## A spectrum of resources: from hot to quantum

How much coal do I need to put into my steam engine to get to Glasgow? How efficient can I make my power station? What’s the best way to cool down a cup of tea? The fundamental answers to these questions lies within the field of thermodynamics.

On the other side of the spectrum, scientists are asking seemingly disparate questions within the field of quantum information. How much entanglement do I need to pass on this message? How can I stop my quantum computer from losing its quantum-ness? What’s the best way I can make a quantum superposition?

Now it may seem that there can be no possible link between the answers to these questions. That a steam engine has nothing to do with quantum entanglement, well that is mostly true. However, all of questions are concerned with the same problem, the extractability and conservation of a given resource. On the thermodynamic scale, engineers want to know how heat and work behave. On the quantum scale, theorists and experimentalists alike are interested in defining and conserving the quantum-ness of a given system.

In order to bridge a gap between these fields, we can start by defining a state. Any system, whether it be a quantum or classical, can be described via a state. There are many different ways in which to write down a state, but in order to emphasize the specific resources under consideration I will be writing my states in matrix form. State in Matrix Form

There are many interesting mathematical and physical motivations for writing states in this way, however I will only be employing two properties of its form: (i) the elements of the matrix correspond to probabilities; (ii) it provides a good pictorial description of the system.

Arguably the most important state in the field of thermodynamics is the thermal state. This is the final state of any interacting thermodynamic system. What do I mean by this?  For example, if I was to leave a hot cup of tea in a cold room, they would eventually reach the same temperature as they exchanged heat. This final state of the overall system would be a thermal state.

When writing the thermal state in matrix form, the system orders its state such that the diagonal values of the state become more or less populated depending on their energy. Where the lower energy levels become more populated in comparison to the higher ones. Thermal state

Given any isolated state, if you temporarily attempt to extract work from this state and then let your system relax into whatever state it wants to, if the final state of the system is a thermal state, then you know that you have completely extracted all possible work.

The next obvious question to ask is, what is the opposite of a thermal state?  The state from which the most amount of work can be extracted from. This state is called a pure state and be written as: Pure State

What makes the pure state so special is the thermodynamic context is that only a single element of the matrix is being populated. It’s as if the components that make up the system have all crowded into the highest energy element possible. Work can then be extracted from this state as the other lesser energy states populate themselves from this one.

Now we have explored the full range of thermodynamic energy states we can now start to think about quantum resources. The quantum resource we will focus on is called quantum coherence. This is a foundational quantum resource that is responsible for a wide range of quantum effects, such as quantum supposition and multipartite entanglement. So how can we possibly grasp any understanding of this complex quantum feature? Well, if you were wondering what happens when the off-diagonal elements are not zero, that’s quantum coherence!

So, the state in thermodynamics whose resource has been fully extracted is the thermal state. What is its equivalent within the resource theory of quantum coherence? It’s called an incoherent state and is written as: Incoherent State

Any state whose elements are entirely concentrated on the diagonal are incoherent states, this includes the thermal state and pure state. Therefore, you know that you have completely extracted all of your available quantum coherence when you end up in an incoherent state.

So what is the state with the maximum amount of extractable coherence, the analogue to the pure state in thermodynamics? It’s called the maximally coherent state and can be written as: Maximally Coherent State

Crucially for the maximally coherent state, every element is identical. As operations are performed on this state that reduce the amount and size of off diagonal elements, the coherence of the state is extracted. This can be repeated until all the coherence is extracted and forms an incoherent state.

So what can we do with all these definitions? Is there some way to bridge the gap between the resources of quantum coherence and extractable work. Well to some degree this is still an active area of research and one with which I’m currently engaged. However, we can at least make a start by attempting to classify the states and attempt to bridge between the resources.

For example, if we order the states from most to least resourceful we produce the following spectrum of states. Ordering the states in this fashion prompts us to ask some questions.

It appears that coherent states exist past the boundary of what states would normally be considered when extracting work from your thermodynamic system. However, recent work suggests that thermodynamic resources can be extracted from the coherence of a state. Does this mean that the full hierarchy of thermodynamic resource states stretch into the quantum realm?

There are several different classifiers that determine where a state appears on this spectrum of extractable resources. For the part of the spectrum considered in thermodynamics we can compare states via a property called majorisation, which determines if one state can be transformed into another without the input of resources. Interestingly, in coherence resource theory, the property of majorisation is used when considering pure to pure state transformations. Could this be because pure states seem to be the boundary states between the two parts of the spectrum?

This is made more interesting when considering that some of my recent work has developed thermodynamic like relations for the resource of coherence for a pure to pure state transformation. Do thermodynamic relations for coherence resource theory only exist when considering the pure states that exist on the boundary?

It is hoped that the answers to these questions will not only help our fundamental understanding of thermodynamic and quantum theory, but also on the boundary between these two fields (if one exists). So perhaps as we extend our thermodynamic theories further and further into the quantum realm, it may not be too long till your train is powered by the quantum realm after all. 