半个量子比特：一种代数分数化
Half a qubit: an algebraic fractionalization
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中文总结 AI 辅助
该研究提出将Székely的半硬币嵌入克雷恩空间的代数分数化方法，构造半量子比特，证明其融合可形成量子比特，推广到1/n量子比特，可通过现有实验方法验证。
中文摘要 AI 辅助
将量子两能级系统分数化通常与基于马约拉纳费米子对的编码相关，这是一种操作性分数化。我们展示了一种替代的代数分数化，方法是将Székely的经典“半硬币”嵌入非埃尔米特克雷恩（Krein）空间中。$(q+pz)^{1/2}$的系数定义了一个带符号序列和一个归一化的非埃尔米特双正交算子，该算子描述了一个双正交半量子比特。我们证明，两个这样的对象通过带符号范德蒙德卷积融合成任意纯量子比特，其中集合$N\ge2$的向量在克雷恩空间中是零的。半量子比特的$L_1$范数以闭式形式给出：$\lVert p\rVert_1 = 2\sqrt{q}-\sqrt{q-p}$。其$L_1$范数随偏置单调增加，且恰好在无偏点$p=q=1/2$处达到其上确界$\sqrt{2}$。我们确定了两个结构结果：第一，宇称和$\eta$-度量生成了一个特殊的可交换$\mathbb Z_2\times\mathbb Z_2$子群；第二，我们发现$\eta$-度量阻碍了宇称的任何局部$\eta$-自伴伙伴，因此半量子比特带有一个$\mathbb{Z}_2$可观测量，但没有局部$SU(2)$。完整的泡利代数仅在融合时出现。我们随后证明，该构造在福克（Fock）基的截断下依然成立：融合后的量子比特在每个截断处都是精确的，且范德蒙德抵消在符号加权的光子数统计中可见，可利用现有的腔和囚禁离子态合成方法进行测试。最后，我们将这种代数分数化推广到$1/n$量子比特，可通过将平方根替换为$n$次方根来实现。
英文摘要
Fractionalizing a quantum two-level system is usually associated with encodings based on pairs of Majorana fermions---an operational fractionalization. We show an alternative algebraic fractionalization by embedding Székely's classical ``half-coin'' into a non-Hermitian Krein space. The coefficients of $(q+pz)^{1/2}$ define a signed sequence and a normalized, non-Hermitian biorthogonal operator describing a biorthogonal half-qubit. We prove that two such objects fuse into an arbitrary pure qubit through the signed Vandermonde convolution that the collective $N\ge2$ vectors are null in Krein space. $L_1$ norm of the half-qubit follows in closed form, $\lVert p\rVert_1 = 2\sqrt{q}-\sqrt{q-p}$. Its $L_1$ norm increases monotonically with the bias and attains its supremum $\sqrt{2}$ precisely at the unbiased point $p=q=1/2$. Interestingly, we identify two structural results as follows. First, number parity and the $η$-metric generate a distinguished commuting $\mathbb Z_2\times\mathbb Z_2$ subgroup. Second, we find the $η$-metric obstructs any local $η$-self-adjoint partner of the parity, so a half-qubit carries a $\mathbb{Z}_2$ observable but no local $SU(2)$. The full Pauli algebra emerges only upon fusion. We then show that the construction survives truncation of the Fock basis: the fused qubit is exact at every cutoff, and the Vandermonde cancellation is visible in sign-weighted photon-number statistics, and can be tested using existing cavity and trapped-ion state-synthesis methods. Finally, we generalize this algebraic fractionalization to a $1/n$-qubit, which can be achieved by replacing the square root with an $n$th root.
