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The degree of entanglement – quantifying an entangled system

go quantum! do Einstein spooky action!
As advertised on TV: “Tangled Blocks” … go quantum! do Einstein spooky action! each block contains a particle entangled with one in the other block.


Remember pet rocks? So, what if someone gave you a gift, claiming it was even better than a Moon rock or Mars pebble. Something almost magical. An expensive novelty item. As advertised on TV: “Tangled Blocks” … go quantum! do Einstein spooky action! each block contains a particle entangled with one in the other block.

This post was inspired by a YouTube video on entanglement. More on that below. But that presentation reminded me that there are degrees of entanglement. And the limits of analogies sans any mathematical framework. And current entanglement-measurement methods.


While AI’s still much in the news regarding chatbots (large language models), more traditional AI neural networks are used for a lot of scientific research. Even “quantumness.”

• > “Using AI to accurately quantify the amount of entanglement in a system” by Bob Yirk, (August 1, 2023)

An international team of physicists has found that deep-learning AI technology can accurately quantify the amount of entanglement in a given system – prior research has shown that the degree of “quantumness” of a given system can be described by a single number. In their paper, published in the journal Science Advances, the group describes their technique and how well it worked when tested in a real-world environment.

Over the past several years, as scientists have learned more about entanglement, they have found that in order for it to be useful in applications, designers of such systems need a way to determine the degree of its entanglement. And that presents a problem, of course, because measuring a quantum state destroys it.

To get around this problem, physicists have developed what is described as quantum tomography, where multiple copies of a state are made and each is measured. This technique can ensure 100% accuracy, but it is exhaustive and requires considerable computing power. Another approach involves making educated guesses using limited information about a system’s state. This involves a trade-off between precision and resource use. In this new effort, the research team brought a new tool to the problem: deep-learning neural networks.

ENTANGLEMENT PER SE – “connected” dots

So, in this YouTube video (below), physicist Katie Mack [The End of Everything: (Astrophysically Speaking)] presents an analogy for quantum entanglement. Explaining what entanglement is about using the familiar macroscopic pair-of-coins model.

Well, while her analogy explains the weirdness, the visualization begs certain questions:

1 . How do you know those two particular “coins” are entangled? – both initially when provided to Alice & Bob and after traveling for 10 years.

Can we produce a single entangled pair on demand? For typical entanglement-measurement methods, there’s only a probability of entanglement for any pair of “particles” (like one in a million).

Can we transfer entangled quantum states on demand?

2 . What does the (often used) term “connection” (rather than “apparent connection” or correlation) imply?

Is this misleading? (Same question for the term “link.”)

Is “shared (quantum) state” too technical a characterization? (For which that “state” remains undisturbed until a measurement – disentanglement – decoherence.)

3 . To what degree is the “system” of the coins entangled?

Evidently, entanglement need not be 100% (or only apply to certain degrees of freedom, in this case side value). [1]

4 . In real-world examples (like entangled photons or electrons), aren’t groups (ensembles) of “particles” involved? And extensive post-hoc analysis of measurements?

There’s a probability of entanglement.

• YouTube > Perimeter Institute for Theoretical Physics > “Quantum 101 Episode 5: Quantum Entanglement Explained” (Aug 1, 2023) – Quantum entanglement allows physicists to create connections between particles that seem to violate our understanding of space and time.


Quantum entanglement is one of the most intriguing and perplexing phenomena in quantum physics. It allows physicists to create connections between particles that seem to violate our understanding of space and time.

This video discusses what quantum entanglement really is, and the experiments that help us understand it. The results of these experiments have applications in new technologies that will forever change our world.

Join Katie Mack, Perimeter Institute’s Hawking Chair in Cosmology and Science Communication, over 10 short forays into the weird, wonderful world of quantum science.

(from transcript)

[As an analogy for two entangled particles prepared in a lab to have opposite (indeterminate) spins: “You can’t know ahead of time if it’s spin up or spin down. All you know is that when you measure it, the other particle will be the opposite.”]

But what if we could do some magic trick to entangle these coins? Sort of.

One comes up, heads, the other will come up tails. No matter how far apart they are.

Let’s say I give this coin to Alice and this one to Bob just before they set off on rocket ships going in opposite directions. Alice and Bob are each under strict instructions not to flip their coins until they’ve been traveling for ten years.

When the time is up, they each flip their coin and immediately send radio signals to each other with the answers.

But while those signals might take years to reach their destinations, the entanglement means that the moment Alice sees heads, she knows that Bob’s coin landed tails and vice versa.

Alice doesn’t have to wait for Bob’s radio signal to arrive to know what it will say. Alice’s coin had a 50/50 chance of coming up heads. But once it does, Bob’s coin is tails 100% of the time.

Because Bob’s signal is still traveling to her when she finds out the answer, it’s as if Alice has predicted the future.

This might sound impossible, but it’s exactly what quantum entanglement does to individual particles in experiments.

  • But how does it work?
  • How do the particles know what to do?
  • Are they passing messages that we can’t see?
  • And why can we entangle particles but not coins?
  • And that is one of the most interesting and puzzling concepts of quantum mechanics.


[1] Re how much entanglement is contained in a quantum state

• Wiki > Quantum entanglement

As a measure of entanglement

Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist. If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems.

• Quantiki > Entanglement-measure

Entanglement measure quantifies how much entanglement is contained in a quantum state. Formally it is any nonnegative real function of a state which can not increase under local operations and classical communication (LOCC) (so called monotonicity), and is zero for separable states.


• Wiki > Quantum tomography – Quantum tomography or quantum state tomography is the process by which a quantum state is reconstructed using measurements on an ensemble of identical quantum states.

• Caltech > Science Exchange > “What Is Entanglement and Why Is It Important?” – Entanglement can also occur among hundreds, millions, and even more particles.

Related posts (and comments)

Imaging at the molecular and atomic level

2022 Nobel Prize – ‘spooky action’ pioneers

Spooky action vs. spooky communication

What the heck is quantum entanglement?

How to create entangled photon pairs

One thought on “The degree of entanglement – quantifying an entangled system

  1. Coherent superconductivity
    Scanning electron microscope image of an ultra-coherent superconducting electro-mechanical system. Credit: Amir Youssefi (EPFL)

    Quantum phenomena in mechanical systems? How does that work? Nanofabrication. And ultra-low quantum decoherence. For 7.7 milliseconds.

    • > “A quantum leap in mechanical oscillator technology” by Ecole Polytechnique Federale de Lausanne (August 11, 2023) – By coupling mechanical oscillators (optomechanical systems) to light photons, scientists have been able to cool them down to their lowest energy level (close to the quantum limit), “squeeze them” to reduce their vibrations even further, and entangle them with each other.

    In order to efficiently operate optomechanical systems in the quantum regime, scientists face a dilemma. On one hand, the mechanical oscillators must be properly isolated from their environment to minimize energy loss; on the other hand, they must be well-coupled to other physical systems such as electromagnetic resonators to control them.

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