Phase transitions for random states and a semicircle law for the partial transpose

For a system of N identical particles in a random pure state, there is a threshold k(0) = k(0)(N) similar to N/5 such that two subsystems of k particles each typically share entanglement if k > k(0), and typically do not share entanglement if k < k(0). By "random" we mean here uniformly distributed on the sphere of the corresponding Hilbert space. The analogous phase transition for the positive partial transpose (PPT) property can be described even more precisely. For example, for N qubits the two subsystems of size k are typically in a PPT state if k < k(1) := N/4 - 1/2 and typically in a non-PPT state if k > k(1). Since, for a given state of the entire system, the induced state of a subsystem is given by the partial trace, the above facts can be rephrased as properties of random induced states. An important step in the analysis depends on identifying the asymptotic spectral density of the partial transposes of such random induced states, a result which is interesting in its own right.

对于处于随机纯态的\(N\)个全同粒子组成的系统，存在一个类似于\(N/5\)的阈值\(k(0)=k(0)(N)\)，使得当\(k > k(0)\)时，每个含\(k\)个粒子的两个子系统通常共享纠缠；而当\(k < k(0)\)时，通常不共享纠缠。这里的“随机”是指在相应希尔伯特空间的球面上均匀分布。正部分转置（PPT）性质的类似相变可以更精确地描述。例如，对于\(N\)个量子比特，如果\(k < k(1):=N/4 - 1/2\)，大小为\(k\)的两个子系统通常处于PPT态；如果\(k > k(1)\)，则通常处于非PPT态。由于对于整个系统的给定状态，子系统的诱导态由偏迹给出，上述事实可以重新表述为随机诱导态的性质。分析中的一个重要步骤取决于确定此类随机诱导态的偏转置的渐近谱密度，这一结果本身就很有趣。