Schedule for: 22w5177 - Mathematical and Conceptual Aspects of Quantum Theory

I address the question of unitary implementation of the dynamics for scalar fields in nonstationary spacetimes describing cosmological scenarios. Together with invariance under spatial isometries, the requirement of a unitary evolution singles out a rescaling of the scalar field and a unitary equivalence class of Fock representations for the associated canonical commutation relations. Moreover, this criterion also provides a privileged quantization for the unscaled field, even though the associated dynamics is not unitarily implementable in that case. Finally, I discuss the relation between the initial data that determine the Fock representations in the rescaled and unscaled descriptions and the relation with adiabatic states.

A quantum combinatorial design is composed of quantum states, arranged with a certain symmetry and balance. They determine distinguished quantum measurements and can be applied for quantum information processing. Negative solution to the famous problem of 36 officers of Euler implies that there are no two orthogonal Latin squares of order six. We show that the problem has a solution, provided the officers are entangled, and construct orthogonal quantum Latin squares of this size. The solution can be visualized on a chessboard of size six, which shows that 36 officers are split in nine groups, each containing four entangled states. It allows us to construct a pure nonadditive quhex quantum error detection code.

Any attempt to formulate a quantum information theory for relativistic systems, must inevitably express informational notions in the language of quantum field theory (QFT). In this talk, we summarize the challenges intrinsic into such an endeavor, focusing in particular on the difficulties involved in providing consistent accounts of quantum measurements in QFT. We present an approach (the quantum temporal probabilities method) which formulates measurements in terms of QFT correlation functions. Finally, we present connections and applications to diverse topics like photo-detection theory, black hole information and non-equilibrium QFTs.

Global properties of N-qubit systems can be described by discrete distributions in a 3-dim space of symmetric measurements. Such distributions contain full and non-redundant information about any collective (invariant under particle permutations) observable in an arbitrary state and can be used for visualization and analysis of macroscopic features of quantum states. We show that the analytical properties of such distributions can be used for characterization of quantum correlations in the limit of large number of particles. In particular, one can establish criteria for quantification of two specific types of quantum correlations, which can be detected by performing collective measurements

Equivariant Functorial Quantum Field Theory (EFQFT) is an axiomatic framework for describing quantum field theories, based on the principles of compositionality and operationalism. EFQFT refines Quantum Field Theory in the General Boundary Formulation (GBFQFT) and it is specifically designed to provide GBFQFT with an interpretation in terms of categories and functors, which in turn provides GBFQFT with a robust mathematical framework under which certain questions admit natural solutions. In this talk I will present a detailed account of the axioms of EFQFT, together with relevant examples. I will discuss in particular the data of EFQFT in dimension 2 and how 2-dimensional quantum Yang-Mills theory with corners fits into this framework. This is joint work with Robert Oeckl.

The theory of causal fermion systems is an approach to describe fundamental physics. It gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. Moreover, causal fermion systems provide a general framework for modelling and analyzing non-smooth spacetime structures. The dynamics of a causal fermion system is described by a nonlinear variational principle, the causal action principle. The aim of the talk is to give a simple introduction, with an emphasis on the underlying concepts. At the end of the talk, I will briefly outline how to get the connection to quantum field theory.

Groupoids provide a powerful and elegant mathematical setting to describe Schwinger’s picture of Quantum Mechanics. We will argue that such is the case discussing a few simple examples and we will elaborate on the notion of elementary quantum systems from this broader perspective.

It has been shown that it is theoretically possible for there to exist quantum and classical processes in which the operations performed by separate parties do not occur in a well-defined causal order. A central question is whether and how such processes can be realised in practice. In order to provide a rigorous argument for the notion that certain such processes have a realisation in standard quantum theory, the concept of time-delocalised quantum subsystem has been introduced. In this talk, I will discuss the concept of time-delocalised subsystem and its relevance to the question of realisability of processes with indefinite causal order in standard quantum theory and quantum gravity. I will explain how, given a description of an experiment in the form of a (generally cyclic) circuit, the experiment can be described with respect to an arbitrary alternative choice of (sub)systems, which is obtained by a transformation akin to a spatio-temporal change of basis. This provides a simple and very general notion of transformation between different equivalent descriptions of an experiment. I will review how the quantum SWITCH can be seen as realisable on time-delocalised systems in standard quantum mechanics and will show that all unitarily extendible tripartite processes with indefinite causal order admit such realisations. Remarkably, this includes processes violating causal inequalities, whose physical realisability has been a central open problem. I will discuss the meaning of causal inequality violation in this setting and argue that it is a meaningful concept to show the absence of a definite causal order between the variables of interest. Finally, I will speculate on the link between time-delocalised systems and quantum reference frames.

Relativistic causality (RC) is the principle that no cause can act outside its future lightcone. It is widely believed that certain observable correlations among spacelike separated events contradict RC, for instance correlations that violate the Bell inequalities. Against this belief, I will argue that the apparent contradiction has arisen only because people have, knowingly or uknowingly, supplemented RC with a further principle of "Local Action" which is not realized in nature. Thus RC itself places no restriction on which spacelike correlations might be possible. Nevertheless, not all such correlations are allowed in current quantum theories, and one can wonder whether these limitations might be reflecting other underlying principles, related to, but distinct from RC. I will tentatively propose "persistence of zero" and "lack of novelty" as two such conditions which play a role in quantum dynamics resembling the role of "factorization" in derivations of the Bell inequalities.

We review the definition of parametrized field theory and pair it with the concept of a cubical omega groupoid. We give a brief picture introduction to cubical omega groupoids and its role in homotopy. The new language lets us state well known features of field theory simply, gives a slightly different interpretation to boundary conditions and gluing, suggests extensions of field theory, and brings a homotopical point of view for discretization and coarse graining.

We provide a setup by which one can recover the geometry of spacetime from local measurements of quantum particle detectors coupled to a quantum field. Concretely, we show how one can recover the field's correlation function from measurements on the detectors. Then, we are able to recover the invariant spacetime interval from the measurement outcomes, and hence reconstruct a notion of spacetime metric. This suggests that quantum particle detectors are the experimentally accessible devices that could replace the classical 'rulers' and 'clocks' of general relativity.

Anticoherent spin states are considered the least classical spin states, as opposed to spin coherent states. In this talk, I will present an overview of the results we have obtained so far on anticoherent spin states. I will discuss their general properties based in particular on their Majorana representation, then I will present their dynamics under depolarisation and finally I will give some insights on how to produce them.

We introduce higher-order partial derivatives of a probability distribution of particle positions as a new object that shares basic properties of quantum mechanical states needed for a quantum algorithm. Discretization of the positions allows one to represent the quantum mechanical state of nbit qubits by 2(nbit+1) classical stochastic bits. Based on this, we demonstrate many-particle interference and representation of pure entangled quantum states via derivatives of probability distributions and find the universal set of stochastic maps that correspond to the quantum gates in a universal gate set. We prove that the propagation via the stochastic map built from those universal stochastic maps reproduces up to a prefactor exactly the evolution of the quantum mechanical state with the corresponding quantum algorithm, leading to an automated translation of a quantum algorithm to a stochastic classical algorithm. We implement several well-known quantum algorithms, analyze the scaling of the needed number of realizations with the number of qubits, and highlight the role of destructive interference for the cost of the emulation.

We present a study of phase diagrams for 2-, 3-, and in general 𝑛-level atoms interacting dipolarly with a radiation field of ℓ modes in a cavity [1]. We show that the super-radiant region of phase space divides itself into monochromatic regions where only one mode of the electromagnetic field dominates [2]. A reduction scheme is presented which, if carried out iteratively, will reduce the general study of 𝑛-levels to that of a collection of 2-level Dicke atoms [1,3]. Furthermore, a truncation scheme for the infinite-dimensional Hilbert space of the system is proposed, as well as a way to judge the goodness of the reduced bases [4,5]. This provides us with a mathematical technique that can be used to solve systems where the number of atoms and excitations grow, yielding a Hilbert space with enormous dimensions, more effectively than with the currently available methods. [1] S. Cordero, E. Nahmad-Achar, O. Castaños and R. López-Peña, Phys. Scr. 92 (4), 044004 (2017). [2] S. Cordero, E. Nahmad-Achar, R. López-Peña and O. Castaños, Phys. Rev. A 92, 053843 (2015). [3] S. Cordero, O. Castaños, R. López-Peña and E. Nahmad-Achar, Phys. Rev. A 94, 013802 (2016). [4] S. Cordero, O. Castaños, R. López-Peña and E. Nahmad-Achar, Phys. Rev. A 99, 033811 (2019). [5] S. Cordero, E. Nahmad-Achar, O. Castaños and R. López-Peña, Phys. Rev. A 100, 053810 (2019).

Information theory is agnostic about the subject matter of the information that it studies and it is, therefore, by its nature very versatile. It is, therefore, unsurprising that information theory provides useful tools throughout engineering and physics including even quantum gravity. But the fact that information theory applies equally to all subject matter may also indicate that information theory may be more than versatile, namely universal. Perhaps, information theory could be universal, with quantum gravity emerging from it. I will start with a brief discussion of tools that information theory can provide to quantum gravity. I will then address the question how spacetime, matter and their dynamics could be emergent from information theory.

In the GFT formulation of quantum gravity, the universe is described as a quantum many-body system with basic entities being quantum simplices, glued to form extended structures by entanglement. Quantum gravity states are then generalised tensor networks, and exhibit a discrete entanglement/geometry correspondence. The emergent cosmological dynamics for the same system takes the form of condensate hydrodynamic equations on superspace, thus a non-linear extension of quantum cosmology. This prompts the exploration of general maps between the hydrodynamics of quantum fluids and cosmology, which had in fact appeared independently in the mathematical physics literature, further corroborated by the discovery of hidden symmetries in cosmological dynamics, which match those of condensate hydrodynamics. A key ingredient is the relational understanding of space and time, which makes superspace the natural arena for gravitational dynamics, as opposed to the "spacetime" manifold. These results, and the perspective they suggest, have also potential implications for analogue gravity systems in the lab.

Consider an operational formulation of Quantum Field Theory in which operators are associated with arbitrary regions of spacetime. If we jiggle some input to one part of the boundary of this region, we want to be sure no information is transmitted to a second part of the boundary if this second part lies outside of the forward light cone of the first part. This leads to a constraint on the allowed operators that can be associated with any given arbitrary region of spacetime. What are these constraints? I will summarise an approach to this building on the work discussed in $\texttt{https://arxiv.org/abs/1807.10980}$ (see Part III and the appendix). This approach takes the iterative conditions of Chribella, D'Ariano, and Perinotti imposed on the operators associated with quantum combs as the basis for causality conditions associated with operators associated with arbitrary regions of spacetime.

We give an introduction to the positive formalism with emphasis on its role in providing a timeless formulation of quantum theory.

The current theories of quantum physics and general relativity alone do not allow us to study situations where the gravitational source is quantum. In my talk, I will propose a strategy to determine the dynamics of probe quantum systems in the presence of mass configurations in superposition, and thus an indefinite spacetime metric, using quantum reference frame (QRF) transformations. In particular, I will establish the formalisms that allow us to move from a QRF where the metric is indefinite to a QRF where the metric is definite. Assuming the covariance of the dynamical laws under the QRF transformation, this will transform the problem of the dynamics of probe quantum systems in indefinite metrics into a physically equivalent problem of the dynamics in a definite metric.

The detection of a rotation, a well-known metrological problem with diverse applications such as in marine navigation or the measurement of small magnetic fields, can be optimized via quantum systems through interference experiments. The most susceptible states of a quantum system under the action of a specific rotation are called optimal quantum rotosensors. For example, these states in the multiqubit systems are the so-called Greenberger-Horne-Zeilinger (GHZ) states. In this talk, we explain how to calculate the optimal quantum rotosensors for a multiqudit-system around a given axis or uniformly averaged over all axes. Making use of a generalization of the Majorana representation for spin (equivalent to a single qudit) systems, we study the cases of fully symmetric or antisymmetric multiqudit systems.

In this talk, three no-go theorems are explored for the existence of functors betweeen categories that are important for fundamental physics. The first concerns spinors: It is shown how spinors cannot and how they can be defined on metric-independent bundles. The second concerns natural metrics on categories of Lorentzian metrics. The third and most central one concerns quantization relations (relations between classical and quantum observables). Here it is shown, in a vast generalization of the Groenewold-van Hove theorem, how minimal assumptions (quantum objects as operators on a Hilbert space and the von Neumann rule) already exclude linearity of any quantization relation. Finally, we suggest an experiment to test the degree of linearity of the quantization relation.