Hoshang Heydari

Quantum speed limits and optimal Hamiltonians for driven systems in mixed states

Inequalities of Mandelstam–Tamm (MT) and Margolus–Levitin (ML) type provide lower bounds on the time that it takes for a quantum system to evolve from one state into another. Knowledge of such bounds, called quantum speed limits, is of utmost importance in virtually all areas of physics, where determination of the minimum time required for a quantum process is of interest. Most MT and ML inequalities found in the literature have been derived from growth estimates for the Bures length, which is a statistical distance measure. In this paper we derive such inequalities by differential geometric methods, and we compare the quantum speed limits obtained with those involving the Bures length. We also characterize the Hamiltonians which optimize the evolution time for generic finite-level quantum systems.

Geometric uncertainty relation for mixed quantum states

In this paper we use symplectic reduction in an Uhlmann bundle to construct a principal fiber bundle over a general space of unitarily equivalent mixed quantum states. The bundle, which generalizes the Hopf bundle for pure states, gives in a canonical way rise to a Riemannian metric and a symplectic structure on the base space. With these we derive a geometric uncertainty relation for observables acting on quantum systems in mixed states. We also give a geometric proof of the classical Robertson-Schrodinger uncertainty relation, and we compare the two. They turn out not to be equivalent, because of the multiple dimensions of the gauge group for general mixed states. We give examples of observables for which the geometric relation provides a stronger estimate than that of Robertson and Schrodinger, and vice versa.

General pure multipartite entangled states and the Segre variety

We construct a measure of entanglement by generalizing the quadric polynomial of the Segre variety for general multipartite states. This measure of entanglement works for any pure state and vanishes on multipartite product states. We give explicit expressions for general pure three-partite and four-partite states, and compare our measure of entanglement with the generalized concurrence.

Operational geometric phase for mixed quantum states

Geometric phase has found a broad spectrum of applications in both classical and quantum physics, such as condensed matter and quantum computation. In this paper we introduce an operational geometric phase for mixed quantum states, based on spectral weighted traces of holonomies, and we prove that it generalizes the standard definition of geometric phase for mixed states, which is based on quantum interferometry. We also introduce higher order geometric phases, and prove that under a fairly weak, generically satisfied, requirement, there is always a well-defined geometric phase of some order. Our approach applies to general unitary evolutions of both nondegenerate and degenerate mixed states. Moreover, since we provide an explicit formula for the geometric phase that can be easily implemented, it is particularly well suited for computations in quantum physics.

Dynamic Distance Measure on Spaces of Isospectral Mixed Quantum States

Distance measures are used to quantify the extent to which information is preserved or altered by quantum processes, and thus are indispensable tools in quantum information and quantum computing. In this paper we propose a new distance measure for mixed quantum states, which we call the dynamic distance measure, and we show that it is a proper distance measure. The dynamic distance measure is defined in terms of a measurable quantity, which makes it suitable for applications. In a final section we compare the dynamic distance measure with the well-known Bures distance measure.

Geometrical Structure of Entangled States and the Secant Variety

We investigate the geometrical structure of entangled and separable bipartite and multipartite states based on the secant variety of the Segre variety. We show that the Segre variety coincides with the space of separable multipartite state and the higher secant variety of the Segre variety coincides with the space of entangled multipartite states.

A geometric framework for mixed quantum states based on a Kähler structure

In this paper we introduce a geometric framework for mixed quantum states based on a Kahler structure. The geometric framework includes a symplectic form, an almost complex structure, and a Riemannian metric that characterize the space of mixed quantum states. We argue that the almost complex structure is integrable. We also in detail discuss a visualizing application of this geometric framework by deriving a geometric uncertainty relation for mixed quantum states. The framework is computationally effective and it provides us with a better understanding of general quantum mechanical systems.

Geometry of quantum evolution for mixed quantum states

The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this work we discuss a geometric formulation of mixed quantum states represented by density operators. Our formulation is based on principal fiber bundles and purifications of quantum states. In our construction, the Riemannian metric and symplectic form on the total space are induced from the real and imaginary parts of the Hilbert-Schmidt Hermitian inner product, and we define a mechanical connection in terms of a locked inertia tensor and moment map. We also discuss some applications of our geometric framework.

Topological quantum gate entanglers for a multi-qubit state

We establish a relation between topological and quantum entanglement for a multi-qubit state by considering the unitary representations of the Artin braid group. We construct topological operators that can entangle a multi-qubit state. In particular we construct operators that create quantum entanglement for multi-qubit states based on the Segre ideal of complex multi-projective space. We also discuss in detail and construct these operators for two-qubit and three-qubit states.

Quantum correlation and Grothendieck's constant

The best upper bound for the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality was first derived by Tsirelson. For increasing number of ±1 valued observables on both sites of the correlation experiment, Tsirelson obtained the Grothendieck’s constant (KG ≈ 1.73±0.06) as a limit for the maximal violation. In this paper, we construct a generalization of the CHSH inequality with four ±1 valued observables on both sites of a correlation experiment and show that the quantum violation approaching 1.58. Moreover, we estimate the maximal quantum violation of a correlation experiment for large and equal number of ±1 valued observables on both sites. In this case, the maximal quantum violation converges to √ 3 ≈ 1.73 for very large n, which coincides with the approximate value of Grothendieck’s constant.The best upper bound for the violation of the Clauser-Home-Shimony-Holt (CHSH) inequality was first derived by Tsirelson. For increasing number of ±1 valued observables on both sites of the correlation experiment, Tsirelson obtained the Grothendiecks constant (K G ≈ 1.73 ± 0.06) as a limit for the maximal violation. In this paper, we construct a generalization of the CHSH inequality with four ±1 valued observables on both sites of a correlation experiment and show that the quantum violation approaches 1.58. Moreover, we estimate the maximal quantum violation of a correlation experiment for large and equal number of ±1 valued observables on both sites. In this case, the maximal quantum violation converges to √3 ≈ 1.73 for very large n, which coincides with the approximate value of Grothendiecks constant.