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Register a new userGeometric interpretation for A-fidelity and its relation with Bures fidelity
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A geometric interpretation for the A-fidelity between two states of a qubit system is presented, which leads to an upper bound of the Bures fidelity. The metrics defined based on the A-fidelity are studied by numerical method. An alternative generalization of the A-fidelity, which has the same geometric picture, to a $N$-state quantum system is also discussed.
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Bures distance holds a special place among various distance measures due to its several distinguished features and finds applications in diverse problems in quantum information theory. It is related to fidelity and, among other things, it serves as a bona fide measure for quantifying the separability of quantum states. In this work, we calculate exact analytical results for the mean root fidelity and mean square Bures distance between a fixed density matrix and a random density matrix, and also between two random density matrices. In the course of derivation, we also obtain spectral density for product of above pairs of density matrices. We corroborate our analytical results using Monte Carlo simulations. Moreover, we compare these results with the mean square Bures distance between reduced density matrices generated using coupled kicked tops and find very good agreement.
Motivated by recent development in quantum fidelity and fidelity susceptibility, we study relations among Lie algebra, fidelity susceptibility and quantum phase transition for a two-state system and the Lipkin-Meshkov-Glick model. We get the fidelity susceptibility for SU(2) and SU(1,1) algebraic structure models. From this relation, the validity of the fidelity susceptibility to signal for the quantum phase transition is also verified in these two systems. At the same time, we obtain the geometric phase in these two systems in the process of calculating the fidelity susceptibility. In addition, the new method of calculating fidelity susceptibility has been applied to explore the two-dimensional XXZ model and the Bose-Einstein condensate(BEC).
We derive several bounds on fidelity between quantum states. In particular we show that fidelity is bounded from above by a simple to compute quantity we call super--fidelity. It is analogous to another quantity called sub--fidelity. For any two states of a two--dimensional quantum system (N=2) all three quantities coincide. We demonstrate that sub-- and super--fidelity are concave functions. We also show that super--fidelity is super--multiplicative while sub--fidelity is sub--multiplicative and design feasible schemes to measure these quantities in an experiment. Super--fidelity can be used to define a distance between quantum states. With respect to this metric the set of quantum states forms a part of a $N^2-1$ dimensional hypersphere.
The bounds of concurrence in [F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98 (2007) 140505] and [C. Zhang textit{et. al.}, Phys. Rev. A 78 (2008) 042308] are proved by using two properties of the fidelity. In two-qubit systems, for a given value of concurrence, the states achieving the maximal upper bound, the minimal lower bound or the maximal difference upper-lower bound are determined analytically.
Geometric phases are robust against certain types of local noises, and thus provide a promising way towards high-fidelity quantum gates. However, comparing with the dynamical ones, previous implementations of nonadiabatic geometric quantum gates usually require longer evolution time, due to the needed longer evolution path. Here, we propose a scheme to realize nonadiabatic geometric quantum gates with short paths based on simple pulse control techniques, instead of deliberated pulse control in previous investigations, which can thus further suppress the influence from the environment induced noises. Specifically, we illustrate the idea on a superconducting quantum circuit, which is one of the most promising platforms for realizing practical quantum computer. As the current scheme shortens the geometric evolution path, we can obtain ultra-high gate fidelity, especially for the two-qubit gate case, as verified by our numerical simulation. Therefore, our protocol suggests a promising way towards high-fidelity and roust quantum computation on a solid-state quantum system.
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