Quantum Information

In this talk we discuss how large classes of classical spin models, such as the Ising and Potts models on arbitrary lattices, can be mapped to the graph state formalism. In particular, we show how the partition function of a spin model can be written as the overlap between a graph state and a complete product state. Here the graph state encodes the interaction pattern of the spin model---i.e., the lattice on which the model is defined---whereas the product state depends only on the couplings of the model, i.e., the interaction strengths. As main examples, we find that the 1D Ising model corresponds to the 1D cluster state, the 2D Ising model without external field is mapped to Kitaev's toric code state, and the 2D Ising model with external field corresponds to the 2D cluster state---but the mappings are completely general in that arbitrary graphs, and also q-state models can be treated. These mappings allow one to make connections between concepts in (classical) statistical mechanics and quantum information theory and to obtain a cross-fertilization between both fields. As a main application, we will prove that the classical Ising model on a 2D square lattice (with external field) is a "complete model", in the sense that the partition function of any other spin model---i.e., for q-state spins on arbitrary lattices---can be obtained as a special instance of the (q=2) 2D Ising partition function with suitably tuned (complex) couplings. This result is obtained by invoking the above mappings from spin models to graph states, and the property that the 2D cluster states are universal resource states for one-way quantum computation. Joint work with Wolfgang Duer and Hans Briegel, see PRL/ 98 117207 (2007)/ and quant-ph/0708.2275. For related work, see also S. Bravyi and R. Raussendorf, quant-ph/0610162.

In this talk, I describe some very simple but significant phenomena predicted by quantum theory. These are described simply in terms of what is observed in the laboratory, without making any presumptions about what sort of microscopic picture of reality might account for these observations. With these phenomena in hand, I demonstrate two simple applications of quantum theory to cryptography: how to build counterfeit-proof money and how to detect eavesdroppers on a channel (and thereby distribute a secret key which can be used to encode messages in a way that cannot be deciphered by one who does not have the key). Moving from quantum information theory to quantum foundations, I show how the qualitative features of these phenomena can be reproduced in a toy theory wherein systems have well-defined properties but observers can only come to know a limited amount about them. This shows the merits of the notion of \"hidden variables\" underlying quantum theory. Finally, I describe a phenomenon that is predicted by quantum theory but that *cannot* be reproduced by a natural class of hidden variable models, namely, those that are *local* in the sense that changes in one region cannot instantaneously affect the state of affairs in another. The phenomenon in question is the existence of certain strange correlations between the measurement outcomes on distant systems. It is illustrated in terms of a two-party game that is played out by a few lucky members of the audience.

The world at the size of individual atoms obeys very different laws of physics from those we are used to in the everyday world around us. Quantum mechanics rules, allowing atoms to be, in some sense, in more than one place at a time. Researchers all over the world are working to build \"quantum computers\" whose memories manipulate an inherently new type of information, \"quantum information.\" Quantum computers have capabilities impossible for a regular computer, no matter how advanced it may be, including the ability to run new kinds of computer programs capable of rapidly solving certain problems, and to make new highly secure codes for secret communication.

One simple way to think about physics is in terms of information. We gain information about physical systems by observing them, and with luck this data allows us to predict what they will do next. Quantum mechanics doesn\'t just change the rules about how physical objects behave - it changes the rules about how information behaves. In this talk we explore what quantum information is, and how strangely it differs from our intuitions. In particular we see how information about quantum particles can become entangled, leading to seemingly impossibly coordinated behaviour for separate objects.

The world at the size of individual atoms obeys very different laws of physics from those we are used to in the everyday world around us. Quantum mechanics rules, allowing atoms to be, in some sense, in more than one place at a time. Researchers all over the world are working to build \"quantum computers\" whose memories manipulate an inherently new type of information, \"quantum information.\" Quantum computers have capabilities impossible for a regular computer, no matter how advanced it may be, including the ability to run new kinds of computer programs capable of rapidly solving certain problems, and to make new highly secure codes for secret communication.

One simple way to think about physics is in terms of information. We gain information about physical systems by observing them, and with luck this data allows us to predict what they will do next. Quantum mechanics doesn\'t just change the rules about how physical objects behave - it changes the rules about how information behaves. In this talk we explore what quantum information is, and how strangely it differs from our intuitions. In particular we see how information about quantum particles can become entangled, leading to seemingly impossibly coordinated behaviour for separate objects.