We prove that any three linearly independent pure quantum states can always be locally distinguished with nonzero probability regardless of their dimension, entanglement or multipartite structure. Almost always, all three states can be unambiguously identified. The only exceptional case, where one state is locally knowable but the other two are not, is found among multi-qubit states.

A subspace of a multipartite Hilbert space is completely entangled if it contains no product states. Such subspaces can be large with a known maximum size, s(max), approaching the full dimension of the system, D. We show that almost all subspaces with dimension s <= s(max) are completely entangled and then use this fact to prove that n random pure quantum states are unambiguously locally distinguishable if and only if n <= D - s(max). This condition holds for almost all sets of states of all multipartite systems and reveals something surprising. The criterion is identical for separable and nonseparable states: entanglement makes no difference.

Fingerprinting enables two parties to infer whether the messages they hold are the same or different when the cost of communication is high: each message is associated with a smaller fingerprint and comparisons between messages are made in terms of their fingerprints alone. In the simultaneous message passing model, it is known that fingerprints composed of quantum information can be made exponentially smaller than those composed of classical information. For small message lengths, we present constructions of optimal classical fingerprinting strategies with one-sided error, in both the one-way and simultaneous message passing models, and provide bounds on the worst-case error probability with the help of extremal set theory. The performance of these protocols is then compared to that for quantum fingerprinting strategies constructed from spherical codes, equiangular tight frames and mutually unbiased bases.

Bell theorems show how to experimentally falsify local realism. Conclusive falsification is highly desirable as it would provide support for the most profoundly counterintuitive feature of quantum theory—nonlocality. Despite the preponderance of evidence for quantum mechanics, practical limits on detector efficiency and the difficulty of coordinating space-like separated measurements have provided loopholes for a classical worldview; these loopholes have never been simultaneously closed. A number of new experiments have recently been proposed to close both loopholes at once. We show some of these novel designs fail in the most basic way, by not ruling out local hidden variable models, and we provide an explicit classical model to demonstrate this. They share a common flaw, which reveals a basic misunderstanding of how nonlocality proofs work. Given the time and resources now being devoted to such experiments, theoretical clarity is essential. Our explanation is presented in terms of simple logic and should serve to correct misconceptions and avoid future mistakes. We also show a nonlocality proof involving four participants which has interesting theoretical properties.

Global operations on a quantum system can process information in ways that local operations on the system’s parts cannot. All uses of entanglement in quantum information theory flow from this one fact, from teleportation [1] to Shor’s factoring algorithm [2]. However a fundamental question remain unanswered. When is global information about a quantum system also available locally? This question can be formally posed as a local state discrimination task. Given one copy of a system in one of a known set of quantum states {|ψi〉}, how much ‘which state’ information can be gleaned by local operations and classical communication (LOCC), and how much more information is revealed by global measurements?

Bell theorems show how to experimentally falsify local realism. Conclusive falsification is highly desirable as it would provide support for the most profoundly counterintuitive feature of quantum theory - nonlocality. Despite the preponderance of evidence for quantum mechanics, practical limits on detector efficiency and the difficulty of coordinating space-like separated measurements have provided loopholes for a classical worldview; these loopholes have never been simultaneously closed. A number of new experiments have recently been proposed to close both loopholes at once. We show some of these novel designs fail in the most basic way, by not ruling out local hidden variable models, and we provide an explicit classical model to demonstrate this. They share a common flaw, which reveals a basic misunderstanding of how nonlocality proofs work. Given the time and resources now being devoted to such experiments, theoretical clarity is essential. Our explanation is presented in terms of simple logic and should serve to correct misconceptions and avoid future mistakes. We also show a nonlocality proof involving four participants which has interesting theoretical properties.

Within the simultaneous message passing model of communication complexity, under a public-coin assumption, we derive the minimum achievable worst-case error probability of a classical fingerprinting protocol with one-sided error. We then present entanglement-assisted quantum fingerprinting protocols attaining worst-case error probabilities that breach this bound.

We examine the phenomenon, of emergence, referring particularly to Arthur Pedcocke's ideas on emergence, the self, and spirituality. He believes that the whole of an emergent structure influences the way its parts cohere and that emergent structures (including minds and persons) and their effects are very important. He thereby hopes to remove the reductionist challenge that seeks to understand a whole fully in terms of its parts. We argue that emergent phenomena are not influential in the above sense. The holistic completeness of these structures at their own theoretical level does not substitute for the causal independence Peacocke suggests by the idea of influence. Some computer simulations that generate emergent complexity follow simple and self-contained sets of rules. Peacocke also adheres to a hierarchical account of reality as a series of levels into which matter is organized, running from atoms through molecules to cells and eventually to whole ecosystems. But influential behavior does not respect this ordering. Further, Peacocke's opposition to reductionism is unnecessary; any "completeness" of lower-level models does not imply the redundancy of higher-level descriptions. Emergence transforms reductionism into a constructive and positive principle.

Entanglement is a useful resource because some global operations cannot be locally implemented using classical communication. We prove a number of results about what is and what is not locally possible. We focus on orthogonal states, which can always be globally distinguished. We establish the necessary and sufficient conditions for a general set of 2 x 2 quantum states to be locally distinguishable, and for a general set of 2 x n quantum states to be distinguished given an initial measurement of the qubit. These results reveal a fundamental asymmetry to nonlocality, which is the origin of "nonlocality without entanglement," and we present a very simple proof of this phenomenon.

The anthropic principle, that the universe exists in some sense for life, has persisted in recent religious and scientific thought because it derives from cosmological fact. It has been unsuccessful in furthering our understanding of the world because its advocates tend to impose final metaphysical solutions onto what is a physical problem. We begin by outlining the weak and strong versions of the anthropic principle and reviewing the discoveries that have led to their formulation. We present the reasons some have given for ignoring the anthropic implications of these discoveries and find these reasons wanting—a real phenomenon demands real investigation. Theological and scientific solutions of the problem are then considered and criticized; these solutions provide dead ends for explanation. Finally, we pursue the path that explanation must follow and look at the physical details of the problem. It seems clear that the anthropic principle has been poorly framed. Removing the ambiguities surrounding the meaning of “life” may lead to more profitable investigations.

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