Boaz Tamir

A Holevo-Type Bound for a Hilbert Schmidt Distance Measure

We prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kullback-Leibler divergence. Suppose Alice is sending classical information to Bob by using a quantum channel while Bob is performing some projective measurements. We bound the classical mutual information in terms of the Hilbert-Schmidt norm by its quantum Hilbert-Schmidt counterpart. This constitutes a Holevo-type upper bound on the classical information transmission rate via a quantum channel. The resulting inequality is rather natural and intuitive relating classical and quantum expressions using the same measure.

Quantum annealing – foundations and frontiers

We briefly review various computational methods for the solution of optimization problems. First, several classical methods such as Metropolis algorithm and simulated annealing are discussed. We continue with a description of quantum methods, namely adiabatic quantum computation and quantum annealing. Next, the new D-Wave computer and the recent progress in the field claimed by the D-Wave group are discussed. We present a set of criteria which can help in testing the quantum features of these computers. We conclude with a list of considerations with regard to future research.

Communication complexity protocols for qutrits

Consider a function where its entries are distributed among many parties. Suppose each party is allowed to send only a limited amount of information to a referee. The referee can use a classical protocol to compute the value of the global function. Is there a quantum protocol improving the results of all classical protocols? In a recent work Brukner et al. showed the deep connection between such problems and the theory of Bell inequalities. Here we generalize the theory to trits. There, the best classical protocol fails whereas the quantum protocol yields the correct answer.

The physics of categorization

In the context of cognition, categorization is the process through which several elements (i.e., words) are grouped into a single set which by naming becomes an abstraction of its elements. For example, tiger, kitty, and max can be categorized as Cats. In this article, we aim to show how the physical, biological and cognitive dimensions are related in the process of categorization or abstraction through the physics of computation. Drawing on Landauer’s principle, we show that the price paid in terms of entropy is higher when grouping elements of low ranking (high probability) than when grouping elements of high ranking (low probability). Therefore, the logic of the cognitive process of abstraction is explained through constraints imposed by memory on the computation of categories. VC 2015 Wiley Periodicals, Inc. Complexity 000: 00–00, 2015

On the power of weak measurements in separating quantum states

We investigate the power of weak measurements in the framework of quantum state discrimination. First, we define and analyze the notion of weak consecutive measurements. Our main result is a convergence theorem whereby we demonstrate when and how a set of consecutive weak measurements converges to a strong measurement. Second, we show that for a small set of consecutive weak measurements, long before their convergence, one can separate close states without causing their collapse. We thus demonstrate a tradeoff between the success probability and the bias of the original vector towards collapse. Next, we use post-selection within the two-state vector formalism and present the non-linear expansion of the expectation value of the measurement device’s pointer to distinguish between two predetermined close vectors.

Minority Game and Kolkata Paise Restaurant Problem

The minority game (MG) is a simple two choice game (see also Appendix D) played between N players, where the players are required to make a choice between two options at each step. The players ending up in the minority, i.e. choice with fewer people, receive a fixed positive pay off.

Zipf’s Law from Kolkata Paise Restaurant Problem

The complexity of human interactions in societies have produced a variety of emergent phenomena, which are often characterized by broad distributions of different quantities. One of the consequences of economic growth is urban agglomeration.

Some Recent Developments: A Brief Discussion

In this chapter we give very brief introduction to some of the interesting very new developments in this field. In particular, we have discussed some of the extensions of Kolkata Paise Restaurant problem for dynamic settings or development of efficient strategies employing reinforced learning or applications of such strategies to other social problems like modelling income distributions in societies etc.

From Classical Games, the Kokata Paise Restuarant Game, to Quantum Games

In this chapter we give a map of classical game theory (see also Appendix C), the main ideas and motivations. The text aims the readers who are familiar of basic notions in game theory. For an elaborate discussion of the topic we refer the reader to Neuman et al. [281], Osborne [212], Fudenberg [106], Heifetz [139].

Reversibility and efficiency in coding protein information.

Why the genetic code has a fixed length? Protein information is transferred by coding each amino acid using codons whose length equals 3 for all amino acids. Hence the most probable and the least probable amino acid get a codeword with an equal length. Moreover, the distributions of amino acids found in nature are not uniform and therefore the efficiency of such codes is sub-optimal. The origins of these apparently non-efficient codes are yet unclear. In this paper we propose an a priori argument for the energy efficiency of such codes resulting from their reversibility, in contrast to their time inefficiency. Such codes are reversible in the sense that a primitive processor, reading three letters in each step, can always reverse its operation, undoing its process. We examine the codes for the distributions of amino acids that exist in nature and show that they could not be both time efficient and reversible. We investigate a family of Zipf-type distributions and present their efficient (non-fixed length) prefix code, their graphs, and the condition for their reversibility. We prove that for a large family of such distributions, if the code is time efficient, it could not be reversible. In other words, if pre-biotic processes demand reversibility, the protein code could not be time efficient. The benefits of reversibility are clear: reversible processes are adiabatic, namely, they dissipate a very small amount of energy. Such processes must be done slowly enough; therefore time efficiency is non-important. It is reasonable to assume that early biochemical complexes were more prone towards energy efficiency, where forward and backward processes were almost symmetrical.