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We introduce the challenges of multi-party quantum entanglement and explain a recent success in learning to take its measure. Given the widely accepted reputation of entanglement as a counter-intuitive feature of quantum theory, we first describe pure-state entanglement itself. We restrict attention to multi-party qubit states. Then we introduce the features that have made it challenging for several decades to extend an entanglement measure beyond the 2-qubit case of Bell states. We finish with a description of the current understanding that solves the 3-qubit entanglement challenge. This necessarily takes into account the fundamental division of the 3-qubit state space into two completely independent sectors identified with the so-called GHZ and $W$ states., 16 pages and 3 figures

Entanglement freezing has been demonstrated existing in various noisy decoherence mechanisms. Here we explore its universality by investigating freezing behavior in a lossless multiparty system, i.e., an $N$-site optical lattice (or equivalently spin-chain) system. Interesting phenomenon of permanent entanglement freezing is found for the infinite-size case ($N\rightarrow\infty$). As the multiparty system size reduces to finite $N$, the interesting behavior of dynamical entanglement "thawing" start to emerge. Especifically, alternative appearances of "freezing" and "thawing" can be identified as time evolves. Our results may provide useful guidance to entanglement control in quantum tasks., Comment: 6 pages

We extend our earlier "virtual detector" method [X. Wang, J. Tian, and J. H. Eberly, Phys. Rev. Lett. 110, 243001 (2013)], a hybrid quantum mechanical and classical trajectory method, to include phases in the classical trajectories. Effects of quantum interferences, lost in the earlier method, are restored. The obtained photoelectron momentum distributions agree well with the corresponding numerical solutions of the time-dependent Schr\"odinger equation.

Ninety years ago in 1927, at an international congress in Como, Italy, Niels Bohr gave an address which is recognized as the first instance in which the term "complementarity", as a physical concept, was spoken publicly [1], revealing Bohr's own thinking about Louis de Broglie's "duality". Bohr had very slowly accepted duality as a principle of physics: close observation of any quantum object will reveal either wave-like or particle-like behavior, one or the other of two fundamental and complementary features. Little disagreement exists today about complementarity's importance and broad applicability in quantum science. Book-length scholarly examinations even provide speculations about the relevance of complementarity in fields as different from physics as biology, psychology and social anthropology, connections which were apparently of interest to Bohr himself (see Jammer [2], Murdoch [3] and Whitaker [4]). Confusion evident in Como following his talk was not eliminated by Bohr's article [1], and complementarity has been subjected to nine decades of repeated examination ever since with no agreed resolution. Semi-popular treatments [5] as well as expert examinations [6-9] show that the topic cannot be avoided, and complementarity retains its central place in the interpretation of quantum mechanics. However, recent approaches by our group [10-13] and others [14-20] to the underlying notion of coherence now allow us to present a universal formulation of complementarity that may signal the end to the confusion. We demonstrate a new relationship that constrains the behavior of an electromagnetic field (quantum or classical) in the fundamental context of two-slit experiments. We show that entanglement is the ingredient needed to complete Bohr's formulation of complementarity, debated for decades because of its incompleteness., Comment: 8 pages, 4 figures

Entanglement is a key resource of quantum science for tasks that require it to be shared among participants. Within atomic, condensed matter and photonic many-body systems the distribution and sharing of entanglement is of particular importance for information processing by progressively larger and larger quantum networks. Here we report a singly-bipartitioned qubit entanglement inequality that applies to any N-party qubit pure state and is completely tight. It provides the first prescription for a direct calculation of the amount of entanglement sharing that is possible among N qubit parties. A geometric representation of the measure is easily visualized via polytopes within entanglement hypercubes., 5pages, 4 figures

For many decades the word "entanglement" has been firmly attached to the world of quantum mechanics, as is the phrase "Bell violation". Here we introduce Shimony-Wolf fields, entirely classical non-deterministic states, as a basis for entanglement and Bell analyses. Such fields are well known in coherence optics and are open to test. We present experimental results showing that Shimony-Wolf states exhibit strong classical Bell violation, in effect opening a way of examining a new sector of the boundary between quantum and classical physics., 15 pages

We find an algebraic formula for the N-partite concurrence of N qubits in an X-matrix. X- matricies are density matrices whose only non-zero elements are diagonal or anti-diagonal when written in an orthonormal basis. We use our formula to study the dynamics of the N-partite entanglement of N remote qubits in generalized N-party Greenberger-Horne-Zeilinger (GHZ) states. We study the case when each qubit interacts with a partner harmonic oscillator. It is shown that only one type of GHZ state is prone to entanglement sudden death; for the rest, N-partite entanglement dies out momentarily. Algebraic formulas for the entanglement dynamics are given in both cases.

We take advantage of the information provided by use of elliptical polarization in a recent two-electron release time experiment [A.N. Pfeiffer et al., Nature Physics 7, 428 (2011)]. This allows a comparative test of the currently dominant conjectures regarding independent-electron tunneling theory vs. fully electron-correlated classical release theory to describe electron release in strong-field double ionization.

We investigate the effects of error correction on non-local quantum coherence as a function of time, extending the study by Sainz and Bj\"ork. We consider error correction of amplitude damping, pure phase damping and combinations of amplitude and phase damping as they affect both fidelity and quantum entanglement. Initial two-qubit entanglement is encoded in arbitrary real superpositions of both \Phi-type and \Psi-type Bell states. Our main focus is on the possibility of delay or prevention of ESD (early stage decoherence, or entanglement sudden death). We obtain the onset times for ESD as a function of the state-superposition mixing angle. Error correction affects entanglement and fidelity differently, and we exhibit initial entangled states for which error correction increases fidelity but decreases entanglement, and vice versa., Comment: 10 pages, 10 figures; references added; journal reference added

We study the dynamics of two qubits interacting with a single mode of a harmonic oscillator beyond the rotating wave approximation in the ideally degenerate regime. Exact analytic expressions are obtained for state properties of interest, including qubit entanglement for a certain class of initial states of the oscillator and the qubits. Qualitative differences and similarities in the evolution of the qubits in the degenerate regime when the oscillator is treated quantum mechanically and classically are discussed.

Multielectron double ionization with elliptically polarized light is examined theoretically using a classical ensemble method going beyond the single-active-electron approximation. A knee structure in the ion signal is found. Our analysis provides an explanation for newly observed SDI phenomena.

The entanglement dynamics of two remote qubits is examined analytically. The qubits interact arbitrarily strongly with separate harmonic oscillators in the idealized degenerate limit of the Jaynes-Cummings Hamiltonian. In contrast to well known non-degenerate RWA results, it is shown that ideally degenerate qubits cannot induce bipartite entanglement between their partner oscillators., Comment: 8 pages, 6 figures

We formulate the classical polarization theory for light by using entanglement analysis. We demonstrate a route to a systematic and consistent measure of ordinary light polarization that extends automatically to a new understanding of the degree of polarization at a point, when a distinct beam direction is absent. We have drawn on recent work [Qian and Eberly, arXiv 1009.5622] dealing with the conservation of quantum information.

We report an approach to quantum open system dynamics that leads to novel nonlinear constant relations governing information flow among the participants. Our treatment is for mixed state systems entangled in a pure state fashion with an unspecified party that was involved in preparing the system for an experimental test, but no longer interacts after $t=0$. Evolution due to subsequent interaction with another party is treated as an amplitude flow channel and uses Schmidt-type bipartite decomposition of the evolving state. We illustrate this with three examples, including both reversible and irreversible information flows, and give formulas for the new nonlinear constraints in each case., 9 pages

The prospect of self-consistent propagation of more than two pulses contemporaneously through multi-resonant media raises open questions: whether soliton solutions exist, and whether a useful generalization of two-level pulse Area can be found. We answer these questions positively for the case of four pulses interacting in combined V and Lambda fashion with an idealized pair of atomic D-lines., Comment: 4 pages, 3 figures

We analyze the separability of the joint state of a collection of two-level systems at finite temperature T. The fact that only separable states are found in the neighborhood of their thermal equilibrium state guarantees that decoherence will destroy any initially arranged entanglement in a finite time., 4 pages, 2 figures

We treat theoretically the two-pulse plane-wave propagation problem associated with the stimulated Raman process. A strong pump pulse and a weak probe pulse interact self-consistently during mutual propagation through a medium in the three-level lambda configuration. We treat the pulse interaction fully coherently, but take initially incoherent mixed ground states to describe the medium. We present analytic pulse envelope solutions to the nonlinear evolution equations, and then test them numerically. We show that the entire process can be interpreted consistently as a three-regime evolution, in which the first and last regimes display behavior quantitatively the same as self-induced transparency. They are connected through a Raman transfer process in the intermediate regime., Submitted in honor of G.S. Agarwal. 11 pages 4 figues

We examine a class of bipartite mixed states which we call X states and report several analytic results related to the occurrence of so-called entanglement sudden death (ESD) under time evolution in the presence of common types of environmental noise. Avoidance of sudden death by application of purely local operations is shown to be feasible in some cases., 11 pages, 2 figures, version 3, some minor changes have been made

Quantum entanglement can manifest itself in the narrowing of wavepackets. We define the phenomenon of phase entanglement and describe its effect on the interpretation of spatial localization experiments., 4 pages, 1 figure, REVTeX

We report the discovery of a "dark area theorem," a new quantum optical relation for propagation of unmatched pulses in thick three-level $\Lambda$-type media. We define dark area and derive the dark area theorem for a coherently prepared and inhomogeneously broadened lambda medium. We also obtain the first equation for the spatial evolution of the dark state amplitude prior to pulse-matching., Comment: 8 pages, 4 figures

Accounting for resources is the central issue in computational efficiency. We point out physical constraints implicit in information readout that have been overlooked in classical computing. The basic particle-counting mode of read-out sets a lower bound on the resources needed to implement a quantum computer. As a consequence, computers based on classical waves are as efficient as those based on single quantum particles., 4 pages, 1 figure