# Publications

**Authors: **Damiano Anselmi, Eugenio Bianchi and Marco Piva

**Year:** 2020

We derive the predictions of quantum gravity with fakeons on the amplitudes and spectral indices of the scalar and tensor fluctuations in inflationary cosmology. The action is R+R2 plus the Weyl-squared term. The ghost is eliminated by turning it into a fakeon, that is to say a purely virtual particle. We work to the next-to-leading order of the expansion around the de Sitter background. The consistency of the approach puts a lower bound (mχ>mϕ/4) on the mass mχ of the fakeon with respect to the mass mϕ of the inflaton. The tensor-to-scalar ratio r is predicted within less than an order of magnitude (4/3

**Authors: **Lucas Hackl and Eugenio Bianchi

**Year:** 2020

We show that bosonic and fermionic Gaussian states (also known as “squeezed coherent states”) can be uniquely characterized by their linear complex structure J which is a linear map on the classical phase space. This extends conventional Gaussian methods based on covariance matrices and provides a unified framework to treat bosons and fermions simultaneously. Pure Gaussian states can be identified with the triple (G,Ω,J) of compatible Kähler structures, consisting of a positive definite metric G, a symplectic form Ω and a linear complex structure J with J2=−11. Mixed Gaussian states can also be identified with such a triple, but with J2≠−11. We apply these methods to show how computations involving Gaussian states can be reduced to algebraic operations of these objects, leading to many known and some unknown identities. We apply these methods to the study of (A) entanglement and complexity, (B) dynamics of stable systems, (C) dynamics of driven systems. From this, we compile a comprehensive list of mathematical structures and formulas to compare bosonic and fermionic Gaussian states side-by-side.

**Authors: **Eugenio Bianchi, Lucas Hackl and Mario Kieburg

**Year:** 2021

In a seminal paper, Page found the exact formula for the average entanglement entropy for a pure random state. We consider the analogous problem for the ensemble of pure fermionic Gaussian states, which plays a crucial role in the context of random free Hamiltonians. Using recent results from random matrix theory, we show that the average entanglement entropy of pure random fermionic Gaussian states in a subsystem of NA out of N degrees of freedom is given by ⟨SA⟩G=(N−12)Ψ(2N)+(14−NA)Ψ(N)+(12+NA−N)Ψ(2N−2NA)−14Ψ(N−NA)−NA, where Ψ is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by ⟨SA⟩G=N(log2−1)f+N(f−1)log(1−f)+12f+14log(1−f)+O(1/N), where f=NA/N. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by Lydzba, Rigol and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant limN→∞(ΔSA)2G=12(f+f2+log(1−f)).

**Authors: **Abhay Ashtekar, Eugenio Bianchi

**Year:** 2021

An outstanding open issue in our quest for physics beyond Einstein is the unification of general relativity (GR) and quantum physics. Loop quantum gravity (LQG) is a leading approach toward this goal. At its heart is the central lesson of GR: Gravity is a manifestation of spacetime geometry. Thus, the approach emphasizes the quantum nature of geometry and focuses on its implications in extreme regimes — near the big bang and inside black holes — where Einstein’s smooth continuum breaks down. We present a brief overview of the main ideas underlying LQG and highlight a few recent advances. This report is addressed to non-experts.

**Authors: **Eugenio Bianchi, Pierre Martin-Dussaud

**Year:** 2021

The metric field of general relativity is almost fully determined by its causal structure. Yet, in spin-foam models for quantum gravity, the role played by the causal structure is still largely unexplored. The goal of this paper is to clarify how causality is encoded in such models. The quest unveils the physical meaning of the orientation of the two-complex and its role as a dynamical variable. We propose a causal version of the EPRL spin-foam model and discuss the role of the causal structure in the reconstruction of a semiclassical spacetime geometry.

**Authors: **Eugenio Bianchi, Lucas Hackl, Mario Kieburg, Marcos Rigol, Lev Vidmar

**Year:** 2021

We introduce and discuss known results for the volume-law entanglement entropy of typical pure quantum states in which the number of particles is not fixed and derive results for the volume-law entanglement entropy of typical pure quantum states with a fixed number of particles. For definiteness, we consider lattice systems of fermions in an arbitrary dimension and present results for averages over all states as well as over the subset of all Gaussian states. For quantum states in which the number of particles is not fixed, the results for the average over all states are well known since the work of Page, who found that in the thermodynamic limit the leading term follows a volume law and is maximal. The associated variance vanishes exponentially fast with increasing system size, i.e., the average is also the typical entanglement entropy. The corresponding results for Gaussian states are more recent. The leading term is still a volume law, but it is not maximal and depends on the ratio between the volumes of the subsystem and the entire system. Moreover, the variance is independent of the system size, i.e., the average also gives the typical entanglement entropy. We prove that while fixing the number of particles in pure quantum states does not qualitatively change the behavior of the leading volume-law term in the average entanglement entropy, it can fundamentally change the nature of the subleading terms. In particular, subleading corrections appear that depend on the square root of the volume. We unveil the origin of those corrections. Finally, we discuss the connection between the entanglement entropy of typical pure states and recent analytical results obtained in the context of random matrix theory, as well as numerical results obtained for physical Hamiltonians.

**Authors: **Richard D.P. East, Pierre Martin-Dussaud, John Van de Wetering

**Year:** 2021

The ZX-calculus, and the variant we consider in this paper (ZXH-calculus), are formal diagrammatic languages for qubit quantum computing. We show that it can also be used to describe SU(2) representation theory. To achieve this, we first recall the definition of Yutsis diagrams, a standard graphical calculus used in quantum chemistry and quantum gravity, which captures the main features of SU(2) representation theory. Second, we show precisely how it embed within Penrose’s binor calculus. Third, we subsume both calculus into ZXH-diagrams. In the process we show how the SU(2) invariance of Wigner symbols is trivially provable in the ZXH-calculus. Additionally, we show how we can explicitly diagrammatically calculate 3jm, 4jm and 6j symbols. It has long been thought that quantum gravity should be closely aligned to quantum information theory. In this paper, we present a way in which this connection can be made exact, by writing the spin-networks of loop quantum gravity (LQG) in the ZX-diagrammatic language of quantum computation.