1,709 research outputs found
Gyrations: The Missing Link Between Classical Mechanics with its Underlying Euclidean Geometry and Relativistic Mechanics with its Underlying Hyperbolic Geometry
Being neither commutative nor associative, Einstein velocity addition of
relativistically admissible velocities gives rise to gyrations. Gyrations, in
turn, measure the extent to which Einstein addition deviates from commutativity
and from associativity. Gyrations are geometric automorphisms abstracted from
the relativistic mechanical effect known as Thomas precession
An Introduction to Hyperbolic Barycentric Coordinates and their Applications
Barycentric coordinates are commonly used in Euclidean geometry. The
adaptation of barycentric coordinates for use in hyperbolic geometry gives rise
to hyperbolic barycentric coordinates, known as gyrobarycentric coordinates.
The aim of this article is to present the road from Einstein's velocity
addition law of relativistically admissible velocities to hyperbolic
barycentric coordinates along with applications.Comment: 66 pages, 3 figure
Distances of qubit density matrices on Bloch sphere
We recall the Einstein velocity addition on the open unit ball \B of
and its algebraic structure, called the Einstein gyrogroup. We
establish an isomorphism between the Einstein gyrogroup on \B and the set of
all qubit density matrices representing mixed states endowed with an
appropriate addition. Our main result establishes a relation between the trace
metric for the qubit density matrices and the rapidity metric on the Einstein
gyrogroup \B.Comment: I thank to my supervisor, Jimmie Lawson. This was accepted in Journal
of Mathematical Physics at September 26, 201
Geometric interpretation for A-fidelity and its relation with Bures fidelity
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 -state quantum system is also discussed.Comment: 4 pages, 1 figure. Phys. Rev.
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