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We present a variety of methods for training complex-valued word embeddings, based on the classical Skip-gram model, with a straightforward adaptation simply replacing the real-valued vectors with arbitrary vectors of complex numbers. In a more “physically-inspired” approach, the vectors are produced by parameterised quantum circuits (PQCs), which are unitary transformations resulting in normalised vectors which have a probabilistic interpretation. We develop a complex-valued version of the highly optimised C code version of Skip-gram, which allows us to easily produce complex embeddings trained on a 3.8B-word corpus for a vocabulary size of over 400k, for which we are then able to train a separate PQC for each word. We evaluate the complex embeddings on a set of standard similarity and relatedness datasets, for some models obtaining results competitive with the classical baseline. We find that, while training the PQCs directly tends to harm performance, the quantum word embeddings from the two-stage process perform as well as the classical Skip-gram embeddings with comparable numbers of parameters. This enables a highly scalable route to learning embeddings in complex spaces which scales with the size of the vocabulary rather than the size of the training corpus. In summary, we demonstrate how to produce a large set of high-quality word embeddings for use in complex-valued and quantum-inspired NLP models, and for exploring potential advantage in quantum NLP models.

Quantum theory and general relativity are about one century old. At present, they are considered the best available explanations of physical reality, and they have been so far corroborated by all experiments realised so far. Nonetheless, the quest to unify them is still ongoing, with several yet untested proposals for a theory of quantum gravity. Here we review the nascent field of information-theoretic methods applied to designing tests of quantum gravity in the laboratory. This field emerges from the fruitful extension of quantum information theory methodologies beyond the domain of applicability of quantum theory itself, to cover gravity. We shall focus mainly on the detection of gravitational entanglement between two quantum probes, comparing this method with single-probe schemes. We shall review the experimental proposal that has originated this field, as well as its variants, their applications, and discuss their potential implications for the quantum theory of gravity. We shall also highlight the role of general information-theoretic principles in illuminating the search for quantum effects in gravity.

A finite duration of cosmic inflation can result in features $mathcal{P}_{mathcal{R}}(k)=|alpha_k-beta_k,mathrm{e}^{mathrm{i}delta_k}|^2 mathcal{P}_{mathcal{R}}^{(0)}(k)$ in the primordial power spectrum that carry information about a quantum gravity phase before inflation. While the almost-scale-invariant power spectrum $mathcal{P}_{mathcal{R}}^{(0)}$ for the quasi-Bunch-Davies vacuum is fully determined by the inflationary background dynamics, the Bogoliubov coefficients $alpha_k$ and $beta_k$ for the squeezed vacuum depend on new physics beyond inflation and have been used to produce phenomenological templates for the features. In this paper, we consider a large class of effective theories of inflation and compute the relative phase $delta_k$. While this phase vanishes in de Sitter space, here we show that it is fully determined by the inflationary background dynamics and we compute it up to the next-to-next-leading order (N2LO) in a Hubble-flow expansion. In particular, for the Starobinsky model of inflation we find that this relative phase can be expressed in terms of the scalar tilt $n_mathrm{s}$ as $delta_k=frac{pi}{2}(n_mathrm{s}-1)-frac{pi}{4}(n_mathrm{s}-1)^2,ln(k/k_*)$. The relative phase results in a negative shift and a running frequency that have been considered in the most studied phenomenological templates for primordial features, thus providing precise theoretical predictions for upcoming cosmological observations.

Recently, it has been proposed a new method [arXiv:2405.21029] to detect quantum gravity effects, based on generating gravitational entanglement between two nano-diamonds with Nitrogen-Vacancy defects, in a magnetically trapped configuration. Here we analyse in detail the proposed experimental setup, with a particular focus on implementing the detection of the gravitationally-induced entanglement using an optical readout based on measuring the position of the nano-diamonds and its complementary basis. We also summarise some of the key theoretical and experimental ideas on which this proposed scheme is based.

Bell scenarios are multipartite scenarios that exclude any communication between parties. This constraint leads to a strict hierarchy of correlation sets in such scenarios, namely, classical, quantum, and nonsignaling. However, without any constraints on communication between the parties, they can realize arbitrary correlations by exchanging only classical systems. Here we consider a multipartite scenario where the parties can engage in at most a single round of communication, i.e., each party is allowed to receive a system once, implement any local intervention on it, and send out the resulting system once. While no global assumption about causal relations between parties is assumed in this scenario, we do make a causal assumption local to each party, i.e., the input received by it causally precedes the output it sends out. We then introduce antinomicity, a notion of nonclassicality for correlations in such scenarios, and prove the existence of a strict hierarchy of correlation sets classified by their antinomicity. Antinomicity serves as a generalization of Bell nonlocality: when all the parties discard their output systems (i.e., in a nonsignaling scenario), it is mathematically equivalent to Bell nonlocality. Like Bell nonlocality, it can be understood as an instance of fine-tuning, one that is necessary in any classical model of cyclic causation that avoids time-travel antinomies but allows antinomic correlations. Furthermore, antinomicity resolves a long-standing puzzle, i.e., the failure of causal inequality violations as device-independent witnesses of nonclassicality. Antinomicity implies causal inequality violations, but not conversely.

This collection of perspective pieces captures recent advancements and reflections from a dynamic research community dedicated to bridging quantum gravity, hydrodynamics, and emergent cosmology. It explores four key research areas: (a) the interplay between hydrodynamics and cosmology, including analog gravity systems; (b) phase transitions, continuum limits and emergent geometry in quantum gravity; (c) relational perspectives in gravity and quantum gravity; and (d) the emergence of cosmological models rooted in quantum gravity frameworks. Each contribution presents the distinct perspectives of its respective authors. Additionally, the introduction by the editors proposes an integrative view, suggesting how these thematic units could serve as foundational pillars for a novel theoretical cosmology framework termed “hydrodynamics on superspace”.

This paper initiates a systematic study of operators arising as integrals of operator-valued functions with respect to positive operator-valued measures and utilizes these tools to provide relativization maps (Yen) for quantum reference frames (QRFs) defined on general homogeneous spaces. Properties of operator-valued integration are first studied and then employed to define general relativization maps and show their properties. The relativization maps presented here are defined for QRFs (systems of covariance) based on arbitrary homogeneous spaces of locally compact second countable topological groups and are shown to be contracting quantum channels, injective for localizable (norm-1 property) frames and multiplicative for the sharp ones (PVMs), extending the existing results.

Modular crossed product algebras have recently assumed an important role in perturbative quantum gravity as they lead to an intrinsic regularization of entanglement entropies by introducing quantum reference frames (QRFs) in place of explicit regulators. This is achieved by imposing certain boost constraints on gravitons, QRFs and other fields. Here, we revisit the question of how these constraints should be understood through the lens of perturbation theory and particularly the study of linearization (in)stabilities, exploring when linearized solutions can be integrated to exact ones. Our aim is to provide some clarity about the status of justification, under various conditions, for imposing such constraints on the linearized theory in the $G_Nto0$ limit as they turn out to be of second-order. While for spatially compact spacetimes there is an essentially unambiguous justification, in the presence of boundaries or the absence of isometries this depends on whether one is also interested in second-order observables. Linearization (in)stabilities occur in any gauge-covariant field theory with non-linear equations and to address this in a unified framework, we translate the subject from the usual canonical formulation into a systematic covariant phase space language. This overcomes theory-specific arguments, exhibiting the universal structure behind (in)stabilities, and permits us to cover arbitrary generally covariant theories. We comment on the relation to modular flow and illustrate our findings in several gravity and gauge theory examples.

One can theoretically conceive of processes where the causal order between quantum operations is no longer well-defined. Certain such causally indefinite processes have an operational interpretation in terms of quantum operations on time-delocalised subsystems — that is, they can take place as part of standard quantum mechanical evolutions on quantum systems that are delocalised in time. In this paper, we formalise the underlying idea that quantum evolutions can be represented with respect to different subsystem decompositions in a general way. We introduce a description of quantum circuits, including cyclic ones, in terms of an operator acting on the global Hilbert space of all systems in the circuit. This allows us to express in a concise form how a given circuit transforms under arbitrary changes of subsystem decompositions. We then explore the link between this framework and the concept of causal perspectives, which has been introduced to describe causally indefinite processes from the point of view of the different parties involved. Surprisingly, we show that the causal perspectives that one can associate to the different parties in the quantum switch, a paradigmatic example of a causally indefinite process, cannot be related by a change of subsystem decomposition, i.e., they cannot be seen as two equivalent descriptions of the same process.

We prove that the generalised second law (GSL), with an appropriate modification, holds in perturbative gravity to all orders beyond the semiclassical limit and without a UV cutoff imposed on the fields. Our proof uses algebraic techniques and builds on the recent work of Faulkner and Speranza, which combined Wall’s proof of the GSL with the identification of generalised entropy as the von Neumann entropy of a boost-invariant crossed product algebra. The key additional step in our approach is to further impose invariance under null translations. Doing so requires one to describe horizon exterior regions in a relational manner, so we introduce `dynamical cuts’: quantum reference frames which give the location of a cut of the horizon. We use idealised dynamical cuts, but expect that our methods can be generalised to more realistic models. The modified GSL that we prove says that the difference in generalised entropies of the regions outside two dynamical cuts is bounded below by the free energy of the degrees of freedom giving the location of the later cut. If one takes a semiclassical limit, imposes a UV cutoff, and requires the cuts to obey certain energy conditions, then our result reduces to the standard GSL.