Hannover 2013 – wissenschaftliches Programm
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Q: Fachverband Quantenoptik und Photonik
Q 67: Quantum information: Concepts and methods V
Q 67.3: Vortrag
Freitag, 22. März 2013, 14:30–14:45, E 214
Stabilizer states are spherical 3-designs – with applications to quantum state discrimination — •Richard Kueng and David Gross — Universität Freiburg
A complex spherical k-design is a configuration of vectors which is “evenly distributed” on a sphere in the sense that it reproduces Haar measure up to kth moments. Here, we show that the set of all n-qubit stabilizer states forms a complex spherical 3-design in dimension 2n. Stabilizer states had previously only been known to constitute 2-designs. The problem is reduced to the task of counting the number of stabilizer states with pre-described overlap with respect to a reference state. This, in turn, reduces to a counting problem in discrete symplectic vector spaces for which we find a simple formula.
We use the finding to answer an open problem posed by in [Matthews, Wehner, Winter, CMP 291 (2008)]: There, the loss of distinguishability suffered by quantum states as a result of a POVM measurement was analyzed. It had been shown that 4-designs (seen as POVMs) perform almost optimally, while 2-designs fall significantly short of this. The performance of 3-designs was left open. Using our explicit example, we find that, unfortunately, 3-designs do not outperform 2-designs.