Some theoretical background on tanglemeter and its steps of derivation

Tanglemeter provides an invariant under local unitary transformations characterization of multipartite entanglement for pure states. The method has been derived in the publication:

Quantum entanglement via nilpotent polynomials, A. Mandilara,V. M. Akulin, A. V. Smilga and L. Viola, Phys. Rev. A 74, 022331 (2006).

The main steps for the derivation of the tanglemeter’s coefficients are the following, assuming that the multipartite state under analysis is expressed in the computational basis:

  • Using local unitary transformations the initial given state is transformed into a canonical state where the “ground state” is maximized and in consequence the amplitudes of the first excited states are zero:

The optimization is performed with the simulated annealing algorithm and this is verified with a random search algorithm.

  • Then, the amplitude of the ground state is factored out as

and using the nilpotent creation operators

the canonical state is be re-written as

Thus the nilpotent polynomial (in the parenthesis) is derived

  • The final step is to take the logarithm of this operator in order to arrive to the tanglemeter

The C{i,j,..} coefficients are the tanglemeter coefficients of the state, they satisfy the condition that |C{i,j,..}|≤1. The output of the program is a list of all coefficients.

What information about the entanglement you may deduct from the tanglemeter’s coefficients?


  • K-separability

Let us consider a state with qubits ordered as {i,j,..k,l,m,..,n}. Two subgroups {i,j,..,k} and {l,m,..,n} of qubits are not entangled iff all tanglemeter coefficients involving indexes from both subgroups are zero. This rule can be extended to three or higher number sub groups as soon as their union reproduces the total assembly of qubits.

  • In first approximation the tanglemeter coefficients can be used to estimate the amount of K-partite entanglement. The more close is the value of a coefficient to unity the better is the estimation on N-partite entanglement provided by the coefficient. We have come to this conclusion by comparing the coefficients with habitually used measures (monotones) of entanglement i.e. concurrence, 3-tangle.

About the programs:

Details on the use of the mathematica function Tangle are included in the .nb file. In short one just need to provide a list of complex numbers describing your state and the number of the qubits. The uploaded programs are made to work for 3, 4, 5 and 6 qubit states but these are easily extendable to a higher number of qubits (up to 10). In case you need the extended version, send us a message.

We also provide the function Gtangle whose output is a graphical representation of the tanglemeter coefficients.

As an example

As an example, the graph below is the output of the Gtangle function for a randomly picked state of six qubits according to the Haar Measure:

In the first column the pairwise tanglementer’s coefficients C{i,j} are plotted, in the second column the tripartite tanglementer’s coefficients C{i,j,k} are plotted, etc.

One can see immediately that this is genuinely entangled state, and one can estimate in first order the bi-partite, tri-partite .. entanglement between the pairs. One of the sub-groups composed by 5 qubits seems to have a higher 5-partite entanglement than others. From the 6th order coefficient we can conclude that this is quite far from a GHZ state where the highest order coefficient would be 1 and all others tanglemeter’s coefficients zero.

Downloads

Gtangle.nb Tangle.nb