【摘 要】
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We report a surprising distinction between measurements in finite and infinite dimensional Hilbert spaces.It is based on the observation that the overlaps of arbitrary Gaussian states in the number ba
【机 构】
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University of Tokyo, Japan
【出 处】
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The XXIX International Colloquium on Group-Theoretical Metho
论文部分内容阅读
We report a surprising distinction between measurements in finite and infinite dimensional Hilbert spaces.It is based on the observation that the overlaps of arbitrary Gaussian states in the number basis are strictly decreasing functions of excitation number,and therefore no convex combination thereof can be proportional to a projection operator.We connect this observation with the construction of t-designs,important ensembles of states that reproduce the t-th moments of the unitarily uniform ensemble.We show that despite the fact that the uniform subensemble of unsqueezed Gaussians forms a 1-design,and despite satisfying the same necessary and sufficient representation theoretic property as in the well understood finite dimensional case for the Heisenberg-Weyl and symplectic groups,the uniform ensemble -- indeed,any ensemble -- of all pure Gaussian states in infinite dimensions cannot comprise a 2-design in this way.This has important consequences for quantum optical tomography,where 2-designs are powerful tools.
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