量子通信过程经典模拟的最优性充要条件
Necessary and sufficient optimality conditions for classical simulations of quantum communication processes
摘要：
我们考虑的过程包括准备，通过量子信道的传输，和随后的测量的量子态。信道的通信复杂度是经典仿真所需的最小经典通信量，最近，我们把这个量的计算归结为一个带线性约束的凸最小化问题。每个解决方案的约束提供了一个上限的通信复杂度。在本文中，我们推导出原来的一个对偶最大化问题。对偶约束的可行点是不等式，给出了通信复杂度的下界，并举例说明。这两个问题的最优值是相等的（零对偶差距）。利用这一性质，我们用一组等式和不等式给出了最优性的充分必要条件。我们使用这些条件和两个合理的，但未经证实的假设，推导出的下限$n 2^{n-1}$的容量等于$n$量子比特的无噪声量子信道。在最近关于量子态真实性的争论中，这个下限可能会产生有趣的结果。
英文摘要：
We consider the process consisting of preparation, transmission through a quantum channel, and subsequent measurement of quantum states. The communication complexity of the channel is the minimal amount of classical communication required for classically simulating it. Recently, we reduced the computation of this quantity to a convex minimization problem with linear constraints. Every solution of the constraints provides an upper bound on the communication complexity. In this paper, we derive the dual maximization problem of the original one. The feasible points of the dual constraints, which are inequalities, give lower bounds on the communication complexity, as illustrated with an example. The optimal values of the two problems turn out to be equal (zero duality gap). By this property, we provide necessary and sufficient conditions for optimality in terms of a set of equalities and inequalities. We use these conditions and two reasonable but unproven hypotheses to derive the lower bound $n 2^{n-1}$ for a noiseless quantum channel with capacity equal to $n$ qubits. This lower bound can have interesting consequences in the context of the recent debate on the reality of the quantum state.
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