On the quantification of entanglement in infinite-dimensional quantum systems

Abstract

We investigate entanglement measures in the infinite-dimensional regime. First, we discuss the peculiarities that may occur if the Hilbert space of a bi-partite system is infinitedimensional, most notably the fact that the set of states with infinite entropy of entanglement is trace-norm dense in state space, implying that in any neighbourhood of every product state lies an arbitrarily strongly entangled state. The starting point for a clarification of this counterintuitive property is the observation that if one imposes the natural and physically reasonable constraint that the mean energy is bounded from above, then the entropy of entanglement becomes a trace-norm continuous functional. The considerations will then be extended to the asymptotic limit, and we will prove some asymptotic continuity properties. We proceed by investigating the entanglement of formation and the relative entropy of entanglement in the infinite-dimensional setting. Finally, we show that the set of entangled states is still tracenorm dense in state space, even under the constraint of a finite mean energy.

Quantum information and relativity theory

“…However, in the general case, infinite-dimensional Hilbert spaces are involved. Recently Parker, Bose, and Plenio (2000), Eisert, Simon, andPlenio (2002), andKeyl, Schlingemann, andWerner (2003) investigated the entanglements of formation and of distillation in infinite-dimensional systems.…”

“…For pure states, a natural measure of entanglement is the von Neumann entropy S = −tr ρ ln ρ of either one of the reduced density matrices. It can be shown (Eisert, Simon and Plenio, 2002) that in an arbitrarily small neighborhood of any state there is an infinity of entangled states. The reason is that in the neighborhood of any state with finite energy, there are states of infinite entropy (Wehrl, 1978).…”

Entanglement and purity of two-mode Gaussian states in noisy channels

“…The negativity has been proved to be convex and monotone under LOCC (local operations and classical communications) [26], but fails to be continuous in trace norm on infinite dimensional Hilbert spaces. Anyway, this problem can be somehow eluded by restricting to states with finite mean energy [28]. For two-mode Gaussian states it can be easily shown that the negativity is a simple function ofñ − , which is thus itself an (increasing) entanglement monotone; one has in fact [12] …”

Quantifying coherence in infinite-dimensional systems

“…Fock states are the eigenstates of the number operatorn :=â †â where we haveâ|n = √ n|n − 1 andâ † |n = √ n + 1|n + 1 . Referring to development of entanglement theory in infinite dimensional systems, the problem of quantification of coherence can addressed by requiring energy constraints [15], which is experimentally relevant. Here and after, we require a new condition for this case (C4): if the first order moment, the average particle number, is finite n < ∞, it should fulfill C(ρ) < ∞.…”