I have just come back from Hong Kong, where I was collaborating with the group of Prof. GuanHua Chen. We are interested in studying mathematical properties of the Hierarchical Equations of Motion from a Master Equation point of view. This was a very productive visit.
I also had a wonderful time exploring the city, and finding great food everywhere.
Our paper, Non-Markovian Open Quantum Systems: System-Environment Correlations in Dynamical Maps has been published!
Abstract: We construct a non-Markovian dynamical map that accounts for systems correlated to the environment. We refer to it as a canonical dynamical map, which forms an evolution family. The relationship between inverse maps and correlations with the environment is established. The mathematical properties of complete positivity is related to classical correlations, according to quantum discord, between the system and the environment. A generalized non-Markovian master equation is derived from the canonical dynamical map.
You can read it for free for the next couple of weeks in: International Journal of Quantum Information Volume: 9, Issues: 7-8 (2011) pp. 1617-1634
My paper on Non-Markovian Open Quantum systems has been published in the Special Issue of International Journal of Quantum Information (IJQI):
Non-Markovian Open Quantum Systems: System-Environment Correlations in Dynamical Maps
We construct a non-Markovian dynamical map that accounts for systems correlated to the environment. We refer to it as a canonical dynamical map, which forms an evolution family. The relationship between inverse maps and correlations with the environment is established. The mathematical properties of complete positivity is related to classical correlations, according to quantum discord, between the system and the environment. A generalized non-Markovian master equation is derived from the canonical dynamical map.
This paper is an updated version of my 2008 preprint.
I had written before about the concept of Lazy States. Questions in non-equilibrium thermodynamics are usually said to be intractable, due to how much they depend on the details of the dynamical equations. We discovered that in quantum non-equilibrium thermodynamics, there was a simple way to separate the role of system-environment states and of the details of their Hamiltonian coupling. For a class of states, $$left[rho^{SE},rho^{S} right]=0$$ , Lazy States, the entropy rate is always zero independent of the interaction Hamiltonian. These Lazy States are rare, which opens the question of how come thermodynamic equilibrium is so common in the universe?
On a new paper in the arXiv titled Almost all states are pretty lazy, Adrian Hutter and Stephanie Wehner, from the Centre for Quantum Technologies in Singapore, tackled exactly this problem. They were able to show that even if states aren’t lazy, almost all states lead to dynamics which is very close to the ones generated by lazy states. Thus, even if systems are away from equilibrium, they cannot be too far away from it. This leads them to conclude that “Almost all states are pretty lazy”.
A news feature in the journal Nature discusses how Quantum Discord is related to Quantum Computation. They interviewed several of my collaborators, Kavan Modi and Animesh Datta, and have a very nice summary of why it has become so fashionable lately.
Quantum Discord was first proposed by Wojciech Zurek as a measure of bipartite quantum correlations different from entanglement. As Wojciech described it to me, he presented this at a conference, and many people did not understand its significance at the time, mainly because it wasn’t clear how it related to entanglement. Meanwhile, Vlatko Vedral independently proposed a similar measure of quantum correlations. These results were both published around 2001, but Zurek’s name stuck.
A few years later, while I was in graduate school, I heard Zurek was coming to visit us in the Sudarshan group. Zurek had been a student of our department decades before, and I was very excited to meet him. I studied some of his papers, and we had a discussion that ended up on the topic of quantum discord. Although at the time I was not thinking too much about measures of quantum correlations, I was interested in the problem of initial system-environment correlations in open quantum systems.
A few months later, while walking around town lake in Austin Tx, I proposed to Kavan Modi (then a graduate student like me) and Prof. Sudarshan that the concept of classical correlations (as defined by quantum discord), might help us understand some of the issues in open quantum systems with initial correlations. That winter, Kavan and I decided to go on a road trip to New Mexico, where we visited our friend Anil Shaji, now a postdoc in Prof. Caves group. We then also met Animesh Datta. During this road trip we also visited Zurek in Los Alamos, and we had further discussions about quantum discord.
All these conversations led to the first paper to use quantum discord, which connected it to the mathematical properties of complete positivity of dynamical maps.
Animesh and Anil took a different direction that ultimate proved to be very useful: they noted that quantum discord was an important resource for some quantum algorithms. It was this result that has led to so many recent publications in the field.
More recently, some of us have shown how quantum discord is a fundamental dynamical characteristic of non-equilibrium thermodynamical systems.
Quantum Discord has led to advances that can be grouped into two areas: as what could become another resource in quantum computation, and as some fundamental property of the dynamics of bipartite states. Could there be a relationship between these?
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Man, you come right out of a comic book. -Williams to Han in: Enter the Dragon
Phys. Rev. Lett. 106, 050403 (2011) [arXiv version]
César A. Rodríguez-Rosario, Gen Kimura, Hideki Imai, and Alán Aspuru-Guzik
We find the necessary and sufficient conditions for the entropy rate of the system to be zero under any system-environment Hamiltonian interaction. We call the class of system-environment states that satisfy this condition lazy states. They are a generalization of classically correlated states defined by quantum discord, but based on projective measurements of any rank. The concept of lazy states permits the construction of a protocol for detecting global quantum correlations using only local dynamical information. We show how quantum correlations to the environment provide bounds to the entropy rate, and how to estimate dissipation rates for general non-Markovian open quantum systems.
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“The more physics you have the less engineering you need.”
-Ernest Rutherford
Our latest paper Lazy states: sufficient and necessary condition for zero quantum entropy rates under any coupling to the environment has been accepted for publication in Physical Review Letters. Woohoo!
Lazy states: sufficient and necessary condition for zero quantum entropy rates under any coupling to the environment
We find the necessary and sufficient conditions for the entropy rate of the system to be zero under any system-environment Hamiltonian interaction. We call the class of system-environment states that satisfy this condition lazy states. They are a generalization of classically correlated states defined by quantum discord, but based on projective measurements of any rank. The concept of lazy states permits the construction of a protocol for detecting global quantum correlations using only local dynamical information. We show how quantum correlations to the environment provide bounds to the entropy rate, and how to estimate dissipation rates for general non-Markovian open quantum systems.
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I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and felt almost giddy at the thought that I now had to probe this wealth of mathematical structure nature had so generously spread out before me.
-Heisenberg
Andy Maloney, a leader in open science, is embracing the open philosophy all the way. He has decided to write his dissertation as it evolves as an open science document. First chapter draft is here. Go there to witness the first fully open dissertation ever.
Yes, every version, every draft, every correction, everything will be posted there. Members of the wiki can also write him comments and suggestions. This is a fantastic idea, as this is the first time others can see how a dissertations actually develops. This will help both experts in the field, and graduate students that want to see how is the dissertation-writing process.
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[Jesus walks out of the lake with a small bottle of lemonade]
Jesus Christ: Lemonade?
Father Eustace: Will there be enough?
Jesus Christ: Oh, there’ll be plenty.
-Jesus Christ Vampire Hunter
This past few months have been intense for me. A lot of exciting collaborations are developing, a lot of interesting breakthroughs in several projects, and a lot of traveling. Here is a summary of the talks I have given:
Quantum coherent resources for exciton transport
Quantum coherence in photosynthetic exciton transport
Dynamical Maps with System-Environment Correlations
Decoherence Rate and System-Environment Correlations
Quantum coherence in photosynthetic exciton transport
Each of those visits have been very productive, and hopefully all will lead to collaborations.
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Kids were very different then. They didn’t have their
heads filled with all this Cartesian Dualism…
-Monty Python on Nostalgia
Minus Two Fish 