Abstract

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.

Highlights

  • There is much interest in quantum entangled states due to various potential applications that use quantum properties of such states

  • A condition for biseparability of 3-qubits maximally disordered subsystems (MDS) density matrices is obtained in the present work by the use of one qubit density matrix multiplied by Bell entangled states [2, 24,25] of the other two qubits

  • The final conclusion from this analysis is that a sufficient condition for biseparability of a 3qubits MDS density matrix is that the sum of l2 norms over 9 triads of HS parameters, is not larger than 1, but the 9 HS triads are different from those used for full separability in Eqs. (4445)

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Summary

Introduction

There is much interest in quantum entangled states due to various potential applications that use quantum properties of such states. A condition for biseparability 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, 24,25] 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. For the general 3-qubits density matrices (non-MDS) which include outer products of σ ' s with unit matrices, a sufficient condition for full separability is given by (6) but we can improve this condition Let us prove the following relation for a 3 qubits MDS density matrix

We note first that
The tensor can be unfolded relative to qubit B by exchanging
Here again one should take into account that the matrix
Performing the summation over i we get
By making the analysis of this example for or
For example we find the relation
By reducing
Then we have
After straightforward calculations we get for this example
By using the SVD for the matrices
This sum is smaller than the sum
Therefore a sufficient condition for separability is given as
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