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arXiv:1302.0709 · Journal of Physics A Mathematical and Theoretical

Area law for random graph states

Benoı̂t Collins, Ion Nechita, Karol Życzkowski

Published Jul 10, 2013• 13 citations
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Abstract

Random pure states of multi-partite quantum systems, associated with arbitrary graphs, are investigated. Each vertex of the graph represents a generic interaction between subsystems, described by a random unitary matrix distributed according to the Haar measure, while each edge of the graph represents a bipartite, maximally entangled state. For any splitting of the graph into two parts we consider the corresponding partition of the quantum system and compute the average entropy of entanglement. First, in the special case where the partition does not cross any vertex of the graph, we show that the area law is satisfied exactly. In the general case, we show that the entropy of entanglement obeys an area law on average, this time with a correction term that depends on the topologies of the graph and of the partition. The results obtained are applied to the problem of distribution of quantum entanglement in a quantum network with prescribed topology.

DOI
10.1088/1751-8113/46/30/305302

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Random pure states of multi-partite quantum systems, associated with arbitrary graphs, are investigated. Each vertex of the graph represents a generic interaction between subsystems, described by a random unitary matrix distributed according to the Haar measure, while each edge of the graph represents a bipartite, maximally entangled state. For any splitting of the graph into two parts we consider the corresponding partition of the quantum system and compute the average entropy of entanglement. First, in the special case where the partition does not cross any vertex of the graph, we show that the area law is satisfied exactly. In the general case, we show that the entropy of entanglement obeys an area law on average, this time with a correction term that depends on the topologies of the graph and of the partition. The results obtained are applied to the problem of distribution of quantum entanglement in a quantum network with prescribed topology.
For any graph consisting of k vertices and m edges we construct an ensemble of random pure quantum states which describe a system composed of 2 m subsystems. Each edge of the graph represents a bipartite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated with a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze the statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko–Pastur) distribution. We derive a discrete family of generalized, Fuss–Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.
In this paper we continue and extend the investigations of the ensembles of random physical states introduced in Hamma et al. [Phys. Rev. Lett. 109, 040502 (2012)]. These ensembles are constructed by finite-length random quantum circuits (RQC) acting on the (hyper)edges of an underlying (hyper)graph structure. The latter encodes for the locality structure associated with finite-time quantum evolutions generated by physical, i.e., local, Hamiltonians. Our goal is to analyze physical properties of typical states in these ensembles; in particular here we focus on proxies of quantum entanglement as purity and $\ensuremath{\alpha}$-Renyi entropies. The problem is formulated in terms of matrix elements of superoperators which depend on the graph structure, choice of probability measure over the local unitaries, and circuit length. In the $\ensuremath{\alpha}=2$ case these superoperators act on a restricted multiqubit space generated by permutation operators associated to the subsets of vertices of the graph. For permutationally invariant interactions the dynamics can be further restricted to an exponentially smaller subspace. We consider different families of RQCs and study their typical entanglement properties for finite time as well as their asymptotic behavior. We find that area law holds in average and that the volume law is a typical property (that is, it holds in average and the fluctuations around the average are vanishing for the large system) of physical states. The area law arises when the evolution time is O(1)O(1)O(1) with respect to the size LLL of the system, while the volume law arises as is typical when the evolution time scales like O(L)O(L)O(L).
We study the ground-state entanglement entropy of free fermions on random graphs. We first establish a general criterion for the volume law on random graphs, based on the spectral characteristics of the Hamiltonian. We then apply this criterion to the Erdős--Rényi random graph, where each of the possible edges is present independently with some probability. Using random matrix theory and asymptotic freeness, we show that the ground-state entanglement entropy obeys an exact volume law in Erdős--Rényi graphs in the thermodynamic limit, with a universal coefficient that is independent of the edge probability of the graph. This coefficient is confirmed numerically to take the value approximately 0.38630.38630.3863 nats, strictly below the Page value. The volume law therefore reflects the absence of geometric locality in the random graph.
We study the average bipartite entanglement entropy of Haar-random pure states in quantum many-body systems with global SU(2)\mathrm{SU}(2)SU(2) symmetry, constrained to fixed total spin JJJ and magnetization Jz=0J_z = 0Jz​=0. Focusing on spin-12\tfrac1221​ lattices and subsystem fractions f<12f < \frac{1}{2}f<21​, we derive a asymptotic expression for the average entanglement entropy up to constant order in the system volume VVV. In addition to the expected leading volume law term, we prove the existence of a 12log⁡V\frac{1}{2}\log V21​logV finite-size correction resulting from the scaling of the Clebsch-Gordon coefficients and compute explicitly the O(1)O(1)O(1) contribution reflecting angular-momentum coupling within magnetization blocks. Our analysis uses features of random matrix ensembles and provides a fully analytical treatment for arbitrary spin densities, thereby extending Page type results to non-Abelian sectors and clarifying how SU(2)\mathrm{SU}(2)SU(2) symmetry shapes average entanglement.
We introduce a mapping between graphs and pure quantum bipartite states and show that the associated entanglement entropy conveys non-trivial information about the structure of the graph. Our primary goal is to investigate the family of random graphs known as complex networks. In the case of classical random graphs, we derive an analytic expression for the averaged entanglement entropy while for general complex networks we rely on numerics. For a large number of nodes n we find a scaling where both the prefactor c and the sub-leading O (1) term g e are characteristic of the different classes of complex networks. In particular, g e encodes topological features of the graphs and is named network topological entropy. Our results suggest that quantum entanglement may provide a powerful tool for the analysis of large complex networks with non-trivial topological properties.
A standard approach to generate random pure quantum states relies on sampling from the Haar measure. However, the entanglement properties of such states present a fundamental challenge for their general applicability. Here, we introduce the σσσ-ensembles $\unicode{x2013}$ a family of random quantum states with only a single control parameter. Crucially, these states are designed such that they can be tuned between volume-law and area-law behavior, which has been a major obstacle thus far. We construct representatives of this ensemble by imposing a probability distribution on the eigenvalues of the successive subsystems, and subsequently reconstructing a compatible global state using the matrix product state (MPS) formalism. Due to their area-law entanglement, our approach circumvents the intractability of Haar-random pure states in classical simulations of quantum systems and is more representative of typical Hamiltonian ground states.