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- W2919219669 abstract "We study the set of two-qubit pure states with real amplitudes and their geometrical representation onto a real projective space. In this representation, we show that the maximally entangled states --those locally equivalent to the Bell States --form two disjoint circles perpendicular to each other. We also show that, taking the natural Riemannian metric on this space, the set of states connected by local gates are equidistant to this pair of circles. Moreover, the unentangled, or so called product states, are $pi/4$ units away to the maximally entangled states. This is, the unentangled states are the farthest away to the maximally entangled states. In this way, if we define two states to be equivalent if they are connected by local gates, we have that there are as many equivalent classes as points in the interval $[0,pi/4]$ with the point $0$ corresponding to the maximally entangled states. The point $pi/4$ corresponds to the unentangled states which geometrically are described by a Klein bottle. Finally, for every $0< d < pi/4$ the point $d$ corresponds to a disjoint pair of Klein bottles. We also show that if a state is $d$ units away from the maximally entangled states, then its entanglement entropy is $S(d) = 1- log_2 sqrt{frac{(1+sin 2 d)^{1+sin 2 d}}{(1-sin 2 d)^{-1+sin 2 d}}}$. Finally, we also show how this geometrical interpretation allows to clearly see the known result that any pair of two-qubit states with real amplitudes can be connected with a circuit that only has single-qubit gates and one controlled-Z gate." @default.
- W2919219669 created "2019-03-11" @default.
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- W2919219669 date "2019-03-05" @default.
- W2919219669 modified "2023-10-18" @default.
- W2919219669 title "Upgrading the Bloch Sphere: Projective Space Foliated by Klein Bottles as a Geometrical Representation of Two-Qubit States and Their Entanglement" @default.
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