

Multiplying both sides of the above equality by �� and taking their Hermitian conjugates, thefollowing useful properties of i� follow from Eq. �10�:

i�2 = id, i�� = i�� . �13�

Finally, a useful relation involves the matrix �2: one can check that

��2 � = �,��2 ⇒ i2�� = � � 2, �14�

where � denotes addition mod 4.The action of the partial transposition can now be recast as

�id � T��P�� =I16

4−

1

2 ��,��L16

��,��2 + �,��2 − 2�,��2�,��2�P��. �15�

This can be proved using the following facts:

1. by transposing the Pauli matrices, one gets

T�� = ��=0

3

��, where � = �� =�1 0 0 0

0 1 0 0

0 0 − 1 0

0 0 0 1�;

2. partially transposing P00= ��+4��+

4� yields the flip F :A � B�B � A, A ,B�M4�C�:

�id � T��P00� =1

4 �i,j=1

4

�i�j� � �j�i� =F

4; and

3. F has the spectral decomposition F=��,��L16��P��.

Thus one derives

�id � T��P�� =1

4 ��,��L16

��I4 � ��P00�I4 � ��

=1

4 ��,�,��,��L16

�� ,

whence Eq. �15� follows since, using Eq. �14�, ��=�,��1−2�,��2�.



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By means of Eq. �15�, the action of the partial transposition on lattice states yields

�id � T��I� =I16

4−

1

2NI�

��,��L16

k��I P��, �16�

with

k��I = �

��2I��,� � 2� + �

��2I�� 2,�� , �17�

whence the partial action of the extended reduction map �Eq. �7�� gives

�id � �V��I� =1

2NI�

��,��L16

k��I �I4 � V�P��I4 � V†� − �I. �18�

From Eq. �16� and �17� the proof of Eq. �4� �and of Eq. �6�� easily follows. Moreover, usingEq. �18� one gets the following.

Proposition 2: Let � denote the minimum of k��I in Eq. �17�. Then, a necessary condition for

�V to reveal the entanglement of any PPT �I is ��1.Proof: Since unitary transformations do not change an operator’s spectrum, it is more conve-

nient to work with the positive, indecomposable map �̃V defined as follows:

�id � �̃V��I� = �I4 � V†��id � �V��I��I4 � V� = − �I4 � V†��I�I4 � V� +1

2NI�

��,��L16

k��I P��.

Then, since ��,��L16P��= I4,

�I �I16

NI⇒ �id � �̃V��I� �

1

2NI�

��,��L16

�k��I − 2�P��.

From the previous result and Eq. �17�, the extended reduction criterion based on �V maydetect entangled PPT lattice states only on the following conditions.

• �=0: namely, if there exists at least one pair �� ,��L16 such that k��I =0. This is possible if

there exist a column C��2 and a row R��2 which do not contribute to I except, possibly, for�� 2,� � 2�.

• �=1: namely, if there exists at least one pair �� ,��L16 with k��I =1. This is possible if there

exist a column C��2 and a row R��2 that contribute to I with only one element except,possibly, for �� 2,� � 2�.

A. Entangled PPT lattice states

In the following, we will consider the second case and postpone the discussion of the first oneto the next section. We start by observing that if �� 2,� � 2� does not belong to I then, accordingto Eq. �5�, the entanglement of �I is revealed by the positive, indecomposable map devised in Ref.12. We shall show that the extended reduction criterion is able to scoop the bound entangled �I inthis case, but also when �� 2,� � 2� does belong to I, thus being stronger than the map used inRef. 12

Proposition 3: If k��I =1 for a column C��2 and a row R��2, then

��id � �̃V��I�� 0, �19�

independently of whether �� 2,� � 2� belongs to I or not.Proof: Since �+

4�A � B��+4�=Tr�ATB�, using the map �Eq. �12�� one gets



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��I4 � V†��I�I4 � V�� =1

NI�

��,��I

��+4�I4 � ��V†��+

4��2

=1

4NI�

��,��I

�Tr��i��i��V��2 =1

NI�

��,��I

�vi��i��2,

where, due to Eq. �8�, either i��=0,1 ,3 with i��=2 or i��=0,1 ,3 with i��=2.Because of Eq. �13� and �14�, the first case corresponds to �=� � 2 and �� 2 and the

second one to �=� � 2 and �� 2, whence

��I4 � V†��I�I4 � V�� =1

NI �

��2I��,� � 2��vi��2�2 + �

��2I�� 2,��v2i��2� ,

whence

��id � �̃V��I�� =1

2NI �

��2I��,� � 2��1 − 2�vi��2�2�

+ ��2

I�� 2,��1 − 2�v2i��2�� .

Suppose that the point of I contributing to k��I =1 be �� ,� � 2� on the row R��2, then V=�i��2

yields the result. A similar argument holds if the contributing point is �� 2,�� on the columnC��2.

Remark 3: The previous result is not sensitive to whether the point �� 2,� � 2� belongingto the column C��2 and row R��2 does contribute to I or not. If it does not, the extended reductioncriterion reveals all the bound entangled lattice states already revealed by the criterion in Ref. 12;if it does belong to I, new bound entangled states not revealed by the latter are seen by theextended reduction map.

B. Separable PPT lattice states

By direct inspection, the argument of the previous proposition, by which all bound entangledlattice states with �=1 are detectable by the extended reduction criterion, cannot be applied when�=0.

It turns out that the case �=0=k��I for some �� ,��L16 with �� 2,� � 2�� I can be studied

by other means. The following observation turns out to be useful.Remark 4: As observed in Ref. 12, one can operate local unitary transformations transform-

ing lattice states into lattice states without changing either their cardinality NI or their entangle-ment properties. Therefore, one can actually subdivide the lattice states into equivalence classes byoperating on them with local unitaries whose only effect is to map the set of subsets I�L16 of asame cardinality into itself. Indeed, given two 4�4 matrices U and V, the structure of ��+

4� is suchthat

U � V�� = I4 � V��UT��+4� .

We distinguish the following cases.

1. Choosing U= I4 and V=��, we get I4 � ��P��I4 � ��= P00, so that we can move anychosen column C� and row R� into C0 and R0.

2. Let U=U1 � U2 and V=U1*

� U2*, where U1 and U2 are two unitary 2�2 matrices that rotate

the Pauli matrices �i, �i� and � j, � j�, respectively, one into the other �apart possibly from a� sign� keeping fixed the third ones, �i� and � j�, respectively �and of course �0�,

U1�iU1† = ± �i�, U1�i�U1

† = � �i, i,i� � 0,

and



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U2� jU2† = ± � j�, U2� j�U2

† = � � j, j, j� � 0.

Then, any two columns Ci, Ci� and rows R j, R j�, respectively, can be changed one into theother, leaving fixed the columns C0, Ci� and the rows R0, R j�, respectively,

V � UPijV†

� U† = I4 � �U1�iU1†

� U2� jU2†�P00I4 � �U1�iU1

†� U2� jU2

†�

= I4 � �i�j�P00I4 � �i�j� = Pi�j�.

3. Finally, choosing U= I4 and V the flip operator �V ��+4�= ��+

4�, V��V=��, one gets

I4 � VP��I4 � V = I4 � �V��V��+4��+

4�I4 � �V��V� = P��.

One can thus exchange columns and rows of a subset I�L16 obtaining a new lattice statein the same equivalence class, hence with the same entanglement properties.

The previous considerations allow us to prove the following result.Proposition 4: If k��

I =0 and �� 2,� � 2� does not belong to I, all the corresponding PPTstates �I are separable.

Proof: The hypothesis means that the column C��2 and the row R��2 do not contribute to I.Thus seven points on the square lattice L16 must be excluded and at the most nine elements of L16

label �I. From Remark 1.2 we know that all states �I with NI=1,2 ,3 ,5 ,7 are NPT. Therefore inorder for �I to be PPT, NI must be equal to 4, 6, 8, or 9. From Remark 1.5, we also know that allPPT �I with NI=4 are separable, in particular, those associated with the following graphs �and allthose in their equivalence classes as in Remark 4�:

3 �

2 �

1 �

0 �

0 1 2 3

,

3

2

1 � �

0 � �

0 1 2 3

.

In the following, we shall use these separable rank-4 lattice states to write other lattice statesof higher rank as their convex combinations which will thus also result separable; for sake ofsimplicity, we shall identify graphs with lattice states and the convex decompositions withweighted sums of graphs.

Because of Remark 4, we can always perform a unitary transformation so that the column androw which are assumed to be completely empty correspond to C0 and R0.

Case NI=9. We have only one possible equivalence class represented by the state �I9 associ-

ated with the graph

�I9:

3 � � �

2 � � �

1 � � �

0

0 1 2 3

.

This lattice state can be convexly decomposed in terms of nine rank-4 PPT �hence separable�lattice states as follows:



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1

9

3 � �

2 � �

1

0

0 1 2 3

+1

9

3

2 � �

1 � �

0

0 1 2 3

+1

9

3 � �

2 � �

1

0

0 1 2 3

+1

9

3 � �

2

1 � �

0

0 1 2 3

+1

9

3

2 � �

1 � �

0

0 1 2 3

+1

9

3 � �

2

1 � �

0

0 1 2 3

+1

9

3

2 � �

1 � �

0

0 1 2 3

+1

9

3 � �

2

1 � �

0

0 1 2 3

+1

9

3 � �

2 � �

1

0

0 1 2 3

.

In decomposing �I9 as done above, each contributing site from the decomposers appears the same

number of times �four in this case� and each decomposer contributes with equal weight �1/9 inthis case�; then, since the decomposers have NI=4, each contributing site appears with weight 4�1/36=1/9 in the resulting convex decomposition. From Proposition 1 we know the state �I

9 isPPT and, as a convex combination of separable states, it is separable.

Case NI=8. With C0 and R0 completely empty there are nine possible states, which can beobtained directly from the above state �I

9 with NI=9 by eliminating one element, i.e.,

3 � � �

2 � � �

1 � �

0

0 1 2 3

,

3 � � �

2 � � �

1 � �

0

0 1 2 3

,

3 � � �

2 � � �

1 � �

0

0 1 2 3

, etc.

By exchanging columns and rows as explained in Remark 4, all of these nine possible statesare unitarily equivalent to the state

�I8:

3 � � �

2 � �

1 � � �

0

0 1 2 3

.

Therefore in order to study these lattice states, it is sufficient to see whether �I8 is separable or

not. From Proposition 1 we know that �I8 is PPT; moreover, it can be convexly decomposed as

follows:



131.230.73.202 On: Sun, 21 Dec 2014 04:01:14

3 � � �

2 � �

1 � � �

0

0 1 2 3

=1

4

3 � �

2

1 � �

0

0 1 2 3

+1

4

3

2 � �

1 � �

0

0 1 2 3

+1

4

3 � �

2

1 � �

0

0 1 2 3

+1

4

3 � �

2 � �

1

0

0 1 2 3

.

In the above decomposition of �I8 each contributing site appears with weight 1 /4 in two different

PPT decomposers of rank 4; therefore, in the resulting convex combination its weight is 2�1/16=1/8. Again, �I

8 is a mixture of separable states and thus separable itself.Case NI=6. We have to fit six elements into nine points of the square lattice L16 �excluding

row R0 and column C0�. According to Proposition 1, in order for the states to be PPT there cannotbe more than two elements in each of the three free columns or rows, i.e., all states such as thefollowing are NPT:

3 � �

2 �

1 � � �

0

0 1 2 3

,

3 �

2 � � �

1 � �

0

0 1 2 3

.

From Proposition 1 it also follows that all states with only two elements in each row �column�but no empty column �row� except for C0 �R0� are NPT, as, for instance, the following ones:

3 � �

2 � �

1 � �

0

0 1 2 3

,

3 �

2 � �

1 � � �

0

0 1 2 3

.

So the only possible way to construct PPT states �I with NI=6 and the row R0 and column C0

completely empty is by putting two elements in each row �column� and leaving another column�row� empty, i.e., lattice states such as

�I6 =

3 � �

2 � �

1 � �

0

0 1 2 3

and all those in its equivalence class obtained by exchanging columns �rows� among themselvesby unitary rotations and rows into columns by means of the flip operator, as described in Remark4.



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This state was already showed to be separable in Ref. 12, where an explicit decomposition hadto be worked out; using the strategy of above, the proof is now much simpler. Indeed, the state �I

6

can be written as follows:

�I6 =

1

3

3

2 � �

1 � �

0

0 1 2 3

+1

3

3 � �

2 � �

1

0

0 1 2 3

+1

3

3 � �

2

1 � �

0

0 1 2 3

.

In the above decomposition of �I6 each element appears in two different rank-4 lattice states, so

that its total weight in the resulting decomposition is 2�1/12=1/6. Thus, since �I6 is a convex

combination of PPT states with NI=4, it is separable. Therefore all PPT lattice states with NI=6and a column and row which are completely empty are separable because they are unitarilyequivalent to �I

6.

C. Lattice states classification

From Remark 1, the only lattice states whose entanglement structure is not under control arethose with NI=6 and 8�NI�14. We now show that the case NI=6 can also be completelycontrolled.

As in the proof of Proposition 4, the clue is Remark 4: lattice states can be grouped intoequivalence classes, each element in a class being obtainable from any of its partners in that classby the local action of unitaries. Members of a class share the same entanglement properties.

This allows us to consider without restriction those graphs where the largest number of sites,that we shall denote by n0, contributing to I by a column are in C0 from bottom to top. We can thusproceed by distinguishing various cases.

Case n0=4. The remaining two contributing sites will necessarily be located on some of theother columns thus violating condition �4�: all lattice states with n0=4 are thus NPT entangled.

Case n0=3.

1. If any of the sites of R3 contribute to I, then the corresponding lattice state violates Eq. �4�and is NPT entangled.

2. If R3� I is empty and one of the rows R0,1,2 contributes with more than one site to I, Eq. �4�is again violated and the lattice state is NPT entangled.

3. If R3� I is empty and the three rows R0,1,2 contribute with only one site to I and there is acolumn, say, C�, ��0, with only one site contributing to I, then the sufficient condition inProposition 1 applies to C� and R3. All these lattice states are thus PPT entangled.

4. If R3� I is empty and there is a column �which we can always suppose to be C1� with twoelements in I, the remaining contributing site makes these states violate the PPT condition inProposition 1. These lattice states are thus NPT entangled. In fact, the row passing throughthe last contributing site, let it be R�, intersects C0 in one site and C1 in none, whence�C1�R�� I contains four sites.

5. If R3� I is empty and the three sites of I from the rows R0,1,2 are on a same column C�,��0, then we are in the situation of Proposition 4. These lattice states are thus separable.

Case n0=2.

1. If one of the rows R2,3 contributes with more than one site to I, condition �4� is violated andthe state �I is NPT entangled.

2. If the rows R2,3 altogether contribute to I with one site at most, then there will be at least oneof the rows R0,1 intercepting I in not less than three sites. In this case, using the flip operatoras in Remark 4.3, rows can be turned into columns and we are back with either n0=3 orn0=4.



131.230.73.202 On: Sun, 21 Dec 2014 04:01:14

3. If each of the rows R2,3 intersects I in only one point and these two elements lie on a samecolumn, again Eq. �4� is violated and �I is NPT entangled. Indeed, the two elements canalways be thought to be along the column C1 which cannot then contain other sites from I.Therefore, Eq. �4� is violated by any remaining contributing site exactly as in point 4 whenn0=3. The following is a representative of such a possibility,

3 �

2 �

1 � �

0 � �

0 1 2 3

.

4. The only remaining case is thus represented by

3 �

2 �

1 � �

0 � �

0 1 2 3

.

This state is separable as it can be decomposed into three PPT lattice states �I with NI=4. Asin Proposition 4, we indicate the weights with which the various graphs contribute to thedecomposition,

3 �

2 �

1 � �

0 � �

0 1 2 3

=1

3

3 �

2 �

1 �

0 �

0 1 2 3

+1

3

3

2

1 � �

0 � �

0 1 2 3

+1

3

3 �

2 �

1 �

0 �

0 1 2 3

.

IV. CONCLUSIONS

Because of the difficulty in finding appropriate entanglement witnesses, the detection ofbipartite entanglement and its qualification as bound or free still represents a challenge from thepoint of view of mathematical physics. Of course, the issue at stake is the lack of a structuralcharacterization of positive, but not completely positive maps.

In order to deepen our actual understanding of positivity of linear maps, in particular, withrespect to their indecomposability and the corresponding phenomenon of bound entanglement, it isstill important to provide classes of states together with maps that are able to detect the differentaspects of their entanglement.

To this end, in this paper we studied a class of bipartite states, called lattice states, each partyconsisting of two-qubit systems, which are nicely associated with subsets I of points on a squarelattice with 16 elements. By using a suitably adapted extended reduction criterion, we exposed



131.230.73.202 On: Sun, 21 Dec 2014 04:01:14

certain relations between bound entanglement and the geometric pattern of the subsets I labelingthe lattice states. More precisely, based on an existing complete characterization of which of thelattice states are PPT, we showed the following.

1. All PPT lattice states labeled by a subset I such that there exist a column C� and a row R�

of the lattice contributing to I with only one point different from �� ,�� are entangled.2. If �� ,�� I, the extended reduction criterion reproduces a result already obtained with a

different indecomposable positive map. If �� ,�� I, new bound entangled lattice states arewitnessed.

3. All PPT lattice states labeled by a subset I that entirely excludes a column and a row areseparable.

However, a thorough classification of the class of lattice states still requires stronger criteriathan the ones presented in this paper. Indeed, the positive map �V �Eq. �7�� is unfortunately not anexhaustive entanglement witness; for instance, it is indecisive in the following cases of PPT latticestates. The first three examples are with �=0 for some �� ,�� I, while the last one, with NI

=11, is a geometric pattern which cannot be controlled by any of the methods employed in thispaper,

NI = 8:

3 � � �

2 � �

1 � �

0 �

0 1 2 3

, NI = 9:

3 � � �

2 � �

1 � � �

0 �

0 1 2 3

,

NI = 10:

3 � � �

2 � � �

1 � � �

0 �

0 1 2 3

, NI = 11:

3 � �

2 � � �

1 � � �

0 � � �

0 1 2 3

.

As a final remark, we note that the structure of the lattice states can naturally be generalizedto higher dimensions. Some results concerning their separability properties have been reported inRef. 18. In principle, the extended reduction criterion elaborated in this paper could also begeneralized to higher dimensional lattice states.

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2 M. D. Choi, Linear Algebr. Appl. 10, 285 �1975�.3 S. L. Woronowicz, Rep. Math. Phys. 10, 165 �1976�.4 E. Störmer, Acta Math. 110, 233 �1963�.5 D. Chruscinski and A. Kossakowski, Phys. Rev. A 74, 022308 �2006�.6 D. Chruscinski and A. Kossakowski, e-print quant-ph/0606211.7 S. Albeverio, K. Chen, and S. M. Fei, Phys. Rev. A 68, 062313 �2003�.8 M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 �1999�.9 H.-P. Breuer, Phys. Rev. Lett. 97, 080501 �2006�.

10 W. Hall, J. Phys. A 39, 14119 �2006�.11 F. Benatti, R. Floreanini, and M. Piani, Phys. Lett. A 326, 187 �2004�.12 F. Benatti, R. Floreanini, and M. Piani, Open Syst. Inf. Dyn. 11, 325 �2004�.13 K. S. Gibbons, M. J. Hoffmann, and W. K. Wootters, Phys. Rev. A 70, 062101 �2004�.14 A. O. Pittenger and M. H. Rubin, Linear Algebr. Appl. 390, 255 �2004�.15 A. O. Pittenger and M. H. Rubin, J. Phys. A 38, 6005 �2005�.16 E. F. Galvao, Phys. Rev. A 71, 042302 �2005�.17 P. Horodecki, M. Lewenstein, G. Vidal, and I. Cirac, Phys. Rev. A 62, 032310 �2000�.18 M. Piani, Phys. Rev. A 73, 012345 �2006�.



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Extended reduction criterion and lattice states