Entanglement of multiparty-stabilizer, symmetric, and antisymmetric states

Abstract

We study various distance-like entanglement measures of multipartite states under certain symmetries. Using group averaging techniques we provide conditions under which the relative entropy of entanglement, the geometric measure of entanglement and the logarithmic robustness are equivalent. We consider important classes of multiparty states, and in particular show that these measures are equivalent for all stabilizer states, symmetric basis and antisymmetric basis states. We rigorously prove a conjecture that the closest product state of permutation symmetric states can always be chosen to be permutation symmetric. This allows us to calculate the explicit values of various entanglement measures for symmetric and antisymmetric basis states, observing that antisymmetric states are generally more entangled. We use these results to obtain a variety of interesting ensembles of quantum states for which the optimal LOCC discrimination probability may be explicitly determined and achieved. We also discuss applications to the construction of optimal entanglement witnesses

The geometric measure of entanglement for a symmetric pure state with non-negative amplitudes

“…[16][17][18][19][20][21] In spite of its importance, its value has only been determined for limited classes of states with large symmetries, such as Greenberger-Horne-Zeilinger ͑GHZ͒ states, generalized W states, and certain families of stabilizer states. 12,22,23 This is because the geometric measure of entanglement is defined in terms of the maximum fidelity between the state and a pure product state, and therefore poses a difficult optimization problem.…”

Classification of Entanglement in Symmetric States

“…Firstly, the definition of the GM (2.1) suggests that the overlap of a symmetric state |ψ s 〉 with a product state will be maximal if the product state is also symmetric. This straightforward conjecture has been actively investigated [21,83], but a proof is far from trivial. After some special cases were proven [146,147], Hübener et al [84] were able to give a proof for the general case of pure symmetric states 5 .…”

“…Symmetric states are known to appear in the Dicke model [80], as eigenstates in various models of solid states physics such as the Lipkin-Meshkov-Glick (LMG) model [22,23], and in the study of macroscopic entanglement of η-paired high T c superconductivity [81]. Furthermore, symmetric states have been actively implemented experimentally [3][4][5][6], and their symmetric properties facilitate the analysis of their entanglement properties [82][83][84][85][86][87]. In experiments with many qubits, it is often not possible to access single qubits individually, necessitating a fully symmetrical treatment of the initial state and the system dynamics [71].…”

“…hold for all pure states [21,83,134,139]. These inequalities do not hold for mixed states 4 , but a generalisation is possible by defining…”

“…In the area of quantum phase transitions the GM has been used to analyse the Lipkin-Meshkov-Glick (LMG) model [23] as well as other spin models [126][127][128]. The survival of entanglement in thermal states was studied with the GM [129], and in quantum information theory the measure has been employed to derive the generalised Schmidt decomposition of Carteret et al [40] and for the study of entanglement witnesses [83,116]. On top of this, the measure has a variety of operational interpretations, including the usability of initial states for Grover's algorithm [130,131], additivity of channel capacities [132] and classification of states as resources for MBQC [17,79,133].…”