Very recently some authors experimentally demonstrate an embedding quantum simulator, using it to efficiently measure two-qubit entanglement. Their embedding quantum simulator  uses three polarization-encoded qubits in a circuit with two concatenated controlled-sign gates. The measurement of only two observables on the resulting tripartite state gives rise to the efficient measurement of bipartite concurrence, which would otherwise need 15 observables [PRL 116, 070503 (2016)]

We devised a very nice method to detect both using just two local measurement bases.

Have a look: https://arxiv.org/abs/1512.05315

negativity, logarithmic negativity, concurrence etc. it is up to your problem situation.

and also u may consider the fidelity with special entangled states.

Dear P. Darabi.   Do you think that each measure is used for a specific issue ?!

I don't think so. I mean it's not a big deal, but for example, when u plot the logarithmic negativity diagram, u see that its behavior is more sensitive than negativity in especial ranges. so, as I said before, it is up to ur problem situation.

let us know about ur problem

thanks a lot.  Must strive to find an efficient measure of entanglement! Bipartite and Multipartite entanglement.

What do you mean by "efficient"?  Efficiently calculable?

reply to: Dear Todd A. Brun

Ideal measure; In this sense, calculate the actual amount of entanglement.

The amount of entanglement doesn't really have a meaning apart from a well-defined measure.  An "ideal" measure of entanglement should have the following characteristics:  it is non-vanishing if and only if the state is entangled; it is maximized by some recognizably "maximally-entangled" states; it has an operational interpretation (i.e., it quantifies the ability to carry out some quantum information protocol); it is monotonic (non-increasing under local operations and classical communication); and it is easy to calculate.  For bipartite pure states there is a measure that satisfies all of those requirements:  the entropy of entanglement, which is monotonic, straightforward to calculate, nonzero for all entangled states and zero for all product states, and which quantifies the number of maximally entangled pairs that can be produced asymptotically from many copies of the given state.

But for mixed states, and multipartite states, no measure that I know of satisfies all of these requirements.  There are a variety of different measures that may satisfy some of these requirements but not others.  Some (like negativity) are widely used in numerical modeling because they are easy to calculate, but in general don't have a direct operational interpretation, and may not be nonzero for all entangled states.  Others have great theoretical importance (like the entanglement of formation) but cannot generally be calculated in closed form for most states--they require difficult optimizations, or regularized expressions, or both.

Dear Todd A. Brun

Thank you for your beautiful and scientific answers. entanglement measures(for bipartite pure states) are invariant under local unitary  (LU) transformations, non-increasing under local operations and classical communication (LOCC). But I mean that which of these are measure to reliable? Based on which index or benchmark?

For illustration, concurrence is better or negativity? Why?!

There are a lot of measure of entanglement that exist nowadays, but we have to focus on the meaning of the term efficient, as mister Todd A. Brun said.

It depends if we want to qualify or quantify the entanglement.

For example, for pure states, we have some algebraic varieties that describe a certain type/class of entaglement, under SLOCC (stochastic local operations with classical communication). Also by using invariant theory, some polynomial can help us to dinstinguish differents classes of entanglement. We can also study the entaglement of pure state by associating, to each quantum state, a geometrical singularity, that can helps to have an information on the entanglement of the state.

There are also some numerical measurement of entanglement that can help us to quantify entanglement, such as Groverian Measure of entanglement, Hyperdeterminant, logarithmic negativity, Von neumann entropy, etc.

So for particular configurations (2, 3 or 4 qubits, 3 qutrits, ...) we can, given any quantum state, determine the class of entanglement that it belongs to. So qualitatively, we have some tools to do that. For other systems, it is not the case, and it may will not be the case ...

For mixed or pure states, we have some numerical measures of entaglement, but each of them give us a certain information about the entanglement, but is not known that a particular measure surpasses all the others and is appliable to any quantum system.

The problem is still open, I think.

Dear Mr. Jaffali

Thanks a lot.

Article A Brief Overview of Bipartite and Multipartite Entanglement Measures

Just to understand your question: Do you ask for a measure through a mathematical criterion (which can only measure "mathematical entanglement") or an experimental measure (which would measure physical entanglement)?

Juan Bulnes Dear Dr. Bulnes,

Mathematical criterion.

Dear Saeed Haddadi,

Let me tell you that, in general, mathematical criteria that identify (directly or indirectly) "mathematical entanglement" do not necessarily identify "physical entanglement." For example, it was shown that the Peres-Horodecki criterion, in the case for 2 qbits, recognizes as entangled a state that cannot be of the physical type; which means that this criterion can identify the mathematical entanglement, but not necessarily the physical entanglement (see the article: "A case of spurious quantum entanglement originated by a mathematical property with a non-physical parameter", by Bulnes and Bonk)

Juan Bulnes

I read your nice paper. Thank you.