Entanglement Measures and Purification Procedures

TL;DR

The paper develops a universal, distance-based framework for quantifying entanglement by measuring how far a state is from the set of disentangled states. It proves general sufficiency conditions (F1–F5) on the chosen distance to guarantee a valid entanglement measure that is zero on separable states, invariant under local unitaries, and non-increasing under local operations with classical communication and post-selection. The authors instantiate this framework with two generators, the Quantum Relative Entropy and the Bures metric, deriving the Relative Entropy of Entanglement (which reduces to the von Neumann entropy for pure states) and a non-pure-state-entropy-based Bures entanglement (which is not E4-satisfying but yields useful purification bounds). They provide an efficient numerical method for computing the measure for two spin-$\tfrac{1}{2}$ systems, and connect the measure to a statistical interpretation via quantum Sanov’s theorem, establishing an upper bound on distillable entanglement and highlighting that entanglement of creation generally exceeds distillable entanglement. The work also extends the framework to multiple subsystems and clarifies the thermodynamic-like aspects of purification, including the irreversibility of typical purification procedures and the role of information loss in distillation efficiency.

Abstract

We generalize previously proposed conditions each measure of entanglement has to satisfy. We present a class of entanglement measures that satisfy these conditions and show that the Quantum Relative Entropy and Bures Metric generate two measures of this class. We calculate the measures of entanglement for a number of mixed two spin 1/2 systems using the Quantum Relative Entropy, and provide an efficient numerical method to obtain the measures of entanglement in this case. In addition, we prove a number of properties of our entanglement measure which have important physical implications. We briefly explain the statistical basis of our measure of entanglement in the case of the Quantum Relative Entropy. We then argue that our entanglement measure determines an upper bound to the number of singlets that can be obtained by any purification procedure and that distillable entanglement is in general smaller than the entanglement of creation.

Figures (2)

• Figure 1: The set of all density matrices, ${\cal T}$ is represented by the outer circle. Its subset, a set of disentangled states ${\cal D}$ is represented by the inner circle. A state $\sigma$ belongs to the entangled states, and $\rho^*$ is the disentangled state that minimizes the distance $D( \sigma || \rho)$, thus representing the amount of quantum correlations in $\sigma$. State $\rho^*_A \otimes \rho^*_B$ is obtained by tracing $\rho^*$ over $A$ and $B$. $D( \rho^* || \rho^*_A \otimes \rho^*_B)$ represent the classical part of the correlations in the state $\sigma$.
• Figure 2: Comparison of the entanglement of creation and the Relative Entropy of Entanglement for the Werner states (these are are Bell diagonal states of the form $W=\hbox{diag}(F,(1-F)/3,(1-F)/3,(1-F)/3.$) One clearly sees that the entanglement of creation is strictly larger than the Relative Entropy of Entanglement for $0 < F < 1$