标题： 高斯2设计的奇特不存在性 显示英文标题

标题： The curious nonexistence of Gaussian 2-designs

摘要： 2-设计——其二阶矩等于均匀哈尔测度集的量子纯态集合——是量子信息科学中多个任务的最优解，尤其是在状态和过程层析成像中。 我们证明高斯态不能形成连续变量（量子光学）希尔伯特空间L2(R)的2-设计。 这令人惊讶，因为仿射辛群HWSp（高斯态的自然对称群）在两个副本的对称子空间上是不可约的。 在有限维希尔伯特空间中，不可约性保证了HWSp协变的集合（如素数维度中的相互正交基）总是2-设计。 这一性质在连续变量中被违反，原因微妙：高斯态的（定义良好的）HWSp不变集合没有平均态，因为平均积分不收敛。 事实上，没有任何高斯集合在精确的意义上接近成为2-设计。 离散与连续量子力学之间的这一惊人差异对光学状态和过程层析成像有重要影响。 显示英文摘要

摘要： 2-designs -- ensembles of quantum pure states whose 2nd moments equal those of the uniform Haar ensemble -- are optimal solutions for several tasks in quantum information science, especially state and process tomography. We show that Gaussian states cannot form a 2-design for the continuous-variable (quantum optical) Hilbert space L2(R). This is surprising because the affine symplectic group HWSp (the natural symmetry group of Gaussian states) is irreducible on the symmetric subspace of two copies. In finite dimensional Hilbert spaces, irreducibility guarantees that HWSp-covariant ensembles (such as mutually unbiased bases in prime dimensions) are always 2-designs. This property is violated by continuous variables, for a subtle reason: the (well-defined) HWSp-invariant ensemble of Gaussian states does not have an average state because the averaging integral does not converge. In fact, no Gaussian ensemble is even close (in a precise sense) to being a 2-design. This surprising difference between discrete and continuous quantum mechanics has important implications for optical state and process tomography.