量子态等距映射的稳定性
Stability of isometries of quantum states
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中文总结 AI 辅助
本文证明正迹类锥上近似等距映射可被Wigner对称一致逼近，给出与维度无关的误差界，并揭示无穷维与有限维稳定性本质差异。
中文摘要 AI 辅助
我们证明了在迹度量与Bures度量下，正迹类锥上的满射近似等距映射具有与维度无关的Hyers-Ulam稳定性。此处，$\varepsilon\geq0$ 是保持距离时加性误差的一致上界。这样的近似等距映射可被一个Wigner对称（酉或反酉共轭）一致逼近，在迹度量下误差至多为 $9\varepsilon/2$，在Bures度量下误差至多为 $2\sqrt2\varepsilon$，即使该映射不固定零点。对于固定零点的映射，在$\delta$-满射的假设下（与$\delta$无关），迹度量下的界可改进为 $3\varepsilon$。在无穷维情形下，我们构造了全状态空间（包括混合态）的双射，其在两种度量下的失真趋于零，但与每个Wigner对称保持固定的均匀距离。对于状态空间，稳定性在每个固定有限维度下成立，且具有随$\varepsilon\to0$趋于零的依赖于维度的模量，但不存在可独立于维度选择的此类模量。
英文摘要
We prove dimension-independent Hyers-Ulam stability for surjective approximate isometries of the positive trace-class cone in both the trace and Bures metrics. Here $\varepsilon\geq0$ is a uniform upper bound on the additive error in preserving distances. Such an approximate isometry is uniformly approximated by a Wigner symmetry (a unitary or antiunitary conjugation), with error at most $9\varepsilon/2$ in the trace metric and $2\sqrt2\varepsilon$ in the Bures metric, even if it does not fix zero. For maps fixing zero, the trace bound improves to $3\varepsilon$ even under the hypothesis of $δ$-surjectivity, independently of $δ$. In infinite dimensions, we construct bijections of the full state space, including mixed states, whose distortions tend to zero in both metrics but which stay a fixed uniform distance from every Wigner symmetry. For state spaces, stability holds in each fixed finite dimension with a dimension-dependent modulus tending to zero as $\varepsilon\to0$, but no such modulus can be chosen independently of dimension.
