# Measuring Quantum Entanglement?

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.

"Research is still ongoing" – sounds like we'll need a follow up :)

What about entanglement swapping? https://vcq.quantum.at/fileadmin/Publications/1993-06.pdfWe can get particles entangled with one another, which never ever interacted with one another in the past…

Seems correct. A bit odd not to mention any of the actual measures, though.

I considered mentioning the Entanglement of Formation:

-(the percentage of Bell states needed on average to synthesize copies of the quantum state in question).

I also considered mentioning the Distillable Entanglement

– (the percentage of Bell states that can on average be synthesized from many copies of the quantum state in question)

In addition, I considered the Relative Entropy of Entanglement

– (the quantum entropy of the state in question, relative to the nearest separable state)

There are at least half a dozen other measures of entanglement with their own definitions.

I wanted to avoid a discussion of density matrices, and keep the conceptual explanation as brief as possible.

Thanks for the insight article!

while reading I sort of jumped to… "The degree to which a pair of particles have shared history" as either a measure or cause of entanglement. Not sure where I got that (misinterpreting Susskind probably). Is there any sense in which that is right?

If so is there any sense in which the degree to which particles have shared future, is meaningful?

If so is there any sense in which the degree to which particles have shared future, is meaningful?

If a pair of isolated particles cannot interact with each other, they will remain unentangled in the past, present, and future.

However, if a pair of particles do interact, all we can be sure of is that it's possible for them to become entangled. Depending on the interaction, they may become entangled in different ways, and indeed, might become unentangled again later on.

The interaction serves as a means for the joint wavefunction to evolve in a way that is not reducible to wavefunctions of individual particles. Particles may become entangled and disentangled again by this interaction.

However, if a pair of particles do interact, all we can be sure of is that it's possible for them to become entangled. Depending on the interaction, they may become entangled in different ways, and indeed, might become unentangled again later on.

The interaction serves as a means for the joint wavefunction to evolve in a way that is not reducible to wavefunctions of individual particles. Particles may become entangled and disentangled again by this interaction.

So if there are two spacelike particles A and C that are initially not entangled, can they become entangled via local interactions (say the Hamiltonian obeys cluster decomposition), say if A interacts with B and B interacts with C, where B is not spacelike separated from A nor spacelike separated from C?

I actually got mislead by the title of the article in a different way. When I read something that says "Measuring Quantum Entanglement", I was expecting description of actual experiments and how the measured quantities on how they are correlated via such entanglement.

Zz.

Let's add one more particle D into the mix…

If A and C are unentangled and spacelike separated..

and A locally interacts with B

and C locally interacts with D

and particles C and D are entangled with each other,

then particles A and B can become entangled with each other even though they are spacelike separated.

This is the basis for quantum teleportation and entanglement swapping.