Canonical and micro-canonical typical entanglement of continuous variable systems

Abstract

Abstract. We present a framework, compliant with the general canonical principle of statistical mechanics, to define measures on the set of pure Gaussian states of continuous variable systems. Within such a framework, we define two specific measures, referred to as 'micro-canonical' and 'canonical', and apply them to study systematically the statistical properties of the bipartite entanglement of n-mode pure Gaussian states (as quantified by the entropy of a subsystem). We rigorously prove the "concentration of measure" around a finite average, occurring for the entanglement in the thermodynamical limit in both the canonical and the micro-canonical approach. For finite n, we determine analytically the average and standard deviation of the entanglement (as quantified by the reduced purity) between one mode and all the other modes. Furthermore, we numerically investigate more general situations, clearly showing that the onset of the concentration of measure already occurs at relatively small n.

Ensemble-averaged entanglement of two-particle states in Fock space

“…Next they calculate the entanglement of one qubit with the others and average over the choice of qubits. Plenio et al [30,31,32] have considered the typical entanglement in ensembles of Gaussian states. These states differ considerably from the two-particle states considered here.…”

Typical entanglement for Gaussian states

“…Let us briefly summarize the work [23]. Analogous to the finite-dimensional case, one considers a bipartite system H A ⊗ H B consisting of k system modes and n − k environment modes (i.e., H A ∼ = L 2 (R) ⊗k and H B ∼ = L 2 (R) ⊗n−k ).…”

“…is simply the energy Φ|H AB |Φ of |Φ . We note that our choice of parameterization differs from [23] by a square (i.e., they use z 2 j instead of z j ), but this is inconsequential. Conversely, every pair (O, Z) defines a Gaussian pure state |Φ obeying the energy constraint (1).…”

Geometry on the manifold of Gaussian quantum channels

“…We provide a rigorous route to investigate how likely it is, among all Gaussian quantum channels, to encounter a channel that is either entanglement breaking [17] or incompatibility breaking [18]. So far, the investigations on the information geometry in the Gaussian domain have been focused on the geometry of the state space [19,20] and the typical properties of quantum correlations [21][22][23]. In [24], first steps are taken to study the geometry of the Gaussian quantum channels.…”