Phase transitions for random states and a semi-circle law for the partial transpose

Abstract

For a system of N identical particles in a random pure state, there is a threshold k_0 = k_0(N) ~ 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.

5 pages, 2 figures. This short note contains a high-level overview of two long and technical papers, arXiv:1011.0275 and arXiv:1106.2264. Version 2: unchanged results, editorial changes, added reference, close to the published article

References in corpus (3)

• Aspects of generic entanglement
• Detecting entanglement of random states with an entanglement witness
• On the comparison of volumes of quantum states

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• Entanglement thresholds for random induced states
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• Eigenvalue Distributions of Reduced Density Matrices
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• Random tensor networks with nontrivial links
• Entanglement transitions as a probe of quasiparticles and quantum thermalization
• Fundamental Limitation on the Detectability of Entanglement
• Fraction of isospectral states exhibiting quantum correlations
• The absolute positive partial transpose property for random induced states
• Observation of entanglement negativity transition of pseudo-random mixed states
• Cut-resistant links and multipartite entanglement resistant to particle loss
• Entanglement phase diagrams from partial transpose moments
• Bound entangled states with extremal properties
• Dvoretzky's Theorem and the Complexity of Entanglement Detection
• Symmetry protected entanglement in random mixed states
• Applications of differential geometry and representation theory to description of quantum correlations
• Markov Gap and Bound Entanglement in Haar Random State
• Entanglement transitions induced by large deviations
• Absolutely maximally entangled pure states of multipartite quantum systems
• Entanglement transition in a cluster spin chain coupled with free spins
• Certifying entanglement dimensionality by $k$-reduction moments