Entangled photon-added coherent states

We study the degree of entanglement of arbitrary superpositions of m, n photon-added coherent states (PACS) |ψ⟩∝u|α,m⟩|β,n⟩+v|β,n⟩|α,m⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathinner {|{\psi }\rangle } \propto u \mathinner {|{{\alpha },m}\rangle }\mathinner {|{{\beta },n }\rangle }+ v \mathinner {|{{\beta },n}\rangle }\mathinner {|{{\alpha },m}\rangle }$$\end{document} using the concurrence and obtain the general conditions for maximal entanglement. We show that photon addition process can be identified as an entanglement enhancer operation for superpositions of coherent states (SCS). Specifically for the known bipartite positive SCS: |ψ⟩∝|α⟩a|-α⟩b+|-α⟩a|α⟩b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathinner {|{\psi }\rangle } \propto \mathinner {|{\alpha }\rangle }_a\mathinner {|{-\alpha }\rangle }_b + \mathinner {|{-\alpha }\rangle }_a\mathinner {|{\alpha }\rangle }_b $$\end{document} whose entanglement tends to zero for α→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \rightarrow 0$$\end{document}, can be maximal if al least one photon is added in a subsystem. A full family of maximally entangled PACS is also presented. We also analyzed the decoherence effects in the entangled PACS induced by a simple depolarizing channel . We find that robustness against depolarization is increased by adding photons to the coherent states of the superposition. We obtain the dependence of the critical depolarization pcrit\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{\text {crit}}$$\end{document} for null entanglement as a function of m,n,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m,n, \alpha $$\end{document} and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}.