Local filtering operations on two qubits

Abstract

We consider one single copy of a mixed state of two qubits and investigate how its entanglement changes under local quantum operations and classical communications (LQCC) of the type $\rho'\sim (A\otimes B)\rho(A\otimes B)^{\dagger}$. We consider a real matrix parameterization of the set of density matrices and show that these LQCC operations correspond to left and right multiplication by a Lorentz matrix, followed by normalization. A constructive way of bringing this matrix into a normal form is derived. This allows us to calculate explicitly the optimal local filterin operations for concentrating entanglement. Furthermore we give a complete characterization of the mixed states that can be purified arbitrary close to a Bell state. Finally we obtain a new way of calculating the entanglement of formation.Comment: 4 page

Quantum Steering Ellipsoids

“…Furthermore, it can be shown [15] that, at most, one eigenvalue of ρ T B e can be negative, and that ρ T B e is full rank for all entangled states [19]. Hence, det ρ T B e < 0 is a necessary and sufficient condition for entanglement.…”

“…Hence, det ρ T B e < 0 is a necessary and sufficient condition for entanglement. Suppose ρ is entangled, then any state in its SLOCC orbit SðρÞ is also entangled [15], including the canonical statẽ ρ ∈ SðρÞ. It follows that ρ is entangled if and only if detðρ T B Þ < 0.…”

“…Construction of the quantum steering ellipsoid.-We now provide an alternative construction of the steering ellipsoid E A to that in [15], which applies even when E A is degenerate.…”

Classification of two-qubit states

“…In [4], it was shown that each of two-qubit states can be generated from one of the two canonical families of two-qubit states by means of transformations (1) (provided that det A = 0, det B = 0). Precisely, each of such states can be generated from a threeparameter family of Bell-diagonal states or from three-parameter rank-deficient states (in [4] rank-deficient states are described by four parameters but one parameter can be eliminated by normalization).…”

“…Precisely, each of such states can be generated from a threeparameter family of Bell-diagonal states or from three-parameter rank-deficient states (in [4] rank-deficient states are described by four parameters but one parameter can be eliminated by normalization). However, as we show below, the classification given in [4] can be refined. Since the Bell-diagonal states are widely discussed in the literature, we focus our attention on the latter family of states.…”