Operational methods in quantum information theory

这个项目的目的是研究量子信息论中的操作方法。特别地，我们研究了(A)量子熵的新的运算特征，以及(B)量子通道识别问题。这些研究的结果总结如下：(A)设p是Hilbert空间H上的概率测度，其支集是相互非平行的单位向量的可数集。设p^&lt；(N)&gt；是第n个I.I.D.定义的H^&lt；[叉积]n&gt；上的概率测度。P的扩张，并考虑H^&lt；[叉积]n&gt；上的随机向量X(1)，…，X(L^&lt；(N)&gt；)服从p^&lt；(N)&gt；我们给出了随机向量“渐近正交性”的几种定义，并研究了相应的正交性容量，即满足每个正交性准则的所有序列{L_n}_n上的LIM_n_nlog L_n的上确界。在弱正交性条件下，表示VEC…此外，我们还阐明了Hausladen等人的无噪量子信道编码定理，证明了对于与度量p对应的密度算符ρ，其正交性容量等于von Neumann熵。是这种刻画的直接后果。在表示向量X(1)，…，X(L^&lt；(N)&gt；)相互几乎正交的强正交性条件下，证明了正交性容量等于2次量子Renyi熵的一半。(B)量子信道识别问题是这样的：给定一个量子信道的参数族{Γ_θ}_θ，找到估计参数θ真值的最佳策略。我们从非对易统计的角度研究了这个问题。特别是，我们已经演示了这个问题的一个重要方面，如下所示。设Γ_θ是作用于二能级量子系统的各向同性退极化通道，其中参数θ表示退极化的大小。通过使用斯托克斯的参数化，它被表示为(x，y，z)→(θx，θy，θz)。由于映射Γ_θ的完全正性要求，参数θ必须位于闭区间[-1/3，1]内。直到量子系统的第二个扩展H[叉积]H，估计各向同性去极化参数θ的最佳策略如下。对于1/(√&lt；3&gt；)[小于等于]θ[小于等于]1，使用Γ_θ[叉积]Γ_θ，输入最大纠缠态；对于1/3[小于等于]θ[小于等于]1/(√；lt；3；>)，使用Γ_θ[叉积]Γ_θ，输入解缠状态；对于-1/3[小于等于]θ[小于等于]1/3，使用Γ_θ[叉积]ID，输入最大纠缠态。令人惊讶的是，看似同质的去极化通道家族{Γ_θ}_θ涉及一种过渡性行为。较少

The purpose of this project is to investigate operational methods in quantum information theory. In particular, we have studied (a) novel operational characterizations of quantum entropies, and (b) quantum channel identification problem. The results of these studies are summarized as follows.(a) Let p be a probability measure on a Hilbert space H the support of which being a countable set of mutually nonparallel unit vectors. Let p^<(n)> be the probability measure on H^<【cross product】n> defined by the nth i.i.d. extension of p, and consider L^<(n)> random vectors X (1) , ..., X (L^<(n)>) on H^<【cross product】n> which are subjected to p^<(n)>. We have introduced several definitions of "asymptotic orthogonality" for the random vectors and have studied the corresponding orthogonality capacity, i.e., the supremum of lim sup_n log L_n/n over all sequences {L_n}_n that satisfy each orthogonality criterion. Under the weak orthogonality condition that represents the situation in which the vec … More tor X (1) is almost orthogonal to the other vectors, the orthogonality capacity has turned out to be identical to the von Neumann entropy for the density operator ρ that corresponds to the measure p. Moreover we have clarified that the noiseless quantum channel coding theorem by Hausladen et al. is a direct consequence of this characterization. Under the strong orthogonality condition that represents the situation in which the vectors X (1), ..., X (L^<(n)>) are mutually almost orthogonal, on the other hand, the orthogonality capacity has turned out to be identical to half the quantum Renyi entropy of degree 2.(b) A quantum channel identification problem is this : given a parametric family {Γ_θ}_θ of quantum channels, find the best strategy of estimating the true value of the parameter θ. We have studied this problem from a noncommutative statistical point of view. In particular, we have demonstrated a nontrivial aspect of this problem as follows. Let Γ_θ be the isotropic depolarization channel acting on the two-level quantum system, in which the parameter θ represents the magnitude of depolarization. By using the Stokes' parametrization, it is represented as (x, y, z) → (θx, θy, θz). Due to the requirement of complete positivity of the map Γ_θ, the parameter θ must lie in the closed interval [-1/3, 1]. Up to the second extension H 【cross product】 H of the quantum system, the best strategy of estimating the isotropic depolarization parameter θ is the following. For 1/(√<3>)【less than or equal】θ【less than or equal】1, use Γ_θ 【cross product】 Γ_θ and input a maximally entangled state ; For 1/3【less than or equal】θ【less than or equal】1/(√<3>) use Γ_θ 【cross product】 Γ_θ and input a disentangled state ; For-1/3【less than or equal】θ【less than or equal】1/3, use Γ_θ 【cross product】 Id and input a maximally entangled state. It is surprising that the seemingly homogeneous family {Γ_θ}_θ of depolarization channels involves a transitionlike behavior. Less