If you have ever asked a quantum researcher about their subject, you will have realised that our first response is to tell you that the quantum world is weird, spooky, and counter-intuitive. It cannot be explained easily because we only just about understand it ourselves! After a bit more conversation, you’ll find us telling you that all these weird features are actually very useful: we talk about a potential for new technologies that can harness the quantum, from quantum computers to cryptography. The idea is to treat these quantum features as useful *resources* that help us to do things in an improved way. To do this, we need to carefully construct a framework that allows us to rigorously characterise these resources. Such a framework is called a quantum* resource theory*.

In this post, I will give a short explanation of resource theories. We begin by outlining the concept generally, but then focus on two particular examples of resources. The first example considers the resource of money in a fictional bank account, while the second example moves properly into the quantum world with the familiar resource of quantum entanglement.

## Resource Theories

Let’s consider a physical system that can possess a given resource under investigation. We describe the system with the notion of a *state*, which is a collection of all the information we have about the system. Whenever the system doesn’t have any resource, it is said to be in a *free state*. The free states are one of the main ingredients of the resource theory. Otherwise, when the system has some resource, it is said to be in a *resource state.*

It is also important to describe how we can cause the system to evolve with time. An *operation* describes such an evolution, giving us an output state of the system from the input state. We want to characterise the operations that do not cost us any resource to carry out. These are called the *free operations*, and form the other main ingredient of the resource theory.

As well as identifying the free states of the system, we also want to be able to compare the resource states so that we can understand when the system has more resource. This can be done by using the free operations, which allow us to place the resource states into a hierarchy. Here, one resource state A is said to be higher up in the hierarchy than another resource state B if we can find a free operation that causes the system to evolve from state A to state B. Indeed, since it does not cost us any resource to implement this free operation, we conclude that the system cannot have gained resource and hence state B is not more resourceful than state A.

The shape of the hierarchy can vary for different types of resources. At the bottom of the hierarchy are the free states. However, the top of the hierarchy does not have to be unique: given two resource states it is not always possible to find a free operation transforming from one resource state to the other. This can lead to a hierarchy that has multiple inequivalent branches, where it is not possible to use free operations to transform the system from a state that is in one branch to a state that is in another branch. The figure below shows both a single-branch and a multi-branch hierarchy.

However, the hierarchy is only qualitative. It allows us to compare resource states of the system, but it does not put a number on how resourceful a state of the system is. We can do this by introducing a *measure* of the resource. A resource measure is a function that condenses the complicated hierarchy into a total quantitative ordering, as can be seen in the figure above. It turns out that there is a myriad of possible ways to choose a resource measure. In fact, the different choices of resource measures are a major topic of conversation for quantum information scientists. It is important to highlight two of the basic requirements of a resource measure. First, we require that a measure takes a value of zero whenever the system doesn’t have any resource, i.e. for all the free states. Second, we require that a measure cannot increase when a free operation is applied to the system.

Finally, as we have already discussed at the start, the idea is to use our resource to help us better carry out a particular task. Hence, one of the objectives in constructing a resource theory is to identify a task whose performance is quantitatively given by one of the resource measures.

### Example 1 – A bank account

We now consider the very simple example of the resource of money in a bank account. Here, our physical system is the bank account, while the state of the system is simply the current balance of the account. We can perform operations on the bank account by visiting the bank and withdrawing or depositing funds.

In this example, it is simple to see that there is just one free state – when the balance of the account is zero (let’s assume that this account cannot go overdrawn!). Whenever the balance of the account is positive, the system is in a resource state. Considering now the operations, depositing funds in the bank account requires us to give money to the bank. Hence, the free operations (i.e., the operations that do not consume the resource) are given by all possible withdrawals of funds from the account.

The free operations allow us to see the hierarchy of resource states. Indeed, if we have two possible balances of the account A and B, we know that balance B is less than balance A if we can get from B to A by withdrawing money from the account. Here the hierarchy is very simple, but we see in the next example that this is not always the case.

One obvious choice of a resource measure is given simply by the balance of the account. However, we can also give alternative measures of the resource. For example, we could measure the amount of money in the account in another currency. Alternatively, we could measure the amount of money in the account by how many pizzas we are able to buy with it. This measure allows us to associate the resource with improved performance of a task – namely, the enjoyment of pizza!

### Example 2 – Quantum entanglement

In this example we cross into the realm of quantum mechanics and consider the resource of quantum entanglement (discussed in more detail in my previous post as well as our post on monogamy and faithfulness). Our physical system now consists of a collection of quantum objects that we call *qubits*. A qubit, short for a quantum bit, is the quantum analogue of a *bit*, which represents a system that can only exist in two distinct states (think, for example, of the faces of a coin). What makes a qubit “quantum” is that it can actually exist in an infinite number of possible states. However, if we then look at the qubit by observing it in a laboratory, we find that it collapses onto one of two possible states. Have a further investigation of quantum superposition, wave function collapse, and the infamous Schrödinger’s cat thought experiment for more information (as well as our short story* Alice and the Zombie Cat*).

So, let’s suppose that we have a collection of these qubits, which can have the resource of quantum entanglement. Without going into too many details, the qubits are entangled when they can only be viewed as a composite and we cannot describe them individually, see here for an explanation. Hence, the system is in a free state whenever the qubits can be described individually (the exact way to write the free states here is not important, suffice to say that the free states are called *separable states*).

We can also look at the free operations of quantum entanglement. Suppose that we distribute each qubit to a different laboratory that can perform *local operations* on that qubit. Furthermore, we allow each laboratory to communicate their actions to the others through *(classical) communication*, see the figure below. The composition of local operations and classical communication (LOCC) are the free operations of quantum entanglement. Indeed, we do not use any of the resource of entanglement to carry out LOCC.

The LOCC impose a hierarchy on the entangled states of the collection of qubits. The form of the hierarchy depends upon the number of qubits in our system. When there are two qubits, the hierarchy has a single branch with a maximally entangled state given by one of the so-called Bell states. On the other hand, for three qubits there are two separate branches, with the GHZ state and the W state at the top of each branch. The number of branches increases with the number of qubits.

There are many ways to measure the amount of entanglement in our multi-qubit system, this review article and this book both provide an excellent summary and also discuss some of the ways that entanglement can be harnessed in the real world. One of the difficulties of using an entanglement measure is that it can be hard to compute. Nevertheless, for two qubit systems one can use the concurrence, which is a measure of entanglement that is very simple to compute.

*This concludes the post. We finally mention that there are other quantum resources than entanglement, such as quantum coherence. One of the objectives of quantum science is to harness these resources to manufacture technologies that outperform those that are currently available. We are in the early stages of developing these technologies, but one thing is for sure: it is an exciting time to be investigating the quantum!*

*If you have any interesting examples of how resource theories can be applied in everyday life, please write a comment below!*