# Entanglement measures

1. Nov 23, 2004

### lalbatros

Hello

I would be interrested in good (web) references dealing with "entanglement measures".
I am looking for not too mathematical references.
I would like to see the connections with physics more than with quantum information theory.
I would be interrested to see the concepts applied to basic experiments, like EPR experiments.

Thanks

2. Nov 23, 2004

### somy

Last edited by a moderator: May 1, 2017
3. Nov 23, 2004

### marlon

4. Nov 23, 2004

### lalbatros

something about "quantitative" measures of entanglement

I am most interresting in how -with a number, a measure- one can say that a state vector is more "entangled" than another.
And maybe first by a discussion of the possibility of that.
Also by the applications of it.

For example, take a 2-particles state vector |s1s2>.
If |s1s2> = |s1>|s2>, then I would expect that the entanglement measure is zero.
I would also expect that with an increasing additional 'mixing' term the entanglement would increase:
If |s1s2> = |s1>|s2> + x |s'1>|s'2> , then the 'measure' should increase with |x| (|x| small).
Of course one could discuss about a change of basis, for example, and other topics.

These are the things that I would like to see discussed in some reference papers or lectures.

Thanks !

5. Nov 23, 2004

### jvangael

Hi,

AFAIK there is no one measure of entanglement. There are different measures which apply well in different situations. I found this article "PRL 78, 12, 2275-2279: Quantifying Entanglement" interesting. Here the authors define a sort of scheme which good entanglement measures should obey. They first show how some entanglement measures conform to their rules and then derive a measure of their own. It is an interesting article as it does not describe one specific measure. Hope it helps,

Jurgen

6. Nov 23, 2004

### lalbatros

Thanks jvangael,

I looks interresting.
Do you know which typical use is made of these measures and why?
Is it essentially related to quantum information theory? For which purpose?
Has it also been of some use for studying other aspects of real physical systems?
Is it also used for fundamental approach of QM?

Thanks again,

Michel

7. Nov 23, 2004

### marlon

Hi

marlon

8. Nov 23, 2004

### marlon

9. Nov 24, 2004

### lalbatros

Hi Marlon

It is probably true that the density matrix could be "diagnosed" with little or with much entanglement.
But -as far a I can guess- entanglement is first concerned with pure states.

A density matrix could represent a statistical mixture of many unentangled stated.
Then, I would say its measure of entanglement would be zero also.

A density matrix could also represent a population of many entangled pure states.
Then, I would expect its entanglement to be an average of the entanglement of each element in the population.

So, in principle, I would expect the theory of entanglement measure to be more concerned with pure states.
However, it might be the density matrix offers some convenient formalism also for the study of entanglement.
If you know of some reference on this topic, it could be useful.

Thanks

Michel

10. Nov 24, 2004

### marlon

Indeed it does...check out the course material on John Preskill's site that i gave earlier...

regards
marlon

11. Nov 24, 2004

### slyboy

There is now an extensive literature on entanglement measures, although most of it is technical in nature. The quantification of entanglement does have more to do with quantum information rather than physics per-se, although there have been quite a few recent attempts to use entanglement measures to discuss features of condensed matter systems, such as phase transitions. However, this is pretty new stuff, so I couldn't find any nontechnical articles about it.

To return to entanglement measures, there is a (pretty much) unique measure of entanglement for pure 2-party states, called the entropy of entanglement. The first thing to note is that a measure of entanglement should be invariant under local reversible operations, by which I mean that Alice and Bob each perform an independent unitary rotation on their system. Then, it is possible to prove that all states are equivalent to ones of the form:

$$| \psi \rangle \sum_{j} \sqrt{p_j} | j \rangle \otimes | j\rangle$$

where the $$| j \rangle$$'s are an orthonormal basis and the $$p_j$$'s form a probability distribution. This is called the Schmidt decomposition of the state.

The entropy of entanglement is defined as the entropy of the probability distribution, i.e.

$$E(| \psi \rangle) = H(p_j) = - \sum_j p_j \mbox{log}_2 (p_j)$$.

With this definition, it is easy to check that all product states of the form $$| \psi \rangle_A \otimes | \eta \rangle_B$$ have zero entanglement and Bell states have 1 unit of entanglement.

OK, that's the math, but what is the physical significance. To understand this, we need to define a class of operations called Local Operations and Classical Communication (LOCC). LOCC operations include everything that Alice and Bob can do by performing independent unitary rotations on their systems, introducing new systems in unentangled states, throwing away parts of their system, performing independent measurements and communicating the results of those measurements to each other via a classical channel (e.g. a telephone line).

Now, suppose that Alice and Bob share a large number, $$n$$, of copies of the state we are interested in. We can ask what is the largest number of maximally entangled Bell states they can make from this via LOCC. It turns out that as $$n \rightarrow \infty$$ this number tends to $$n E(|\psi\rangle)$$. This is called the "distillable entanglement" of the state.

Conversely, starting with $$n$$ maximally entangled states, we can ask how many copies of the state we are interested in can be made via LOCC. It turns out that this is $$n / E(|\psi\rangle)$$ in the asyptotic limit. This is called the "entanglement cost" of the state.

This gives physical meaning to the entropy of entanglement, and we can see that it is essentially unique because the process of converting back and forth between the state we are interested in and maximally entangled states is reversible in the asymptotic limit. I say "essentially" because this is not true for a strictly finitie number of copies of the state. Then we have to look at the "entanglement monotones" considered by Nielsen and Vidal in order to get a complete classification of entangled states, but I won't go into that here.

Not quite, the problem is that a density matrix can be written as a convex sum of pure states in an infinite number of ways. To get a good measure of entanglement you have to take the minimum of the average entropy of entanglement over all such decompositions. This is called the entanglement of formation. However, this minimisation is incredibly hard to perform in general and the only case for which there is a known analytic formula is for two-qubit states. This was discovered by Wootters.

We can also ask about converting back and forth between maximally entangled states in the mixed case, and define a distillable entanglement and entanglement cost for mixed states. It has been proved that these are not equal, so there is no unique measure for mixed states. Interestingly, there are even states with zero distillable entanglement and nonzero entanglement cost, which have been called "bound entangled states".

The situation gets even more complicated for multiparty (>2) states, since in that case there is no single state that all states (even pure ones) can be converted to by LOCC. It is not even known if there is a set of such states (a so-called Mimimally Reversible Entanglement Generating Set (MREGS)). A few results are known for special cases, e.g. 3 or 4 qubits, but the general problem is still completely open and is probably intractible.

OK, what about the relation to EPR and other areas of physics. Well, one might wonder if the amount of entanglement is related to the degree of violation of a Bell inequality for instance. It does not apprear that there is such a simple relation and there are even known examples of bound entangled states that violate Bell inequalities. Related quatifications of "nonlocality" have been developed, such as the amount of instantaneous communication you would need to generate quantum correlations in a Bell experiment if otherwise restricted to local hidden variables. It is not known if these are simply related to entanglement measures or not.

As for condensed matter physics, well they tend to use the entanglement of formation (or other related measures) to investigate the entanglement between different particles in the ground state of some Hamiltonian. They look for evidence of universality results (i.e. similar scaling of entanglement over coarse grainings of the system in large classes of Hamiltonians) and look for evidence of thing like phase transitions. There is also a program of taking entanglement into account in quantum renormalization group calculations, which may lead to better convergence in general. I am not an expert myself, but it looks like there are some pretty exciting recent results in this area. Related speculations concern the role of entanglement in understanding black hole entropy and information-loss in black holes, on which there have been a few speculative calculations.

12. Nov 24, 2004

### jvangael

Hi All,

Let me explain some things I learned lately on entanglement measures. First of all, there is entanglement for pure states and entanglement for mixed states. A pure state is unentangled if it can be written as a productstate. A mixed state is unentangled if it can be written as a mixture of products.

Take for example the equal mixture of the two maximally entangled states $$|00> + |11>, |00>-|11>$$. Just calculating the density matrix shows you that it is an equal mixture of $$|00>, |11>$$ which is not entangled at all. I haven't looked into this too much so i cannot tell you why this happens but it has very profound implications for quantum computation. (I can elaborate on this if anyone is interested)

If you want an easy measure for entanglement use the Schmidt decomposition. This only works for bipartite entanglement but it is very easy to calculate. The density matrix has the amount of entnanglement in it somewhere but I wouldn't call it a convenient formalism to study entanglement. Measures like the mutual VN entropy use the density matrix but they are not very complete (see paper I mentioned previously) measures of entanglement.

By the way, I forgot to mention Michael Nielsens measure of entanglement which sure has a very intuitive meaning and is relativey easy to calculate. His measures asks how many (asympotically) Bell states can be extracted out of a certain state. The explanation in his book is very good.

Within QIT, entanglement is considered as a resource that can be used for example to teleport or to do superdense coding. These phenomena consume entanglement and thus the amount of entanglement specifies how much teleportation, ... can be done. I am interested in knowing if it is used in any fundamental questions to QM and I will investigate it a little. I vagely recall that for example the CHSH inequality deviates maximally in a Quantum setting if the particles are Bell states. This suggests that there is a link between the entanglement measures and Bell inequalities but I do not know of any formal link.

I hope this helps a bit, take care,

Jurgen

13. Nov 24, 2004

### jvangael

Hi Slyboy,

Isn't there a possibility that some relation between entanglement measures and Bell inequalities is possible? Look at section 4.3.4 (4.1.8 in old edition) in Chapter 4 of Preskill's notes. A state $$\alpha |00> + \beta |11>$$ can violate the CHSH equalities. The amount of violation is dependent on $$\alpha, \beta$$. Because these two values also relate to the entropy of entanglement there is some connection in this specific case.

If you know any papers on the relation between entanglement measures and violation of Bell Inequalities, let me know. I am very interested in studying them.

Jurgen

14. Nov 26, 2004

### slyboy

This isn't Nielsen's measure of entanglement. It was invented in this paper:

http://www.arxiv.org/abs/quant-ph/9511030

However, you might be talking about the single copy entanglement monotones discussed in the book, which were invented by him.

There are certainly relations, but they are not very clear and seem to be more qualitative than quantitative at the moment. In fact, before the modern operational approach to entanglement was invented, the degree of violation of a Bell inequality was regarded by some people to be a sort of measure of entanglement.

The central difficulty in relating the two concepts is that a Bell inequality refers to a particular experimental setup, in the sense that we have to specify the number of measurements available and the number of outcomes of each measurement in order to arrive at a Bell inequality. On the other hand, entanglement measures refer to properties of states, and a priori they don't seem to be so closely connected to the setup of particular Bell experiments. Different states may violate different Bell inequalities by different amounts and even in the bipartite case there are Bell inequalities that are maximally violated by states that are not maximally entangled.

Having said that, we might consider the totality of all Bell inequalities that can be derived for a particular physical system and see how entanglement is related to the violation of all of them. The problem is that there are infinite inequalities for a given physical system. This could be resolved if we could identify complete classes of "relevant" inequalities, such that the violation of all other inequalities could be derived from then. It seems rather unlikely that this can be done, as the task of even listing all Bell inequalities is NP-complete in the number of measurements considered and the number of settings in each measurement (see the book of Pitowsky for this result).

To me, it seems more natural to try and identify operational measures of nonlocality, such as the amount of instantaneous classical communication you need to supplement a local hidden variable theory with in order to generate all possible quantum correlations for a given state.

Anyway, you might be interested in some of these papers on nonlocality:

http://www.arxiv.org/abs/quant-ph/0410027
http://www.arxiv.org/abs/quant-ph/0306129
http://www.arxiv.org/abs/quant-ph/0310166
http://www.arxiv.org/abs/quant-ph/0304076
http://www.arxiv.org/abs/quant-ph/0307125
http://www.arxiv.org/abs/quant-ph/9709026
http://www.arxiv.org/abs/quant-ph/0404097
http://www.arxiv.org/abs/quant-ph/0407221
http://www.arxiv.org/abs/quant-ph/0404076

15. Nov 26, 2004

### jvangael

Thanks sliboy,

It will take me a while to go through them but I sure am interested in some of these results.

Thanks,

Jurgen