Separability and distillability in composite quantum systems -a primer-

Abstract

Quantum mechanics is already 100 years old, but remains alive and full of challenging open problems. On one hand, the problems encountered at the frontiers of modern theoretical physics like Quantum Gravity, String Theories, etc. concern Quantum Theory, and are at the same time related to open problems of modern mathematics. But even within non-relativistic quantum mechanics itself there are fundamental unresolved problems that can be formulated in elementary terms. These problems are also related to challenging open questions of modern mathematics; linear algebra and functional analysis in particular. Two of these problems will be discussed in this article: a) the separability problem, i.e. the question when the state of a composite quantum system does not contain any quantum correlations or entanglement and b) the distillability problem, i.e. the question when the state of a composite quantum system can be transformed to an entangled pure state using local operations (local refers here to component subsystems of a given system). Although many results concerning the above mentioned problems have been obtained (in particular in the last few years in the framework of Quantum Information Theory), both problems remain until now essentially open. We will present a primer on the current state of knowledge concerning these problems, and discuss the relation of these problems to one of the most challenging questions of linear algebra: the classification and characterization of positive operator maps.

11 pages latex, 1 eps figure. Final version, to appear in J. Mod. Optics, minor typos corrected, references added

References in corpus (2)

• Evidence for Bound Entangled States with Negative Partial Transpose
• Construction of quantum states with bound entanglement

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• A Note on Invariants and Entanglements
• Generalizations of entanglement based on coherent states and convex sets
• Detecting multipartite entanglement
• Complete hierarchies of efficient approximations to problems in entanglement theory
• Generalized spin squeezing inequalities in $N$ qubit systems: theory and experiment
• Quantifying Quantumness and the Quest for Queens of Quantum
• A few steps more towards NPT bound entanglement
• Correlation Effects in the Quench-Induced Phase Separation Dynamics of a Two-Species Ultracold Quantum Gas
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• Security bound of two-bases quantum key-distribution protocols using qudits
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• Many-Body Expansion Dynamics of a Bose-Fermi Mixture Confined in an Optical Lattice
• A Family of Norms With Applications In Quantum Information Theory
• An Introduction to Quantum Entanglement: a Geometric Approach
• Realizing Physical Approximation of the Partial Transpose
• Separability of Rank Two Quantum States
• Foundations of quantum theory and quantum information applications
• Entanglement in quantum information theory
• Diagonal unitary and orthogonal symmetries in quantum theory
• Channel kets, entangled states, and the location of quantum information
• Tensor of coherences parameterization of multiqubit density operators for entanglement characterization
• Entanglement Distillation; A Discourse on Bound Entanglement in Quantum Information Theory
• Ancilla-assisted sequential approximation of nonlocal unitary operations
• Norms and Cones in the Theory of Quantum Entanglement
• Distillation Protocols for Mixed States of Multilevel Qubits and the Quantum Renormalization Group
• Computational tools for solving a marginal problem with applications in Bell non-locality and causal modeling
• Nonlinear entanglement witnesses for four qubits in mutually unbiased bases
• Convexity and the Separability Problem of Quantum Mechanical Density Matrices
• New Partial Trace Inequalities and Distillability of Werner States
• Some aspects of the interplay between bipartite correlations and quantum channels
• Anomalous quantum correlations in the motion of a trapped ion
• The entanglement or separability of mixed quantum states as a matter of the choice of observables
• Generalized distillability conjecture and generalizations of Cauchy-Bunyakovsky-Schwarz inequality and Lagrange identity