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Quantifying Quantumness and the Quest for Queens of Quantum

par Daniel Braun - 1er octobre 2010

Toutes les versions de cet article : English , français

What are the "most quantum" states that one can possibly produce and what are their properties ? Experimental progress, which has led to the coherent manipulation of a small number of quantum systems, such as trapped ions, or superconducting qubits, has rendered these questions highly relevant not only for the foundations of physics, but also for practical implementation of quantum information processing. In an article that has now appeared in the New Journal of Physics, we make the notion of "quantumness" precise and quantitative, and use it to identify the "Queens of Quantum", i.e. the "most quantum" states, up to 11-dimensional Hilbert spaces. We find that these states are highly symmetric, beautiful objects, with Majorana representations that coincide in several cases with Platonic bodies.

Key to this development was a generalization of the notion of the most "classical states" - the opposite end of the spectrum of quantumness. Classical states are well known in quantum optics. There, a state is considered "classical" if it is a mixture of coherent states, i.e. states of uncertainty as small as possible as allowed by Heisenberg’s uncertainty principle. In 2008, we extended this notion to finite dimensional systems (O. Giraud, P. Braun, and D. Braun, Phys. Rev. A 78, 042112). We found that these classical states, which are in general a subset of separable (i.e. non-entangled) states, form a convex set. It was then natural to define "quantumness" as a "distance" from the set of classical states, thus generalizing a well-known entanglement measure based on the distance of a state from the set of separable states. In our new article in NJP we show that quantumness defined this way has the desired properties, such as never increasing under classical mixing of states. Thermal states always loose their quantumness beyond a certain critical temperature. And a large amount of "quantumness" can be obtained from "Schroedinger cat states", where widely separated components are superposed. However, the answers to the question of what are the "Queens of Quantum" bear a few surprises. The Schroedinger cats obtained do not necessarily correspond to the largest difference in angular momentum, and not necessarily to equal amplitudes of the superposed states, as states with slightly smaller angular momentum can show larger quantum uncertainty than states with maximal angular momentum. In terms of their Majorana representations, the "Queens of Quantum" states correspond to highly symmetric figures. This translates in particular in 5 and 7 dimensional Hilbert spaces to the Platonic bodies "tetrahedron" and "octahedron", which Plato identified with "fire" and "air". But sometimes a slightly lower symmetry is preferable, such as in the case of 9 dimensions - where the Platonic body (the cube), is NOT realized. Future work will have to identify the physical tasks for which these states are optimal. The "Queens of Quantum" in 5 and 7 dimensions coincide with states known to be optimal for reference frame alignment, and other exciting applications might be uncovered in the future. (Contact : D. Braun, groupe "Quantum Coherence").

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Queen of Quantum

The "Queens of Quantum", the "most quantum" states possible, have now been identified for quantum systems with up to 11-dimensional Hilbert space. For details, see New Journal of Physics 12 063005 (2010)