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arXiv quant-ph/0207024quant-ph
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纠缠见证的几何结构和纠缠的局域检测

The geometry of entanglement witnesses and local detection of entanglement

Arthur O. Pittenger, Morton H. Rubin

发表信息：
Phys. Rev. A 67, 012327 (2003).
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摘要：

令$H^{[ N]}=H^{[ d_{1}]}\otimes ... \otimes H^{[ d_{n}]}$是希尔伯特空间的张量积，并且令$τ_{0}$是在希尔伯特-施密特范数下最接近纠缠态$ρ_{0}$的可分态。令$\tildeτ_{0}$表示沿着从$I/N$到$ρ_{0}$的线段与$ρ_{0}$最接近的可分离状态，其中$I$是单位矩阵。在[pitrubmat]之后，可以根据$I, τ_{0}$和$\tildeτ_{0}$构造检测$ρ_{0}$纠缠的见证$W_{0}$。如果已知$τ_{0}$和$\tildeτ_{0}$作为可分离投影的凸组合的表示，则可以通过局部测量来检测$ρ_{0}$的纠缠。Gühne \textit{et. al.}在[bruss 1]中获得了一类两个量子比特状态所需的最小测量设置数。我们利用几何方法将其结果推广到$d$为素数时相应的两量子点情形，并得到了最少的测量设置数。在这些特殊的两方病例中，$τ_{0}=\tildeτ_{0}$。我们说明了我们的一般方法与两个参数的家庭的三个量子比特束缚纠缠态的$τ_{0} \neq \tildeτ_{0}$，我们表明我们的方法适用于$n$量子比特。在[pitt]中，我们详细阐述了可分离态的“远面”相对于从正交不可扩展乘积基构造的束缚纠缠态$ρ_{0}$的作用。在本文中的几何方法导致的纠缠见证表示为一个常数倍$I$和可分离的密度$μ_{0}$的远面从$ρ_{0}$。直到归一化，这与[bruss 1]中针对那里分析的特定示例获得的证据一致。

英文摘要：

Let $H^{[ N]}=H^{[ d_{1}]}\otimes ... \otimes H^{[ d_{n}]}$ be a tensor product of Hilbert spaces and let $τ_{0}$ be the closest separable state in the Hilbert-Schmidt norm to an entangled state $ρ_{0}$. Let $\tildeτ_{0}$ denote the closest separable state to $ρ_{0}$ along the line segment from $I/N$ to $ρ_{0}$ where $I$ is the identity matrix. Following [pitrubmat] a witness $W_{0}$ detecting the entanglement of $ρ_{0}$ can be constructed in terms of $I, τ_{0}$ and $\tildeτ_{0}$. If representations of $τ_{0}$ and $\tildeτ_{0}$ as convex combinations of separable projections are known, then the entanglement of $ρ_{0}$ can be detected by local measurements. Gühne \textit{et. al.} in [bruss1] obtain the minimum number of measurement settings required for a class of two qubit states. We use our geometric approach to generalize their result to the corresponding two qudit case when $d$ is prime and obtain the minimum number of measurement settings. In those particular bipartite cases, $τ_{0}=\tildeτ_{0}$. We illustrate our general approach with a two parameter family of three qubit bound entangled states for which $τ_{0} \neq \tildeτ_{0}$ and we show our approach works for $n$ qubits. In [pitt] we elaborated on the role of a ``far face'' of the separable states relative to a bound entangled state $ρ_{0}$ constructed from an orthogonal unextendible product base. In this paper the geometric approach leads to an entanglement witness expressible in terms of a constant times $I$ and a separable density $μ_{0}$ on the far face from $ρ_{0}$. Up to a normalization this coincides with the witness obtained in [bruss1] for the particular example analyzed there.

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