Inductive entanglement classification of four qubits under stochastic local operations and classical communication

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Inductive entanglement classification of four qubits under stochastic local operations and classical communication

L. Lamata,^1,*_{J. León,}^1,†D. Salgado,^2,‡ and E. Solano^3,4,§

1Instituto de Matemáticas y Física Fundamental, CSIC, Serrano 113-bis, 28006 Madrid, Spain

2Dpto. Física Teórica, Universidad Autónoma de Madrid, 28049 Cantoblanco, Madrid, Spain

3Physics Department, ASC, and CeNS, Ludwig-Maximilians-Universität, Theresienstrasse 37, 80333 Munich, Germany

4Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Apartado Postal 1761, Lima, Peru 共Received 3 November 2006; published 21 February 2007兲

Using an inductive approach to classify multipartite entangled states under stochastic local operations and classical communication introduced recently by the authors 关Phys. Rev. A 74, 052336共2006兲兴, we give the complete classification of four-qubit entangled pure states. Apart from the expected degenerate classes, we show that there exist eight inequivalent ways to entangle four qubits. In this respect, permutation symmetry is taken into account and states with a structure differing only by parameters inside a continuous set are considered to belong to the same class.

DOI:10.1103/PhysRevA.75.022318 PACS number共s兲: 03.67.Mn, 03.65.Ud, 02.10.Yn

I. INTRODUCTION

Entanglement is the distinguishing feature of quantum systems关1,2兴and is especially useful in using such systems to process information关3,4兴. Despite its relevance this prop- erty is not yet fully understood and different questions re- main open关5兴. Among them a complete classification of entangled states under adequately chosen criteria of equivalence stands as a formidable task. Two of these criteria appear as outstanding: namely the so-called local unitary 共LU兲and stochastic local operations and classical communi- cation共SLOCC兲equivalences. The former is mathematically expressed by the relation 兩␺典⬃LU兩␾典⇔兩␺典=U^关1兴_丢¯

丢U^关^N^兴兩␾典 for unitary matrices U^关^k^兴, whereas the latter is mathematically translated as 兩␺典⬃SLOCC兩␾典⇔兩␺典=F^关1兴

丢¯丢F^关^N^兴兩␾典for nonsingular matricesF^关^k^兴. Although these entanglement equivalence definitions can be straightfor- wardly generalized to include mixed states, we will restrict ourselves to the pure state case. Furthermore, we will center upon the identification of entanglement classes under the second criterion.

The SLOCC equivalence 共also known as local filtering operations兲was introduced in关6兴and enjoys a clear physical motivation: two multipartite pure states 兩␺典 and 兩␾典 are equivalent under SLOCC if one can be obtained from the other with non-null probability using only local operations upon each subsystem and classical communication among the different parties. In this manner, the complete classification of entanglement forN= 3 qubits can be derived; namely, any genuinely entangled three-qubit systems is equivalent either to the Greenberger-Horne-Zeilinger共GHZ兲state or to theW state. The difficulties for more general cases—i.e. for Nⱖ4—was also pointed out in Ref.关6兴, where a nonenumer- able set of entanglement classes would be required.

In this work, using our preceding inductive approach关7兴, we give the complete classification of entanglement for N

= 4 qubits. Although this was already apparently performed in Ref.关8兴, we argue that their enumeration does not follow the same philosophy as in the seminal paper by Dür et al.

关6,9兴. In Sec. II, we recopile the preceding relevant results which will be later used in Sec. III to obtain in detail all entanglement classes under SLOCC. Our concluding remarks are presented in Sec. IV.

II. REVIEW OF THE INDUCTIVE METHOD The forthcoming entanglement classification forNqubits is based on the inductive approach which the present authors introduced in Ref. 关7兴 and in some auxiliary results which were proved therein. The main idea of this approach was to establish a classification of the right singular subspace of the coefficient matrix of each pure state兩␺典expressed in an ad- equate product basis 共typically the canonical—

computational—basis兲. This classification must be carried out according to the entanglement classes for N− 1 qubits 共hence the term inductive兲. Consequently, knowing in ad- vance that there are only two entanglement classes for two qubits—namely, the product class 00 and the entanglement class ⌿—the right singular subspace W of the coefficient matrix of the state of any three-qubit system results to be of six possible types: i.e. two one-dimensional subspaces W= span兵兩␾␺典其 andW= span兵兩⌿典其 driving us to the degenerate classes 000 and 0₁⌿23, respectively; two two- dimensional product subspaces W=兩␾典丢C² and W=C²

丢兩␺典, driving us to the degenerate classes 02⌿13and 0₃⌿12, respectively; and, finally, another two two-dimensional classesW= span兵00, 00其 andW= span兵00,⌿其, driving us to the well-known GHZ and W genuine entanglement classes.

Let us recall thatW= span兵00,⌿其stands for the fact that the subspaceWcontains共up to normalization兲only one product vector共i.e., only one of type 00兲. This will sometimes be also expressed as W= span兵兩␾␸典,兩⌿典其. This convention will be profusely followed in the next section. We will employ these

*Electronic address: lamata@imaff.cfmac.csic.es

†Electronic address: leon@imaff.cfmac.csic.es

‡Electronic address: david.salgado@uam.es

§Electronic address: enrique.solano@physik.lmu.de

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six tripartite entanglement classes to find those ofN= 4 qubits.

Moreover, in the proof, we will make use of the following results, already proved in关7兴.

Proposition 1. Let W be a two-dimensional subspace in C²丢C². Then W= span兵兩␾␸典,兩⌿典其 if, and only if, W= span兵兩␾␸典,兩␾␸^¯典+兩␾^¯␸典其, where the overbar denotes linear independence.

Let us also recall that only two options are left for the structure of any two-dimensional subspace 关10兴—namely, W= span兵00, 00其andW= span兵00,⌿其, where the above convention has been used. Complementarily, the following criterion to discern whether a given tripartite pure state belongs to any of the six entanglement classes will be also necessary.

We previously needed the following definition.

Definition 1.Let

C^共1兲⬅C_1兩23=

冉

^c^c¹¹¹211 ^cc¹¹²₂₁₂ ^cc¹²¹₂₂₁ ^cc¹²²₂₂₂

冊

^, ^共^1a^兲

C^共2兲⬅C_2兩13=

冉

^cc¹¹¹₁₂₁ ^cc¹¹²₁₂₂ ^cc²¹¹₂₂₁ ^cc²¹²₂₂₂

冊

^, ^共1b兲

and

C^共3兲⬅C_3兩12=

冉

^c^c¹¹¹112 ^cc¹²¹₁₂₂ ^cc²¹¹₂₁₂ ^cc²²¹₂₂₂

冊

^共1c兲

be the coefficient matrices of the partitions 1兩23, 2兩13, and 3兩12, respectively, of the tripartite pure state 兩⌿典=兺ijkcijk兩i− 1典丢兩j− 1典丢兩k− 1典. We will define the matri- cesW₁ andW₂ as

W₁=

冉

^cc¹¹¹₁₂₁ ^cc¹¹²₁₂₂

冊

^, ^W²⁼

冉

^cc²¹¹₂₂₁ ^cc²¹²₂₂₂

冊

^. ^共2兲

With these definitions and remembering that the spectrum of a matrixA is denoted as␴共A兲, in关7兴 we proved the following.

Theorem 1. Let 兩⌿典 denote the pure state of a tripartite system andC^共ⁱ^兲 its coefficient matrix according to the parti- tioni兩jk. Then,

共1兲兩⌿典 belongs to the 000 class if, and only if,

r共C^共ⁱ^兲兲= 1 for all i= 1 , 2 , 3.

共2兲兩⌿典 belongs to the 0₁⌿23 class if, and only if, r共C^共1兲兲= 1 andr共C^共^k^兲兲= 2 fork= 2 , 3.

共3兲兩⌿典 belongs to the 0₂⌿13 class if, and only if, r共C^共²^兲兲= 1 andr共C^共^k^兲兲= 2 fork= 1 , 3.

共4兲兩⌿典 belongs to the 0₃⌿12 class if, and only if, r共C^共3兲兲= 1 andr共C^共^k^兲兲= 2 fork= 1 , 2.

共5兲兩⌿典 belongs to the GHZ class if, and only if, one of the following situations occurs:

共i兲 r共C^共ⁱ^兲兲= 2 for alli= 1 , 2 , 3 andr共W1兲=r共W2兲= 1.

共ii兲 r共C^共ⁱ^兲兲= 2 for all i= 1 , 2 , 3,r共Wj兲= 2, r共Wk兲= 1, and␴共W_j⁻¹W_k兲is nondegenerate.

共iii兲 r共C^共ⁱ^兲兲= 2 for alli= 1 , 2 , 3,r共W₁兲= 2,r共W₂兲= 2, and␴共W₁⁻¹W₂兲 is nondegenerate.

共6兲兩⌿典belongs to theW class if, and only if, one of the following situations occurs:

共i兲 r共C^共ⁱ^兲兲= 2 for all i= 1 , 2 , 3, r共W_j兲= 2, r共W_k兲= 1, and␴共Wj

−1Wk兲is degenerate.

共ii兲 r共C^共ⁱ^兲兲= 2 for all i= 1 , 2 , 3,r共W1兲= 2,r共W2兲= 2, and␴共W₁⁻¹W₂兲is degenerate.

III. FOUR-QUBIT CLASSIFICATION

The classification inductive procedure for N= 4 qubits is long, although systematic. We will first detect the degenerate classes, which furthermore are predictable in advance: 0000, 0i₁0i₂⌿i₃i₄, 0i₁ GHZ, 0i₁W, and ⌿i₁i₂⌿i₃i₄, withik= 1 , 2 , 3 , 4.

Then, we will work out the genuine classes. The classification is performed according to the structure of the right singular subspace of the coefficient matrix of the four-partite pure states: W= span兵⌿i,⌿j其, where

⌿i,j= 000, 0k⌿, GHZ,W.

A. Degenerate classes

As stated we will revise in turns each of the possible structures of the right singular subspaceWof the coefficient matrix. Remember that nowW傺C²丢C²丢C².

共1兲 W= span兵000其. In this case, we can write W= span兵兩␾␸␺典其. Mimicking the same arguments as in 关7兴共see also below for the genuine classes兲, we obtain the canonical vector兩0000典, corresponding to the degenerate class 0000.

共2兲 W= span兵01⌿23其. This drives us to the canonical vector兩0000典+兩0011典, corresponding to the degenerate class 0₁0₂⌿34.

共3兲 W= span兵02⌿13其. This drives us to the canonical vector兩0000典+兩0101典, corresponding to the degenerate class 0₁0₃⌿24.

共4兲W= span兵03⌿12其. This drives us to the canonical vector 兩0000典+兩0110典, corresponding to the degenerate class 0₁0₄⌿23.

共5兲W= span兵GHZ其. This drives us to the canonical vector 兩0000典+兩0111典, corresponding to the degenerate class 0₁GHZ.

共6兲 W= span兵W其. This drives us to the canonical vector 兩0001典+兩0010典+兩0100典, corresponding to the degenerate class 0₁W.

共7兲 W= span兵000,000其. When W=兩␾␸典丢C², we obtain the canonical vector 兩0000典+兩1001典, corresponding to the degenerate class 0₂0₃⌿14.

共8兲 W= span兵000,000其. When W=兩␾典丢C²丢兩␺典, we obtain the canonical vector 兩0000典+兩1010典, corresponding to the degenerate class 0₂0₄⌿13.

共9兲 W= span兵000,000其. When W=C²丢兩␸␺典, we obtain the canonical vector兩0000典+兩1100典, corresponding to the degenerate class 0₃0₄⌿12.

共10兲 W= span兵000,000其. When W=兩␾典

丢span兵兩␸␺典,兩␸^¯␺^¯典其, we obtain the canonical vector 兩0000典 +兩1011典, corresponding to the degenerate class 0₂GHZ.

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共11兲 W= span兵000,000其. When W= span兵兩␾␸␺典,兩␾^¯␸␺^¯典其, we obtain the canonical vector兩0000典+兩1101典, corresponding to the degenerate class 0₃GHZ.

共12兲 W= span兵000,000其. When W= span兵兩␾␸␺典,兩␾^¯␸^¯␺典其, we obtain the canonical vector兩0000典+兩1110典, corresponding to the degenerate class 0₄GHZ.

共13兲 W= span兵000, 01⌿23其. When W=兩␾典

丢span兵兩␸1␺1典,兩␸^¯2␺^¯2典其, we obtain the canonical vector 兩0001典+兩0010典+兩1000典, corresponding to the degenerate class 0₂W.

共14兲 W= span兵000, 02⌿13其. When W

= span兵兩␾1␸␺1典,兩␾2␸␺2典+兩␾^¯2␸␺^¯2典其, we obtain the canonical vector兩0001典+兩0100典+兩1000典, corresponding to the degenerate class 0₃W.

共15兲 W= span兵000, 03⌿12其. When W

= span兵兩␾1␸1典,兩␾2␸2典+兩␾^¯2␸^¯2典其丢兩␺典, we obtain the canonical vector兩0010典+兩0100典+兩1000典, corresponding to the degenerate class 0₄W.

共16兲 W= span兵01⌿13, 0₁⌿13其. When W=C²丢span兵兩⌿典其, we obtain the canonical vector 兩0000典+兩0011典+兩1100典 +兩1111典=共兩00典+兩11典兲共兩00典+兩11典兲, corresponding to the degenerate class⌿12⌿34.

共17兲 W= span兵02⌿13, 0₂⌿13其. When W= span兵兩␾␸␺典 +兩␾^¯␸␺^¯典,兩␾␸^¯␺典+兩␾^¯␸^¯␺^¯典其, we obtain the canonical vector 兩0000典+兩0101典+兩1010典+兩1111典=共兩00典+兩11典兲13共兩00典+兩11典兲24, corresponding to the degenerate class⌿13⌿24.

共18兲W= span兵03⌿13, 0₃⌿13其. WhenW=兩⌿典丢C², we obtain the canonical vector 兩0000典+兩0110典+兩1001典+兩1111典

=共兩00典+兩11典兲14共兩00典+兩11典兲23, corresponding to the degenerate class⌿14⌿23.

No further degenerate classes can be found within the rest of possible structures forW. Notice how all of them can be obtained from the first 6 and 16th cases by just permuting the qubit indices. We have preferred to make use of a systematic approach in order to illustrate its usage.

B. Genuine classes

In order to be exhaustive we must check out all possible structures that the right singular subspaceWmay have. Ana priori list is given by span兵000,000其, span兵000, 0i₁⌿其, span兵000, GHZ其, span兵000,W其, span兵0i₁⌿, 0i₂⌿其, span兵0i₁⌿, GHZ其, span兵0i₁⌿,W其, span兵GHZ, GHZ其, span兵GHZ,W其, and span兵W,W其, wherei_k= 1 , 2 , 3. The same convention as above has been followed when we say that, e.g., span兵0i₁⌿, GHZ其indicates that only one vector belong- ing to the class 0_i

1⌿ is contained in W, the rest being all GHZ共and possiblyW兲. In other words, in span兵⌿i,⌿k其it is understood that⌿iand⌿kare those vectors inW with the lowest degree of entanglement degeneracy, according to the scale 000⬍0_i

k⌿⬍GHZ,W.

1.W=span{000,000}

All cases leading to degenerate cases have been considered above. However, whenW= span兵兩␾␸␺^,␾^¯^¯␸␺^¯典其, a well- known entanglement class arises: namely, the GHZ class. We

choose nonsingular transformationsF^关2兴 so that兩0典=F^关2兴兩␾典, 兩1典=F^关2兴兩␾^¯典and similarly for F^关3,4兴. We choose alsoF^关1兴 so that关F^关1兴兩v₁典F^关1兴兩v₂典兴=关␮ij*兴⁻¹

共

^␴⁰¹⁻¹ ^␴⁰²⁻¹

兲

^{, where}^兩v^j^典^denotes

the left singular vectors of the coefficient matrixC_⌽ of the state 兩⌽典苸C²丢C²丢C²丢C² and ␮ij are the coefficients expressing the right singular vectors 兩wk典 of C_⌽ in terms of 兩␾␸␺典and兩␾^¯␸^¯␺^¯典:兩wi典=␮i1兩␾␸␺典+␮i2兩␾^¯␸^¯␺^¯典. Then, we can write关7兴

CF关1兴丢¯丢F关4兴兩⌽=共␮ij

*兲⁻¹

冉

^␴⁰¹ ␴⁰2

冊

⁻¹

冉

^␴⁰¹ ␴⁰2

冊

⫻

冉

^␮¹¹^* 0 0 0 0 0 0 ␮12*

␮21* 0 0 0 0 0 0 ␮22*

冊

=

冉

1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1

冊

^. ^共3兲

This matrix corresponds to the canonical state 兩0000典 +兩1111典—i.e., to the so-called GHZ state.

2.W=span{000, 0₁⌿23}

In general, in this caseW= span兵兩␾1␸1␺1典,兩␾2典丢共兩␸2␺2典 +兩␸¯2␺^¯2典兲其. The degenerate case takes place when 兩␾1典 and 兩␾2典 are linearly dependent, driving us to the class 0₂W, already stated above.

Thus, let us consider the case when 兵兩␾1典,兩␾2典其 are linearly independent. Now the two-dimensional subspace span兵兩␸1␺1典,兩␸2␺2典+兩␸^¯2␺^¯2典其 can be either of type span兵00, 00其or of type span兵00,⌿其:

共i兲When span兵兩␸1␺1典,兩␸2␺2典+兩␸^¯2␺^¯2典其= span兵00, 00其, then there exist linearly independent vectors 兩␾典, 兩␾^¯典, 兩␸␺典, and 兩␸^¯␺^¯典 such that 兩wk典=␮k1兩␾␸␺典+␮k2兩␾^¯典丢共兩␸␺典 +兩␸^¯␺^¯典兲. Thus, it is immediate to realize that there exist nonsingular transformationsF^关^k^兴,k= 1 , 2 , 3 , 4, such that

C_F关1兴丢...丢F^关4兴兩⌽=共␮ij*兲⁻¹

冉

^␴⁰¹ ␴⁰2

冊

⁻¹

冉

^␴⁰¹ ␴⁰2

冊

⫻

冉

^␮␮¹¹^*21* ^{0 0 0}0 0 0 ^␮␮¹²^*22* ^{0 0}0 0 ^␮␮¹²^*22*

冊

=

冉

1 0 0 0 0 0 0 0

0 0 0 0 1 0 0 1

冊

^. ^共4兲

This canonical coefficient matrix corresponds to the state 兩0000典+兩1100典+兩1111典.

共ii兲 When span兵兩␸1␺1典,兩␸2␺2典+兩␸^¯2␺^¯2典其= span兵00,⌿其, then there exist vectors兩␾典, 兩␾^¯典,兩␸典,兩␸^¯典, 兩␺典, and 兩␺^¯典such that兩wk典=␮k1兩␾␸␺典+␮k2兩␾^¯典丢共兩␸␺^¯典+兩␸^¯␺典兲. Thus, it is immediate to realize that there exist nonsingular transforma- tionsF^关^k^兴,k= 1 , 2 , 3 , 4, such that

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CF关1兴丢...丢F关4兴兩⌽=共␮ij

*兲⁻¹

冉

^␴⁰¹ ␴⁰2

冊

⁻¹

冉

^␴⁰¹ ␴⁰2

冊

⫻

冉

^␮␮¹¹^*₂₁^* ^{0 0 0 0}^{0 0 0 0} ^␮␮¹²^*₂₂^* ^␮␮¹²^*₂₂^* ⁰⁰

冊

=

冉

1 0 0 0 0 0 0 0

0 0 0 0 0 1 1 0

冊

^. ^共⁵^兲

This canonical coefficient matrix corresponds to the state 兩0000典+兩1101典+兩1110典.

3.W=span{000, 0₂⌿13}

Although it is always possible to reproduce a similar argument to the one above, we will resort to permutation symmetry, thus leading to the degenerate class 0₃W 共already stated兲and the genuine classes with canonical states given by 兩0000典+兩1010典+兩1111典and兩0000典+兩1011典+兩1110典.

4.W=span{000, 0₃⌿12}

With analogous arguments, in this case the degenerate class is 0₄W 共already stated兲 and the genuine classes are identified by the canonical states given by 兩0000典+兩1001典 +兩1111典and兩0000典+兩1011典+兩1101典.

5.W=span{000, GHZ}

This class is highly richer than those above. In this case W= span兵兩␾1␸1␺1典,兩␾2␸2␺2典+兩␾^¯2␸^¯2␺^¯2典其, with the restriction that no product state other than兩␾1␸1␺1典 and no 0k⌿ state belong toW共otherwise we would be in one of the preceding classes兲. We define now nonsingular transformations F^关^k^兴 such thatF^关²^兴兩␾2典=兩0典,F^关²^兴兩␾^¯2典=兩1典, and similarly for兩␸2典, 兩␸^¯2典, 兩␺2典, and 兩␺^¯2典. We define also F^关¹^兴 such that 关F^关1兴兩v₁典F^关1兴兩v₂典兴=关␮ij

*兴⁻¹

共

^␴⁰¹⁻¹ ^␴⁰²⁻¹

兲

, with the same conven- tions as above. Then, the canonical vector can be written as 关11兴

兩0␾␸␺典+兩1000典+兩1111典. 共6兲

The factor vectors兩␾典,兩␸典, and兩␺典are arbitrary up to the restriction of not producing more than one 000 vector and no 0k⌿vector inW. Although this recipe embraces all possible cases, we will single out two different subsets within this class: namely,共i兲that of states leading to right singular subspaces W with no W state in it—i.e., containing only one product vector and GHZ vectors—and共ii兲that of states leading to right singular subspacesWwith at least oneWvector in it. Indeed, we will show that there will be either one or twoW vectors inW.

共i兲The set of states leading toWcontaining noWvectors is that identified by the canonical ones given by 兩001␺典+兩1000典+兩1111典, 兩00␸1典+兩1000典+兩1111典 and 兩0␾01典 +兩1000典+兩1111典, with no restrictions upon the factor vectors.

Moreover, if we denote the components of兩␾典in the 共non- necessarily orthonormal兲basis兵兩␾2典,兩␾^¯2典其by␾0and␾1and similarly for 兩␸典 and 兩␺典, then whenever 0⫽

冑

␾0␸0␺0

= ±

冑

_␾₁_␸₁_␺₁_⫽0, there will also be a unique product vector,

the rest being all GHZ in the right singular vector of the generic state共6兲.

This is proved resorting to theorem 1. We will illustrate the calculation with the first canonical vector 兩001␺典 +兩1000典+兩1111典, whose right singular subspace is spanned by兩01␺典and _冑¹₂兩000典+_冑¹₂兩111典. Thus, a generic vector inW will be of the form ␣共兩000典+兩111典兲+␤兩01␺典, whose coefficient matrix is关cf. Eqs.共1a兲–共1c兲兴

C^共¹^兲=

冉

^␣^{0 0}⁰ ^␤␺⁰⁰^* ^␤␺␣¹^*

冊

^. ^共⁷^兲

For any values of ␺k

* it is clear that r共C^共ⁱ^兲兲= 2 for i= 1 , 2 , 3 共except for the trivial case ␣= 0兲. Then, if ␺1= 0, r共W1兲

=r共W2兲= 1; thus, it is a GHZ vector; if␺1⫽0,r共W1兲= 2 and r共W2兲= 1 and the spectrum ofW₁⁻¹W₂ is never degenerate. It is again a GHZ vector for any value of兩␺典.

The other canonical vectors are handled in a similar fashion, and they can additionally be obtained under permutations among qubits 2, 3, and 4. Among them we single out the particular cases 兩0000典+兩1101典+兩0111典, 兩0000典+兩1011典 +兩0111典 and兩0000典+兩1110典+兩0111典, which can also be obtained from those in the 0_k⌿classes by permutations among thefourqubits.

For the generic state 共6兲, let us write, as explicitly supposed, W= span兵兩␾␸␺典,兩000典+兩111典其, with no vector 兩␾典, 兩␸典, and兩␺典equal to 兩0典 or 兩1典. If we denote again by sub- indices 0 and 1 the components of each vector兩␾典,兩␸典, and 兩␺典 in the 兵兩0典,兩1典其 basis, then the coefficient matrix of a generic vector inWwill be

C=

冉

^␣␾␣␾⁰^␸1⁰␸^␺0⁰␺⁺0^{␤ ␣␾}␣␾⁰1^␸␸⁰0^␺␺¹1 ^␣␾␣␾⁰1^␸␸¹1^␺␺⁰0 ␣␾^␣␾1␸⁰1^␸␺¹1^␺+¹␤

冊

^,

共8兲 where no coefficient is null, as supposed. Then it should be clear that r共C^共ⁱ^兲兲= 2 for all i= 1 , 2 , 3. Furthermore, since detW₁=␣␤␾0␸1␺1 and detW₂=␣␤␾1␸0␺0, we will have r共W₁兲=r共W₂兲= 2 共except for the trivial cases ␣␤= 0兲; thus, we must check the degeneracy of the spectrum of W₁⁻¹W₂:

␴共W₁⁻¹W₂兲 is degenerate whenever the discriminant of the characteristic polynomial is null—i.e., whenever 关tr共W₁⁻¹W₂兲兴²− 4 det共W₁⁻¹W₂兲= 0, which drives us to the con- dition ␤⁺␣共

冑

_␾₀_␸₀_␺₀±

冑

_␾₁_␸₁_␺₁兲²= 0, which immediately yields the above stated conditions.

共ii兲The set of states leading toWcontaining at least one W vector comprises the rest of possibilites. To prove this assertion we need to show that all possibilities forWhaving noW vector has been already considered. The most generic case occurs when W= span兵兩␾1␸1␺1典,兩␾2␸2␺2典+兩␾^¯2␸^¯2␺^¯2典其, which we detach in the following particular cases.

共a兲If兩␾1␸1␺1典=兩␾2␸2␺2典,兩␾^¯2␸^¯2␺^¯2典, then two 000 vectors belong toW. This case is ruled out.

共b兲 If 兩␾1␸1␺1典=兩␾2␸2␺典,兩␾^¯2␸^¯2␺典, with 兩␺典⫽兩␺2典,兩␺^¯2典, respectively, then there will be a 0k⌿ in W unless 兩␺典

=兩␺^¯2典,兩␺2典, respectively. We give the proof for the first case.

Let兩␾1␸1␺1典=兩␾2␸2␺典, with 兩␺典⫽兩␺2典. A generic vector in W will be of the form ␣兩␾2␸2␺典+␤共兩␾2␸2␺2典+兩␾^¯2␸^¯2␺^¯2典兲

(5)

=兩␾2␸2典共␣兩␺典+␤兩␺2典兲+␤兩␾^¯2␸^¯2␺^¯2典. It is clear that, provided that 兩␺典and 兩␺^¯2典 are linearly independent, it is always possible to find␣^,␤ ^{such that} ␣兩␺典+␤兩␺^¯2典=␤^¯兩␺^¯2典, driving us to a 0₃⌿ vector in W, against the general hypothesis. By symmetry, all other cases can be accounted for in a similar fashion, leaving us with the cases that whenever two vectors in兩␾␸␺典 are equal to two vectors in兩␾2␸2␺2典 or 兩␾^¯2␸^¯2␺^¯2典, then the other must be the third one corresponding to the opposite. For instance, if兩␾2␸2␺2典=兩000典, 兩␾^¯2␸^¯2␺^¯2典=兩111典, and兩␾1␸1␺1典=兩00␺典, it must be 兩␺典=兩1典.

共c兲 If 兩␾1␸1␺1典=兩␾2␸␺典, with 兩␸典⫽兩␸2典, 兩␺典⫽兩␺2典, and 兩␸␺典⫽兩␸^¯2␺^¯2典, use the same nonsingularF^关^k^兴as in the generic case共6兲to arrive at the coefficient matrix for a general vector inW:

冉

^␣␸⁰^␺⁰⁰⁺^{␤ ␣␸}⁰⁰^␺¹ ^␣␸⁰¹^␺⁰ ^␣␸␤¹^␺¹

冊

^, ^共⁹^兲

with the above conditions translated as 兩␺典⫽兩0典⇔␸1⫽0, 兩␺典⫽兩0典⇔␺1⫽0, and 兩␸␺典⫽兩11典⇔共␸0,␺0⫽0兲 simultaneously. It is then clear that 共except for the trivial cases

␣␤= 0兲,r共C^共ⁱ^兲兲= 2 for all i= 1 , 2 , 3, r共W1兲= 2 and r共W2兲= 1.

The spectrum␴共W₂⁻¹W₁兲is degenerate only if␣␸0␺0+␤= 0.

Thus there will be no W vector in W only when

␸0␺0= 0—i.e., when兩␸典=兩1典 or 兩␺典=兩1典, which, going back before the application of the nonsingular F^关^k^兴, means that 兩␸1典=兩␸^¯2典and兩␺1典=兩␺^¯2典. By using␴xupon qubits 2, 3, and 4 the symmetric case 兩␾1␸1␺1典=兩␾^¯2␸␺典, with 兩␸典⫽兩␸^¯2典, 兩␺典⫽兩␺^¯2典, and兩␸␺典⫽兩␸2␺2典, is also included in this analysis. By permutations among the qubits 2, 3, and 4 all preceding cases are accounted for in a similar fashion.

共d兲 Finally the case 兩␾1␸1␺1典=兩␾␸␺典, with 兩␾典⫽兩␾2典,兩␾^¯2典, 兩␸典⫽兩␸2典,兩␸^¯2典, and 兩␺典⫽兩␺2典,兩␺^¯2典, drives us, after application of the nonsingularF^关^k^兴, to the coefficient matrix共8兲, already accounted for.

No more options are left; thus, we have scrutinized all possibleW’s with structure span兵000, GHZ其.

6.W=span{000, W}

By span兵000,W其we indicate that only one 000 vector and W vectors belong to W. The generic case will be that expressed by W= span兵兩␾1␸1␺1典,兩␾2␸2␺^¯2典+兩␾2␸^¯2␺2典 +兩␾^¯2␸2␺2典其. We will show that there will be no GHZ state in Wonly if兩␾1␸1␺1典=兩␾2␸2␺2典, which after application of the nonsingular transformationsF^关^k^兴 will drive us to the canonical state 兩0001典+兩0010典+兩0100典+兩1000典—i.e., the W state for four qubits. As before, we will be exhaustive.

共a兲 If 兩␾1␸1␺1典=兩␾2␸2␺^¯2典,兩␾2␸^¯2␺2典,兩␾^¯2␸2␺2典, then it is clear that a 0k⌿ can be found in W against the general hypothesis. This case is then ruled out.

共b兲If兩␾1␸1␺1典=兩␾2␸2␺典, with兩␺典⫽兩␺^¯2典, after application of the nonsingularF^关^k^兴, we obtain the coefficient matrix for a generic vector inW:

冉

^␣␺␤⁰ ^␣␺¹⁰⁺^{␤ ␤}^{0 0}⁰

冊

^. ^共10兲

If␣␺1+␤= 0, then there will exist non-null␣^,␤such that a 0₃⌿vector will belong toW, against the hypothesis. Thus it must be␺1= 0—i.e.,兩␺典=兩0典. Under this restriction and after using theorem 1 upon the coefficient matrix共10兲, all vectors will be of type W, since r共C^共ⁱ^兲兲= 2 for all i= 1 , 2 , 3, r共W1兲= 2, r共W2兲= 1, and the spectrum ␴共W₁⁻¹W₂兲 is always degenerate. By permutation symmetry the cases 兩␾1␸1␺1典

=兩␾2␸␺2典,兩␾␸2␺2典, with the corresponding restrictions, are also considered in this analysis.

共c兲If兩␾1␸1␺1典=兩␾2␸␺^¯2典, a similar argument leads to the coefficient matrix for a generic vector inWgiven by

冉

⁰␤ ^␣␸⁰⁰⁺^{␤ ␤}⁰ ⁰^␣␸⁰ ¹

冊

^. ^共11兲

After application of theorem 1, it should be clear that it is always possible to find non-null ␣^,␤ ^{such that} 共11兲 corresponds to a GHZ vector in W, against the hypothesis. By permutation symmetry, the rest of cases with the 000 generator having two factors in common with some of components of theW generator is also contained in this analysis.

共d兲 If 兩␾1␸1␺1典=兩␾1␸␺典, with 兩␸典⫽兩␸2,␸^¯2典 and 兩␺典⫽兩␺2,␺^¯2典, after a similar argument the coefficient matrix of a generic vector in the right singular subspace will be

冉

^␣␸⁰⁰^␺⁰ ^␣␸⁰^␺⁰¹⁺^{␤ ␣␸}¹^␺⁰⁰⁺^{␤ ␣␸}␤¹^␺¹

冊

^. ^共12兲

It should be clear that r共C^共ⁱ^兲兲= 2 for all i= 1 , 2 , 3. Now, if

␸1␺0+␸0␺1⫽0, there will always exist non-null ␣^,␤ ^such that r共W₁兲=r共W₂兲= 1, hence a GHZ vector, against the hypothesis. On the contrary, if ␸1␺0+␸0␺1= 0, then r共W1兲= 2 andr共W2兲= 1, but it is always possible to find non-null␣,␤ such that␴共W₁⁻¹W₂兲is nondegenerate; hence, GHZ苸W. No- tice that at most anotherW vector, apart from the generator, can be found. By permutation symmetry and by using ␴x

upon each qubit, the rest of cases in which the 000 generator contains one common factor vector with theW generator is also considered in this analysis.

共e兲 Finally, if 兩␾1␸1␺1典=兩␾␸␺典, with no common factor vector with theWgenerator, a similar argument leads to the coefficient matrix for a generic vector in the right singular subspace given by

冉

␣␾^␣␾1␸⁰0^␸␺⁰0^␺+⁰␤ ^␣␾␣␾⁰^␸1⁰␸^␺0¹␺⁺1^{␤ ␣␾}␣␾⁰^␸1¹␸^␺1⁰␺⁺0^{␤ ␣␾}␣␾⁰1^␸␸¹1^␺␺¹1

冊

^.

共13兲 A systematic application of theorem 1 makes it clear that there always exist␣,␤such that this matrix corresponds to a GHZ state, against the hypothesis.

7.W=span{0₁⌿23, 0₁⌿23}

The generic case will be W= span兵兩␾1⌿1典,兩␾2⌿2典其, where 兩⌿k典 denotes a bipartite entangled vector. Several cases appear.

共i兲 When 兩␾1典 and 兩␾2典 are linearly dependent, then W=兩␾典丢span兵兩⌿1典,兩⌿2典其, driving us to the already known degenerate classes 0₁0₂⌿, 02GHZ and 0₂W, respectively,

(6)

when it adopts the structures span兵兩␾⌿典其, 兩␾典

丢span兵00, 00其, and 兩␾典丢span兵00,⌿其. In the forthcoming cases, we will then impose the linear independence of兩␾1典 and兩␾2典.

共ii兲If 兩⌿1典and兩⌿2典 are linearly dependent, then W=C²

丢兩⌿典, corresponding to the degenerate class ⌿12⌿34, as stated above. Therefore, we will also impose the linear independence of兩⌿1典 and兩⌿2典.

共iii兲 When span兵兩⌿1典,兩⌿2典其= span兵00, 00其, then it is always possible to find linear independent factor vectors such that兩⌿j典=a_j1兩␸␺典+a_j2兩␸^¯␺^¯典, witha_j1a_j2⫽0. Defining nonsingular F^关^k^兴 such that F^关2兴关a₁₁兩␾1典兴=兩0典, F^关2兴关a21兩␾2典兴=兩1典, F^关3兴关兩␸典兴=兩0典, F^关3兴关兩␸^¯典兴=兩1典, F^关4兴关兩␺典兴

=兩0典,F^关4兴关兩␺^¯典兴=兩1典, andF^关1兴as in preceding cases, we obtain the canonical matrix

冉

^{1 0 0}^{0 0 0} ^␭⁰¹ ^{0 0 0}^{1 0 0} ␭⁰2

冊

^, ^共14兲

where we have defined ␭i⬅^a_aⁱ²^*

i1*. This matrix corresponds to the canonical state 兩0000典+兩1100典+␭1兩0011典+␭2兩1111典, with␭1⫽␭2.

共iv兲 When span兵兩⌿1典,兩⌿2典其= span兵00,⌿其, then it is always possible to find linear independent factor vectors such that 兩⌿j典=a_j1兩␸␺典+a_j2共兩␸␺^¯典+兩␸^¯␺典兲, with a_j1a_j2⫽0. Defin- ing nonsingular F^关^k^兴 such that F^关²^兴关a11兩␾1典兴=兩0典, F^关²^兴关a21兩␾2典兴=兩1典, F^关³^兴关兩␸典兴=兩0典, F^关³^兴关兩␸^¯典兴=兩1典, F^关⁴^兴关兩␺典兴

=兩0典,F^关4兴关兩␺^¯典兴=兩1典, andF^关1兴as in preceding cases, we obtain the canonical matrix

冉

¹⁰ ^␭⁰¹ ^␭⁰¹ ^{0 0}^{0 1} ␭⁰2 ␭⁰2 ⁰0

冊

^, ^共15兲

where we have defined ␭i⬅^a_aⁱ²^*

i1*. This matrix corresponds to the canonical state 兩0000典+兩1100典+␭1兩0001典+␭1兩0010典 +␭2兩1101典+␭2兩1110典, with ␭1⫽␭2.

All possible cases have been considered. Furthermore, notice that several states obtained by permutations involving qubit 1 in preceding cases are enclosed in this class. This will also happen in forthcoming classes: some permutations involving qubit 1 usually involve a change of entanglement class, although the general structure of the canonical state shows permutation symmetry关12兴.

8.W=spanˆ0₁⌿23, 0₂⌿13‰

The generic case is given by W= span兵兩␾1⌿1典,兩␾2␸2␺2典 +兩␾^¯2␸2␺^¯2典其. Two possibilities arise.

共i兲 span兵兩⌿1典,兩␸2␺2典其= span兵00, 00其= span兵兩␸2␺2典, 兩␸^¯2␺^¯^¯2典其, where the double overbar also indicates linear independence with respect to 兩␺2典, although possible linear dependence with 兩␺^¯2典. Then a generic vector in Wwill be given by ␣兩␾1典共a兩␸2␺2典+b兩␸^¯2␺^¯^¯2典兲+␤共兩␾2␸2␺2典 +兩␾^¯2␸2␺^¯2典兲,with

ab⫽0. Defining nonsingular F^关^k^兴 such that F^关2兴关兩␾2典兴=兩0典, F^关2兴关兩␾^¯2典兴=兩1典, F^关3兴关兩␸2典兴=兩0典, F^关3兴

关

^b_a

兩

␸^¯2

典

_兴=兩1典, F^关4兴关兩␺2典兴

=兩0典, F^关4兴关兩␺^¯2典兴=兩1典, and F^关1兴 as in preceding cases, we obtain the canonical vector 兩0␾00典+兩0␾¹␺典+兩1000典+兩1101典, with the restrictions that 兩␺典⫽0 and 兩␾典⫽兩0 , 1典, simultaneously 共otherwise we would be reconsidering previous classes兲. With the use of ␴xupon qubit 4, we are also em- bracing the case in which span兵兩⌿1典,兩␸2␺^¯2典其= span兵00, 00其

= span兵兩␸2␺^¯2典,兩␸^¯2␺^¯^¯2典其, where now the double overbar denotes linear independence with respect to兩␺^¯2典.

共ii兲 span兵兩⌿1典,兩␸2␺2典其= span兵00,⌿其= span兵兩␸2␺2典,兩␸2␺^¯^¯2典 +兩␸^¯2␺2典其, with the same convention as before for the double overbar. Expressing 兩⌿1典=a兩␸2␺2典+b共兩␸2␺^¯^¯2典+兩␸^¯2␺2典兲 and with a similar argument as the preceding case, we arrive at the canonical state 兩0␾⁰␺典+兩0␾10典+兩1000典+兩1101典, with the restrictions that 兩␺典⫽0 and兩␾典⫽兩0 , 1典, simultaneously 共otherwise we would be reconsidering previous classes兲. By using ␴x upon qubit 4, we are also considering the case span兵兩⌿1典,兩␺2␾^¯2典其= span兵00,⌿其.

9.W=span{0₁⌿23, 0₃⌿12}

Although this case can also be analyzed with similar arguments, we will use permutation symmetry among qubits 2 and 3 in the case span兵01⌿23, 0₂⌿13其to arrive at the canonical states 兩0␾00典+兩0␾␸1典+兩1000典+兩1110典 and 兩0␾␸0典 +兩0␾01典+兩1000典+兩1110典.

10.W=span{0₂⌿13, 0₂⌿13}

Under permutation symmetry between qubits 2 and 3 in the case span兵01⌿23, 01⌿23其, we rapidly find two genuinely entangled canonical states—namely, 兩0000典+兩1010典 +␭1兩0101典+␭2兩1111典 and 兩0000典+兩1010典+␭1兩0101典 +␭1兩0001典+␭2兩1011典+␭2兩1110典—and one degenerate canonical state of type⌿13⌿24共apart from the already consid- ered兲.

11.W=span{0₂⌿13, 0₃⌿12}

Under permutation symmetry between qubits 2 and 4 in the case span兵0₁⌿23, 0₃⌿12其 we rapidly find two genuinely entangled canonical states: namely,兩000␺典+兩0␾¹␺典+兩1000典 +兩1101典 and兩0␾⁰␺典+兩001␺典+兩1000典+兩1101典.

12.W=span{0₃⌿12, 0₃⌿12}

Under permutation symmetry between qubits 2 and 4 in the case span兵01⌿23, 0₁⌿23其, we rapidly find two genuinely entangled canonical states—namely, 兩0000典+兩1001典 +␭1兩0110典+␭2兩1111典 and 兩0000典+兩1001典+␭1兩0010典 +␭1兩0100典+␭2兩1011典+␭2兩1101典—and one degenerate canonical state of type⌿14⌿23共apart from that already consid- ered兲.

13.W=span{0₁⌿23, GHZ}

The generic case will be of the form W

= span兵兩␾1⌿1典,兩␾2␸2␺2典+兩␾^¯2␸^¯2␺^¯2典其. Finding nonsingular transformations F^关ⁱ^兴 as before, we can write W

= span兵兩␾⌿典,兩000典+兩111典其. A generic vector inWwill then

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be of the form␣兩␾⌿典+␤共兩000典+兩111典兲. The restrictions on 兩␾典and兩⌿典共thus on兩␾1典and兩⌿1典兲will be that no other 0i_k⌿ must belong toW. For instance, a particular restriction will be that 兩⌿典苸span兵00, 11其, since otherwise 兩⌿典=a兩00典 +b兩11典,ab⫽0, and then a generic state inWwill adopt the form 共␣^a兩␾典+␤兩0典兲兩00典+共␣^b兩␾典+␤兩1典兲兩11典. It is always possible to find non-null␣^,␤such that the preceding state is a 0₁⌿23vector different to兩␾⌿典—i.e., another 01⌿23vector:

we would be in the span兵0₁⌿23, 0₁⌿23其case. The canonical state will be given by兩0␾⌿典+兩1000典+兩1111典.

14.W=span{0₂⌿13, GHZ}

By permutation symmetry between qubits 2 and 4, the canonical state will be兩0⌿24␾3典+兩1000典+兩1111典, where the first factor denotes an entangled state between qubits 2 and 4 and two factor states involving state 1 and 3.

15.W=span{0₃⌿12, GHZ}

By permutation symmetry between qubits 2 and 3, the canonical state will be兩0⌿23␾4典+兩1000典+兩1111典, where the first factor denotes an entangled state between qubits 2 and 3 and two factor states involving state 1 and 4.

16.W=span{0₁⌿23, W}

By W= span兵01⌿23,W其 it is meant that only one degenerate tripartite vector of type 0₁⌿23belongs toW, all the rest being of type W. It is immediate to prove that this is impossible—i.e., that there is always possible to find a GHZ vector in W. The generic case will be given by W

= span兵兩␾1␸1␺1典,兩␾2␸2␺^¯2典+兩␾2␸^¯2␺2典+兩␾^¯2␸2␺2典其. Defining F^关^k^兴 so that F^关²^兴兩␺2典=兩0典, F^关²^兴兩␾^¯2典=兩1典 and similarly for k

= 3 , 4 and using the same F^关1兴 as in preceding cases, the coefficient matrix for a generic 关13兴 vector 兩␾⌿典+␭共兩000典 +兩111典兲inWwill be

冉

␾1^␾⌿⁰^⌿00⁰⁰+␭ ^␾⁰␾^⌿1⌿⁰¹01⁺^␭ ^␾⁰␾^⌿1⌿¹⁰10⁺^␭ ^␾␾⁰1^⌿⌿¹¹11

冊

^. ^共16兲

Now there is a continuous range of values␭苸Csuch that r共C^共ⁱ^兲兲= 2. Focus upon the set of which such that detW₂⫽0 共again a continuous range of them兲; then, it is clear that the spectrum ofW₂⁻¹W₁will be in general nondegenerate, driving us to a GHZ vector. Thus we are in fact in the case span兵01⌿23, GHZ其.

17.W=span{0₂⌿13, W}

By permutation symmetry the same analysis as in the preceding case allows us to conclude that we are in the span兵02⌿13, GHZ其 case.

18.W=span{0₃⌿12, W}

By permutation symmetry the same analysis as in the preceding case allows us to conclude that we are in the span兵03⌿12, GHZ其 case.

19.W=span{GHZ,W}

In this class there exist no degenerate tripartite vectors;

i.e., all of them are either of type GHZ or of typeW. With the same arguments as before, the canonical state is found to be of the form 兩1␾␸␺典+兩1␾^¯␸^¯␺^¯典+兩0001典+兩0010典+兩0100典, where the product vectors兩␾典,兩␾^¯典,兩␸典,兩␸^¯典,兩␺典, and兩␺^¯典 are restricted to produce no degenerate state inW.

20.W=span{GHZ,GHZ}

The notation is meant to describe the situation in which only GHZ vectors are contained in W. Apart for particular instances in which a degenerate tripartite vector may appear as a linear combination of both GHZ generators共which we rule out, since they have been already dealt with兲, we will show that there will always exist at least oneWvector inW.

The generic case is given by span兵兩␾1␸1␺1典 +兩␾^¯1␸^¯1␺^¯1典,兩␾2␸2␺2典+兩␾^¯2␸^¯2␺^¯2典其. Then, after defining nonsingular transformations F^关^k^兴 in a similar fashion as in all preceding cases, the coefficient matrix for a generic vector in Wwill be given by

TABLE I. Genuine entanglement classes for four qubits.

Class共W兲 Canonical states Name Proposed notation

span兵000,000其兩0000典+兩1111典 GHZ W_000,000

span兵000, 0_k⌿其兩0000典+兩1100典+兩1111典 W_000,0

k⌿

兩0000典+兩1101典+兩1110典

span兵000, GHZ其兩0␾␸␺典+兩1000典+兩1111典 W_000,GHZ

span兵000,W其兩1000典+兩0100典+兩0010典+兩0001典 W W_000,W

span兵0_k⌿, 0_k⌿其兩0000典+兩1100典+␭1兩0011典+␭2兩1111典 ⌽4 W₀

k⌿,0_k⌿

兩0000典+兩1100典+␭1兩0001典+␭1兩0010典+␭2兩1101典 +␭2兩1110典

span兵0_i⌿, 0_j⌿其兩0␾00典+兩0␾1␺典+兩1000典+兩1101典 W₀

i⌿,0j⌿

兩0␾0␺典+兩0␾10典+兩1000典+兩1101典

span兵0_k⌿, GHZ其兩0␾⌿典+兩1000典+兩1111典 W₀

k⌿,GHZ

span兵GHZ,W其兩0001典+兩0010典+兩0100典+兩1␾␸␺典+兩1␾¯␸¯␺¯典 W_GHZ,W