Scientific Series

Information theory establishes the fundamental limits on data transmission, storage, and processing. Quantum information theory unites information theoretic ideas with an accurate quantum-mechanical description of reality to give a more accurate and complete theory with new and more powerful possibilities for information processing. The goal of both classical and quantum information theory is to quantify the optimal rates of interconversion of different resources. These rates are usually characterized in terms of entropies. However, nonadditivity of many entropic formulas often makes finding answers to information theoretic questions intractable. In a few auspicious cases, such as the classical capacity of a classical channel, the capacity region of a multiple access channel and the entanglement assisted capacity of a quantum channel, additivity allows a full characterization of optimal rates. Here we present a new mathematical property of entropic formulas, uniform additivity, that is both easily evaluated and rich enough to capture all known quantum additive formulas. We give a complete characterization of uniformly additive functions using the linear programming approach to entropy inequalities. In addition to all known quantum formulas, we find a new and intriguing additive quantity: the completely coherent information. We also uncover a remarkable coincidence---the classical and quantum uniformly additive functions are identical; the tractable answers in classical and quantum information theory are formally equivalent.

In this talk I will review the construction of space starting purely from quantum mechanics and without assuming that the notion of space is attached to a preconceived notion of classical reality. I will show that if one start with the simplest notion of a quantum system encoded into the Heisenberg group algebra one naturally obtain a  notion of space that generalizes the usual notion of Euclidean space. Along the way I will try to illustrate how this notion is in a way going back to the roots of the discovery of QM by Heisenberg-Born-Jordan which never made it past the original papers and giving a critical reading of the subsequent interpretation of space and QM that were put forward by Schrodinger and von Neumann. The notion of space that emerge from quantum mechanics is naturally modular in the sense of Aharonov, it also naturally possess a built-in length scale and renders possible to assign a new notion of locality to non-local superpositions. I will illustrate how  such space can allow reconciliation of relativity with the presence of a fundamental scale.

I will show how to construct such spaces following the original analysis by Mackay and also show that such modular spaces possess a beautiful geometrical structure that generalizes Riemanian geometry to phase space. A geometry we have named Born geometry. I hope this will open wild speculations on the nature of locality in the presence of quantum mechanics and more broadly the nature of classical reality viewed from a quantum perspective.

When the wavefunction of a macroscopic system unitarily evolves from a low-entropy initial state, there is good circumstantial evidence it develops "branches", i.e., a decomposition into orthogonal components that can't be distinguished from the corresponding incoherent mixture by feasible observations, with each component a simultaneous eigenstate of preferred macroscopic observables.  Is this decomposition unique?  Can the number of branches decrease in time?

Answering these questions is hard because branches are defined only intuitively, much like early investigations of algorithms prior to the Church–Turing thesis.  A rigorous definition could give a speed up to certain non-stationary matrix-product state numerical simulations, as well as solve Kent's Set Selection problem in the consistent histories formalism, a 20-year-old open challenge in the foundations of quantum mechanics. I introduce some tentative definitions based on the idea of redundant information, and establishing a uniqueness theorems. A key counterexample is provided by the Shor error-correction code, which demonstrates that branch structure with robust, redundant records on macroscopic scales can hide incompatible (noncommuting) structure on microscopic scales.

We study the effective twisted superpotential of 3d N=2 gauge theories compactified on a circle.  This is a rich object which encodes much of the protected information in these theories.  We review its properties, and survey some applications, including the algebra of Wilson loops, computation of supersymmetric partition functions on S^1 bundles, and the reduction of 3d dualities to two dimensions.

We introduce a simple and modern discussion of rotational superradiance based on quantum field theory. We work with an effective theory valid at scales much larger than the size of the spinning object responsible for superradiance. Within this framework, the probability of absorption by an object at rest completely determines the superradiant amplification rate when that same object is spinning. We first discuss in detail superradiant scattering of spin 0 particles with orbital angular momentum l = 1, and then extend our analysis to higher values of orbital angular momentum and spin. Along the way, we provide a simple derivation of vacuum friction—a “quantum torque” acting on spinning objects in empty space. Our results apply not only to black holes but to arbitrary spinning objects. We also discuss superradiant instability due to formation of bound states and, as an illustration, we calculate the instability rate Γ for bound states with massive spin 1 particles.

Quantum groups from character varieties
Abstract: The moduli spaces of local systems on marked surfaces enjoy many nice properties. In particular, it was shown by Fock and Goncharov that they form examples of cluster varieties, which means that they are Poisson varieties with a positive atlas of toric charts, and thus admit canonical quantizations. I will describe joint work with A. Shapiro in which we embed the quantized enveloping algebra U_q(sl_n) into the quantum character variety associated to a punctured disk with two marked points on its boundary. The construction is closely related to the (quantized) multiplicative Grothendieck-Springer resolution for SL_n. I will also explain how the R-matrix of U_q(sl_n) arises naturally in this topological setup as a (half) Dehn twist. Time permitting, I will describe some potential applications to the study of positive representations of the split real quantum group U_q(sl_n,R)

It is a common expectation in quantum gravity that the fundamental nature of space-time would be radically different from the smooth continuum of classical general relativity. In this talk it shall be shown that a quantum modification from loop quantum gravity crucial for singularity resolution is also responsible for deforming the underlying space-time in a manner which cannot be realized using classical geometric structures. With the minimum requirement that the quantum theory satisfies a well-defined notion of covariance explicit midisuperspace models manifesting such non-Riemannian space-times shall be shown to exhibit the physical phenomenon of non-singular signature change. Robustness and implications of this effect shall also be briefly discussed.

A central question in quantum computation is to identify which problems can be solved faster on a quantum computer. A Holy Grail of the field would be to have a theory of quantum speed-ups that delineates the physical mechanisms sustaining quantum speed-ups and helps in the design of new quantum algorithms. In this talk, we present such a toy theory for the study of a class of quantum algorithms for algebraic problems, including Shor’s celebrated factoring algorithm. Our theory is an extension of Gottesman’s stabilizer formalism based on elements of group and hypergroup theory. Using our methods, we develop classical simulation algorithms for Clifford-like circuits containing quantum Fourier transforms as well as new quantum algorithms for hidden symmetries and hyper-symmertry problems. During the talk, we will discuss the role of resources such as entanglement, interference and contextuality within our formalism and connect quantum speed-ups therein to the presence of precise algebraic structures.

Based on the following works:

[1] https://arxiv.org/abs/1210.3637
[2] https://arxiv.org/abs/1409.3208
[3] https://arxiv.org/abs/1409.4800
[4] https://arxiv.org/abs/1509.05806

Tensor networks offer an efficient representation of many-body wave-functions in an exponentially large Hilbert space by exploiting the area law of ground state quantum entanglement. I will start with a gentle introduction to the tensor network formalism. Then I will describe its application to realizing Wilson's renormalization group directly on quantum lattice models (e.g. quantum spin chains), with emphasis on the RG fixed points corresponding to conformal field theories. We will see how to define both global and local scale transformations on the lattice in such a way that, intriguingly, conformal invariance can be tested (and conformal data extracted) right at the UV cut-off scale.

Can we decompose the information of a composite system into terms arising from its parts and their interactions?
For a bipartite system (X,Y), the joint entropy can be written as an algebraic sum of three terms: the entropy of X alone, the entropy of Y alone, and the mutual information of X and Y, which comes with an opposite sign. This suggests a set-theoretical analogy: mutual information is a sort of "intersection", and joint entropy is a sort of "union".
The same picture cannot be generalized to three or more parts in a straightforward way, and the problem is still considered open. Is there a deep reason for why the set-theoretical analogy fails?
Category theory can give an alternative, conceptual point of view on the problem. As Shannon already noted, information appears to be related to symmetry. This suggests a natural lattice structure for information, which is compatible with a set-theoretical picture only for bipartite systems.
The categorical approach favors objects with a structure in place of just numbers to describe information quantities. We hope that this can clarify the mathematical structure underlying information theory, and leave it open to wider generalizations.

Abstract: We present the foundation for a holographic dictionary with depth perception. The dictionary consists of natural operators associated with CFT bilocals whose duals are simple, diffeomorphism-invariant bulk operators.  These objects admit a description as fields in kinematic space, a phase space for such probes.  The framework of kinematic space allows for conceptually simple derivations of many results known in the literature, including linearized Einstein’s equations, the relationship between conformal blocks and geodesic Witten diagrams, and the CFT representation of bulk local operators.   Reference: https://arxiv.org/abs/1604.03110

Soft theorems for the scattering of low energy photons and gravitons and cosmological consistency conditions on the squeezed-limit correlation functions are both understood to be consequences of invariance under large gauge transformations. I apply the same method used in cosmology -- based on the identification of an infinite set of "adiabatic modes" and the corresponding conserved currents -- to derive flat space soft theorems for electrodynamics and gravity. I discuss how the recent derivations based on the asymptotic symmetry groups (BMS) can be continued to a finite size sphere surrounding the scattering event. Finally, I comment on Hawking-Perry-Strominger proposal of "soft hair on black holes".