March 2023

We write explicitly the complete Lorentzian metric of a singularity-free spacetime where a black hole transitions into a white hole located in its same asymptotic region. In particular, the metric interpolates between the black and white horizons. The metric satisfies the Einstein field equations up to the tunneling region. The matter giving rise to the black hole is described by the Oppenheimer-Snyder model, corrected with loop-quantum-cosmology techniques in the quantum region. The interior quantum geometry is fixed by a local Killing symmetry, broken at the horizon transition. At large scale, the geometry is determined by two parameters: the mass of the hole and the duration of the transition process. The latter is a global geometrical parameter. We give the full metric outside the star in a single coordinate patch.

The ZX-calculus is a universal graphical language for qubit quantum computation, meaning that every linear map between qubits can be expressed in the ZX-calculus. Furthermore, it is a complete graphical rewrite system: any equation involving linear maps that is derivable in the Hilbert space formalism for quantum theory can also be derived in the calculus by rewriting. It has widespread usage within quantum industry and academia for a variety of tasks such as quantum circuit optimisation, error-correction, and education. The ZW-calculus is an alternative universal graphical language that is also complete for qubit quantum computing. In fact, its completeness was used to prove that the ZX-calculus is universally complete. This calculus has advanced how quantum circuits are compiled into photonic hardware architectures in the industry. Recently, by combining these two calculi, a new calculus has emerged for qubit quantum computation, the ZXW-calculus. Using this calculus, graphical-differentiation, -integration, and -exponentiation were made possible, thus enabling the development of novel techniques in the domains of quantum machine learning and quantum chemistry. Here, we generalise the ZXW-calculus to arbitrary finite dimensions, that is, to qudits. Moreover, we prove that this graphical rewrite system is complete for any finite dimension. This is the first completeness result for any universal graphical language beyond qubits.

This paper concerns the structure of meanings within natural language. Earlier, a framework named DisCoCirc was sketched that (1) is compositional and distributional (a.k.a. vectorial); (2) applies to general text; (3) captures linguistic `connections’ between meanings (cf. grammar) (4) updates word meanings as text progresses; (5) structures sentence types; (6) accommodates ambiguity. Here, we realise DisCoCirc for a substantial fragment of English. When passing to DisCoCirc’s text circuits, some `grammatical bureaucracy’ is eliminated, that is, DisCoCirc displays a significant degree of (7) inter- and intra-language independence. That is, e.g., independence from word-order conventions that differ across languages, and independence from choices like many short sentences vs. few long sentences. This inter-language independence means our text circuits should carry over to other languages, unlike the language-specific typings of categorial grammars. Hence, text circuits are a lean structure for the `actual substance of text’, that is, the inner-workings of meanings within text across several layers of expressiveness (cf. words, sentences, text), and may capture that what is truly universal beneath grammar. The elimination of grammatical bureaucracy also explains why DisCoCirc: (8) applies beyond language, e.g. to spatial, visual and other cognitive modes. While humans could not verbally communicate in terms of text circuits, machines can. We first define a `hybrid grammar’ for a fragment of English, i.e. a purpose-built, minimal grammatical formalism needed to obtain text circuits. We then detail a translation process such that all text generated by this grammar yields a text circuit. Conversely, for any text circuit obtained by freely composing the generators, there exists a text (with hybrid grammar) that gives rise to it. Hence: (9) text circuits are generative for text.

These are the lecture notes of guest lectures for Artur Ekert’s course Introduction to Quantum Information at the Mathematical Institute of Oxford University, Hilary Term 2023. Some basic familiarity with Dirac notation is assumed. For the readers of Quantum in Pictures (QiP) who have some basic quantum background, these notes also constitute the shortest path to an explanation of how what they learn in QIP relates to the traditional quantum formalism.

To obtain Bell statistics from hybrid states composed of finite- and infinite-dimensional systems, we propose a hybrid measurement scheme, in which the continuous mode is measured using the generalized pseudospin operators, while the finite (two)-dimensional system is measured in the usual Pauli basis. Maximizing the Bell expression with these measurements leads to the violations of local realism which is referred to as hybrid nonlocality. We demonstrate the utility of our strategy in a realistic setting of cavity quantum electrodynamics, where an atom interacts with a single mode of an electromagnetic field under the Jaynes-Cummings Hamiltonian. We dynamically compute the quenched averaged value of hybrid nonlocality in imperfect situations by incorporating disorder in the atom-cavity coupling strength. In the disordered case, we introduce two kinds of measurement scenarios to determine the Bell statistics — in one situation, experimentalists can tune the optimal settings according to the interaction strength while such controlled power is absent in the other case. In contrast to the oscillatory behavior observed in the ordered case, the quenched averaged violation saturates to a finite value in some parameter regimes in the former case, thereby highlighting an advantage of disordered systems. We also examine the connection between Wigner negativity and hybrid nonlocality.

A popular interpretation of classical physics assumes that every classical system is in a well-defined pure state, which may be unknown to the observer, but is nevertheless part of the physical reality. Here we show that this interpretation is not always tenable. We construct a toy theory that includes all possible classical systems, alongside with another set of systems, called anti-classical, which are dual to the classical ones in a similar way as anti-particles are dual to particles. In the world of our toy theory, every classical system can be entangled with an anti-classical partner, and every classical mixed state can be obtained from a pure entangled state by discarding the anti-classical part. In the presence of such entanglement, it is impossible to assign a well-defined pure state to classical systems alone. Even more strongly, we prove that entangled states of classical/anti-classical composites exhibit activation of Bell nonlocality, and we use this fact to rule out every ontological model in which individual classical systems are assigned well-defined local states.

Unextendible product bases (UPBs) play a key role in the study of quantum entanglement and nonlocality. A famous open question is whether there exist genuinely unextendible product bases (GUPBs), namely multipartite product bases that are unextendible with respect to every possible bipartition. Here we shed light on this question by providing a characterization of UPBs and GUPBs in terms of orthogonality graphs. Building on this connection, we develop a method for constructing UPBs in low dimensions, and we derive a lower bound on the size of any GUPB, significantly improving over the state of the art. Moreover, we show that every minimal GUPB saturating our bound must be associated to regular graphs. Finally, we discuss a possible path towards the construction of a minimal GUPB in a tripartite system of minimal local dimension.

We show how the fixed-spin asymptotics of the EPRL model can be used to perform the spin-sum for spin foam amplitudes defined on fixed two-complexes without interior faces and contracted with coherent spin-network states peaked on a discrete simplicial geometry with macroscopic areas. We work in the representation given in Ref. 1. We first rederive the latter in a different way suitable for our purposes. We then extend this representation to 2-complexes with a boundary and derive its relation to the coherent state representation. We give the measure providing the resolution of the identity for Thiemann’s state in the twisted geometry parametrization. The above then permit us to put everything together with other results in the literature and show how the spin sum can be performed analytically for the regime of interest here. These results are relevant to analytic investigations regarding the transition of a black hole to a white hole geometry. In particular, this work gives detailed technique that was the basis of estimate for the black to white bounce appeared in Ref. 2. These results may also be relevant for applications of spinfoams to investigate the possibility of a ‘big bounce’.

An analysis is given of the local phase space of gravity coupled to matter to second order in perturbation theory. Working in local regions with boundaries at finite distance, we identify matter, Coulomb, and additional boundary modes. The boundary modes take the role of reference frames for both diffeomorphisms and internal Lorentz rotations. Passing to the quantum level, we identify the constraints that link the bulk and boundary modes. The constraints take the form of a multi-fingered Schr”odinger equation

We summarize recent developments at the interface of quantum gravity and quantum information, and discuss applications to the quantum geometry of space in loop quantum gravity. In particular, we describe the notions of link entanglement, intertwiner entanglement, and boundary spin entanglement in a spin-network state. We discuss how these notions encode the gluing of quanta of space and their relevance for the reconstruction of a quantum geometry from a network of entanglement structures. We then focus on the geometric entanglement entropy of spin-network states at fixed spins, treated as a many-body system of quantum polyhedra, and discuss the hierarchy of volume-law, area-law and zero-law states. Using information theoretic bounds on the uncertainty of geometric observables and on their correlations, we identify area-law states as the corner of the Hilbert space that encodes a semiclassical geometry, and the geometric entanglement entropy as a probe of semiclassicality.