Separability of 3-qubits density matrices, related to l1 and l2 norms and to unfolding of tensors into matrices

We treat separability of 3 (and more) qubits states. Especially we discuss density matrices with maximally disordered subsystems (MDS), by using Hilbert-Schmidt (HS) decompositions, where in the general case these density matrices include 27 HS parameters. By using ‘unfolding methods’, the MDS tensors are converted into matrices and by applying singular values decompositions (SVD) to these matrices the number of the parameters for treating full separability, in the general MDS case, is reduced to 9, and under the condition that the sum of the absolute values of these parameters is not larger than 1, we conclude that the density matrix is fully separable. In order to know if density matrices with MDS are separable, one needs to check with 9 parameters at a time and not with all 27 parameters. We use also Frobenius (l2) norms. For treating bi-separability of 3-qubits MDS density matrices, the 27 HS parameters are divided into 9 triads. If the sum of the nine l2 norms for these triads is not larger than 1, we conclude that the density matrix is bi-separable. We analyze the relations between 3 qubits MDS density matrices and the method of high order singular value decomposition (HOSVD). We demonstrate the use of our methods in examples. For 3-qubits states which are non-MDS the HS decomposition includes up to 63 parameters. If the sum of the absolute values of all the HS parameters is not larger than 1, we conclude that the density matrix is fully separable, and we have explicit expressions for their separability. For the systems of GHZ and W states mixed with white noise we find a simple way to reduce the sum of the absolute values of the HS parameters and get better conditions for their full separability.


Introduction
Entanglement of qubits systems is at the core of the quantum computation field. There is much interest in quantum entangled states due to various potential applications that use the quantum properties of such states. The most famous application is the use of quantum systems for a new generation of computers that will be based on principles of quantum computation (QC). Therefore it is of utmost importance to quantify entanglement in such systems and to have a definite criterion when such systems are separable. The definition of full separability of a three-partite system is: A density matrix ρ on Hilbert space H A ⊗H B ⊗H C where A, B and C are the three parts of a three-partite system is defined as non-entangled/separable if there exist density operators r r r The interpretation of such definition is that for three-partite separable state these parts are completely independent of each other. In a pictorial description: Assuming 3-qubits fully separable state, we send the 3 qubits to Alice, Charles and Jacob, respectively, which are far from each other. Any measurement made by one of them will not affect the quantum properties of the qubits belonging to others. In the present work we develop sufficient separability conditions for three (and more) qubits. We develop also conditions for bi-separability where in a pictorial description one of the three qubits (e.g., that which belongs to Jacob) is separated from the others two qubits (belonging to Alice and Charles) where the latter two qubits are entangled. In a pictorial description any measurement made by Jacob will not affect the quantum properties of the qubits belonging to Alice and Charles and vice versa.
However, measurement of a qubit belonging to Alice (Charles) can affect the quantum properties of the qubit belonging to Charles (Alice) (obtaining e.g. EPR effects). The quantum analysis of full separability and biseparability of three (and more) qubits is therefore very important both from its theoretical interest and from the quantum computation applications.
In an old paper [1] we analyzed the use of Hilbert-Schmidt (HS) decompositions in relation to information theory. The outer products in the HS decompositions are of Pauli matrices where we relate the correlations of 2-qubits systems to one and two qubits measurements, and the correlations for 3-qubits systems to certain one, two and three-qubits measurements, etc. One advantage of such description is that it is valid for both pure and mixed states. In later works [2][3][4] we used such decompositions for treating separability and entanglement properties of qubits systems. The separability and entanglements properties of two qubits systems were treated by us in [2]. Our aim in the present article is to develop new methods for treating separability of three qubits (and more) which were not used in our previous works [3,4]. In our previous work [3] we treated bi-separability problems for 3-qubits systems with maximally disordered subsystems (MDS) i.e., density matrices which by tracing over any subsystem it gives the unit matrix [5]. While in the previous work [3] we obtained explicit expressions for bi-separability only for very special MDS cases, we succeed in the present work to give explicit biseparability expressions for any MDS state, including up to 27 parameters (see later in the article equation (44)).
In [4] we have treated several topics which are not treated in the present work (e.g. GHZ diagonal states, Braid states mixed with white noise, qubit and a qudit, etc). In [4] we developed also explicit full separability expressions for 3-qubits MDS states and noticed that such expressions are related to unitary transformation of the l 1 norm [6] condition  where R a,b,c are the HS parameters of the MDS density matrices,. This fact leads us to the idea in the present work that the analysis of three qubits (and more) MDS states becomes complicated due to its tensor expressions which cannot be diagonalized and therefore the methods of 'unfolding of tensors into matrices', which have been described in the literature [7][8][9][10][11], should be useful for treating MDS density matrices. We analyze in the present article various properties of the MDS density matrices and extend the analysis to the use of l 2 norm and to High Order Singular Values Decompositions (HOSVD). The HOSVD method has been developed in various mathematical works [7][8][9][10][11], but the use of such method to density matrices was not the concern of these authors. We find improvements in the condition for full separability for MDS density matrices by the use of this method. We demonstrate the use of our methods by examples. In the previous work [4] we treated also the separability problem for density matrices which are non-MDS including GHZ and W density matrices mixed with white noise with probability p. We found in this work [4] that while for the GHZ case p1/5 is a sufficient and necessary condition for separabiliy the condition p1/9 can be used as a sufficient condition for separability of the W state mixed with white noise. We emphasized in this work [4]: 'it is not obviously necessary and perhaps may be improved by other separability methods'. In the present work we improve these methods and obtain the condition for full separability of the W state mixed with white noise: p1/5 which is the same as that of GHZ mixed with white noise.
While concurrence is an efficient criterion for entanglement for two-qubits density matrices, in general it is not possible numerically to calculate the concurrence for more than two qubits, especially for the case of 27 MDS parameters (see, e.g., [12] where only lower bounds for quantification of entanglement by concurrence were found). Non-vanishing negativity [13] is a sufficient condition for inseparability of ρ but not necessary. For 3-qubits MDS density matrices (and all such odd n-qubits) we prove in the paper that the eigenvalues of the partial-transpose (PT) are the same as those of ρ. Therefore the negativity for these states is zero so that we cannot conclude if it is separable or inseparable. Our work gives sufficient (not necessary) conditions for separability of such density matrices, It is important to know if a density matrix is separable or entangled. Concurrence of 3-qubits density matrix is very difficult to calculate and it can be done only for special cases. The condition of Negativity >0 is a sufficent condition for entanglement but if Negativity=0 the result is not known. We treat this condition in a converse direction, giving a sufficient condition (not necessary) for separability. In a certain sense these two criterions complement each other. The proofs of our conditions look quite complicated but at the end, the results are quite simple: For getting the final conditions for separability one needs to calculate only the SVD decompositions of 3×3 and 3×9 matrices and such calculations are standard mathematical routine.
For MDS density matrices, by definition Here the 3-qubits are denoted by A, B, C, ρ is the density matrix, I denotes the unit 2×2 matrix, Tr A represents the trace over qubit A and ⊗ denotes the outer product. The density matrix of the 3 C will turn R a,b,c into, say, S p,q,r . The crucial point in treating the special case of 3-qubits MDS density matrices by the l 1 norm is that the condition for separability, given by: is not invariant under orthogonal transformations and we can improve the condition for separabilty by using orthogonal transformations which will reduce this 'separability form'.
Compared to the separability problem of two qubits the separability problem for 3 qubits MDS density matrix becomes very complicated as tensors cannot be diagonalized. In order to overcome the nondiagonalization problem of tensors, methods of 'unfolding of tensors into matrices' have been described in the literature [7][8][9][10][11]. These unfolding methods help us in the analysis of 3-qubits MDS systems, but as we are interested in application of such methods to density matrices special unfolding methods for this purpose are developed in the present work. While conditions for entanglement of 3-qubits have been treated by various other authors (see e.g. [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]) they have not applied the unfolding methods. Also the other authors have not given explicitly separable forms for the density matrices, which are studied in the present work.
The general idea of getting explicit separability for n-qubits system is that we get a sufficient condition, but it might be not necessary as the condition for separabilty might be improved by using different methods. This method is different from those methods in which one gets the condition for entanglement [14][15][16][17][18][19][20][21][22][23]25]. Both methods are important and in the present work we concentrate on finding explicit separability forms for n-qubits systems.
We analyze full separability properties of the 3-qubits MDS density matrices given by equation (2) by using unfolding of the tensor R a,b,c . One way of unfolding the 3-qubits tensor relative to A means that you keep the parameter a fixed as 1 or 2 or 3, (with the corresponding Pauli matrices (σ 1 ) A , or (σ 2 ) A , or (σ 3 ) A ) and then the parameters R 1,b,c , or R 2,b,c , or R 3,b,c are considered, respectively, as matrices with 3×3 dimension. Then, by using the singular-value-decomposition (SVD) [7,8] for the matrices: R 1,b,c , R 2,b,c , and R 3,b,c we get for each of them 3 singular values (SV's). We show that if the sum of these 9 SV's is not larger than 1, the density matrix is fully separable and we have an explicitly separable form for the density matrix. Similar unfolding methods can be made relative to B, or C.
In another method we develop fully separable forms for 3 qubits MDS density matrices which are related to Frobenius (l 2 ) norms [6] of 9 triads of HS parameters. We show that if the sum of the l 2 norms of the 9 HS triads is not larger than 1 then the 3-qubits MDS density matrix is fully separable, and we have another explicitly separable form for it.
For 3-qubits the density matrix may not be fully separable but may be bi-separable, i.e., not genuinely entangled [15]. A condition for bi-separability of 3-qubits MDS density matrices is obtained in the present work by the use of one qubit density matrix multiplied by Bell entangled states [2,29,30] of the other two qubits. By using this method the 27 HS parameters are divided into 9 triads which are different from those used for full separability. If the sum of the nine l 2 norms for these different triads is not larger than 1, then we conclude that the density matrix is (at least) bi-separable.
We apply the method of high order singular value decomposition (HOSVD) [7][8][9][10][11] for treatment of sufficient conditions for separability of 3-qubits MDS states. We demonstrate the application of this method in examples.
For the general 3-qubits density matrices, the HS decomposition includes 63 parameters which in a 4 dimensional notation may be written as m n k = n k R ; , , 0, 1, 2, 3.
, Such terms include products of Pauli matrices σ i (i=1, 2, 3), and the unit operator σ 0 =1. In various actual cases some of these parameters vanish. A sufficient condition for full separability is given by [4]: , , 0 , , 0,0,0 3 , , 0,0,0 but this condition may be improved. We demonstrate improvements in the condition for full separability by analyzing the system of GHZ state mixed with white noise and of W state mixed with white noise and get better conditions for full separability.

Fundamental properties of 3-qubits MDS density matrices
We try to use the Peres-Horodecki (PH) criterion [31,32] to obtain information about the eigenvalues of the MDS density matrix. We show now that the 3-qubits MDS density matrix and its PT have the same eigenvalues.
For showing this result we write the 3-qubits MDS density matrix as Here R, given in a short notation, includes all the terms in the summation of equation (2). By performing the (full) transpose of ρ into ρ T , every σ y in (2) is transformed to −σ y . This transformation does not change the eigenvalues (ρ and ρ T have the same eigenvalues). By a 180°unitary rotation of all qubits around the y axis the eigenvalues of the density matrix are not changed, but s s s s  - -, .
x x z z We denote the resulting density matrix by ρ TU . Here the superscript TU represents transpose of (2), i.e. of the whole density matrix, plus a unitary transformation. We emphasize that ρ TU and ρ have the same eigenvalues. However, since we assumed an odd number of σ we get R→−R by the TU transformation. Hence . 6

TU
On the other hand, the partial transpose plus a 180°rotation around y for one qubit (say qubit A) also yields ; . 7 We find therefore that r ( ) PTU A ; has the same eigenvalues as ρ. This proof can easily be generalized for any odd number of qubits with MDS, where the eigenvalues of ρ are equal to the eigenvalues of its PT transformation so that for such systems the PH criterion does not give information about entanglement.
A further conclusion comes from the fact (using the same argument) that for odd n MDS density matrices the eigenvalues of + ( ) ( ) I R n are the same as those of -( ) ( ) I R; n it follows that the eigenvalues of ρ can be written as The eigenvalues of R come in pairs ±r i for any odd n MDS density matrix (including the 3-qubits as a special case for n=3) and are bounded by -.
The separability problem can also be related to Frobenius (l 2 ) norms which are given by the square root of sums of squared HS parameters [7,8]. Let us prove the following relation for a 3 qubits MDS density matrix: We note first that On the other hand Here, λ i are the 8 eigenvalues of ρ. Since   l 0 1 4 i (recalling that the 8 r i come in 4 pairs | | r , i as given by (8)) we write: By using (10-13), we get (9). Equation (9) may be generalized to any MDS density matrix with odd-n. Note that the equality in (9) According to (9), a necessary condition for (2) to be a density matrix is that the Frobenius norm of the sum of the 27 parameters, represented by the left side of (9), should not be larger than 1.

Unfolding of the 3-qubits MDS tensor R a,b,c into matrices
In this section we describe unfolding processes by which tensors are unfolded into matrices. Such processes have been described in the literature [7][8][9][10][11] but the use of such unfolding processes becomes different in the present paper as we relate the analysis to density matrices, which was not the concern of the other works [7][8][9][10][11].

Explicitly separable forms for 3-qubits MDS density matrices related to the l 1 norm
A fully separable-like form for the density matrix (2) related to the l 1 norm can be given as: I  I  I  s i g nR  I  I  I  s i g nR  I  I  I  s i g nR  I  I  I  s i g nR   R  I  I  I   8  1 , , Each expression in the curly brackets of (16) represents a pure state density matrix multiplied by 2. We get according to (16) that a sufficient condition for full separability is given by the relation This seems the simplest sufficient condition for full separability but it is not necessary and may be improved. However before that we would like to explain how we obtain (16) and how it can be generalized to any n-qubit MDS system. For each R a b c , , in (16) expanding the products in the curly brackets which multiply | | R , , , one is left with the product of the unit operators and the relevant MDS terms (all other terms obtained in this expansion cancel out). A similar construction can be made for any number of qubits, say n. The number of products in the curly bracket will be 2 n−1 . We demonstrate the construction for 5-qubits A, B, C, D, E by starting with the product: s s s s s , , , . and then by expanding (18) we get the products of the s s ' corresponding to the , , , . but we get also products which are not MDS (In addition to a product of the unit matrices multiplied by | | R a b c d e , , , . ). In order to avoid the wrong terms, change an even number of pluses into minuses for all products in (18), e.g. s s s s s and add all the terms in analogy to (16).
One should notice that in such expansion we get 16 rows instead of the 4 rows used in (16)  The condition for full separability of (3) is changed to Here we used the SVD relation , U ( a) and V ( a) are 3×3 real orthogonal matrices, and etc. Taking absolute values in (23) we get Performing the summation over i we get , Since usually (26) is a strict inequality we expect a corresponding improvement in the sufficient condition.
Using the right hand side of (22) in (21) and using the general criterion (3) we find that under the condition (relative to A), an explicitly fully separable form for the 3-qubits MDS density matrix is obtained. In a similar way by using this procedure relative to B or C one gets, respectively, One can choose the optimal condition for explicit full separability from the three conditions given by (28) and (29).
We demonstrate the present method, for improving the condition for separability by decreasing the l 1 norm, in the following example: This result is in agreement with (8).
As a sufficient condition (3) for separability, we get for the R a b c , , parameters of (30): One can choose the optimal condition for explicit full separability from these 3 possibilities.

Full separability for 3-qubits MDS density matrices related to the l 2 norm
A fully separable-like form for the density matrix (2) related to l 2 norms can be given as: Each term in the curly brackets of (35) represents a pure state density matrix multiplied by 2. It is straightforward to show that the complicated separable form of (35) is reduced to the density matrix (2) by manipulating all the cross products in this equation. According to (35) a sufficient condition for full separability is given by We find that the density matrix in the above example can be presented by the explicitly separable form of equation (35).

Bi-separability for 3-qubits MDS density matrices
In our previous work [2] we treated bi-separability of 3-qubits MDS density matrix by using one qubit density matrix multiplied by entangled Bell states [2,29,30], of the other two qubits, but we treated only very special cases. In the present section we generalize the analysis to any 3-qubits MDS density matrix including up to 27 HS parameters.
Let us show the bi-separability obtained for the following simple 3-qubits MDS density matrix: r s s s s s s s s s     (44) below. Therefore the sufficient condition for bi-separability of equation (2) becomes that the sum of the Frobenius norms of the 9 (at most) triads of MDS-parameters is not larger than 1. Such condition is sufficient for bi-separability but the sufficient condition for bi-separability may perhaps be improved by other methods.

Full separability of 3-qubits MDS density matrices improved by the use of the high order singular value decomposition (HOSVD)
It is interesting to see how the present methods can be extended by relating them to the method of high order singular value decomposition (HOSVD) [7][8][9][10][11].
In the HOSVD method we use the SVD for the matrices ( ) ( ) R R , ; .
are equal, respectively, to the singular values, ( ) ( ) ( ) Therefore a sufficient condition for full separability is now given by  a ( ) 1 9. 55 The eigenvalues of the density matrix ρ in this example are given by: We have a density matrix under the condition:  a | | 3 3 1 .In the region:  a < 1 9 1 3 3(55) for full separability does not hold. We will show now that this condition can be greatly improved by the use of HOSVD, so that in the whole region that we have a density matrix it is fully separable.
The unfolded matrices: ( ) R 1 of (15) (and equal ones for ( ) ( ) R R ,  a a a a a a a a a  a a a a a a a a a  a a a a a a a  The high order transformed matrix ( ) S 1 is calculated by equation (49) where å 1 and V 1 are obtained by the SVD of ( ) R , 1 U 3 and U 2 are calculated by the SVD of ( ) R 3 and ( ) R , 2 respectively. After straightforward calculations we get for this example So, the condition for full separability is  a | | 3 3 1 ,which is equivalent to the condition for the present example to be a density matrix.
Example 3: Three-qubits MDS density matrices with 27 HS parameters Let us assume that we have 3-qubits MDS density matrix with the following unfolding matrix ( ) R 1 of equation (15) relative to qubit A:  One should notice that the p th ' row = ( ) p 1, 2, 3 of equation (60) includes 9 terms where each 3 of them are inserted in the first, second and third row, of the matrix ( ) S , p q r , respectively. By calculating the singular values of (60) we get: Sum of singular values=0.997<1 so that the condition of HOSVD for full separability is satisfied. On the other hand we get: å , 1   3 3 . , 2 1 2 so that, respectively, the l 1 norm condition (equation (28)) and the l 2 norm condition (equation (36)) are not satisfied.
We find that the use of HOSVD for obtaining ( ) S 1 leads to sufficient condition for full separability, in the above two examples, while the original matrix ( ) R 1 does not give a sufficient condition for full separability. The HOSVD is found to be useful for improving the condition for full separability in the above two examples, but we would like to emphasize that the HOSVD condition gives only sufficient condition (but not necessary).  I  I  I  I  I  I   8 .

Explicitly
This density matrix with probability p mixed with white noise is given by Therefore a sufficient condition for full separability is given as An explicitly separable form for the density matrix (64) was given in [33]. The relation (68) has been discussed in various works [25,27], showing that this condition is both sufficient and necessary for full separability.
We treat now the sufficient condition for full separability for W state with probability p mixed with white noise.
The W state is given by: I  I  I  I  I  I  I  I  I  I  I  I   I  I  I   3 8  2  2  2   2  2  2  2  2  2  2  2 2 7 0 Except for the first and second row of (72), the terms in the rows 3, 4, 5 need to be written as outer products of density matrices of qubits A, B, C. As an example it is easy to see that  which is similar to that of GHZ mixed with white noise. By using the PT transformation, we find [27] that under the condition > + » ( ) / p 3 3 8 2 0.209589 the density matrix of W state mixed with white noise is not fully separable. Under the condition .. p<0.2 this density matrix is fully separable so only in a very small region the full separability problem is not clarified.

Conclusions
We have shown that the PH and Negativity criterions are inconclusive for 3-qubits MDS states as these density matrices and their PT have the same eigenvalues. These eigenvalues come in 4 pairs where the sum of eigenvalues for each pair is 1/4. These results can be generalized to any odd-n MDS density matrix where in the general case the sum of eigenvalues in one pair is given by The main results for MDS density matrices can be summarized as follows. Tensors related to 3-qubits can be converted to matrices: ( ) R 1 given by equation (15) and similar ones for ( ) R 2 and ( ) R . 3 A fully separable-like form for the 3-qubits MDS density matrix related to the l 1 norm is given by (16). This equation may be generalized to any n-qubit density matrix. By using unitary transformations of the l 1 norm for the MDS 3-qubits matrices the condition for full separability is given by (28) or (29). Such conditions give much better criterions for full separability relative to that of (3). A fully separable form for the 3-qubits MDS density matrix related to the l 2 norm is given by equation (35). This separable form leads to full separability condition given by equation (36). This separable form can be generalized to any n-qubit MDS density matrix. Biseparability of 3-qubits MDS density matrix was obtained in the present work by using one qubit density matrix multiplied by entangled Bell states [2,29,30] of the other two qubits. Previous conditions for special biseparability cases [4] were generalized to any 3-qubits MDS state given by equation (44) including up to 27 HS parameters. HOSVD method for obtaining the condition for full separabilty has been analyzed and its use was demonstrated in examples.
For GHZ and W states mixed with white noise with probability p we found explicit fully separable forms for their density matrix showing by improving previous calculations [4] that for both cases the density matrix is fully separable under the condition 5p1.