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Kolmogorov complexity is a measure of the information contained in a binary string. We investigate the notion of quantum Kolmogorov complexity, a measure of the information required to describe a quantum state. We show that for any definition of quantum Kolmogorov complexity measuring the number of classical bits required to describe a pure quantum state, there exists a pure n-qubit state which requires exponentially many bits of description. This is shown by relating the classical communication complexity to the quantum Kolmogorov complexity. Furthermore we give some examples of how quantum Kolmogorov complexity can be applied to prove results in different fields, such as quantum computation and communication.

Inferring a quantum system\'s state, from repeated measurements, is critical for verifying theories and designing quantum hardware. It\'s also surprisingly easy to do wrong, as illustrated by maximum likelihood estimation (MLE), the current state of the art. I\'ll explain why MLE yields unreliable and rank-deficient estimates, why you shouldn\'t be a quantum frequentist, and why we need a different approach. I\'ll show how operational divergences -- well-motivated metrics designed to evaluate estimates -- follow from quantum strictly proper scoring rules. This motivates Bayesian Mean Estimation (BME), and I\'ll show how it fixes most of the problems with MLE. I\'ll conclude with a couple of speculations about the future of quantum state and process estimatio

In stochastic treatments of the ERRB set-up, it is equivalent to impose Bell\'s inequalities, a local causality condition, or a certain \"non-contextual hidden variables\" condition. But these conditions are violated by quantum mechanics. On the other hand, it is possible to view quantum mechanics as part of \"quantum measure theory\", a generalization of probability measure theory that allows pair wise interferences between histories whilst banning higher order interference. In this setting, is may be possible find quantum analogues of the three stochastic conditions. Following this line of inquiry, we will see that quantum measure theory allows no stronger violations of Bell\'s inequalities than does standard quantum theory. We also gain some insights into how to define causality in quantum theory.

Quantum information theory has two equivalent mathematical conjectures concerning quantum channels, which are also equivalent to other important conjectures concerning the entanglement. In this talk I explain these conjectures and introduce recent results.

It is a fundamental property of quantum mechanics that non-orthogonal pure states cannot be distinguished with certainty, which leads to the following problem: Given a state picked at random from some ensemble, what is the maximum probability of success of determining which state we actually have? I will discuss two recently obtained analytic lower bounds on this optimal probability. An interesting case to which these bounds can be applied is that of ensembles consisting of states that are themselves picked at random. In this case, I will show that powerful results from random matrix theory may be used to give a strong lower bound on the probability of success, in the regime where the ratio of the number of states in the ensemble to the dimension of the states is constant. I will also briefly discuss applications to quantum computation (the oracle identification problem) and to the study of generic entanglement.

Ancillary state construction is a necessary component of quantum computing. Ancillae are required both for error correction and for performing universal computation in a fault-tolerant way. Computation to an arbitrary accuracy, however, is effectively achieved by increasing the number of qubits in order to suppress the variance in the expected number of errors. Thus, it is important to be able to construct very large ancillary states. Concatenated quantum coding provides a means of constructing ancillae of any size, but, this fact aside, concatenation is not a particularly efficient form of coding. More efficient codes exist, but these codes lack the substructure of concatenated codes that enables fault-tolerant preparation of large ancillae. In this talk I will discuss the advantages of coding in large blocks, both from the perspective of efficiency and analysis, and I will describe my progress in developing construction procedures for moderately large ancillae.

The Everett (many-worlds) interpretation has made great progress over the past 20-30 years, largely due to the role of decoherence in providing a solution to the preferred basis problem. This makes it a serious candidate for a realist solution to the measurement problem. A remaining objection to the Everett interpretation (and one that is often considered fatal) is that that interpretation cannot make adequate sense of quantum probabilities. Dvaid Deutsch and David Wallace have argued that, by applying decision theory to the case of a rational agent who believes in the many-worlds interpretation, we can prove that such agents _act as if_ the theory predicted objective probabilities in the sense of fundamental indeterminism, or ignorance of initial conditions. I raise the issue of whether or not this, if true, is all that the many-worlds theorist needs from \'probability\'. I first suggest a reason for thinking that the answer might be \'no\': the reason is that knowing how to act on the assumption that a given theory is true is prima facie irrelevant to the question of whether we have any reason to believe the theory in the first place. I then go on to offer a solution to this problem, drawing on resources from Bayesian confirmation theory. My conclusion is that the problem of probability in the Everett interpretation has been solved.

We explore the role of rotational symmetry of quantum key distribution (QKD) protocols in their security. Specifically, in the first part of the talk, we consider a generalized QKD protocol with discrete rotational symmetry. Note that, before our work, each QKD protocol seems to have a different security proof. Given that the techniques of those proofs are similar, it will be interesting to have a unified proof for QKD protocols with symmetry (e.g., the BB84 protocol and the SARG04 protocol). This is exactly what we achieve in our work. We show that rotational symmetry plays an important role in the unified security proof of QKD protocols with symmetry, leading to simple and structural security relations. In the second part, we consider a QKD protocol that does not possess rotational symmetry and analyze its security. Interestingly, even without any rotational symmetry, this protocol can still be proven secure. However, the security relation is not as simple as those in the first part, due to the lack of symmetry. Therefore, although rotational symmetry is not required in a QKD protocol to ensure its security, rotational symmetry does provide significant simplification in the security analysis, leading to simple security relations.