Wigner function for qutrit system

[ashish arya]

Hi;

     I am working on a project where I was required to calculate wigner function for qudits. I started with d=3 (qutrits) and implemented following steps in qutip:

1. Generated random matrices using rand_dm with dim=[3,3], used low density to have non-zero elements only at the diagonal.

2. Generated wigner function for the above using qutip.wigner(random_dm, xvec, xvec,method='laguerre').

I was expecting to get the following results:

A. random_dm= [1, 0, 0

                               0, 0, 0

                               0, 0, 0] => wigner function positive everywhere 

B. random_dm= [0, 0, 0

                              0, 1 ,0

                              0, 0, 0] => wigner function positive everywhere 

C. random_dm= [0, 0, 0

                              0,  0, 0

                              0, 0, 1] => wigner function positive everywhere 

What I got was following:

A.=> wigner function positive everywhere 

B.=> wigner function negative somewhere 

C.=> wigner function negative somewhere 

I tried doing the same using basis() and plot_wigner() but the results are same i.e. negative wigner for B and C.

Now given that the A, B and C are basis states with no magic, getting a negative wigner function for B and C is astonishing and in variance with literature (victor veitch thesis).

Can someone kindly let me know where is the mistake? 

With regards,

Ashish Arya

 

[Neill Lambert]

Hi Ashish,

I am not very familiar with discrete Wigner functions, but my guess is that they are not completely equivalent to what you get by giving a truncated Fock space to the continuous Wigner functions as defined in Qutip. 

In other words, the definition in qutip is essentially assuming you are attempting to get the continuous phase space representation of a continuous system, and uses the states you give it like a truncated Fock Space representation of that system. Hence the first state you gave in your examples is equivalent to a ground-state harmonic oscillator, which is Gaussian with positive Wigner function. the second state you give it is equivalent to a single photon state, and hence gives a wigner function with negative parts, etc.

Looking at the references you mentioned the discrete wigner function is a little different from this (it seems to treat every state like a discrete point in an effective phase space). if so, it seems like an interesting function to implement if you wanted to contribute to qutip!

all the best

neill

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[ashish arya]

Thanks Neill for the clarification on specifics of the wigner module of qutip.

Ashish Arya

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