If you are vaguely interested in quantum mechanics, you must check out the game Quantum Minesweeper. You might want to start with the video tutorial before you play online.

The game differs from classical Minesweeper in the following ways:

• The board is really a quantum superposition of two boards. It is your goal to figure out the superpositions. It is simplified, as only one kind of phase is allowed.
• There are three different kind of measurements that you can do, each one a limited number of times. The measurements are:

1. classical measurement – collapse that can trigger a mine probabilistically. Very risky!
2. entropy measurement – it indicates if there is a superposition or not, but doesn’t tell you if there is a mine or not!
3. interaction-free measurements – it is very magical, doesn’t collapse the wave function, actually gives you the phase information. Very powerful!

This game is fantastic!

Technical digression:

I have a question that might be a good undergraduate research project for someone interested in quantum information. What is the optimal strategy for the game? That is, if you thought of this game as a kind of state tomography problem, is there a general protocol to extract the state with high fidelity, given the constrains of the number of measurements? To make it more interesting, imagine a version of quantum minesweeper where the boards could have between them any kind of phase, how much harder would solving it be?

—

Give it one last try
til the next
one more
last try.
-A Wilhelm Scream

A few days ago we posted a new paper.

General Bound on the Rate of Decoherence [arXiv:10045405]

Cesar A. Rodriguez-Rosario, Gen Kimura, Hideki Imai, Alan Aspuru-Guzik

We establish the necessary and sufficient conditions for a quantum system to be stable under any general system-environment interaction. Quantum systems are stable when the time-derivative of their purity is zero. This stability provides a dynamical explanation of the classicality of measurement apparatus. We also propose a protocol to detect global quantum correlations using only local dynamical information. We show how quantum correlations to the environment provide bounds to the purity rate, which in turn can be used to estimate dissipation rates for general non-Markovian open quantum systems.

[SciRate]

The paper could have been alternatively titled: “Necessary and Sufficient Conditions for System Stability Under Any Coupling to the Environment”. In this post, I want to discuss briefly our first result of the paper:

.

We were interested in finding universal decoherence stability criteria that depended on the structure of the system-environment state, but was independent of the particular Hamiltonian dynamics. We focused on the measure of decoherence called “Purity”, in particular the rate of change of purity. We found that there exist system-environment states that preserve the purity of the system independent of the details of the interaction Hamiltonian. These states are given by the commutator in the equation above vanishing, and we call them “Stable System States” or SSS for lack of a better name.

SSS states are sparse topologically and not-dense: they are quite rare. But, at the same time, they include states whose system part looks very classical. On first sight, since they are rare, this would raise the question of why does the world looks classical to us. However, the equation above also implies that these states are stable under decoherence, and thus can be long-lived.

In other words, we can prove how classical states emerge naturally in the world without any assumptions of the dynamics! This provides a non-equilibrium thermodynamical explanation to why our universe looks classical.

Reuter’s ScienceWatch recently posted an interview with Prof. Daniel Lidar from USC where they discuss Daniel’s most important papers, focusing on decoherence free subspaces. Daniel was very kind to mention my own work on Open Quantum Systems with initial correlations during the interview.

===

“The theory of its operation is rudimentary and attempts to improve its performance are still made in an almost haphazard way”
-Sadi Carnot on engines

It took some time with the printing proofs, but finally, the paper has been published.

Quantum stochastic walks: A generalization of classical random walks and quantum walks

We introduce the quantum stochastic walk (QSW), which determines the evolution of a generalized quantum-mechanical walk on a graph that obeys a quantum stochastic equation of motion. Using an axiomatic approach, we specify the rules for all possible quantum, classical, and quantum-stochastic transitions from a vertex as defined by its connectivity. We show how the family of possible QSWs encompasses both the classical random walk (CRW) and the quantum walk (QW) as special cases but also includes more general probability distributions. As an example, we study the QSW on a line and the glued tree of depth three to observe the behavior of the QW-to-CRW transition.

Phys. Rev. A 81, 022323 (2010)

Previously: video abstract

—

Man, you come right out of a comic book. -Enter the Dragon

2010 has been awesome so far. I’m having a hard time keeping up with blogging all the good news.

Talks

I was in invited The Winter Meeting on Statistical Mechanics in Taxco, Mexico. What a fantastic conference! I learned a lot about many different areas in Statistical Physics, got to meet many awesome researchers, and the keynote talks were in a natural amphitheater inside the Cacahuamilpa caves. Stunning! This was one of the best conferences I’ve been to.

I was also invited to give a talk at Reed College last week. This was my first time ever in Portland, Oregon, and I fell in love with the city. It felt like a mixture of Austin, Northern California and Seattle that I really liked. The academic culture at Reed is something that should be emulated everywhere: students honestly don’t care about grades, just about learning. One thing is to hear it, and another is to witness how true it is! The physics department at Reed has the most motivated and energetic physicists I’ve ever met. Wow.

Papers:

Finally, the paper that I had mentioned before appeared in PRL:

Time-Dependent Density Functional Theory for Open Quantum Systems with Unitary Propagation

Also, the PRA on assignment maps is out in the published wild.

Linear assignment maps for correlated system-environment states

My friend and collaborator Kavan Modi had been posting on his blog his musings about Linear Assignments Maps, Correlations and Not-Completely Positive Maps. His original posts can be found here:

• Linear Independence
• Pechuka’s Theorem
• Linear Embedding
• An argument for not-completely positive maps

This was an experiment testing the possibilities of doing Open Science in theoretical research. It helped us to publicly discuss the issues, and after some discussion face to face, and private discussions using Google Wave (and the watexy robot for equations) we posted a paper in the arXiv!

Linear Assignment Maps for Correlated System-Environment States

An assignment map is a mathematical operator that describes initial system-environment states for open quantum systems. We reexamine the notion of assignments, introduced by Pechukas, and show the conditions assignments can account for correlations between the system and the environment, concluding that assignment maps can be made linear at the expense of positivity or consistency is more reasonable. We study the role of other conditions, such as consistency and positivity of the map, and show the effects of relaxing these. Finally, we establish a connection between the violation of positivity of linear assignments and the no-broadcasting theorem.

Very promptly, the paper was accepted for publication on Physical Review A, and should appear in the journal in a few weeks.

I’ll comment on my experiences of this clumsy and incomplete Open Science and remote collaboration attempt soon, hoping that the Open Science community will give me ideas of how to streamline this process.

—
When a reporter asked Asher [Asher Peres] if quantum teleportation could teleport the soul as well as the body, Asher answered, characteristically, “No, not the body, just the soul.”

This is the Video Abstract for our paper: Quantum stochastic walks: A generalization of classical random walks and quantum walks

Quantum Stochastic Walks – Video Abstract – arXiv:0905.2942 from Minus Two Fish on Vimeo.

We just posted a paper in the arXiv.

Quantum stochastic walks: A generalization of classical random walks and quantum walks

We introduce the quantum stochastic walk (QSW), which determines the evolution of generalized quantum mechanical walk on a graph that obeys a quantum stochastic equation of motion. Using an axiomatic approach, we specify the rules for all possible quantum, classical and quantum-stochastic transitions of a vertex as defined from its connectivity. We show how the family of possible QSW encompasses both the classical random walk (CRW) and the quantum walks (QW) as special cases, but also includes more general probability distributions. As an example, we study the QSW on the line, its QW to CRW transition and transitions to genearlized QSWs that go beyond the CRW and QW. QSWs provide a new framework to the study of quantum walks with environmental effects as well as quantum algorithms.

I promise a simple explanation of Classical Random Walks soon!

Time-Dependent Density Functional Theory looked like a mess when it was first explained to me. I probably made the face of a person who smells sushi for their first time, wondering if this is some sort of bad joke.

After all, quantum mechanics is supposed to be a theory of non-commuting observables that evolve in a linear fashion. TD-DFT is nothing like that, yet it claims to reproduce all the same effects. Fishy indeed.

TD-DFT first focuses on the density of the wave function, in particular, the position basis of it. This is relevant for chemical calculations where it is very important to know where are the electrons. Of course, the density is one of many observables that are relevant, but TD-DFT makes it stand out by letting this observable evolve by means of a functional of itself. In other words, you don’t fully evolve the wave function by means of an operator, but instead you have a very complicated, non-linear functional that takes as its input the density and lets it evolve. In practice, since the functional is non-linear, in practice, the evolution is done iteratively.

Runge and Gross proved that if you only cared about the evolution of one observable, the density, this procedure is equivalent to the full quantum mechanical evolution. In other words, you can map the evolution of a particular observable the wavefunction under Schrodinger’s equation into a functional of the same observable.

What you gain from this approach is a computational speedup. The prize paid is that writing the exact functional is actually a very hard problem, at least as hard as doing the full quantum mechanical evolution. However, in practice, approximated functional can be written down and used for real calculations that can predict properties for real materials. This technique is widely used, mostly as a black box toolkit used by many physical chemists around the world.

In our latest papers, we were able to show that this mapping can be also performed for open quantum systems instead of just Schrodinger’s equation. First, we developed the general theory of how the Runge-Gross theorem can be generalized, placing it in context of previous incomplete attempts. This paper was published in PCCP as a Hot Article. In it, we discuss how the theorem works even in the highly non-Markovian regime of an open quantum system.

In our second paper, we take this even further. The evolution of an observable of an open quantum system can be mapped to a functional for a close system. At first, this seemed counter-intuitive. After all, you cannot map the evolution of an open system into a closed system.

O te peinas o te haces rolos.

However, if you only care about one observable, and you are willing to use non-linear functionals, this can be done consistently, for just that observable. Since most of the code written for TD-DFT was for closed systems, our results shows that those techniques could be used to model open quantum systems. We feel that new chemical calculations with thermodynamic effects can now be explored with this theory.

—
Dr. Strangelove: It is not only possible, it is essential.

Kiyoshi Ito, a Stochastic Man of Longevity

Mathematician Kiyoshi Ito died at the young age of 93 this past month. Ito was the inventor of calculus for stochastic processes, known as the Ito Calculus.

Calculus, as invented by Isaac Newton and Gottfried Leibniz, studied the rate of change of nice smooth variables, in terms of their differentials, infinitesimal quantities described by . To properly define a Leibniz differential, the variable must be nicely behaved. Words that are often associated with nice variables are smooth, differentiable and/or continuous.

This limited the scope of applications of calculus. In particular, it does not apply to a random process. A random process, such as rolling a dice, is not nicely behaved, each roll of the dice being very different from the one before, its values literally jumping around a lot. A processes given by probabilistic, random, rules is called a stochastic processes.

My favorite stochastic process is the random walk, and is defined as follows.

Imagine a drunk guy, who can either take a step forward or backwards. Each direction has an equal probability, so you can think of the drunk guy carrying out a random walk, where the direction of each step is determined by a coin toss, heads giving a step forward while tail signifying a step backwards.

This class of problems are very common in statistical physics, finance and biology. The difficulty with doing calculations of stochastic processes is that the variables are not nice, and thus their differentials are not well defined.

Ito invented his own type of differential for exactly this purpose. Although the rules he computed were inspired by traditional calculus, they are on a different class of their own. It’s impact is so broad that is difficult to think of a field with a component of applied math where Ito calculus does not play a role.

New York Times has the story.