Quantum Information

The basic problem of much of condensed matter and high energy physics, as well as quantum chemistry, is to find the ground state properties of some Hamiltonian. Many algorithms have been invented to deal with this problem, each with different strengths and limitations. Ideas such as entanglement entropy from quantum information theory and quantum computing enable us to understand the difficulty of various problems. I will discuss recent results on area laws and use these to prove that we can use matrix product states to efficiently represent ground states for one-dimensional systems with a spectral gap, while certain other one-dimensional problems, without the gap assumption, almost certainly have no efficient way for us to even represent the ground state on a classical computer. I will also discuss recent results on higher-dimensional matrix product states, in an attempt to extend the remarkable success of matrix product algorithms beyond one dimension.

In this talk, we will investigate the distinguishability of quantum operations from both discrete and continuous point of view. In the discrete case, the main topic is how we can identify quantum measurement apparatuses by considering the patterns of measurement outcomes. In the continuous case, we will focus on the efficiency of parameter estimation of quantum operations. We will discuss several methods that can achieve Heisenberg Limit and prove in some other cases the impossibility of breaking the Standard Quantum Limit. The general treatment of estimation of quantum operations also allows an investigation of the effect of noise on estimation efficiency.

This is a talk in two parts. The first part is on evolution of a system under a Hamiltonian. First, a general method for implementing evolution under a Hamiltonian using entanglement and classical communication is presented. This method improves on previous methods by requiring less entanglement and communication, as well as allowing more general Hamiltonians to be implemented. Next, a method for simulating evolution under a sparse Hamiltonian using a quantum computer is presented. When H acts on n qubits, and has at most a constant number of nonzero entries in each row/column, we may select any positive integer k such that the simulation requires O((log*n)t^(1+1/2k)) accesses to matrix entries of H. The second part of the talk is on adaptive measurements of optical phase. Standard measurement schemes, using each resource independently, lead to a phase uncertainty that scales as 1/sqrt(N). It has long been conjectured that it should be possible to achieve a precision limited only by the Heisenberg uncertainty principle, dramatically improving the scaling to 1/N. I present a Heisenberg-limited phase estimation procedure which has been demonstrated experimentally. We use multiple applications of the phase shift on unentangled single-photon states, and generalize Kitaev\\\'s phase estimation algorithm using adaptive measurement theory to achieve a standard deviation scaling at the Heisenberg limit.

We define a measure of the quantumness of correlations, based on the operative task of local broadcasting of a bipartite state. Such a task is feasible for a state if and only if it corresponds to a joint classical probability distribution, or, in other terms, it is strictly classically correlated. A gap, defined in terms of quantum mutual information, can quantify the degree of failure in fulfilling such a task, therefore providing a measure of how non-classical a given state is. We are led to consider the asymptotic average mutual information of a state, defined as the minimal per-copy mutual information between parties, when they share an infinite amount of broadcast copies of the state. We analyze the properties of such quantity, and find that it satifies many of the properties required for an entanglement measure. We show that it lies between the quantum- and the classical-conditioned versions of squashed entanglement. The non-vanishing of asymptotic average mutual information for entangled states may be interpreted as a signature of monogamy of entanglement.

We use a Bayesian approach to optimally solve problems in noisy binary search. We deal with two variants: 1. Each comparison can be erroneous with some probability 1 - p. 2. At each stage k comparisons can be performed in parallel and a noisy answer is returned. We present a (classic) algorithm which optimally solves both variants together, up to an additive term of O(log log (n)), and prove matching information theoretic lower bounds. We use the algorithm with the results of Farhi et al. (FGGS99)presenting a quantum search algorithm in an ordered list of expected complexity less than log(n)/3, and some improved quantum lower bounds on noisy search, and search with an error probability. Joint work with Michael Ben-Or.

Roughly speaking, the more Alice is entangled with Bob, the harder it is for her to send her state to Charlie. In particular, it will be shown that the squashed entanglement, a well known entanglement measure, gives the fastest rate at which a quantum state can be sent between two parties who share arbitrary side information. Likewise, the entanglement of formation and entanglement cost is shown to be the fastest rate at which a quantum state can be sent when the parties have access to side-information which is maximally correlated. A further restriction on the type of side-information implies that the rate of state transmission is given by the quantum mutual information. This suggests a new paradigm for understanding entanglement and other correlations in terms of quantum Shannon theroy. Different types of side-information correspond to different types of correlations with the squashed entanglement and the mutual information being two extremes. Furthermore, there is a dual paradigm: if one distributes the side-information as aliciously as possible so as to make the sending of the state as difficult as possible, one finds maximum rates which give interpretations to known quantities as well as new ones.

Do ideas about information and reality inspire fruitful new approaches to the hardest problems of modern physics? What can we learn about the paradoxes of quantum mechanics, the beginning of the universe and our understanding of black holes by thinking about the very essence of information? The answers to these questions are surprising and enlightening, but also controversial. The topic of information within physics has involved some of the 20th century\'s greatest scientists in long-running intellectual battles that continue to the present day. In this special debate, hosted by the CBC\'s Bob McDonald of \'Quirks and Quarks\', you will enjoy a lively discussion between four prominent physicists who have thought long and hard about these questions. information, quantum mechanics, quantum cryptology, entropy, everything computes, properties equals information, uncertainty principle, quantum computer, hologram, black hole, event horizon, coherence, Schrodinger, interference and predictability, quantum state, teleportation, entanglement

One of the most significant questions in quantum information is about the origin of the computational power of the quantum computer; namely, from which feature of quantum mechanics and how does the quantum computer obtain its superior computational potential compared with the classical computer? In my talk, I address this open question more concisely through the study of measurement-based quantum computer, in which all the quantum resource is attributed to entanglement since computation is carried through its consumption by local measurements. I also show a simple model of the phase transition of quantum computer occurring at some threshold, below which the quantum computer comes to allow an efficient classical simulation in accordance with an exponential drop in the amount of entanglement.

Quantum fields in the Minkowski vacuum are entangled with respect to local field modes. This entanglement can be swapped to spatially separated quantum systems using standard local couplings. A single, inertial field detector in the exponentially expanding (de Sitter) vacuum responds as if it were bathed in thermal radiation in a Minkowski universe. Using two inertial detectors, interactions with the field in the thermal case will entangle certain detector pairs that would not become entangled in the corresponding de Sitter case.The two universes can thus be distinguished by their entangling power.

First, a brief description of Bell\'s theorem will be given. It states states that there is no classical-like (local realistic) description of all quantum predictions. Next, a plausible class of non-local realistic models will be presented which is incompatible with quantum mechanics, as first shown by Leggett. Experiments confirming the incompatibility will be described. Finally, it will be argued that quantum mechanics can be seen as a theory of systems with limited information resources.

We attempt at characterizing the correlations present in the quantum computational model DQC1, introduced by Knill and Laflamme [Phys. Rev. Lett. 81, 5672 (1998)]. The model involves a collection of qubits in the completely mixed state coupled to a single control qubit that has nonzero purity. Although there is little or no entanglement between two parts of this system, it provides an exponential speedup in certain problems. On the contrary, we find that the quantum discord across the most natural split is nonzero for typical instances of the DQC1 ciruit. Nonzero values of discord indicate the presence of nonclassical correlations. We propose quantum discord as figure of merit for characterizing the resources present in this computational model. This might be a complementary measure for counting resources in quantum information science.

A class of operations distinct to entangled states shared between more than two parties is their conversion to entangled states shared between fewer parties. The extent to which these can be achieved in the regime of local operations and classical communication provides an operational characterisation of multiparty states, for example in the \"entanglement of assistance\" and related quantities. I will give a brief overview of this topic and discuss our results showing qualitatively different behaviour when the parties receiving the final state are not chosen beforehand.