Mathematical models use information from past observations to generate predictions about the future. If two models make identical predictions the one that needs less information from the past to do this is preferred. It is already known that certain classical models (certain Hidden Markov Models called ε-machines which are often optimal classical models) are not in general the preferred ones. We extend this result and show that even optimal classical models (models with minimal internal entropy) in general are not the best possible models (called ideal models). Instead of optimal classical models we can construct quantum models which are significantly better but not yet the best possible ones (i.e. they have a strictly smaller internal entropy). In this paper we show conditions when the internal entropies between classical models and specific quantum models coincide. Furthermore it turns out that this situation appears very rarely. An example shows that our results hold only for the specific quantum model construction and in general not for alternative constructions. Furthermore another example shows that classical models with minimal internal entropy need not to be related to quantum models with minimal internal entropy.
Mathematical models use information from past observations to generate predictions about the future. If two models make identical predictions the one that needs less information from the past to do this is preferred. It is already known that certain classical models (certain Hidden Markov Models called ε-machines which are often optimal classical models) are not in general the preferred ones. We extend this result and show that even optimal classical models (models with minimal internal entropy) in general are not the best possible models (called ideal models). Instead of optimal classical models we can construct quantum models which are significantly better but not yet the best possible ones (i.e. they have a strictly smaller internal entropy). In this paper we show conditions when the internal entropies between classical models and specific quantum models coincide. Furthermore it turns out that this situation appears very rarely. An example shows that our results hold only for the specific quantum model construction and in general not for alternative constructions. Furthermore another example shows that classical models with minimal internal entropy need not to be related to quantum models with minimal internal entropy.
Among the predictive hidden Markov models that describe a given stochastic process, the ϵ-machine is strongly minimal in that it minimizes every Rényi-based memory measure. Quantum models can be smaller still. In contrast with the ϵ-machine ’s unique role in the classical setting, however, among the class of processes described by pure-state hidden quantum Markov models, there are those for which there does not exist any strongly minimal model. Quantum memory optimization then depends on which memory measure best matches a given problem’s circumstance.
To make sense of the world around us, we develop models, constructed to enable us to replicate, describe, and explain the behaviours we see. Focusing on the broad case of sequences of correlated random variables, i.e., classical stochastic processes, we tackle the question of determining whether or not two different models produce the same observable behavior. This is the problem of identifiability. Curiously, the physics of the model need not correspond to the physics of the observations; recent work has shown that it is even advantageous -- in terms of memory and thermal efficiency -- to employ quantum models to generate classical stochastic processes. We resolve the identifiability problem in this regime, providing a means to compare any two models of a classical process, be the models classical, quantum, or `post-quantum', by mapping them to a canonical `generalized' hidden Markov model. Further, this enables us to place (sometimes tight) bounds on the minimal dimension required of a quantum model to generate a given classical stochastic process.
Isolating past information relevant for future prediction is central to quantitative science. Quantum models offer a promising approach, enabling statistically faithful modeling while using less past information than any classical counterpart. Here we introduce a class of phase-enhanced quantum models, representing the most general means of simulating a stochastic process unitarily in causal order. The resulting constructions surpass previous state-of-art methods---both in reducing the information they need to store about the past and in the minimal memory dimension they require to store this information. Moreover, these two features are generally competing factors in optimization---leading to an ambiguity in optimal modeling that is unique to the quantum regime. Our results simultaneously offer quantum advantages for stochastic simulation and illustrate further qualitative differences between classical and quantum notions of complexity.
We show that the main difference between classical and quantum systems can be understood in terms of information entropy. Classical systems can be considered the ones where the internal dynamics can be known with arbitrary precision while quantum systems can be considered the ones where the internal dynamics cannot be accessed at all. As information entropy can be used to characterize how much the state of the whole system identifies the state of its parts, classical systems can have arbitrarily small information entropy while quantum systems cannot. This provides insights that allow us to understand the analogies and differences between the two theories.
In stochastic modeling, the excess entropy – the mutual information shared between a process's past and future – represents the fundamental lower bound of the memory needed to simulate its dynamics. However, this bound cannot be saturated by either classical machines or their enhanced quantum counterparts. Simulating a process fundamentally requires us to store more information in the present than is shared between the past and the future. Here, we consider a generalization of hidden Markov models beyond classical and quantum models, referred to as n-machines, that allow for negative quasiprobabilities. We show that under the collision entropy measure of information, the minimal memory of such models can equal the excess entropy. Our results suggest that negativity can be a useful resource for achieving nonclassical memory advantage.
Cryptographic protocols, though used in a wide range of everyday tasks, are typically built for certain unrealistic ideal conditions in which the user is assumed to work. For example, one such condition that is implicitly assumed is that the user has perfect randomness that she can use as her cryptographic secret key. Another commonly made assumption is that the user runs her cryptographic tasks (such as encryption) on a device that is perfectly secure; that is, its internals are not tampered with or even observed by an adversary.
In this thesis, we focus on designing cryptographic techniques that are secure under less-than-ideal conditions. In order to remove the assumption that users share perfectly random secret keys, we study the problem of privacy amplification: key agreement between two parties who both know a weak secret w, such as a password. The goal of the protocol is to convert this non-uniform secret w into a uniformly distributed string R that is fully secret from the adversary. R may then be used as a key for running symmetric cryptographic protocols (such as encryption or authentication). Because we make no computational assumptions, the entropy in R can come only from w. Thus such a protocol must minimize the entropy loss during its execution, so that R is as long as possible. In this thesis, we present the first protocol for information-theoretic key agreement that has entropy loss linear in the security parameter. The result is optimal up to constant factors. We also extend our results to the case where the users share correlated secrets w and w' . This setting occurs, for example, when the secrets are biometric readings taken at different points of time and is more generally known as information-reconciliation.