高斯协方差几何的黄金交汇点：扇区各向异性、相对熵响应与局部惩罚

中文摘要 AI 辅助

我们在保持操作意义区分的前提下，比较了居中单模玻色子高斯协方差的四种局部几何。在共同的迹正交归一化下，Bures 和协方差 Fisher--Rao 余度量行列式在 \\(x^2=\varphi\\) 处本质性地交叉。引入常数作用量匹配参数 \\(\eta\\) 后，Bures--Wasserstein 行列式仅在 \\((x,\eta)=(\sqrt\varphi,\sqrt\varphi)\\) 处与它们交汇。行列式密度相等并不意味着张量相等：相对于 Fisher--Rao，交汇处的 Bures 余度量为 \\[ Q_{\rm gold}=\varphi^{-2}\Pi_0+\varphi\Pi_2, \qquad \det Q_{\rm gold}=1, \\] 因此其逆具有精确的标量到权重二二次惩罚比 \\(\varphi^3\\)。归一化的 Bogoliubov--Kubo--Mori 度量提供了独立的相对熵诊断。它在径向上与 Bures 一致，但形状余度量特征值为 \\(4\hbar^2x/\log[(x+1)/(x-1)]\\)；因此它不形成第四个行列式交汇点，并在纯态边界处变得切向刚性。操作上，权重二运动由仅系统的高斯幺正生成，而径向运动需要高斯信道或辅助膨胀。已知的 Uhlmann--Ruan 纯化商实现了局部 Bures 态度量，但显式的 \\(\mathfrak{sp}(4,\mathbb R)\\) 测试表明，未加权的右不变 Frobenius 门范数不能复现黄金惩罚。条件分支选择和有限速率 Bures--Wasserstein 代价分离作为次要陈述呈现，而非动力学相变。

英文摘要

Information distance and circuit complexity are both obtained by minimizing lengths, but they minimize over different objects. We make this distinction explicit for faithful one-mode Gaussian states. First, invariant-form uniqueness implies that no positive-definite quadratic gate cost can be invariant under the full adjoint action of the noncompact symplectic group; a positive Cartan majorant necessarily introduces additional reference data. The Uhlmann purification quotient realizes the Bures metric, and the radial covariance direction requires a system-ancilla coupling because system-only Gaussian unitaries preserve the Williamson eigenvalue. We then minimize fixed right-invariant quadratic norms on the minimal two-mode Gaussian gate algebra \(\mathfrak{sp}(4,\mathbb R)\). For the unweighted Frobenius norm, the quotient coefficients for radial and traceless covariance tangents are $G_0=[\hbar^2(u-1)]^{-1}$ and $G_2=[\hbar^2(3u-1)]^{-1}$, where $u=(2ν/\hbar)^2$. Their ratio does not equal the Bures ratio. The radial coefficient, however, reproduces the Bures value exactly at every $u$; the mismatch is confined to the traceless sector. More generally, a constant block-diagonal two-weight schedule gives $G_0/G_2=1+2(β/α)u/(u-1)$; matching Bures throughout the isotropic family would require the state-dependent relation $β/α=1/u$. At the Bures-Fisher determinant crossing \(u=φ\), pointwise matching is possible only by inserting $β/α=φ^{-1}$. Thus the Bures purification quotient is an exact state-geometric cost, but it is neither an unweighted symplectic gate cost nor a member of this fixed two-weight Nielsen family. The existence of a more general fixed positive gate norm realizing the quotient remains open.

发表机构

• Trinity College Institute of Neuroscience, Trinity College Dublin（都柏林三一学院神经科学研究所）

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