Weilenmann M, Dive B, Trillo D, Aguilar E A, Navascués M
Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmangasse 3, 1090 Vienna, Austria.
Phys Rev Lett. 2020 May 22;124(20):200502. doi: 10.1103/PhysRevLett.124.200502.
One of the most widespread methods to determine if a quantum state is entangled, or to quantify its entanglement dimensionality, is by measuring its fidelity with respect to a pure state. In this Letter, we find a large class of states whose entanglement cannot be detected in this manner; we call them unfaithful. We find that unfaithful states are ubiquitous in information theory. For small dimensions, we check numerically that most bipartite states are both entangled and unfaithful. Similarly, numerical searches in low dimensions show that most pure entangled states remain entangled but become unfaithful when a certain amount of white noise is added. We also find that faithfulness can be self-activated, i.e., there exist instances of unfaithful states whose tensor powers are faithful. To explore how the fidelity approach limits the quantification of entanglement dimensionality, we generalize the notion of an unfaithful state to that of a D unfaithful state, one that cannot be certified as D-dimensionally entangled by measuring its fidelity with respect to a pure state. For describing such states, we additionally introduce a hierarchy of semidefinite programming relaxations that fully characterizes the set of states of Schmidt rank at most D.
确定一个量子态是否纠缠，或者量化其纠缠维度，最广泛使用的方法之一是测量它相对于纯态的保真度。在本信函中，我们发现了一大类态，其纠缠无法通过这种方式检测到；我们称它们为非忠实态。我们发现非忠实态在信息论中无处不在。对于低维度情况，我们通过数值验证发现大多数二分态既是纠缠的又是非忠实的。类似地，低维度的数值搜索表明，大多数纯纠缠态在添加一定量的白噪声后仍保持纠缠但变为非忠实态。我们还发现忠实性可以自我激活，即存在非忠实态的张量幂是忠实态的情况。为了探究保真度方法如何限制纠缠维度的量化，我们将非忠实态的概念推广到(D)非忠实态，即通过测量其相对于纯态的保真度无法被认证为(D)维纠缠的态。为了描述此类态，我们额外引入了一系列半定规划松弛，它们完全刻画了施密特秩至多为(D)的态集。
