Erratum: Quantifying Entanglement of Maximal Dimension in Bipartite Mixed States [Phys. Rev. Lett. 117, 190502 (2016)].

This corrects the article DOI: 10.1103/PhysRevLett.117.190502.

Quantifying Entanglement of Maximal Dimension in Bipartite Mixed States.

The Schmidt coefficients capture all entanglement properties of a pure bipartite state and therefore determine its usefulness for quantum information processing. While the quantification of the corresponding properties in mixed states is important both from a theoretical and a practical point of view, it is considerably more difficult, and methods beyond estimates for the concurrence are elusive. In particular this holds for a quantitative assessment of the most valuable resource, the forms of entanglement that can only exist in high-dimensional systems. We derive a framework for lower bounding the appropriate measure of entanglement, the so-called G-concurrence, through few local measurements. Moreover, we show that these bounds have relevant applications also for multipartite states.

Monogamy equalities for qubit entanglement from Lorentz invariance.

A striking result from nonrelativistic quantum mechanics is the monogamy of entanglement, which states that a particle can be maximally entangled only with one other party, not with several ones. While there is the exact quantitative relation for three qubits and also several inequalities describing monogamy properties, it is not clear to what extent exact monogamy relations are a general feature of quantum mechanics. We prove that in all many-qubit systems there exist strict monogamy laws for quantum correlations. They come about through the curious relationship between the nonrelativistic quantum mechanics of qubits and Minkowski space. We elucidate the origin of entanglement monogamy from this symmetry perspective and provide recipes to construct new families of such equalities.

Negativity as an estimator of entanglement dimension.

Among all entanglement measures negativity arguably is the best known and most popular tool to quantify bipartite quantum correlations. It is easily computed for arbitrary states of a composite system and can therefore be applied to discuss entanglement in an ample variety of situations. However, as opposed to logarithmic negativity, its direct physical meaning has not been pointed out yet. We show that the negativity can be viewed as an estimator of how many degrees of freedom of two subsystems are entangled. As it is possible to give lower bounds for the negativity even in a device-independent setting, it is the appropriate quantity to certify quantumness of both parties in a bipartite system and to determine the minimum number of dimensions that contribute to the quantum correlations.

A quantitative witness for Greenberger-Horne-Zeilinger entanglement.

Along with the vast progress in experimental quantum technologies there is an increasing demand for the quantification of entanglement between three or more quantum systems. Theory still does not provide adequate tools for this purpose. The objective is, besides the quest for exact results, to develop operational methods that allow for efficient entanglement quantification. Here we put forward an analytical approach that serves both these goals. We provide a simple procedure to quantify Greenberger-Horne-Zeilinger-type multipartite entanglement in arbitrary three-qubit states. For two qubits this method is equivalent to Wootters' seminal result for the concurrence. It establishes a close link between entanglement quantification and entanglement detection by witnesses, and can be generalised both to higher dimensions and to more than three parties.

Quantifying tripartite entanglement of three-qubit generalized Werner states.

Multipartite entanglement is a key concept in quantum mechanics for which, despite the experimental progress in entangling three or more quantum devices, there is still no general quantitative theory that exists. In order to characterize the robustness of multipartite entanglement, one often employs generalized Werner states, that is, mixtures of a Greenberger-Horne-Zeilinger (GHZ) state and the completely unpolarized state. While two-qubit Werner states have been instrumental for various important advancements in quantum information, as of now there is no quantitative account for such states of more than two qubits. By using the GHZ symmetry introduced recently, we find exact results for tripartite entanglement in three-qubit generalized Werner states and, moreover, the entire family of full-rank mixed states that share the same symmetries.

Entanglement of three-qubit Greenberger-Horne-Zeilinger-symmetric states.

The first characterization of mixed-state entanglement was achieved for two-qubit states in Werner's seminal work [Phys. Rev. A 40, 4277 (1989)]. A physically important extension concerns mixtures of a pure entangled state [such as the Greenberger-Horne-Zeilinger (GHZ) state] and the unpolarized state. These mixed states serve as benchmark for the robustness of multipartite entanglement. They share the symmetries of the GHZ state. We call such states GHZ symmetric. Here we give a complete description of the entanglement in the family of three-qubit GHZ-symmetric states and, in particular, of the three-qubit generalized Werner states. Our method relies on the appropriate parametrization of the states and on the invariance of entanglement properties under general local operations. An application is the definition of a symmetrization witness for the entanglement class of arbitrary three-qubit states.

Influence of classical resonances on chaotic tunneling.

Dynamical tunneling between symmetry-related stable modes is studied in the periodically driven pendulum. We present strong evidence that the tunneling process is governed by nonlinear resonances that manifest within the regular phase-space islands on which the stable modes are localized. By means of a quantitative numerical study of the corresponding Floquet problem, we identify the trace of such resonances not only in the level splittings between near-degenerate quantum states, where they lead to prominent plateau structures, but also in overlap matrix elements of the Floquet eigenstates, which reveal characteristic sequences of avoided crossings in the Floquet spectrum. The semiclassical theory of resonance-assisted tunneling yields good overall agreement with the quantum-tunneling rates, and indicates that partial barriers within the chaos might play a prominent role.

Resonance-assisted decay of nondispersive wave packets.

We present a quantitative semiclassical theory for the decay of nondispersive electronic wave packets in driven, ionizing Rydberg systems. Statistically robust quantities are extracted combining resonance-assisted tunneling with subsequent transport across chaotic phase space and a final ionization step.

Resonance- and chaos-assisted tunneling in mixed regular-chaotic systems.

We present evidence that nonlinear resonances govern the tunneling process between symmetry-related islands of regular motion in mixed regular-chaotic systems. In a similar way as for near-integrable tunneling, such resonances induce couplings between regular states within the islands and states that are supported by the chaotic sea. On the basis of this mechanism, we derive a semiclassical expression for the average tunneling rate, which yields good agreement in comparison with the exact quantum tunneling rates calculated for the kicked rotor and the kicked Harper.