Session F65: Quantum Field Theories and Foundations

Quantum Cellular Automata as Quantum Fields: Interactions, Locality, and Negative Energy States

It has been shown that quantum walks (QWs) on a lattice can give rise to relativistic wave equations like the Dirac equation for fermions and the Klein-Gordon or Maxwell equations for bosons in the long wavelength limit. In the many-particle case, quantum cellular automata (QCAs) can produce free quantum field theories (fermionic or bosonic) in the long-wavelength limit, at the cost of allowing the local subsystems to become high-dimensional. However, adding a local interaction between bosonic and fermionic QCAs reveals a tension between locality (in the strong sense of QCAs) and positive energy: QCAs generically have both positive and negative energy solutions, and local interactions will in general excite both, making the ``vacuum'' unstable to cascades of positive- and negative-energy particle creation. We discuss the source of this conflict, and possible methods to overcome it, as well as the relationship between the strong notion of locality in QCAs and weaker notions in quantum field theories.   

Entanglement Entropy in Timelike Slices: a Free Fermion Study

We define the entanglement entropy of a quantum state in a discrete set of points in an arbitrary spacetime slice, and give the explicit formula for free fermions. We investigate timelike (causal) slices specifically. For 1D lattice free fermions, we calculated the time-direction entanglement entropy in a set of times t_n=nτ (1≤n≤K) on the same site, and identified a stabilizing transition at τ=τ₀=2π/E₀, where E₀ is the energy range of single-fermion spectrum. At zero temperature, the time-like entanglement entropy of the lattice fermion with τ<τ₀ resembles the Cardy formula for one flavor of chiral fermion. For generic slices, the entanglement entropy shows a clear transition between spacelike and timelike slices. We conjecture τ₀ is the upper bound of time period for consecutive local observations to retrieve information from a quantum state, and conjecture a similar τ₀ exists in interacting models.   

Spacetime mutual information

Quantum mutual information is the measure of correlations between two regions of a system, and is known to bound all the spatial correlations between two such regions. However, in experiments we also measure temporal correlations; where the latter are associated to dynamical properties such as thermalization, transport, and causal propagation of information. An immediate obstacle in generalizing quantum mutual information to time-like separated regions is that such regions are described by two different Hilbert spaces, which makes it less obvious to define what entanglement is. In this talk I will introduce a spacetime generalization of mutual information that overcomes this obstacle and generalizes several of the properties of standard mutual information; in particular it bounds spacetime correlation functions. Moreover, this quantity can be framed as a constrained version of the channel relative entropy used in quantum channel discrimination. I will also show that, in the context of many-body systems, the spacetime mutual information displays different asymptotic behaviors in time depending on whether the system thermalizes and has conserved quantities.   

The Quantum Space-Time Loophole in Quantum Foundations: An Interferometric Probe with Quantum Metrology

The core tenet of quantum mechanics is its rejection of local realism, exemplified by Bell tests. Despite impressive recent constraints on the locality and freedom-of-choice loopholes, including the "cosmic Bell test," one significant loophole remains entirely unaddressed: Every Bell test so far assumed a definite background fabric of space-time, which is in foundational conflict with the background independence of general relativity, our standard theory of space-time and gravity. This intersection of QM and GR is perhaps best characterized by the holographic principle, in which the 2D entropy of black holes, the densest objects possible, sets a universal 2D upper bound on the quantum information of 3D empty space-time. Because information density decreases with scale, there must be some nonlocal correlations in the background space-time, reformulating the locality loophole. Such a bandwidth limit is predicted to cause a precise ruler or clock to see "pixelation," an irreducible indeterminacy of space-time itself, at the scale of a 10m lab fluctuating by a billionth of a single atom. This incredible level of precision is now reachable by use of novel quantum metrology in laser interferometers, which can improve on the latest LIGO technology by as much as an order of magnitude. We present a research program to study this frontier, and a brief overview of our experiment currently being commissioned at Cardiff University.   

Numerical Investigations of Symmetric Informationally Complete POVMs

Symmetric informationally complete quantum measurements or SIC-POVMs are interesting from a number of perspectives. For instance, in QBism, where quantum states are understood as degrees belief rather than objective features of nature, SIC-POVMs give a way of representing the Born rule so that the distinction between it and the classical law of total probability is the minimum possible [DeBrota, Fuchs, and Stacey, Phys. Rev. Res. 2, 013074 (2020)]. SIC-POVMs are also optimal for linear quantum-state tomography and a number of other quantum information applications. In the search for SIC-POVMs, we are most interested in the group covariant case, where the problem boils down to finding a single fiducial vector for generating the whole structure. For finite dimensions d, this amounts to finding a solution to d^2 simultaneous fourth-order polynomial equations generated by the discrete Weyl-Heisenberg group. However, it has been conjectured that it is already enough to satisfy only 3d/2 of the defining equations to find a solution [Appleby, Dang, and Fuchs, Entropy 16, 1484 (2014)]. In this talk, I will present the most up-to-date numerical findings with regard to this conjecture.   

The Group Inverse for Quantum Information Theory

In the tradition of subjectivist probability theory, QBism views quantum states as beliefs on the part of users of quantum theory. What quantum theory adds is a new normative rule, equivalent to the Born rule, for transferring beliefs from one experiment to another. In the case of a so-called minimal informationally complete reference device, implementing this rule amounts to inverting a full-rank conditional probability matrix. In the more general case, frame theory offers a framework for constructing the appropriate linear transformation. We show that this construction turns out to be equivalent to taking the group inverse of a conditional probability matrix, and thus can be calculated without invoking a prior Hilbert space representation. We discuss the group inverse more generally, its history, its properties, and methods of calculation from the point of view of quantum information theory.   

Efficient Quantum Description of Probabilistic Data

Contextuality plays an important role in quantum computing as a potential resource for quantum speedup. Here we investigate contextuality as a resource for quantum communication. Finding the most efficient quantum representation of probabilistic data is known to solve a certain communication complexity problem. We explore the link between contextuality and the most efficient quantum representation.   

No-Go Theroems in Modal Quantum Theory

Modal quantum theory (MQT) is a "toy model" based on finite fields that lacks the probability structure of actual quantum theory (AQT). In essence, MQT retains the principle of superposition for quantum states but not unitarity or state normalization. Many familiar "no-go" results in AQT have analogues in MQT, including the no-cloning, no-deleting and no-broadcasting theorems. Other results, such as the no-hiding theorem, are actually false in MQT. This talk gives a brief overview of these results.   

Interpretation of Quantum Theory: the quantum "grue-bleen" problem

DeWitt described the many-worlds interpretation by stating that "The mathematical formalism of the quantum theory is capable of yielding its own interpretation."  Can this be true?  We introduce a general framework applicable to many kinds of dynamical theories, asking whether a system's state and time evolution alone can support non-trivial interpretational statements.  For many theories, including both quantum mechanics and classical Hamiltonian mechanics, the answer is no.  Wallace and others have noted the need for an additional "frame" structure to support even a many-worlds interpretation of quantum mechanics.  We argue that such structure cannot (for example) have its origin in the dynamics of the system.  

 

 

    

Properties of the Probability Current in Purely Probabilistic Representations of Quantum Theory

Braasch and Wootters have recently explored properties of the quasi-probability current for discrete Wigner-function (quasi-probability) representations of finite dimensional quantum theory. In this talk, I will consider analogous properties for a purely probabilistic representation of the same setting, i.e., one with true-blue probabilities. Of most interest is a representation induced by the Symmetric Informationally Complete Positive Operator-Valued Measure (SIC-POVMs), which is particularly important for QBism. I will then generalize the findings there to any Informationally Complete (IC) POVM representation and motivate a notion of gauge invariance for the von Neumann equation.   

The Grasshopper Problem

A unit sphere is covered by a lawn such that one of every pair of antipodal points belongs to the lawn. A grasshopper lands on the lawn and then makes a jump with a fixed distance in a random direction. What is the optimal lawn shape such that the grasshopper will have the best chance of landing on the lawn again? This problem has surprising connections to statistical physics and quantum information, specifically Bell’s inequalities. In this setup two parties measure spins about randomly chosen axes and obtain correlations for pairs of axes separated by a fixed angle. By discretizing the lawn to a set of spins it is possible to use numerical methods to optimize the lawn shape. We find that at smaller jump distances the optimal lawns resemble cogwheels, similarly to the planar case. At larger jumps we find shapes such as stripes and increasingly complex labyrinths. We will discuss analytical and numerical results for the spherical grasshopper problem, as well as the connection to Bell’s inequalities involving random measurement choices.