Though any pair of particles, given their history, might happen to be entangled, some pairs are more entangled than others.
Since at least the 1990s, physicists have been honing their understanding and treatment of quantum entanglement. In the past few decades, entanglement has been used as a resource in groundbreaking new techniques giving rise to quantum cryptography, quantum superdense coding, quantum teleportation, and even quantum computing.
However, just as a better understanding of entanglement has given rise to these feats of quantum engineering, these new techniques have also informed our understanding of quantum entanglement.
With this in mind, it is crucial to understand just what we are measuring when we say we’re measuring entanglement. Indeed, quantum entanglement is one of those scientific buzzwords that sounds terribly impressive, but what is it, really?
The distinction between whether or not a pair of particles (i.e., the state describing them) is entangled at all is surprisingly straightforward, and makes for a good starting point.
Let’s assume we have a pair of independent particles A and B. Independent particles can each be described by their own independent quantum wavefunctions—one for A and another for B. If these particles are isolated from any external disturbances, and they do not interact with each other, then at all times they will remain independent; the state describing the pair AB, [itex]\psi(x_A,x_B)[/itex], will always factor into a product of states [itex]\psi_A (x_A)\psi_B (x_B)[/itex] for A and B, respectively.
However, if the pair of isolated particles do interact with each other, say, (like a pair of isolated electrons repelling each other), then the wavefunction describing the pair AB over time might no longer factor into a product (one for A and another for B). When the joint wavefunction is non-separable like this, we say (i.e., define) that it is “entangled”. Based on this idea, we can make very general definitions of what makes for a separable state, and from that what an entangled state is supposed to be.
So, a state is entangled if it is not separable, and a separable state is one that could’ve come from independent pairs of particles. What does this have to do with measuring entanglement?
There’s actually (at least) two schools of thought on what one’s measuring when measuring entanglement. On the one hand, entanglement is a resource consumed in the techniques mentioned above; The more entanglement a pair of particles has, the better they can be used to, say convey more information in super-dense coding. Indeed, different measures of entanglement are defined based on these different techniques. On the other hand, entanglement is thought of as a property of the quantum state itself, a geometric property. There are other measures of entanglement based on this idea such as, say, the “distance” to the nearest separable state, but they all agree on the same set of assumptions defining them.
A measure of entanglement must:
- Be zero (i.e., a minimum) for separable states
- Be non increasing under local operations ( since you can’t make more entanglement without more interaction)
- Be a continuous function of the quantum state (since an infinitesimally different quantum state ought to have at most an infinitesimally different amount of entanglement).
There’s still a fair amount of debate over what other axioms might uniquely define a measure of entanglement. The measures we do have are many, and each has its own uses. With new results, we may find that the many different measures of entanglement actually capture different aspects of the same quality. Research is still ongoing.