Proof of Page's conjecture on the average entropy of a subsystem

Quantum Ergodicity in the Many-Body Localization Problem

We generalize Page's result on the entanglement entropy of random pure states to the many-body eigenstates of realistic disordered many-body systems subject to long-range interactions. This extension leads to two principal conclusions: first, for increasing disorder the "shells" of constant energy supporting a system's eigenstates fill only a fraction of its full Fock space and are subject to intrinsic correlations absent in synthetic high-dimensional random lattice systems. Second, in all regimes preceding the many-body localization transition individual eigenstates are thermally distributed over these shells. These results, corroborated by comparison to exact diagonalization for an SYK model, are at variance with the concept of "nonergodic extended states" in many-body systems discussed in the recent literature.

Exact variance of von Neumann entanglement entropy over the Bures-Hall measure

The Bures-Hall distance metric between quantum states is a unique measure that satisfies various useful properties for quantum information processing. In this work, we study the statistical behavior of quantum entanglement over the Bures-Hall ensemble as measured by von Neumann entropy. The average von Neumann entropy over such an ensemble has been recently obtained, whereas the main result of this work is an explicit expression of the corresponding variance that specifies the fluctuation around its average. The starting point of the calculations is the connection between correlation functions of the Bures-Hall ensemble and those of the Cauchy-Laguerre ensemble. The derived variance formula, together with the known mean formula, leads to a simple but accurate Gaussian approximation of the distribution of von Neumann entropy of finite-size systems. This Gaussian approximation is also conjectured to be the limiting distribution for large dimensional systems.

Generic Entanglement Entropy for Quantum States with Symmetry

When a quantum pure state is drawn uniformly at random from a Hilbert space, the state is typically highly entangled. This property of a random state is known as generic entanglement of quantum states and has been long investigated from many perspectives, ranging from the black hole science to quantum information science. In this paper, we address the question of how symmetry of quantum states changes the properties of generic entanglement. More specifically, we study bipartite entanglement entropy of a quantum state that is drawn uniformly at random from an invariant subspace of a given symmetry. We first extend the well-known concentration formula to the one applicable to any subspace and then show that 1. quantum states in the subspaces associated with an axial symmetry are still highly entangled, though it is less than that of the quantum states without symmetry, 2. quantum states associated with the permutation symmetry are significantly less entangled, and 3. quantum states with translation symmetry are as entangled as the generic one. We also numerically investigate the phase-transition behavior of the distribution of generic entanglement, which indicates that the phase transition seems to still exist even when random states have symmetry.

Entanglement in coupled kicked tops with chaotic dynamics

The entanglement of eigenstates in two coupled, classically chaotic kicked tops is studied in dependence of their interaction strength. The transition from the noninteracting and unentangled system toward full random matrix behavior is governed by a universal scaling parameter. Using suitable random matrix transition ensembles we express this transition parameter as a function of the subsystem sizes and the coupling strength for both unitary and orthogonal symmetry classes. The universality is confirmed for the level spacing statistics of the coupled kicked tops and a perturbative description is in good agreement with numerical results. The statistics of Schmidt eigenvalues and entanglement entropies of eigenstates is found to follow a universal scaling as well. Remarkably, this is not only the case for large subsystems of equal size but also if one of them is much smaller. For the entanglement entropies a perturbative description is obtained, which can be extended to large couplings and provides very good agreement with numerical results. Furthermore, the transition of the statistics of the entanglement spectrum toward the random matrix limit is demonstrated for different ratios of the subsystem sizes.

On the Exact Variance of Tsallis Entanglement Entropy in a Random Pure State

The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy.

Generalized Entanglement Entropies of Quantum Designs

The entanglement properties of random quantum states or dynamics are important to the study of a broad spectrum of disciplines of physics, ranging from quantum information to high energy and many-body physics. This Letter investigates the interplay between the degrees of entanglement and randomness in pure states and unitary channels. We reveal strong connections between designs (distributions of states or unitaries that match certain moments of the uniform Haar measure) and generalized entropies (entropic functions that depend on certain powers of the density operator), by showing that Rényi entanglement entropies averaged over designs of the same order are almost maximal. This strengthens the celebrated Page's theorem. Moreover, we find that designs of an order that is logarithmic in the dimension maximize all Rényi entanglement entropies and so are completely random in terms of the entanglement spectrum. Our results relate the behaviors of Rényi entanglement entropies to the complexity of scrambling and quantum chaos in terms of the degree of randomness, and suggest a generalization of the fast scrambling conjecture.

Proof of Vivo-Pato-Oshanin's conjecture on the fluctuation of von Neumann entropy

It was recently conjectured by Vivo, Pato, and Oshanin [Phys. Rev. E 93, 052106 (2016)2470-004510.1103/PhysRevE.93.052106] that for a quantum system of Hilbert dimension mn in a pure state, the variance of the von Neumann entropy of a subsystem of dimension m≤n is given by-ψ_{1}(mn+1)+m+n/mn+1ψ_{1}(n)-(m+1)(m+2n+1)/4n^{2}(mn+1),where ψ_{1}(·) is the trigamma function. We give a proof of this formula.

Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians

In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)PRLTAO0031-900710.1103/PhysRevLett.71.1291], Page proved that the average entanglement entropy of subsystems of random pure states is S_{ave}≃lnD_{A}-(1/2)D_{A}^{2}/D for 1≪D_{A}≤sqrt[D], where D_{A} and D are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy ⟨S⟩ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models lnD_{A}-(lnD_{A})^{2}/lnD≤⟨S⟩≤lnD_{A}-[1/(2ln2)](lnD_{A})^{2}/lnD. Consequently, we prove that (i) if the subsystem size is a finite fraction of the system size, then ⟨S⟩<lnD_{A} in the thermodynamic limit; i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal; i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.

Novel quantum phase transition from bounded to extensive entanglement

The nature of entanglement in many-body systems is a focus of intense research with the observation that entanglement holds interesting information about quantum correlations in large systems and their relation to phase transitions. In particular, it is well known that although generic, many-body states have large, extensive entropy, ground states of reasonable local Hamiltonians carry much smaller entropy, often associated with the boundary length through the so-called area law. Here we introduce a continuous family of frustration-free Hamiltonians with exactly solvable ground states and uncover a remarkable quantum phase transition whereby the entanglement scaling changes from area law into extensively large entropy. This transition shows that entanglement in many-body systems may be enhanced under special circumstances with a potential for generating "useful" entanglement for the purpose of quantum computing and that the full implications of locality and its restrictions on possible ground states may hold further surprises.

Random pure states: Quantifying bipartite entanglement beyond the linear statistics

We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.

Characterizing eigenstate thermalization via measures in the Fock space of operators

The eigenstate thermalization hypothesis (ETH) attempts to bridge the gap between quantum mechanical and statistical mechanical descriptions of isolated quantum systems. Here, we define unbiased measures for how well the ETH works in various regimes, by mapping general interacting quantum systems on regular lattices onto a single particle living on a high-dimensional graph. By numerically analyzing deviations from ETH behavior in the nonintegrable Ising model, we propose a quantity that we call the n-weight to democratically characterize the average deviations for all operators residing on a given number of sites, irrespective of their spatial structure. It appears to have a simple scaling form, which we conjecture to hold true for all nonintegrable systems. A closely related quantity, which we term the n-distinguishability, tells us how well two states can be distinguished if only n-site operators are measured. Along the way, we discover that complicated operators on average are worse than simple ones at distinguishing between neighboring eigenstates, contrary to the naive intuition created by the usual statements of the ETH that few-body (many-body) operators acquire the same (different) expectation values in nearby eigenstates at finite energy density. Finally, we sketch heuristic arguments that the ETH originates from the limited ability of simple operators to distinguish between quantum states of a system, especially when the states are subject to constraints such as roughly fixed energy with respect to a local Hamiltonian.

Large deviation bounds for k -designs

We present a technique for de-randomizing large deviation bounds of functions on the unitary group. We replace the Haar measure with a pseudo-random distribution, a k -design. k -Designs have the first k moments equal to those of the Haar measure. The advantage of this is that (approximate) k -designs can be implemented efficiently, whereas Haar random unitaries cannot. We find large deviation bounds for unitaries chosen from a k -design and then illustrate this general technique with three applications. We first show that the von Neumann entropy of a pseudo-random state is almost maximal. Then we show that, if the dynamics of the universe produces a k -design, then suitably sized subsystems will be in the canonical state, as predicted by statistical mechanics. Finally we show that pseudo-random states are useless for measurement-based quantum computation.

Most quantum States are too entangled to be useful as computational resources

It is often argued that entanglement is at the root of the speedup for quantum compared to classical computation, and that one needs a sufficient amount of entanglement for this speedup to be manifest. In measurement-based quantum computing, the need for a highly entangled initial state is particularly obvious. Defying this intuition, we show that quantum states can be too entangled to be useful for the purpose of computation, in that high values of the geometric measure of entanglement preclude states from offering a universal quantum computational speedup. We prove that this phenomenon occurs for a dramatic majority of all states: the fraction of useful n-qubit pure states is less than exp(-n;{2}). This work highlights a new aspect of the role entanglement plays for quantum computational speedups.

Generic entanglement can be generated efficiently

We find that generic entanglement is physical, in the sense that it can be generated in polynomial time from two-qubit gates picked at random. We prove as the main result that such a process generates the average entanglement of the uniform (unitarily invariant) measure in at most O(N3) steps for N qubits. This is despite an exponentially growing number of such gates being necessary for generating that measure fully on the state space. Numerics furthermore show a variation cutoff allowing one to associate a specific time with the achievement of the uniform measure entanglement distribution. Various extensions of this work are discussed. The results are relevant to entanglement theory and to protocols that assume generic entanglement can be achieved efficiently.

Average entropy of a subsystem from its average Tsallis entropy

In the non-extensive Tsallis scenario, Page's conjecture for the average entropy of a subsystem [Phys. Rev. Lett. 71, 1291 (1993)] as well as its demonstration are generalized, i.e., when a pure quantum system, whose Hilbert space dimension is mn, is considered, the average Tsallis entropy of an m-dimensional subsystem is obtained. This demonstration is expected to be useful to study systems where the usual entropy does not give satisfactory results.