Generalizations of entanglement based on coherent states and convex sets

Abstract

Unentangled pure states on a bipartite system are exactly the coherent states with respect to the group of local transformations. What aspects of the study of entanglement are applicable to generalized coherent states? Conversely, what can be learned about entanglement from the well-studied theory of coherent states? With these questions in mind, we characterize unentangled pure states as extremal states when considered as linear functionals on the local Lie algebra. As a result, a relativized notion of purity emerges, showing that there is a close relationship between purity, coherence and (non-)entanglement. To a large extent, these concepts can be defined and studied in the even more general setting of convex cones of states. Based on the idea that entanglement is relative, we suggest considering these notions in the context of partially ordered families of Lie algebras or convex cones, such as those that arise naturally for multipartite systems. The study of entanglement includes notions of local operations and, for information-theoretic purposes, entanglement measures and ways of scaling systems to enable asymptotic developments. We propose ways in which these may be generalized to the Lie-algebraic setting, and to a lesser extent to the convex-cones setting. One of our motivations for this program is to understand the role of entanglement-like concepts in condensed matter. We discuss how our work provides tools for analyzing the correlations involved in quantum phase transitions and other aspects of condensed-matter systems. * barnum@lanl.gov † knill@lanl.gov ‡ g ortiz@lanl.gov § lviola@lanl.gov 2 Contents

Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

“…3 can be expressed in terms of generalized entanglement [1,14,21]. A pure state is generalized unentangled (GU) with respect to a preferred set O of observables if it is extremal among states considered as linear functionals on O, otherwise it is generalized entangled.…”

“…Let L = r k=1 w(h k )h k and define ω bŷ L|hw = ω|hw . Then ψ|L|ψ < ω for |ψ = |hw , from which it follows that |hw hw| = lim t→∞ e −tω e tL [14]. Because the eigenvalues ofL are integral, convergence is exponentially fast in t. Let…”

Generalized state spaces and nonlocality in fault-tolerant quantum-computing schemes

“…In a manner similar to [17], we will consider a redefinition of the single-particle-state space away from the "quantum" version, thereby giving a different notion of "separability" under which one can apply the algorithm of [10]. As a starting point for our generalized state spaces we will use the following characterization of the quantum set Q d of dimension d singleparticle states.…”

“…The aim of the present work is to use a generalized notion of entanglement [17] (in a sense described in the next section) to apply a modified version of the Harrow and Nielsen method to devices where there are restrictions on the measurements available. Such restrictions arise naturally in fault tolerant quantum computation for two reasons.…”

“…It also motivates the construction of generalized probabilistic theories [18] that are inspired by fault tolerant quantum computation schemes (although in our primary case the generalized theory turns out to be very close to a classical theory), and provides regimes of classically tractable noisy quantum evolution that appear to fall outside the scope of existing methods. It also falls under the program initiated in [17] to investigate so-called generalized notions of entanglement, and motivates some interesting questions regarding the contrast between different forms of generalized multiparty entanglement.…”