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arXiv:1510.07222 · International Journal of Quantum Information

Explicit constructions of all separable two-qubits density matrices and related problems for three-qubits systems

Y. Ben-Aryeh, A. Mann

Published Dec 1, 2015• 10 citations
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Abstract

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).

DOI
10.1142/s0219749915500616

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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).
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.
Explicit separability of general two qubits density matrices is related to Lorentz transformations. We use the 4-dimensional form [Formula: see text] of the Hilbert–Schmidt (HS) decomposition of the density matrix. For the generic case, in which Lorentz transformations diagonalize [Formula: see text] into [Formula: see text], we give relations between the [Formula: see text] and the [Formula: see text]. In particular, we consider two cases: (a) Two qubits density matrices with one pair of linear terms in the HS decomposition. (b) Two qubits density matrices with two or three symmetric pairs of linear terms. Some of the theoretical results are demonstrated by numerical calculations. The four non-generic cases (which may be reduced to case (a) are analyzed and the non-generic property is related explicitly to Lorentz velocity [Formula: see text] which is not reachable physically.
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.
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.
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.
Separability and entanglement for n-qubits systems are quantified by using Hilbert-Schmidt (HS) decompositions in which the density matrices are decomposed into various terms representing certain one qubit, two-qubits,and larger qubits measurements. The present method is more general than previous methods for bipartite systems as it can be used for quantification of entanglement for large n-qubits systems.We demonstrate the use of such method by analyzing three qubits GHZ states, and three qubits general Bell states produced by a certain multiplications of Braid operators, operating on the computational basis of states. Quantum correlations are obtained by measuring all qubits of these systems, while a measurement of a part of the system gives only classical correlations. Quantification of entanglement for these systems is given by the use of HS decompositions.