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Physics of Particles and Nuclei Letters

Separability of two-qubit state in terms of local invariants

Vladimir P. Gerdt, Yu. G. Palii, A. Khvedelidze

Published Sep 1, 2011• 1 citation
Publisher

Abstract

The separability condition for the two-qubit mixed state is formulated in terms of a system of inequalities in invariants of the adjoint SU (2) ⊗ SU (2) action on the space of density matrices, i.e., positive semidefinite 4 × 4 Hermitian matrices.
DOI
10.1134/s1547477111050074

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In the present paper few steps are undertaken towards the description of the qubit-qutrit pair - quantum bipartite system composed of two and three level subsystems. The computational difficulties with the construction of the local unitary polynomial invariants are discussed. Calculations of the Molien functions and Poincare series for the qubit-qubit and qubit-qutrit local unitary invariants are outlined and compared with the known results. The requirement of positive semi-definiteness of the density operator is formulated explicitly as a set of inequalities in five Casimir invariants of the algebra su(6).
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.
The entanglement properties of a two-qubit system are analyzed from the point of view of Lie algebras and geometry. We deal with $4\ifmmode\times\else\texttimes\fi{}4$ matrices belonging to a particular class having seven free parameters, constructed from the imposed symmetries on two-qubit states expanded in terms of a basis whose elements belong to a subalgebra of su(4). The entanglement or separability character of the state is determined and measured by analyzing the eigenvalues of an auxiliary matrix obtained from the original state after performing on it a nonunitary operation. In terms of its eigenvalues, we define two squared distances having a Minkowski metric, ${s}^{2}={t}^{2}\ensuremath{-}{\stackrel{P\vec}{V}}^{2}$, which become a measure of the entanglement. The Minkowski metric is a signature of that class of two-qubit states (matrices) and also of the auxiliary matrices to be constructed. The squared distance is invariant by any unitary transformation on the two-qubit state. We illustrate the theory with examples.
Entanglement in a multipartite quantum system represents non-local correlations between the system parts. This phenomenon is not affected by the action of a symmetry group, which acts separately on each part of the multipartite composite system. The space of the invariants of the action is a natural domain of the definition for all possible measures of the entanglement. Allowed values of the invariants are restricted by a set of inequalities, which is analyzed in this paper. The inequalities ensue from the semipositivity and Hermicity of the density matrix for the system as well as from the orbit space description in terms of the invariant theory.
We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices P+\mathfrak{P}_ +P+​ . Since elements of P+\mathfrak{P}_ +P+​ are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, P+∈R15\mathfrak{P}_ + \in \mathbb{R}^{15}P+​∈R15 . We define P+\mathfrak{P}_ +P+​ explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.
We propose that the entanglement of mixed states is characterized properly in terms of a probability density function ${\mathcal{P}}_{\ensuremath{\rho}}(\mathcal{E})$. There is a need for such a measure since the prevalent measures (such as concurrence and negativity) for two-qubit systems are rough benchmarks and not monotones of each other. Focusing on the two-qubit states, we provide an explicit construction of ${\mathcal{P}}_{\ensuremath{\rho}}(\mathcal{E})$ and show that it is characterized by a set of parameters of which concurrence is but one particular combination. ${\mathcal{P}}_{\ensuremath{\rho}}(\mathcal{E})$ is manifestly invariant under $\text{SU}(2)\ifmmode\times\else\texttimes\fi{}\text{SU}(2)$ transformations. It can, in fact, reconstruct the state up to local operations---with the specification of at most four additional parameters. Finally, the measure resolves the controversy regarding the role of entanglement in quantum computation in NMR systems.