Gustavo Rigolin

Quantum teleportation of an arbitrary two-qubit state and its relation to multipartite entanglement

We explicitly show a protocol in which an arbitrary two qubit state vertical bar {phi}>=a vertical bar 00>+b vertical bar 01>+c vertical bar 10>+d vertical bar 11> is faithfully and deterministically teleported from Alice to Bob. We construct the 16 orthogonal generalized Bell states that can be used to teleport the two qubits. The local operations Bob must perform on his qubits in order to recover the teleported state are also constructed. They are restricted only to single-qubit gates. This means that a controlled-NOT gate is not necessary to complete the protocol. A generalization where N qubits are teleported is also shown. We define a generalized magic basis, which possesses interesting properties. These properties help us to suggest a generalized concurrence from which we construct a measure of entanglement that has a clear physical interpretation: A multipartite state has maximum entanglement if it is a genuine quantum teleportation channel.

Beyond the quantum adiabatic approximation: Adiabatic perturbation theory

Aside from interpretation, Quantum Mechanics (QM) is undoubtedly one of the most successful and use- ful theories of modern Physics. Its practical impor- tance is evidenced at microscopic and nano scales where Schrodingers Equation (SE) dictates the evolution of the systems state, i.e., its wave function, from which all the properties of the system can be calculated and confronted against experimental data. However, SE can only be ex- actly solved for a few problems. Indeed, there are many reasons that make the solution of such a differential equa- tion a difficult task, such as the large number of degrees of freedom associated with the system one wants to study. Another reason, the one we want to address in this pa- per, is related to an important property of the systems 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 dimen- sional Hilbert spaces. For time dependent Hamiltonians, on the other hand, things are more mathematically in- volved. Even for a two-level system (a qubit) we do not, in general, obtain a closed-form solution given an arbi- trary time dependent Hamiltonian, although a general statement can be made for slowly varying Hamiltonians. If a systems 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 re-

Fighting noise with noise in realistic quantum teleportation

We investigate how the efficiency of the quantum teleportation protocol is affected when the qubits involved in the protocol are subjected to noise or decoherence. We study all types of noise usually encountered in real world implementations of quantum communication protocols, namely, the bit flip, phase flip (phase damping), depolarizing, and amplitude damping noise. Several realistic scenarios are studied in which a part or all of the qubits employed in the execution of the quantum teleportation protocol are subjected to the same or different types of noise. We find noise scenarios not yet known in which more noise or less entanglement lead to more efficiency. Furthermore, we show that if noise is unavoidable it is better to subject the qubits to different noise channels in order to obtain an increase in the efficiency of the protocol.

Adiabatic theorem for quantum systems with spectral degeneracy

Department of Physics, Indiana University, Bloomington, IN 47405, USA(Dated: November 24, 2011)By stating the adiabatic theorem of quantum mechanics in a clear and rigorous way, we estab-lish necessary and sufficient conditions for its validity employing our recently developed adiabaticperturbation theory. We present results for the most general case of quantum systems, i.e., thosewith degenerate energy spectra. These conditions are of upmost importance to assess the validityof practical implementations of non-Abelian braiding and adiabatic quantum computation. To il-lustrate the degenerate adiabatic approximation, and the necessary and sufficient conditions for itsvalidity, we analyze in depth an exactly solvable time-dependent degenerate problem.

Adiabatic perturbation theory and geometric phases for degenerate systems.

We introduce an adiabatic perturbation theory for quantum systems with degenerate energy spectra. This perturbative series enables one to rigorously establish conditions for the validity of the adiabatic theorem of quantum mechanics for degenerate systems. The same formalism can be used to find nonadiabatic corrections to the non-Abelian Wilczek-Zee geometric phase. These corrections are relevant to assess the validity of the practical implementation of the concept of fractional exchange statistics. We illustrate the formalism with an exactly solvable time-dependent problem.

Dynamically Disordered Quantum Walk as a Maximal Entanglement Generator

We show that the entanglement between the internal (spin) and external (position) degrees of freedom of a qubit in a random (dynamically disordered) one-dimensional discrete time quantum random walk (QRW) achieves its maximal possible value asymptotically in the number of steps, outperforming the entanglement attained by using ordered QRW. The disorder is modeled by introducing an extra random aspect to QRW, a classical coin that randomly dictates which quantum coin drives the systems time evolution. We also show that maximal entanglement is achieved independently of the initial state of the walker, study the number of steps the system must move to be within a small fixed neighborhood of its asymptotic limit, and propose two experiments where these ideas can be tested.

Probabilistic quantum teleportation in the presence of noise

We extend the research program initiated in [Phys. Rev. A 92, 012338 (2015)], where we restricted our attention to noisy deterministic teleportation protocols, to noisy probabilistic (conditional) protocols. Our main goal now is to study how we can increase the fidelity of the teleported state in the presence of noise by working with probabilistic protocols. We work with several scenarios involving the most common types of noise in realistic implementations of quantum communication tasks and find many cases where adding more noise to the probabilistic protocol increases considerably the fidelity of the teleported state, without decreasing the probability of a successful run of the protocol. Also, there are cases where the entanglement of the channel connecting Alice and Bob leading to the greatest fidelity is not maximal. Moreover, there exist cases where the optimal fidelity for the probabilistic protocols are greater than the maximal fidelity (2/3) achievable by using only classical resources, while the optimal ones for the deterministic protocols under the same conditions lie below this limit. This result clearly illustrates that in some cases we can only get a truly quantum teleportation if we use probabilistic instead of deterministic protocols.

Entangling power of disordered quantum walks

We investigate how the introduction of different types of disorder affects the generation of entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional quantum random walks (QRW). Disorder is modeled by adding another random feature to QRW, i.e., the quantum coin that drives the systems evolution is randomly chosen at each position and/or at each time step, giving rise to either dynamic, fluctuating, or static disorder. The first one is position-independent, with every lattice site having the same coin at a given time, the second has time and position dependent randomness, while the third one is time-independent. We show for several levels of disorder that dynamic disorder is the most powerful entanglement generator, followed closely by fluctuating disorder. Static disorder is the less efficient entangler, being almost always less efficient than the ordered case. Also, dynamic and fluctuating disorder lead to maximally entangled states asymptotically in time for any initial condition while static disorder has no asymptotic limit and, similarly to the ordered case, has a long time behavior highly sensitive to the initial conditions.

Degenerate Adiabatic Perturbation Theory: Foundations and Applications

We present details and expand on the framework leading to the recently introduced degenerate adiabatic perturbation theory [Phys. Rev. Lett. 104, 170406 (2010)], and on the formulation of the degenerate adiabatic theorem, along with its necessary and sufficient conditions given in [Phys. Rev. A 85, 062111 (2012)]. We start with the adiabatic approximation for degenerate Hamiltonians that paves the way to a clear and rigorous statement of the associated degenerate adiabatic theorem, where the non-abelian geometric phase (Wilczek-Zee phase) plays a central role to its quantitative formulation. We then describe the degenerate adiabatic perturbation theory, whose zeroth order term is the degenerate adiabatic approximation, in its full generality. The parameter in the perturbative power series expansion of the time-dependent wave function is directly associated to the inverse of the time it takes to drive the system from its initial to its final state. With the aid of the degenerate adiabatic perturbation theory we obtain rigorous necessary and sufficient conditions for the validity of the adiabatic theorem of quantum mechanics. Finally, to illustrate the power and wide scope of the methodology, we apply the framework to a degenerate Hamiltonian, whose closed form time-dependent wave function is derived exactly, and also to other non-exactly-solvable Hamiltonians whose solutions are numerically computed.

Probabilistic quantum teleportation via thermal entanglement

We study the probabilistic (conditional) teleportation protocol when the entanglement needed for its implementation is given by thermal entanglement, i.e., when the entangled resource connecting Alice and Bob is an entangled mixed state described by the canonical ensemble density matrix. Specifically, the entangled resource we employ here is given by two interacting spin-