Alonso Botero

Revisiting Hardy's Paradox: Counterfactual Statements, Real Measurements, Entanglement and Weak Values

Hardys paradox is revisited. Usually the paradox is dismissed on grounds of counterfactuality, i.e., because the paradoxical effects appear only when one considers results of experiments which do not actually take place. We suggest a new set of measurements in connection with Hardys scheme, and show that when they are actually performed, they yield strange and surprising outcomes. More generally, we claim that counterfactual paradoxes point to a deeper structure inherent to quantum mechanics.

Quantum averages of weak values

We re-examine the status of the weak value of a quantum mechanical observable as an objective physical concept, addressing its physical interpretation and general domain of applicability. We show that the weak value can be regarded as a definite mechanical effect on a measuring probe specifically designed to minimize the back reaction on the measured system. We then present an interesting framework for general measurement conditions where the back reaction on the system may not be negligible in which the measurement outcomes can still be interpreted as quantum averages of weak values. We show that in the classical limit, there is a direct correspondence between quantum averages of weak values and posterior expectation values of classical dynamical properties according to the classical inference framework.

Spatial structures and localization of vacuum entanglement in the linear harmonic chain

We study the structure of vacuum entanglement for two complimentary segments of a linear harmonic chain, applying the modewise decomposition of entangled Gaussian states discussed by Boteno and Reznik [Phys. Rev. A 67, 052311 (2003)]. We find that the resulting entangled mode-shape hierarchy shows a distinctive layered structure with well-defined relations between the depth of the modes, their characteristic wavelength, and their entanglement contribution. We rederive in the strong coupling (diverging correlation length) regime, the logarithmic dependence of entanglement on the segment size predicted by conformal field theory for the boson universality class and discuss its relation with the mode structure. We conjecture that the persistence of vacuum entanglement between arbitrarily separated finite-size regions is connected with the localization of the highest-frequency innermost modes.

BCS-like modewise entanglement of fermion Gaussian states

Abstract We show that with respect to any bipartite division of modes, pure fermion Gaussian states display the same type of structure in its entanglement of modes as that of the BCS wave function, namely, that of a tensor product of entangled two-mode squeezed fermion states. We show that this structure applies to a wider class of “isotropic” mixed fermion states, for which we derive necessary and sufficient conditions for mode entanglement.

The difference between two random mixed quantum states: exact and asymptotic spectral analysis

We investigate the spectral statistics of the difference of two density matrices, each of which is independently obtained by partially tracing a random bipartite pure quantum state. We first show how a closed-form expression for the exact joint eigenvalue probability density function for arbitrary dimensions can be obtained from the joint probability density function of the diagonal elements of the difference matrix, which is straightforward to compute. Subsequently, we use standard results from free probability theory to derive a relatively simple analytic expression for the asymptotic eigenvalue density (AED) of the difference matrix ensemble, and using Carlsons theorem, we obtain an expression for its absolute moments. These results allow us to quantify the typical asymptotic distance between the two random mixed states using various distance measures; in particular, we obtain the almost sure asymptotic behavior of the operator norm distance and the trace distance.

Exploring the local elastic properties of bilayer membranes using molecular dynamics simulations.

Membrane mechanical elastic properties regulate a variety of cellular processes involving local membrane deformation, such as ion channel function and vesicle fusion. In this work, we used molecular dynamics simulations to estimate the local elastic properties of a membrane. For this, we calculated the energy needed to extract a DOPE lipid molecule, modified with a linker chain, from a POPC bilayer membrane using the umbrella sampling technique. Although the extraction energy entails several contributions related not only to elastic deformation but also to solvation, careful analysis of the potential of mean force (PMF) allowed us to dissect the elastic contribution. With this information, we calculated an effective linear spring constant of 44 ± 4 kJ·nm(-2)·mol(-1) for the DOPC membrane, in agreement with experimental estimates. The membrane deformation profile was determined independently during the stretching process in molecular detail, allowing us to fit this profile to a previously proposed continuum elastic model. Through this approach, we calculated an effective membrane spring constant of 42 kJ·nm(-2)·mol(-1), which is in good agreement with the PMF calculation. Furthermore, the solvation energy we derived from the data is shown to match the solvation energy estimated from critical micelle formation constants. This methodology can be used to determine how changes in lipid composition or the presence of membrane modifiers can affect the elastic properties of a membrane at a local level.

The classical limit of quantum optics: not what it seems at first sight

What light is and how to describe it has always been a central subject in physics. As our understanding has increased, so have our theories changed: geometrical optics, wave optics and quantum optics are increasingly sophisticated descriptions, each referring to a larger class of phenomena than its predecessor. But how exactly are these theories related? How and when wave optics reduces to geometric optics is a rather simple problem. Similarly, how quantum optics reduces to wave optics has also been considered to be a very simple business. It is not so. As we show here the classical limit of quantum optics is a far more complicated issue; it is in fact dramatically more involved and it requires a complete revision of all our intuitions. The revised intuitions can then serve as a guide to finding novel quantum effects.

Geometric phase and modulus relations for probability amplitudes as functions on complex parameter spaces

We investigate general differential relations connecting the respective behaviors of the phase and modulus of probability amplitudes of the form 〈ψf|ψ〉, where |ψf〉 is a fixed state in Hilbert space and |ψ〉 is a variable state, treated as a section of a U(1) bundle over a complex subspace of the corresponding ray space R=CPn. Amplitude functions on such holomorphic line bundles, while not strictly holomorphic, nevertheless satisfy generalized Cauchy–Riemann conditions involving the U(1) Berry–Simon connection on the parameter space. These conditions entail invertible relations between the gradients of the phase and modulus, therefore allowing for the reconstruction of the phase from the modulus (or vice versa) and other conditions on the behavior of either polar component of the amplitude. As a special case, we consider amplitude functions valued on the space of pure states, the ray space R=CPn, where transition probabilities have a geometric interpretation in terms of geodesic distances as measured with t...

Entanglement and Weak Values: A Quantum Miracle Cookbook

The concept of the weak value has proved to be a powerful and operationally grounded framework for the assignment of physical properties to a quantum system at any given time. More importantly, this framework has allowed us to identify a whole range of surprising quantum effects, or “miracles”, which are readily testable but which lie buried “under the noise” when the results of measurements are not post-selected. In all cases, these miracles have to do with the fact that weak values can take values lying outside the conventional ranges of quantum expectation values. We explore the extent to which such miracles are possible within the weak value framework. As we show, given appropriate initial and final states, it is generally possible to produce any set of weak values that is consistent with the linearity of weak values, provided that the states are entangled states of the system with some external ancillary system. Through a simple constructive proof, we obtain a recipe for arbitrary quantum miracles, and give examples of some interesting applications. In particular, we show how the classical description of an infinitely-localized point in phase-space is contained in the weak-value framework augmented by quantum entanglement. [Editor’s note: for a video of the talk given by Prof. Botero at the Aharonov-80 conference in 2012 at Chapman University, see quantum.chapman.edu/talk-27.]

Exploring the Elastic Properties of Bilayer Membranes using Molecular Dynamics Simulations

Local membrane deformation has been implicated in regulating a variety of cellular processes, such as ion channel function and vesicle fusion. In this work, we show how molecular dynamic simulations can be used alone or in conjunction with the continuum elastic model to estimate membrane elastic properties. Detail analysis allowed us to divide the energetic cost associated with the partial or complete extraction of a DOPE lipid from a POPC bilayer into two main contributions: a) the elastic deformation of the membrane, involving displacement of neighboring lipids, and b) the solvation energy associated to the exposure of the acyl chains to the water phase. Membrane elastic deformation was observed in molecular detail, and structural information from the simulations was used with the continuum elastic model to estimate an effective membrane spring constant independently from the energy parameters of the simulations. The membrane spring constant was also calculated from the potential of mean force and a good agreement was found between the two methods. Additionally, we confirmed that the calculated solvation energy matches that estimated by critical micelle formation constants. This methodology provides a computational tool for determining membrane elastic properties as a function of composition and in the presence of membrane modifiers.