Classification of two-qubit states

Abstract

Verstraete, Dehaene and DeMoor showed that each of the two-qubit states can be generated from one of two canonical families of two-qubit states by means of transformations preserving the tensor structure of the state space. Precisely, each of such states can be generated from a three-parameter family of Bell-diagonal states or from three-parameter rank-deficient states. In this paper, we show that this classification of two-qubit states can be refined. In particular, we show that the latter canonical family of states can be reduced to three fixed states and a two-parameter family of two-qubit states. For this family of states, we provide a simple parametrization that guarantees positive semidefiniteness of the states and enables easier calculation of the Wootters concurrence and quantum discord. Moreover, we present a new general parametrization of all two-qubit states generated from the canonical families of states using sets of (pseudo)orthogonal four-vectors (frames). An advantage of the presented approach lies in the fact that the standard conditions for positive semidefiniteness of states are equivalent to (pseudo)orthogonality conditions for four-vectors serving as This work has been supported by the Polish National Science Centre under the contract 2014/15/B/ST2/00117 and by the University of Lodz.

SLOCC orbit of rank-deficient two-qubit states: quantum entanglement, quantum discord and EPR steering

“…We have extended an introductory analysis of this orbit given in [3]. In particular, we have explicitly calculated entanglement of formation and quantum discord for all states from the considered SLOCC orbit of rank-deficient two-qubit states.…”

“…Classification of all two-qubit states into orbits with respect to transformations (3), so-called SLOCC orbits, was considered in [2]. In our previous work [3] we have refined this classification. In [2,3] it was shown that each of two-qubit states can be generated from one of the following distinct states:…”

“…In our previous work [3] we have refined this classification. In [2,3] it was shown that each of two-qubit states can be generated from one of the following distinct states:…”

Majorana representation for mixed states

“…Hence, the proof is done for any root ξ = 0 but the statement also holds in the case of ξ = 0 and its corresponding antipodal point ξ = ∞: Let us suppose that the constant term in p (σ) ρ (ζ) is zero, and hence there is a root ξ = 0. The hermiticity property (33) implies that the coefficient of the highest exponent ζ 2σ is also zero, implying that p…”

Separability and entanglement of two qubits density matrices using Lorentz transformations

“…In our previous work 10 we studied the explicit constructions of separable two qubits density matrices based on the study of Lorentz transformations developed by Verstraete et al 4 6 − Following this approach an arbitrary two qubits state can be written in the form ( ) ( ) real. Detailed analysis for the two qubits density matrix which is of the form (1.4) has been given by Caban et al 11,12 Verstraete et al, 4 6 − have shown that the 4 4 × matrix , R µ ν can be written as 1 Then, for this special case we get:…”