Explicit separable density matrices, for mixed two qubits states, are derived by the use of Hilbert Schmidt decompositions and Peres Horodecki criterion. A strongly separable two qubits mixed state is defined by multiplications of two density matrices given with pure states while weakly separable two qubits state is defined by multiplications of two density matrices which includes non-pure states. We find the sufficient and necessary condition for separability of two-qubits density matrices and show that under this condition the two-qubit density matrices are strongly separable.
Explicit separable density matrices, for mixed two qubits states, are derived by the use of Hilbert Schmidt decompositions and Peres Horodecki criterion. A strongly separable two qubits mixed state is defined by multiplications of two density matrices given with pure states while weakly separable two qubits state is defined by multiplications of two density matrices which includes non-pure states. We find the sufficient and necessary condition for separability of two-qubits density matrices and show that under this condition the two-qubit density matrices are strongly separable.
Hilbert-Schmidt (HS) decompositions are employed for analyzing systems of n-qubits, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary transformations (PTU) for one qubit from the whole system, are used for indicating inseparability. A sufficient criterion for full separability of the n-qubits and qubit-qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and n-qubits systems, with emphasis on maximally disordered subsystems (MDS) (i.e., density matrices rho(MDS) for which tracing over any subsystem gives the unit density matrix). A sufficient condition that rho(MDS) is not separable is that it has an eigenvalue larger than 1/d for a qubit and a qudit, and larger than 1/2^(n-1) for n-qubits system. The PTU transformation does not change the eigenvalues of the n-qubits MDS density matrices for odd n. Thus the Peres-Horodecki criterion does not give any information about entanglement of these density matrices, but this criterion is useful for indicating inseparability for even n. The changes of the entanglement and separability properties of the GHZ state, the Braid entangled state and the W state by mixing them with white noise are analyzed by the use of the present methods. The entanglement and separability properties of the GHZ-diagonal density matrices, composed of mixture of 8 GHZ density matrices with probabilities p(i), is analyzed as function of these probabilities. In some cases we show that the Peres-Horodecki criterion is both sufficient and necessary.
Explicitly separable density matrices are constructed for all separable two-qubits states based on Hilbert–Schmidt (HS) decompositions. For density matrices which include only two-qubits correlations the number of HS parameters is reduced to 3 by using local rotations, and for two-qubits states which include single qubit measurements, the number of parameters is reduced to 4 by local Lorentz transformations. For both cases, we related the absolute values of the HS parameters to probabilities, and the outer products of various Pauli matrices were transformed to pure state density matrices products. We discuss related problems for three-qubits. For n-qubits correlation systems ([Formula: see text]) the sufficient condition for separability may be improved by local transformations, related to high order singular value decompositions (SVDs).
Hilbert–Schmidt (HS) decompositions are employed for analyzing systems of [Formula: see text]-qubit, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary (PTU) transformations for one qubit from the whole system, are used for indicating entanglement/separability. A sufficient criterion for full separability of the [Formula: see text]-qubit and qubit–qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and [Formula: see text]-qubit systems, with emphasis on maximally disordered subsystems (MDS) (i.e. density matrices for which tracing over any subsystem gives the unit density matrix). A sufficient condition that [Formula: see text] (MDS) is not separable is that it has an eigenvalue larger than [Formula: see text] for a qubit and a qudit, and larger than [Formula: see text] for [Formula: see text]-qubit system. The PTU transformation does not change the eigenvalues of the [Formula: see text]-qubit MDS density matrices for odd [Formula: see text]. Thus, the Peres–Horodecki (PH) criterion does not give any information about entanglement of these density matrices. The PH criterion may be useful for indicating inseparability for even [Formula: see text]. The changes of the entanglement and separability properties of the GHZ state, the Braid entangled state and the [Formula: see text] state by mixing them with white noise are analyzed by the use of the present methods. The entanglement and separability properties of the GHZ-diagonal density matrices, composed of mixture of 8[Formula: see text]GHZ density matrices with probabilities [Formula: see text], is analyzed as function of these probabilities. In some cases, we show that the PH criterion is both sufficient and necessary.
Hilbert–Schmidt (HS) decompositions and Frobenius norms are used to analyze biseparability of 3-qubit systems, with particular emphasis on density matrices with maximally disordered subsystems (MDS) and on the [Formula: see text] state mixed with white noise. The biseparable form of a MDS density matrix is obtained by using the Bell states of a 2-qubit subsystem, multiplied by density matrices of the third qubit, which include the relevant HS parameters. Using our methods, a sufficient condition and explicit biseparability of the [Formula: see text] state mixed with white noise are given. They are compared with the sufficient condition for explicit full separability given in a previous work.
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 ( l 2 ) 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 l 2 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.
We treat 3-qubits states with maximally disordered subsystems, by using Hilbert-Schmidt decompositions.By using unfolding methods, the tensors are converted into matrices and by applying singular values decompositions to these matrices the number 27 of the Hilbert Schmidt parameters, is reduced to 9 and under the condition that the sum of absolute values of these parameters is not larger than 1, we conclude that the density matrix is fully separable . In another method we divide the 27 Hilbert Schmidt parameters into 9 triads where for each triad we calculate the Frobenius norm of 3 Hilbert-Schmidt parameters. If the sum of nine norms is not larger than 1 then we conclude that the density matrix is fully separable . The condition for biseparability of maximally disordered density matrices is obtained by the use of one qubit density matrix multiplied by Bell entangled states of the other two qubits. If the sum of the nine Frobenius norms for these different triads is not larger than 1, we conclude that the density matrix is biseparable. We analyze the relations between three qubits maximally disordered subsystems density matrices and the method of high order singular value decomposition (HOSVD) and show that this method may improve the sufficient condition for full separability. For three qubits states which include multiplication with the unit matrix we find a simple way to reduce the sum of the absolute values of the Hilbert Schmidt parameters absolute values and get better conditions for their full separability for special cases.