Nonlocality and Conflicting Interest Games

[From Left to Right] Anna Pappa, Niraj Kumar, Thomas Lawson, Miklos Santha, Shengyu Zhang, Eleni Diamanti, Iordanis Kerenidis

Authors: Anna Pappa^1,2, Niraj Kumar^1,3, Thomas Lawson¹, Miklos Santha^2,4, Shengyu Zhang⁵, Eleni Diamanti¹, Iordanis Kerenidis^2,4

Affiliation:
¹LTCI, CNRS–Télécom ParisTech, Paris, France, 
²LIAFA, CNRS–Université Paris 7, France, 
³Indian Institute of Technology, Kanpur, India, 
⁴CQT, National University of Singapore, Singapore, 
⁵Department of Computer Science and Engineering and ITCSC, The Chinese University of Hong Kong, Shatin, Hong Kong.

Nonlocality is a fundamental property of quantum mechanics that has puzzled researchers since the early formulations of quantum theory. Consider two parties, Alice and Bob, with inputs x_A and x_B respectively, who are positioned far from each other, and are asked to produce one output each (y_A for Alice and y_B for Bob). Even if the two players have pre-agreed on some local hidden variables, there exist quantum correlations that cannot be reproduced by any such set of variables [1,2]. These correlations allow the two parties to perform several computational tasks more efficiently, e.g. they can win specific games with probabilities strictly higher than allowed by any classical theory.

Till now, all known examples of quantum games considered players that have common interests, meaning that they either jointly win or lose the game. A famous such example is the CHSH game [3; CHSH stands for first letters of last names of the authors of this paper], where the players win if their outputs are different when both input bits are equal to 1, and if they are the same otherwise. It can be shown that classical resources provide a winning probability of 0.75, while the sharing of a maximally entangled pair can boost the winning probability to approximately 0.85. Another important type of games is conflicting interest games. A typical example is the “Battle of the Sexes”, where Alice and Bob want to meet, but Alice wants to go to the ballet, while Bob prefers theater. In case both go to the ballet, Alice is very pleased and Bob is fine with it; if they go to the theater, Bob is very pleased and Alice is fine with it, while if they go to different places, they are both very unhappy.

In our recent work [4], we examine whether the nonlocal feature of quantum mechanics can offer an advantage similar to the one observed in the CHSH game, but for games with conflicting interests. In order to observe a quantum advantage, we will study games with incomplete information (or Bayesian games), where each party receives some input unknown to the other party [5]. We present a Bayesian game with conflicting interests, and we show that there exist quantum strategies with average payoff for the two players strictly higher than that allowed by any classical strategy. The payoffs of the players for different inputs can be viewed as a table: the rows correspond to the outputs/actions of Alice (y_A), while the columns to the outputs/actions of Bob (y_B).

The players are interested in maximizing their average payoff over the probability distribution of their inputs, and they may use some advice from a third party (source) in order to achieve their goal. This advice can be in the form of classical bits or quantum states. In general, the classical bits received by the two players can be correlated between them (for example they can be either 00 or 11), and the quantum states may be entangled. By examining all possible strategies with classical advice, it is not difficult to verify that in our game, the sum of the average payoffs of the two players cannot be more than 1.125.

On the other hand, if we consider the case where quantum advice is given to the two players in the form of a maximally entangled state (Bell pair), the players can use projective measurements on their part of the state [6], in order to boost the sum of their average payoffs to 1.28, which is higher than any strategy with classical advice can achieve. It is very interesting to note here that the strategy that attains this payoff is also a quantum equilibrium, meaning that no player can gain a higher payoff by choosing a different strategy unilaterally.

Finally, we have demonstrated our game in practice, using the commercial entangled photon source quED by QuTools and taking a large number of independent runs of the game, in order to estimate each player’s average payoff. We found that the joint payoff is 1.246, which is well above the classical bound of 1.125, and slightly below the maximum value allowed by quantum strategies (1.28), because of experimental noise.

In conclusion, we demonstrated that the nonlocal feature of quantum mechanics can offer an advantage in a scenario where the two parties have conflicting interests. We examined a Bayesian game that attains higher payoffs for both players when using quantum advice compared to any classical strategy, and we experimentally verified the quantum advantage, by playing the game using a commercial photon source.

References:
[1] John Bell, "On the Einstein Podolsky Rosen paradox". Physics, 1, 195-200 (1964). Full Article.
[2] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, Stephanie Wehner, “Bell nonlocality”, Review of Modern Physics, 86, 419 (2014). Abstract.
[3] John F. Clauser, Michael A. Horne, Abner Shimony, Richard A. Holt, “Proposed Experiment to Test Local Hidden-Variable Theories”. Physical Review Letters, 23, 880–884 (1969). Abstract.
[4] Anna Pappa, Niraj Kumar, Thomas Lawson, Miklos Santha, Shengyu Zhang, Eleni Diamanti, Iordanis Kerenidis, "Nonlocality and Conflicting Interest Games", Physical Review Letters, 114, 020401 (2015). Abstract.
[5] J. C. Harsanyi, Management Science, 14 (3), 159-183 (Part I), 14 (5): 320-334 (Part II), 14 (7): 486-502 (Part III) (1967/1968).
[6] Richard Cleve, Peter Høyer, Ben Toner, John Watrous, "Consequences and limits of nonlocal strategies", Proceedings of the 19th Annual IEEE Conference on Computational Complexity, pages 236–249 (2004). Full Article.

Homeostasis and dynamic phase transition in a simple model of cells with chemical signaling: Can renormalization group teach us something nontrivial about biology?

Anatolij Gelimson (left) and Ramin Golestanian

Authors: Anatolij Gelimson, Ramin Golestanian

Affiliation: Rudolf Peierls Centre for Theoretical Physics, University of Oxford, United Kingdom.

The motility of bacteria or cells in response to chemicals (chemotaxis) has attracted a lot of interest in biological and medical research [2]. It plays a crucial role in cancer metastasis [3], the early stages of bacterial colony formation, wound healing and development of embryos [2]. However, the underlying mechanisms of these important processes are not fully understood due to the high complexity of these living many-body systems.

Figure 1: The interplay between the two general processes growth and chemotaxis (figure 1) leads to a variety of collective phenomena.

In our recent publication [1] in Physics Review Letters we have developed a simple model to shed some light on these interacting cells, also taking into account cellular growth and death. To study it, we have applied the method of so-called Dynamical Renormalization Groups common for the theory of phase transitions [4]. Similar to physical systems, it turns out that details of the microscopic behavior of cells do not impact the collective behavior on a large scale, whereas the interplay between the two general processes growth and chemotaxis (figure 1) leads to a variety of collective phenomena, which includes a sharp transition from a phase that has moderate controlled growth and death, and regulated chemical interactions, to a phase with strong uncontrolled growth/death and no chemical interactions [1]. Remarkably, for a range of parameters, the transition point shows nontrivial collective motion, which can even be described analytically. [1]

Bacteria such as E. coli have developed an elaborate run-and-tumble search strategy for the needed chemicals by coupling sensing of the chemicals to their motility machinery [5]. In eukaryotic cells, the chemotaxis mechanism is even more complex, often involving thousands of molecular motors or actin polymerization [6].

However, if one regards the effects of these microscopic mechanisms on a more macroscopic level, the resulting motion of bacteria and cells can effectively be modeled as a directed motion towards (or away from) increasing concentrations of chemicals [7]. On this coarse-grained level of description, the motion of bacteria in a field of chemicals is therefore somewhat analogous to the motion of particles in a gravitational or electrical field [8, 9].

But other than in non-active matter, distinctive features of a living system are also growth and death, which we need to take into account in a generic model for the formation of cellular or bacterial aggregations [10]. Interestingly, it turns out that the interplay between chemotactic interactions and growth-death processes leads to a range of different collective behaviors of cells.

We have studied our cellular model with the method of Dynamical Renormalization groups [4]. The basic idea behind it is simple: while microscopically a large number of particles, cells or bacteria might show very complicated behavior with a variety of different interactions, on a more macroscopic level only very few of these interactions will actually determine the collective effects. The so-called renormalization is basically a systematic way of observing a many-particle system from a coarser and coarser level. Coarsening the system will result in make some interactions disappear, whereas others will become stronger. In Physics, this development is called a flow in parameter space. [4]

Figure 2

In our model we have found a threshold in growth and chemotactic strength at which the flow in parameter space changes, which corresponds to a critical change of the macroscopic behavior of cells (figure 2). Below the threshold, the bacteria show jamming and aggregation due to chemotaxis. But above the threshold, chemotaxis becomes irrelevant and the behavior of cells is dominated by uncontrolled growth and death [1].

This threshold could potentially be tested experimentally and also contribute towards answering of fundamentally challenging questions in metastatic growth or bacterial colony formation. The hope is that our research will help understand what controls the communication between strongly dividing cells that are far apart and their collective behavior. The method of Dynamical Renormalization groups we have applied is very generic and could be powerful to shed light on more complex scenarios, like for example adhesive metastatic cells or chemical-dependent cell growth.

References:
[1] Anatolij Gelimson, Ramin Golestanian, "Collective Dynamics of Dividing Chemotactic Cells", Physical Review Letters, 114, 028101 (2015). Abstract.
[2] S.J. Singer, Abraham Kupfer, "The Directed Migration of Eukaryotic Cells", Annual Review of Cell Biology, 2, 337 (1986).
 Abstract.
[3] Douglas Hanahan, Robert A. Weinberg, Cell, 144, 646 (2011). Full Article.
[4] Ernesto Medina, Terence Hwa, Mehran Kardar, Yi-Cheng Zhang, "Burgers equation with correlated noise: Renormalization-group analysis and applications to directed polymers and interface growth", Physical Review A, 39, 3053 (1989). Abstract.

[5] Howard C. Berg, "E. coli in Motion" (Springer-Verlag, New York, 2004).
[6] Herbert Levine, Wouter-Jan Rappel, "The physics of eukaryotic chemotaxis", Physics Today, 66 (issue 2), 24 (2013). Abstract.

[7] Evelyn F. Keller, Lee A. Segel, "Traveling bands of chemotactic bacteria: A theoretical analysis", Journal of Theoretical Biology, 30, 235 (1971). 
Abstract.
[8] Pierre-Henri Chavanis, Carole Rosier, Clément Sire, "Thermodynamics of self-gravitating systems", Physical Review E, 66, 
036105 (2002). Abstract. 

[9] Pierre-Henri Chavanis, Clément Sire, "Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions", Physical Review E, 69, 016116 (2004). 
Abstract.
[10] Martin Nowak, "Evolutionary Dynamics", Harvard University Press (2006).

Building A Better Quantum Interface

From left to right: (top row) Bernardo Casabone, Konstantin Friebe, Birgit Brandstätter, (bottom row) Klemens Schüppert, Rainer Blatt, Tracy E. Northup.

Author: Konstantin Friebe

Affiliation: Institut für Experimentalphysik, Universität Innsbruck, Austria.

One of the largest challenges in quantum information science stems from the fact that to date no truly scalable quantum computer has been built, i.e., current devices only contain a few quantum bits (qubits). However, in order to access regimes where the power of quantum computation really comes into play, one would need scalable architectures with many qubits.

One approach for solving this issue is called distributed quantum computing. It consists of many small-scale quantum computers that are linked via photonic channels in close analogy to the internet or the “cloud” [1]. In this way, many small quantum computers can be linked together to make one large computer.

In this approach, the challenge consists of faithfully transferring quantum information between remote small-scale quantum computers via photonic channels. Quantum interfaces for this purpose can be built using cavity quantum electrodynamics systems. In such a system, the stationary qubits for computation are kept inside a cavity, also known as an optical resonator, i.e., between two mirrors. These mirrors enhance the interaction between the stationary qubits and photons (“flying qubits”), so that faithful transfer of quantum information from matter to light becomes possible. In fact, the mirrors make it possible to generate a single information-carrying photon from a stationary qubit and to be able to send this photon to another quantum computer with high efficiency [2].

Figure 1: Schematic of the setup. Two calcium ions (green spheres) are trapped inside an optical resonator (mirrors). By addressing the ions with laser beams at wavelengths 729 nm (global 729, addressing 729) and 393 nm, it is possible to prepare them in an entangled state with controllable phase, and a single photon can be generated in the cavity (red standing wave profile). The polarisation of the photon carries one qubit of information. After the photon has left the resonator, it is analysed using waveplates (λ/2, λ/4) and a polarising beam splitter (PBS), which splits up the two orthogonal polarisations H (horizontal) and V (vertical). The photon is then detected at one of two avalanche photodiodes (APD1 and APD2). (This Figure is from reference [3]).

In our recent experiment at the University of Innsbruck, Austria, we trapped two calcium ions inside an optical resonator [3]. In this case, the ions constitute the small-scale quantum computer. Ions are well-suited for this task, as an extensive toolbox for their preparation, manipulation and readout exists. The two ions were laser-cooled and prepared in an entangled state by manipulating their electronic and motional states with a laser field. Entanglement means that the two ions have “lost their individuality” and have to be described as a collective system with collective qualities. In this case, it is the electronic states of the two ions that are entangled with one another. This entangled state can be characterized by a phase, i.e., a number between zero and 2π. By controlling the phase of the entangled state of the two ions, it was possible to either enhance the probability to generate a photon in the cavity (phase 0) or to suppress the generation of a photon (phase π). The first case is called superradiance, while the suppression is called subradiance.

Figure 2: Probability of detecting a photon as a function of the time after the photon generation is started. The blue circles show the photon detection probability for the superradiant case (entangled state with phase 0), while the brown diamonds represent the subradiant state (entangled state with phase π). For comparison, the case of the two individual ions is shown (open triangles). For the superradiant case, the photon is produced faster than for a single ion, while in the subradiant case, photon generation is suppressed. Lines are simulations. (This Figure is from reference [3]).

We next encoded one qubit of information in the state of two entangled ions, that is, we used two “physical qubits” as a single “logical qubit”. The information stored in this qubit was then mapped onto the polarisation state of a single photon. By analysing the polarisation of the photon after it had left the resonator, we were able to show that the transfer of information was more faithful if the two ions were in the state with phase 0 than for a single ion. The efficiency of the process was higher, too.

Figure 3: Process fidelity (upper panel), the measure for the faithfulness of the transfer of quantum information, and efficiency (lower panel) as a function of the time after the photon generation is started. Blue filled circles are data from the superradiant entangled state, while open black circles are data from a single ion. Both process fidelity and efficiency are higher for the case of two entangled ions in the superradiant state. Lines are simulations. (This Figure is adapted from reference [3]).

In conclusion, by encoding quantum information in entangled states of more than one qubit, a better, i.e., more faithful quantum interface can be built, which is important for distributed quantum computing. Additionally, by choosing the phase π instead, qubits can be completely decoupled from the interface. This is interesting for future quantum computers consisting of long arrays of ions, where one might need a selective interface for just certain qubits, while the others remain undisturbed.

References:
[1] H. J. Kimble, "The quantum internet", Nature, 453, 1023 (2008). Abstract. 
[2] T.E. Northup, R. Blatt, "Quantum information transfer using photons", Nature Photonics, 8, 356 (2014). Abstract. 
[3] B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt, T. E. Northup, "Enhanced quantum interface with collective ion-cavity coupling", Physical Review Letters, 114, 023602 (2015). Abstract.
[4] C. Russo, H. G. Barros, A. Stute, F. Dubin, E. S. Phillips, T. Monz, T. E. Northup, C. Becher, T. Salzburger, H. Ritsch, P. O. Schmidt, R. Blatt, "Raman spectroscopy of a single ion coupled to a high-finesse cavity", Applied Physics B, 95, 205 (2009). Abstract.