Entanglement of projection and a new class of quantum erasers

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Abstract
We define a new measurement of entanglement, the entanglement of projection, and find that it is natural to write the entanglements of formation and assistance in terms of it. Our measure allows us to describe a new class of quantum erasers which restore entanglement rather than just interference. Such erasers can be implemented with simple quantum computer components. We propose realistic optical versions of these erasers. PACS: 03.65.Bz,03.67.Lx Entanglement is the degree to which the wave function does not factorize. For example, anS = 0 two particle system |+−i−|− +i is maximally entangled: measurement of the spins reveals they are completely anticorrelated. The concept of entanglement goes to the very heart of quantum mechanics, and understanding its nature is a prerequisite to understanding quantum mechanics itself. Two-particle entanglement was used by Einstein, Podolsky and Rosen [1] to argue that quantum mechanics could not be a complete description of reality— that there had to be an underlying local theory. But J. S. Bell used such entangled states to show that any local underlying theory would have to satisfy certain inequalities, which quantum mechanics explicitly violates [2]. Experiments on such entangled states have shown 1 that these inequalities are violated just as quantum mechanics predicts [3]. Modern research on entanglement includes proposals for providing cleaner demonstrations of this nonlocality using three-particle entangled states [4], and on quantifying entanglement [5–8]. The goal of this Letter is to define a new class of quantum,erasers which restore entanglement of a multistate subsystem, rather than just interference, and to quantify that restoration with a new measure of entanglement [9]. A quantum eraser [10] is a device in which coherence appears to be lost in a subset of the system, but in which that coherence can be restored by erasing the tagging information which originally “destroyed” it. Traditional erasers [11,12] need only two distinct subsystems. For example, if one sends particle A through two slits, and if one “tags” which slit A goes through via the interaction with a tagging particle, T, then the interference pattern will disappear. But if one makes the “which slit” information in T unobservable, even in principle, then one can restore the
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