Probabilistic pure state conversion on the majorization lattice

Abstract

Entanglement is among the most fundamental-and at the same time puzzling-properties of quantum physics. Its modern description relies on a resource-theoretical approach, which treats entangled systems as a means to enable or accelerate certain informational tasks. Hence, it is of crucial importance to determine whether-and how-different entangled states can be converted into each other under free operations (those which do not create entanglement from nothing). Here, we show that the majorization lattice provides an efficient framework in order to characterize the allowed transformations of pure entangled states under local operations and classical communication. The underlying notions of meet $\wedge$ and join $\vee$ in the majorization lattice lead us to define, respectively, the optimal common resource and optimal common product states. Based on these two states, we introduce two optimal probabilistic protocols for the (single-copy) conversion of incomparable bipartite pure states, which we name greedy and thrifty. Both protocols reduce to Vidal's protocol [G. Vidal, Phys. Rev. Lett. 83, 1046 (1999)] if the initial and final states are comparable, but otherwise the thrifty protocol can be shown to be superior to the greedy protocol as it yields a more entangled residual state when it fails (they both yield the same entangled state with the same optimal probability when they succeed). Finally, we consider the generalization of these protocols to entanglement transformations involving multiple initial or final states.

Minor corrections to match the published version of the paper; 10 pages, 3 figures

References in corpus (20)

• Quantum entanglement
• Three qubits can be entangled in two inequivalent ways
• Quantum Resource Theories
• Fundamental limitations for quantum and nano thermodynamics
• The second laws of quantum thermodynamics
• Entanglement of pure states for a single copy
• Entanglement-assisted local manipulation of pure quantum states
• Minimal conditions for local pure-state entanglement manipulation
• Quantum complex networks
• Approximate transformations and robust manipulation of bipartite pure state entanglement
• Tripartite entanglement transformations and tensor rank
• A Simple Algorithm for Local Conversion of Pure States
• Detecting coherence via spectrum estimation
• Optimal common resource in majorization-based resource theories
• Structure of the thermodynamic arrow of time in classical and quantum theories
• Approximate transformations of bipartite pure-state entanglement from the majorization lattice
• Partial order on passive states and Hoffman majorization in quantum thermodynamics
• Local Transformations of Multiple Multipartite States
• Deterministic transformations of bipartite pure states
• Extremal elements of a sublattice of the majorization lattice and approximate majorization

Cited by in corpus (1)

• Probabilistic Coherence Transformation Under Strictly Incoherent Operation