Non-additive entropies and quantum statistics

Jorge David Castaño‐Yepes, Cristian Felipe Ramirez-Gutierrez

In this paper, the impact of temperature fluctuations in the entanglement of two qubits described by the Heisenberg $XY$ model is studied. To model the out-of-equilibrium situation, super-statistics is used with fluctuations given by a $\chi^2$ distribution function, and its free parameters are chosen in such a way that resembles the non-additive Tsallis thermodynamics. In order to preserve the Legendre structure of the thermal functions, particular energy constraints are imposed on the density operator and the internal energy. Analytical results are obtained using an additional set of constraints after a parametrization of the physical temperature. We show that the well-known parametrization may lead to undesirable values of the physical temperature so that by analyzing the entropy as a function of energy, the correct values are found. The quantum entanglement is obtained from the Concurrence and is compared with when the non-additive prescription is not imposed.

Masanori Ohya

Sarah Chehade, Anna Vershynina

S. S. Apostolov, Z. А. Maizelis, OLEG V. USATENKO +1 more

We study the statistical properties of Ising spin chains with a finite (although arbitrarily large) range of interaction among the elements. We examine mesoscopic subsystems (fragments of an Ising chain) with lengths comparable with the interaction range. The equivalence of the Ising chains and the multi-step Markov sequences is used for calculating different non-additive statistical quantities of the chain and its fragments. In particular, we study the variance of the fluctuating magnetization of fragments, magnetization of the chain in an external magnetic field, etc. Asymptotical expressions for the non-additive energy and entropy of mesoscopic fragments are derived in the limiting cases of weak and strong interactions.

Huangjun Zhu, Lin Chen, Masahito Hayashi

We study the additivity property of three multipartite entanglement measures, i.e. the geometric measure of entanglement (GM), the relative entropy of entanglement and the logarithmic global robustness. Firstly, we show the additivity of GM of multipartite states with real and non-negative entries in the computational basis. Many states of experimental and theoretical interest have this property, e.g. Bell diagonal states, maximally correlated generalized Bell diagonal states, generalized Dicke states, the Smolin state and the generalization of Dür's multipartite bound entangled states. We also prove the additivity of the other two measures for some of these examples. Secondly, we show the non-additivity of GM of all antisymmetric states of three or more parties. We also provide a unified explanation of the non-additivity of the three measures of the antisymmetric projector states. In addition, we derive analytical formulae for the three measures of one copy and two copies of the antisymmetric projector states, respectively. Thirdly, we show, with a statistical approach, that almost all multipartite pure states with a sufficiently large number of parties are nearly maximally entangled with respect to GM and relative entropy of entanglement, and they have non-additive GM. Hence, more states may be suitable for universal quantum computation if measurements can be performed on two copies of the resource states. We also show that almost all the multipartite pure states cannot be produced reversibly with the combination of multipartite GHZ states under asymptotic LOCC, unless the relative entropy of entanglement is non-additive for generic multipartite pure states.

Fabio Benatti