Approximate transformations of bipartite pure-state entanglement from the majorization lattice

Abstract

We study the problem of deterministic transformations of an \textit{initial} pure entangled quantum state, $|\psi ⟩$, into a \textit{target} pure entangled quantum state, $|\varphi ⟩$, by using \textit{local operations and classical communication} (LOCC). A celebrated result of Nielsen [Phys. Rev. Lett. \textbf{83}, 436 (1999)] gives the necessary and sufficient condition that makes this entanglement transformation process possible. Indeed, this process can be achieved if and only if the majorization relation $\psi \prec \varphi$ holds, where $\psi$ and $\varphi$ are probability vectors obtained by taking the squares of the Schmidt coefficients of the initial and target states, respectively. In general, this condition is not fulfilled. However, one can look for an \textit{approximate} entanglement transformation. Vidal \textit{et. al} [Phys. Rev. A \textbf{62}, 012304 (2000)] have proposed a deterministic transformation using LOCC in order to obtain a target state $|{\chi }^{\mathrm{opt}}⟩$ most approximate to $|\varphi ⟩$ in terms of maximal fidelity between them. Here, we show a strategy to deal with approximate entanglement transformations based on the properties of the \textit{majorization lattice}. More precisely, we propose as approximate target state one whose Schmidt coefficients are given by the supremum between $\psi$ and $\varphi$. Our proposal is inspired on the observation that fidelity does not respect the majorization relation in general. Remarkably enough, we find that for some particular interesting cases, like two-qubit pure states or the entanglement concentration protocol, both proposals are coincident.

arXiv commentRevised manuscript close to the accepted version in Physica A (10 pages, 1 figure)

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• The Optimal Uncertainty Relation
• 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
• Local Transformations of Multiple Multipartite States
• Extremal elements of a sublattice of the majorization lattice and approximate majorization
• The Complementary Information Principle of Quantum Mechanics
• Probabilistic pure state conversion on the majorization lattice
• Probabilistic Coherence Transformation Under Strictly Incoherent Operation
• A Universal Quantum Certainty Relation for Arbitrary Number of Observables