## Lectures on Quantum Field Theory for Quantum Information

Oct-Dec 2013

Quantum Information theory has been originally formulated in terms of qubits (and qudits), which are entities best described in ordinary Quantum Mechanics. However, there are natural extensions for them in the context of Quantum Field Theory. As a matter of fact, usual concepts in Quantum Info such as entanglement or teleportation, accept formulations in terms of Quantum Fields. Nevertheless, much more work is needed in order to place Quantum Information in the more rigorous framework of Quantum Field Theory.

The aim of these seminars is mainly to emphasize the importance of Quantum Field theory in the study of Quantum Information, such as its role in understanding the new physical implementations of Quantum Processing tasks or how Local Quantum Theory can be used to better model detector interactions. The lectures will provide the audience with the tools required to better understand in which way special and general relativitic features can be introduced in Quantum Information, giving rise to interesting, fundamental questions regarding Quantum Fields, Localization of Quantum States and Relativistic Quantum Information.

### Organizing Committee

Juan Leon, CSIC

Hans Westman, CSIC

### Local Organizing Committee

Marco del Rey Zapatero, CSIC

Matías Rodríguez Vázquez, CSIC

# Week 1

**Quantum Field Theory, Groundwork for Quantum Technologies**

22nd, 23rd and 24th October 2013

**Carlos Sabín (U. Nottingham, UK)**

*Relativistic Quantum Metrology*

In this lecture, we will present a formalism for relativistic quantum metrology that may be useful to develop future space-‐based measurement devices. In particular, possible implementations with circuit quantum electrodynamics and Bose-‐Einstein condensates will be discussed.

**Lucas Lamata (University of the Basque Country, Bilbao, Spain)**

*Quantum Field Theories for Future Quantum Technologies*

In this lecture, we will identify and discuss the key ingredients of quantum field theories for the sake of their implementations in current quantum technologies. In particular, we will concentrate on relevant features of quantum electrodynamics and interacting fermionic and bosonic models.

**Enrique Solano (University of the Basque Country & Ikerbasque, Bilbao, Spain)**

*Quantum Technologies for Quantum Field Theories*

In these lectures, we will consider different quantum technologies that may be able to implement key features of quantum field theories. The main goal will be to identify relevant features of trapped ions and circuit quantum electrodynamics that may allow the realization of fermionic and bosonic models, approaching the continuum, emulating condensed matter, high-‐energy physics, and quantum chemistry models.

**Charles Bamber (National Research Council, Ottawa, Canada)**

*Lifting the Lid on Quantum Mechanics*

It used to be that a quantum system could never be completely measured with high precision. Conventional wisdom said that this was forbidden by the Heisenberg uncertainty principle. We now have a class of experiments and a methodology for completely characterizing quantum systems experimentally. All of the information about the system can be extracted in principle to arbitrary precision. We can know everything. As it turns out quantum mechanics can be reformulated to describe the evolution and dynamics of this measured information set. In other words, measurement embodies the system. This framework may enable new tests of quantum mechanics.

# Week 2

**Quantum Information in Relativistic Scenarios**

12th, 13th, 14th and 15th November 2013

**Hans Halvorson (Institute for Advanced Study, Princeton, US )**

*Quantum Theory of Infinite Systems: from Fields to Information*

It has long been known that the traditional Hilbert space formalism for quantum mechanics is unable to provide an adequate description of field systems. For a more flexible approach, we turn to the theory of operator algebras (C* algebras) and their representations (von Neumann algebras). We introduce algebraic quantum theory, and describe some of its applications such as the Rindler-‐Fulling effect, non-‐localizability of particles, and the information-‐theoretic characterization of quantum mechanics.

**David Jennings (Imperial College, London, UK)**

*What is the `Physical Corner’ of the Fock Space?*

I will start by explaining why the physics we observe only occupies an exponentially small manifold of the full space of states, and introduce the idea of entanglement regulation via the matrix product state representation (MPS) of quantum states. Building on this, I will then explain the extension of these ideas to quantum field systems, initially for one spatial dimension and then for higher dimensions. The resultant physical states will automatically obey entropy area laws, and the expectation values of field observables are determined by the dissipative dynamics of a lower dimensional virtual field. The construction provides powerful new analytical and computational tools to describe the physics of quantum field systems, and I will finish by discussing potential future applications.

**Göran Johannson (U. Chalmers, Göteborg, Sweden)**

*Studying relativistic effects in superconducting circuits*

In this lecture, I will start by briefly describing circuit quantization, i.e. the basic method to describe the quantum dynamics of the electromagnetic field confined to superconducting circuits. I’ll also introduce the key non-linear element, i.e. the Josephson junction (JJ). The JJ can be regarded as a nonlinear dissipationless inductance. Two JJs in a loop function as a tunable inductance, which can then be used for ultrafast modulation of the circuit parameters, i.e. the boundary conditions for the electromagnetic field.The ultrafast, dissipationless modulation of the boundary condition for the quantized electromagnetic field is the key element for simulating relativistic effects in quantum field theory. I’ll discuss how this setup has been used to study the dynamical Casimir effect, and also briefly discuss a proposal to realize the twin paradox, which in its extension could include quantum clocks.

**Terry Rudolph (Imperial College, London, UK)**

*Quantum Information in a Relativistic Setting*

There are three complementary motivations for studying extensions of quantum information theory to relativistic settings. The first is that we live in a relativistic world and so ultimately we may need to deal with relativistic effects in practice. The second is that sophisticated mathematical tools are available within the non-‐relativistic theory and carrying these over into the relativistic setting has recently proven very powerful. The last is possibly the most exciting but certainly the least well understood -‐ perhaps there are information processing tasks achievable within relativistic settings that are not doable in standard non-‐relativistic quantum information theory. In these lectures I will summarize where we currently are at with the first and third motivations.

**Ralf Schützhold (U. Duisburg-Essen, Germany)**

*Quantum Correlations: from Hawking Radiation to Ultra-Cold Atoms*

Quantum correlations are relevant for many physical systems ranging from black holes (e.g., Hawking radiation) to condensed matter and quantum optics (e.g., ultra-cold atoms in optical lattices). After a brief introduction into the basics of these two (seemingly very different) extremal cases, I will point out some of their common features — in particular regarding the important role of quantum correlations.

# Week 3

**Local Quantum Physics**

2nd, 3rd and 4th December 2013

**Hal Haggard (Centre de Physique Theorique, Marseille, France)**

*Entanglement and Thermality in Finite Spacetime Regions*

Entanglement compellingly explains the thermal properties of quantum black holes: it naturally encodes an area law for entropy that is independent of the matter species surrounding the black hole and of the cutoff on these quantum fields. Surprisingly, this turns out to be just one example of an entire formalism for treating the thermodynamics of quantum isolated systems-‐-‐-‐an entanglement thermodynamics. In these lectures I will introduce entanglement thermodynamics, examine hot finite regions, and discuss how entanglement is being used to expose the architecture of spacetime.

**Carlo Rovelli (Centre de Physique Theorique, Marseille, France)**

*What is a Particle?*

Gravity makes the notion of particle problematic in quantum field theory. In general Poincaré invariance is not available, and the standard notion of quantum particles is ill-‐ defined on curved spacetime and quantum gravity. I observe that already on flat space there exist two distinct notions of particles: globally defined $n$-‐particle Fock-‐states and *local particle states*. The last describe the physical objects detected by finite-‐size particle detectors, are eigenstates of local field operators. In an appropriate limit, global and local particle states converge in a weak topology (not in norm). Unlike conventional global particle states, local particle states remain meaningful in the presence of classical and quantum gravity.

**Jakob Yngvason (Institute for Mathematical Physics, Vienna, Austria)**

*An Invitation to Local Quantum Physics (Slides)*

The combination of (special) relativity and quantum theory leads to mathematical structures that differ in several respects from those familiar from quantum mechanics of systems with a finite number of degrees of freedom. The mini-‐course will survey a selection of insights into the structure of relativistic quantum physics that have accumulated through the efforts of many people over more than 50 years. A central concept is that of the localization of observables in space and time, and the name “Local Quantum Physics” has been coined to emphasize this aspect. Topics: Relativistic symmetries; problems with position operators in relativistic quantum mechanics and their resolution; relativistic causality and local algebras of observables; from local algebras to scattering of particles.