Quantum Physics

The purpose of this talk is to describe bosonic fields and their Lagrangians in the causal set context. Spin-0 fields are defined to be real-valued functions on a causal set. Gauge fields are viewed as SU(n)-valued functions on the set of pairs of elements of a causal set, and gravity is viewed as the causal relation itself. The purpose of this talk is to come up with expressions for the Lagrangian densities of these fields in such a way that they approximate the Lagrangian densities expected from regular Quantum Field Theory on a differentiable manifold in the special case where the causal set is a random sprinkling of points in the manifold. I will then conjecture that that same expression is appropriate for an arbitrary causal set.

I should like to show how particular mathematical properties can limit our metaphysical choices, by discussing old and new theorems within the statistical-model framework of Mielnik, Foulis & Randall, and Holevo, and what these theorems have to say about possible metaphysical models of quantum mechanics. Time permitting, I should also like to show how metaphysical assumptions lead to particular mathematical choices, by discussing how the assumption of space as a relational concept leads to a not widely known frame-invariant formulation of classical point-particle mechanics by Föppl and Zanstra, and related research topics in continuum mechanics and general relativity.

Mutually unbiased bases (MUBs) have attracted a lot of attention the last years. These bases are interesting for their potential use within quantum information processing and when trying to understand quantum state space. A central question is if there exists complete sets of N+1 MUBs in N-dimensional Hilbert space, as these are desired for quantum state tomography. Despite a lot of effort they are only known in prime power dimensions. I will describe in geometrical terms how a complete set of MUBs would sit in the set of density matrices and present a distance between bases–a measure of unbiasedness. Then I will explain the relation between MUBs and Hadamard matrices, and report on a search for MUB-sets in dimension N=6. In this case no sets of more than three MUBs are found, but there are several inequivalent triplets.

The \\\"frequency comb\\\" defined by the eigenmodes of an optical resonator is a naturally large set of exquisitely well defined quantum systems, such as in the broadband mode-locked lasers which have redefined time/frequency metrology and ultra precise measurements in recent years. High coherence can therefore be expected in the quantum version of the frequency comb, in which nonlinear interactions couple different cavity modes, as can be modeled by different forms of graph states. We show that is possible to thereby generate states of interest to quantum metrology and computing, such as multipartite entangled cluster and Greenberger-Horne-Zeilinger states.

The Achilles\\\' heel of quantum information processors is the fragility of quantum states and processes. Without a method to control imperfection and imprecision of quantum devices, the probability that a quantum computation succeed will decrease exponentially in the number of gates it requires. In the last ten years, building on the discovery of quantum error correction, accuracy threshold theorems were proved showing that error can be controlled using a reasonable amount of resources as long as the error rate is smaller than a certain threshold. We thus have a scalable theory describing how to control quantum systems. I will briefly review some of the assumptions of the accuracy threshold theorems and comment on experiments that have been done and should be done to turn quantum error correction into an experimental reality.

We construct a simple translationally invariant, nearest-neighbor Hamiltonian on a chain of 10-dimensional qudits that makes it possible to realize universal quantum computing without any external control during the computational process, requiring only initial product state preparation. Both the quantum circuit and its input are encoded in an initial canonical basis state of the qudit chain. The computational process is then carried out by the autonomous Hamiltonian time evolution. After a time greater than a polynomial in the size of the quantum circuit has passed, the result of the computation can be obtained with high probability by measuring a few qudits in the computational basis. This result also implies that there cannot exist efficient classical simulation methods for generic translationally invariant nearest-neighbor Hamiltonians on qudit chains, unless quantum computers can be efficiently simulated by classical computers (or, put in complexity theoretic terms, unless BPP=BQP). This is joint work with Daniel Nagaj.

The renormalization group (RG) is one of the conceptual pillars of statistical mechanics and quantum field theory, and a key theoretical element in the modern formulation of critical phenomena and phase transitions. RG transformations are also the basis of numerical approaches to the study of low energy properties and emergent phenomena in quantum many-body systems. In this colloquium I will introduce the notion of \\\"entanglement renormalization\\\" and use it to define a coarse-graining transformation for quantum systems on a lattice [G.Vidal, Phys. Rev. Lett. 99, 220405 (2007)]. The resulting real-space RG approach is able to numerically address 1D and 2D lattice systems with thousands of quantum spins using only very modest computational resources. From the theoretical point of view, entanglement renormalization sheds new light into the structure of correlations in the ground state of extended quantum systems. I will discuss how it leads to a novel, efficient representation for the ground state of a system at a quantum critical point or with topological order.

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.

As physicists, we have become accustomed to the idea that a theory\\\'s content is always most transparent when written in coordinate-free language. But sometimes the choice of a good coordinate system is very useful for settling deep conceptual issues. Think of how Eddington-Finkelstein coordinates settled the longstanding question of whether the event horizon of a Schwarzschild black hole corresponds to a real spacetime singularity or not. Similarly we believe for an information-oriented or Bayesian approach to quantum foundations: That one good coordinate system may (eventually!) be worth more than a hundred blue-in-the-face arguments. This talk will motivate and chronicle the search for one such candidate coordinate system---the so-called Symmetric Informationally Complete Measurement---which has caught the attention of a handful of us here at PI and a handful of our visitors.

Bell\\\'s theorem is commonly understood to show that EPR correlations are not explainable via a local hidden variable theory. But Bell\\\'s theorem assumes that the initial state of the particles is independent of the final detector settings. It has been proposed that this independence assumption might be undermined by a relativistically-allowed form of \\\"backward causation\\\", thereby allowing construction of a local hidden-variable model after all. In this talk, I will show that there is no backward causation model which yields the desired correlations. However, there are other physical scenarios yielding nontrivial nonlocal correlations which violated Bell\\\'s independence assumption. I will present two.

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.