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.

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.

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”.

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.

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.

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?

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.