Quantum Information

I will outline our recent approach to a theory of quantum networks, which we base on the graphical tensor calculus of Penrose. We aim to use this approach as a means to unite quantum computation, condensed matter physics and tensor network states with a long history of existing methods and techniques. I will sketch the status of our approach and focus on methods we have developed to reason and control the states created by specific networks of contracted tensors, as well as tensor contractions capturing the key properties of algebraic invariants of operators and states.

Continuous-variable SICPOVMS seem unlikely to exist, for a variety of reasons. But that doesn't rule out the possibility of other 2-designs for the continuous-variable Hilbert space L2(R). In particular, it would be nice if the coherent states -- which form a rather nice 1-design -- could be generalized in some way to get a 2-design comprising *Gaussian* states. So the question is: "Can we build a 2-design out of Gaussian states?". The answer is "No, but in a very surprising way!" Like coherent states, Gaussian states have a natural transitive symmetry group. For coherent states, it's the Heisenberg group. For Gaussian states, it's the affine symplectic group -- the Heisenberg group plus squeezings and rotations. And this group acts irreducibly on the symmetric subspace of L2(R) x L2(R)... which, by Schur's Lemma, implies that the Gaussian states *should* be a 2-design. Yet a very simple explicit calculation shows that they are not! The resolution is fascinating -- it turns out that the "symplectic twirl" involves an integral that does not quite converge, and this provides a loophole out of Schur's Lemma. So, in the end, we: (1) Show that Gaussian 2-designs do not exist, (2) Demonstrate a major stumbling block to *any* symplectic-covariant 2-designs for L2(R), (3) Gain a pretty complete understanding of *one* of the [formerly] mysterious discrepancies between discrete and continuous Hilbert spaces.

This talk presents sufficient conditions for equilibration and thermalization of subsystems within closed many body quantum systems. That is, we identify when the local properties of the equilibrium state resemble those of a thermal state. With this aim, the recent progress in this field is reviewed and we introduce a novel perturbation technique for a realistic weak coupling between the subsystem and its environment. Unlike the standard perturbation theory, our technique is robust in the thermodynamic limit. Based on our thermalization results, we construct a simple and fully general quantum algorithm for preparing Gibbs states with a certified runtime and error bonds.

In my talk I raise the question of the fundamental limits to the size of thermal machines - refrigerators, heat pumps and work producing engines - and I will present the smallest possible ones. I will also discuss the issue of a possible complementarity between size and efficiency and show that even the smallest machines could be maximally efficient. Finally I will present a new point of view over what is work and what do thermal machines actually do.

One of the key features of the quantum Hall effect (QHE) is the fractional charge and statistics of quasiparticles. Fractionally charged anyons accumulate non-trivial phases when they encircle each other. In some QHE systems an unusual type of particles, called non-Abelian anyons, is expected to exist. When one non-Abelian particle makes a circle around another anyon this changes not only the phase but even the direction of the quantum-state vector in the Hilbert space. This property makes non-Abelian anyons promising for fault-tolerant quantum computation. Several experiments allowed an observation of fractional charges. Probing exchange statistics is more difficult and has not been accomplished for identical anyons so far. We will discuss how the statistics can be probed with Mach-Zehnder interferometry, tunneling experiments and far-from-equilibrium fluctuation-dissipation theorem.

In many physical scenarios, close relations between the bulk properties of quantum systems and theories associated to their boundaries have been observed. In this work, we provide an exact duality mapping between the bulk of a quantum spin system and its boundary using Projected Entangled Pair States (PEPS). This duality associates to every region a Hamiltonian on its boundary, in such a way that the entanglement spectrum of the bulk corresponds to the excitation spectrum of the boundary Hamiltonian. We study various specific models, like a deformed AKLT , an Ising-type , and Kitaev's toric code, both in finite ladders and infinite square lattices. In the latter case, some of those models display quantum phase transitions. We find that a gapped bulk phase with local order corresponds to a boundary Hamiltonian with local interactions, whereas critical behavior in the bulk is reflected on a diverging interaction length of the boundary Hamiltonian. Furthermore, topologically ordered states yield non-local Hamiltonians. As our duality also associates a boundary operator to any operator in the bulk, it in fact provides a full holographic framework for the study of quantum many-body systems via their boundary. Work done in collaboration with Didier Poilblanc, Norbert Schuch, and Frank Verstraete.

A symmetric informationally complete positive-operator-valued measure (SIC POVM) is a special POVM that is composed of d^2 subnormalized pure projectors with equal pairwise fidelity. It may be considered a fiducial POVM for reasons of its high symmetry and high tomographic efficiency. Most known SIC POVMs are covariant with respect to the Heisenberg-Weyl (HW) group. We show that in prime dimensions the HW group is the unique group that may generate a SIC POVM. In particular, in prime dimensions not equal to three, each group covariant SIC POVM is covariant with respect to a unique HW group. In addition, the symmetry group of the SIC POVM is a subgroup of the Clifford group, which is the normalizer of the HW group. Hence, two SIC POVMs covariant with respect to the HW group are unitarily equivalent if and only if they are on the same orbit of the Clifford group. In dimension three, each group covariant SIC POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC POVM is a subgroup of at least one of the Clifford groups of these HW groups respectively. There may exist two or three orbits of equivalent SIC POVMs for each group covariant SIC POVM, depending on the order of its symmetry group.

Two-party Bell correlation inequalities (that is, inequalities involving only correlations between dichotomic observables at each site, such as the CHSH inequality) are well-understood: Grothendieck's inequality stipulates that the quantum bias can only be a constant factor larger than the classical bias, and the maximally entangled state is always the most nonlocal resource. In part due to the complex nature of multipartite entanglement, tripartite inequalities are much more unwieldy. In a recent breakthrough result, Perez-Garcia et. al. (quant-ph/0702189) showed using tools originating from the study of operator algebras that in this setting the quantum-classical violation could be arbitrarily large. Moreover, they showed that GHZ states could only lead to bounded violations, so that they were not the most non-local states. We extend and simplify their results in a number of ways: - We show that large families of states, including generalizations of GHZ states and stabilizer states, can only lead to bounded violations. - We prove bounds on the maximal quantum-classical violation as a function both of the local dimension (this was already shown in Perez-Garcia et. al., but we give a much simpler proof), and of the number of settings per site. - We provide a simple probabilistic construction of an inequality for which there is an unbounded quantum-classical gap. Our construction is simpler, and has better parameters, than the one in Perez-Garcia et.al. It is essentially optimal in terms of the local dimension of one of the parties, and off by a quadratic factor in terms of the number of settings. In this talk I will survey some of these results, focusing on the tools that have so far been useful for their analysis (and do not involve operator algebras!). Based on joint works with Jop Briet, Harry Buhrman, and Troy Lee. Some of this work is available at arXiv:0911.4007.

Many fundamental results in quantum foundations and quantum information theory can be framed in terms of information-theoretic tasks that are provably (im)possible in quantum mechanics but not in classical mechanics. For example, Bell's theorem, the no-cloning and no-broadcasting theorems, quantum key distribution and quantum teleportation can all naturally be described in this way. More generally, quantum cryptography, quantum communication and quantum computing all rely on intrinsically quantum information-theoretic advantages. Much less attention has been paid to the information-theoretic power of relativistic quantum theory, although it appears to describe nature better than quantum mechanics. This talk describes some simple information-theoretic tasks that distinguish relativistic quantum theory from quantum mechanics and relativistic classical physics, and a general framework for defining tasks that includes all previously known (im)possibility theorems and raises many open questions. This suggests a new way of thinking about relativistic quantum theory, and a possible new approach to defining non-trivial relativistic quantum theories rigorously. I also describe some simple and surprisingly powerful applications of these ideas to cryptography, including a new secure scheme for simultaneously committing to and encrypting a prediction and ways of securely "tagging" an inaccessible object so as to guarantee its position.

We review situations under which a standard quantum adiabatic condition fails. We reformulate the problem of adiabatic evolution as the problem of Hamiltonian eigenpath traversal, and give convergence conditions in terms of the length of the eigenpath and the minimum energy gap of the Hamiltonians. We introduce a randomized evolution method that can be used to traverse the eigenpath and prove its convergence and cost. We then describe more efficient methods for the same task and show that their implementation complexity is close to optimal.