Research Interests

Quantum Entanglement and Quantum Correlations

Quantum entanglement is just one sort of quantum correlation, where by quantum correlation one means correlations between two or more systems that has a pure quantum origin. An important extension to the concept of entanglement is quantum discord (QD). Its origin can be traced back to two inequivalent ways of extending the mutual information/correlation to the quantum world. More intuitively, and not so rigorous, QD is directly related to the superposition principle manifesting itself within and/or between subsystems while entanglement is the superposition principle manifesting itself only between subsystems. These quantum correlations can be used to improve the execution of or create many communication and computational tasks unattainable in the classical world, such as quantum teleportation, cryptography, and superdense coding.

But we can also use these quantum correlations as tools to investigate the behavior of matter under quantum phase transitions (QPTs). QPTs are macroscopic changes occurring in the ground state of a many-body system driven solely by quantum fluctuations (Heisenberg uncertainty principle). It is a truly quantum effect that occurs for temperatures near the absolute zero.

An interesting property of entanglement and QD is their ability to detect many QPTs by an abrupt, sometimes non-analytical, change in their behavior at the critical point of a QPT. However, it is a difficult experimental task to bring the system under investigation to very small temperatures where a QPT sets in. Therefore, a great theoretical challenge is to determine to what extent entanglement and QD are robust enough to "feel" a QPT at finite temperatures. One of my recent goals is to understand the full usefulness of these quantum correlations to detect QPTs at zero and finite temperatures.

Quantum Communication via Partially Entangled States

It is known that almost all quantum communication protocols only work with maximally entangled states, which are difficult to be produced and stored. But in real-life situations only partially entangled states are available. The standard solution to this limitation is based on entanglement concentration or distillation, where a few maximally entangled states are obtained out of an ensemble of partially entangled ones. However, this approach only works when many partially entangled states are available and it does not use directly the partially entangled states. Therefore, the following questions naturally arise: What sort of tasks can be done using directly partially entangled states, in special when only a few of them are available? Is there any quantum communication task where partially entangled states are more appropriate than maximally entangled ones? Is there a situation where the direct use of partially entangled states give better results than the use of standard entanglement concentration techniques?

Continuous Variable Systems

Most of quantum communication protocols and the description of quantum computation are based on discrete systems, i.e., finite dimensional systems such as qubits (two-level systems). Qubits are a natural extension of the classical bit. It is possible, nevertheless, to do quantum computation and quantum communication using systems described by pairs of canonical variables. These systems lie in an infinite dimensional Hilbert space and are usually described by the standard position and momentum operators. They are called CV-systems for short.

The concept of logical gates, teleportation, and entanglement, to name just a few key concepts of quantum information theory, are easily extended from qubits to CV-systems. As well as for qubits, one is interested in the quantification of the degree of entanglement between two CV-systems. I have been working for awhile with the development of theoretical tools and experimental proposals that aim to quantify the entanglement content of such systems acting only locally. I have also developed strategies to locally measure the global density matrix of entangled CV-systems.

I am also interested in improving the efficiency of CV-systems quantum communication protocols, such as quantum teleportation, and also in new ways of employing CV-systems in order to do quantum computation.

Perturbation Theories

Quantum mechanics (QM) is undoubtedly one of the most successful and useful theories of modern Physics. Its practical importance is evidenced at microscopic and nano scales where the Schroedinger Equation (SE) dictates the evolution of the system's state, i.e., its wave function, from which all the properties of the system can be calculated and confronted against experiment. However, SE can only be exactly solved for a few problems. Indeed, there are many reasons that make the solution of such a differential equation a difficult task, such as the large number of degrees of freedom associated with the system one wants to study. Another reason is related to an important property of the system's Hamiltonian: its time dependence.

For time independent Hamiltonians the solution to SE can be cast as an eigenvalue/eigenvector problem. This allows us to solve SE in many cases exactly, in particular when we deal with systems described by finite dimensional Hilbert spaces. For time dependent Hamiltonians, on the other hand, things are more mathematically involved. Even for a two-level system (a qubit) we do not, in general, obtain a closed-form solution given an arbitrary time dependent Hamiltonian, although a general statement can be made for slowly varying Hamiltonians. If a system's Hamiltonian H changes slowly during the course of time, say from t=0 to t=T, and the system is prepared in an eigenstate of H at t=0, it will remain in the instantaneous (snapshot) eigenstate of H(t) during the whole evolution. This is the content of the well-known adiabatic theorem.

But what happens if H(t) is not slowly enough changed? For how long can we still consider the system to be in a snapshot eigenstate of H(t), i.e., for how long the adiabatic approximation is reliable? What are the corrections to the adiabatic approximation? What changes to the Berry phase occur when one is away from the adiabatic approximation? One of my research goals is to provide practical and useful answers to these questions by means of a perturbative expansion about the adiabatic approximation, named adiabatic perturbation theory (APT).

Furthermore, the control of the dynamics of micro and nanoscopic systems is at the heart of experimental and theoretical schemes in quantum information and nanosciences. When, in a given physical process, there is a clear separation of time scales into fast and slow that fact helps in understanding its dynamics. This is the core motivation behind the well-known adiabatic theorem of QM for non-degenerate systems. But, how fast is fast and how slow is slow? It is crucial to quantify the validity of any time-dependent approximation in a general context since what is at stake is the practical dynamical implementation of concepts such as holonomic quantum computation or the detection of fractional (adiabatic) statistics.

Another goal of mine is to asses the validity of the adiabatic approximation/theorem for degenerate spectra by means of the extension of the adiabatic perturbation theory explained above for degenerate subspaces (DAPT). DAPT is a genuine perturbative expansion in the ''velocity'' v to drive the system from one particular configuration to another, with the perturbative corrections to the time-dependent SE given about the adiabatic approximation (AA). Also, DAPT is a practical and operational formulation to test the validity of a broad range of concepts such as non-adiabatic fractional statistics and the extent to which AA is valid for a degenerate physical system, furnishing, as a bonus, the corrections to DAPT. Finally, the development of DAPT also helps in the determination of corrections to the Wilczek-Zee (WZ) non-Abelian geometric phase.

The key ingredients in the construction of APT and DAPT, differentiating it from all standard perturbation theories, are three fold: (a) the rescaling of the real time t to s=vt, allowing a consistent perturbative expansion to all orders in terms of v. Here v=1/T, being T the total time spent to drive the system from its initial to its final configuration; (b) the correct setting of the initial condition at t=0, i.e., one should recover the zeroth order not only when v tends to zero but also at t=0 for any v; and (c) a powerful vectorial ansatz/notation that factored out singular terms and permitted the construction of simple matrix recursive relations between higher order and lower order coefficients.

Quantum Computation and Quantum Simulation

A great challenge in modern Physics is the understanding of the collective behavior of a many-body system looking at the elementary interactions of its constituents. However, it is known that the greater the size of the system the greater the complexity of the simulations, where the latter becomes intractable in many important physical systems. Actually, the computational complexity for most of the problems scales non-polynomially with the size of the system, mainly because one uses classical computers to perform such simulations. Since quantum mechanics rules the microscopic world, it is expected that a quantum computer could do a better job. And the most promising way to handle those problems is the use of a well controlled quantum system to simulate another one. In other words, we need a quantum simulator. But what kinds of quantum systems can be simulated in this fashion? What are the minimum requirements a given quantum system must have to be faithfully mapped onto another one? How can one assess that the quantum simulator are working properly? What are the errors introduced in this mapping and how can they be corrected? These are questions that must be answered before any quantum simulator is built.

Entanglement Properties of Many-Body and Condensed Matter Systems

The techniques used to understand multipartite entanglement can also be employed to study few- and many-body physics. A recent successful example of this approach is the particular behavior of entanglement during a quantum phase transition (QPT), where discontinuities or divergences of certain entanglement measures occur at the critical point, indicating that entanglement may be a useful QPT indicator. Also, it is believed that some concepts coming from many-body physics can help in a deeper understanding of multipartite entanglement. It is the exploration of this intersection of concepts and ideas that defines one of my research goals and brings us to the following important questions: What is the real importance of entanglement to the transition from the quantum to the classical world? Can we better understand some macroscopic properties of matter by understanding its entanglement properties?

Conceptual Aspects of Quantum Mechanics

Quantum mechanics (QM) is the most "counter-intuitive" theory of theoretical physics. For some, it "peacefully coexists" with the Einstein theory of relativity (ETR). For others, there may be an "apparent incompatibility" between QM and ETR. This incompatibility is mainly due to a sort of "spooky action at a distance" and to the lack of "elements of reality" that is inherent to QM. In spite of these uncomfortable properties, so far no experiment ever contradicted QM. As Jonh S. Bell rightly put it, "for all practical purposes" QM works well. However, when one deeply thinks about its theoretical framework and sets of postulates, one cannot fully grasp their intuitive meaning. One is always feeling that something is missing and that a more robust theoretical framework is needed.

One of my long term goals is to dig deep into the foundations of QM by assuming that it is an information theory, one that best describes the microscopic world. With that assumption, it is possible that the theoretical tools of classical information theory help us in the understanding of QM from a different perspective, one which may open the way to elucidate some of its "paradoxes". Also, it may be that QM is just the tip of the iceberg of a more fundamental theory. It may be that QM is related to some new theory or new level of understanding very much in the same way that thermodynamics is connected to statistical physics. This new level of understanding will only be useful if it ultimately bridges the gap between the quantum/microscopic world and the classical/macroscopic one.

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