Infinitesimal local operations and differential conditions for entanglement monotones

Much of the theory of entanglement concerns the transformations that are possible to a state under local operations with classical communication (LOCC); however, this set of operations is complicated and difficult to describe mathematically. An idea which has proven very useful is that of the {\it entanglement monotone}: a function of the state which is invariant under local unitary transformations and always decreases (or increases) on average after any local operation. In this paper we look on LOCC as the set of operations generated by {\it infinitesimal local operations}, operations which can be performed locally and which leave the state little changed. We show that a necessary and sufficient condition for a function of the state to be an entanglement monotone under local operations that do not involve information loss is that the function be a monotone under infinitesimal local operations. We then derive necessary and sufficient differential conditions for a function of the state to be an entanglement monotone. We first derive two conditions for local operations without information loss, and then show that they can be extended to more general operations by adding the requirement of {\it convexity}. We then demonstrate that a number of known entanglement monotones satisfy these differential criteria. Finally, as an application, we use the differential conditions to construct a new polynomial entanglement monotone for three-qubit pure states. It is our hope that this approach will avoid some of the difficulties in the theory of multipartite and mixed-state entanglement.Comment: 21 pages, RevTeX format, no figures, three minor corrections, including a factor of two in the differential conditions, the tracelessness of the matrix in the convexity condition, and the proof that the local purity is a monotone under local measurements. The conclusions of the paper are unaffecte

Measuring non-linear functionals of quantum harmonic oscillator states

Using only linear interactions and a local parity measurement we show how entanglement can be detected between two harmonic oscillators. The scheme generalizes to measure both linear and non-linear functionals of an arbitrary oscillator state. This leads to many applications including purity tests, eigenvalue estimation, entropy and distance measures - all without the need for non-linear interactions or complete state reconstruction. Remarkably, experimental realization of the proposed scheme is already within the reach of current technology with linear optics.Comment: 5 pages, 2 figures. Minor corrections and some new references adde

Quantum state diffusion with a moving basis: computing quantum-optical spectra

Quantum state diffusion (QSD) as a tool to solve quantum-optical master equations by stochastic simulation can be made several orders of magnitude more efficient if states in Hilbert space are represented in a moving basis of excited coherent states. The large savings in computer memory and time are due to the localization property of the QSD equation. We show how the method can be used to compute spectra and give an application to second harmonic generation.Comment: 8 pages in RevTeX, 1 uuencoded postscript figure, submitted to Phys. Rev.

Quantum state diffusion, localization and computation

Numerical simulation of individual open quantum systems has proven advantages over density operator computations. Quantum state diffusion with a moving basis (MQSD) provides a practical numerical simulation method which takes full advantage of the localization of quantum states into wave packets occupying small regions of classical phase space. Following and extending the original proposal of Percival, Alber and Steimle, we show that MQSD can provide a further gain over ordinary QSD and other quantum trajectory methods of many orders of magnitude in computational space and time. Because of these gains, it is even possible to calculate an open quantum system trajectory when the corresponding isolated system is intractable. MQSD is particularly advantageous where classical or semiclassical dynamics provides an adequate qualitative picture but is numerically inaccurate because of significant quantum effects. The principles are illustrated by computations for the quantum Duffing oscillator and for second harmonic generation in quantum optics. Potential applications in atomic and molecular dynamics, quantum circuits and quantum computation are suggested.Comment: 16 pages in LaTeX, 2 uuencoded postscript figures, submitted to J. Phys.

An observable entanglement measure for unknown mixed quantum states

We show how an unknown mixed quantum state's entanglement can be quantified by a suitable, local parity measurement on its two-fold copy.Comment: in press in PR

Quantum Walks driven by many coins

Quantum random walks have been much studied recently, largely due to their highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum random walk on the line: the use of multiple quantum ``coins'' in order to diminish the effects of interference between paths. We find solutions to this system in terms of the single coin random walk, and compare the asymptotic limit of these solutions to numerical simulations. We find exact analytical expressions for the time-dependence of the first two moments, and show that in the long time limit the ``quantum mechanical'' behavior of the one-coin walk persists. We further show that this is generic for a very broad class of possible walks, and that this behavior disappears only in the limit of a new coin for every step of the walk.Comment: 36 pages RevTeX 4.0 + 5 figures (encapsulated Postscript). Submitted to Physical Review

Entanglement Witnesses from Single-Particle Interference

We describe a general method of realizing entanglement witnesses in terms of the interference pattern of a single quantum probe. After outlining the principle, we discuss specific realizations both with electrons in mesoscopic Aharonov-Bohm rings and with photons in standard Young's double-slit or coherent-backscattering interferometers.Comment: 5 pages, 3 figures, epl2, uses pstricks.st

Comment on "Probabilistic Quantum Memories"

This is a comment on two wrong Phys. Rev. Letters papers by C.A. Trugenberger. Trugenberger claimed that quantum registers could be used as exponentially large "associative" memories. We show that his scheme is no better than one where the quantum register is replaced with a classical one of equal size. We also point out that the Holevo bound and more recent bounds on "quantum random access codes" pretty much rule out powerful memories (for classical information) based on quantum states.Comment: REVTeX4, 1 page, published versio