Lorentz正映射的重合性
Concurrence of Lorentz-positive maps
摘要：
设L_n是n维Lorentz锥。从R^m到R^n的线性映射M称为Lorentz-正的，如果M[L_m]包含在L_n中.本文将最初用来量化二分密度矩阵纠缠的并发度概念推广到Lorentz正映射，并给出了并发度的一个显式公式，从而得到了以2 × 2复厄米矩阵为输入的任意正算子的并发度公式，进而得到了秩为2的任意二分密度矩阵的并发度公式.即，设P：H（2）\to H（d）为正算子，λ_1，.，λ_4是谱束σ_2（P（X））- λdet X的广义特征值，按降序排列，其中σ_2是谱的第二对称函数。则并发性由表达式C（P;X）= 2 \sqrt{σ_2(P(X)) - λ_2 det X}给出。作为应用，我们计算了所有2边图的密度矩阵的并发性。类似的结果适用于我们称之为I-保真度的函数，其中第二大广义特征值λ_2被最小特征值λ_4代替。
英文摘要：
Let L_n be the n-dimensional Lorentz cone. A linear map M from R^m to R^n is called Lorentz-positive if M[L_m] is contained in L_n. We extend the notion of concurrence, which was initially introduced to quantify the entanglement of bipartite density matrices, to Lorentz-positive maps and provide an explicite formula for it. This allows us to obtain formulae for the concurrence of arbitrary positive operators taking 2 x 2 complex hermitian matrices as input and consequently of arbitrary bipartite density matrices of rank 2. Namely, let P: H(2) \to H(d) be a positive operator, and let λ_1,...,λ_4 be the generalized eigenvalues of the pencil σ_2(P(X)) - λdet X, in decreasing order, where σ_2 is the second symmetric function of the spectrum. Then the concurrence is given by the expression C(P;X) = 2\sqrt{σ_2(P(X)) - λ_2 det X}. As an application, we compute the concurrences of the density matrices of all graphs with 2 edges. Similar results apply for a function which we call I-fidelity, with the second largest generalized eigenvalue λ_2 replaced by the smallest eigenvalue λ_4.
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