Error in matrix exponentiation?

[Analabha Roy]

Hi all,

Unless I'm missing something, a hermitian operator H and exp(-iH) have the same eigenvectors. So, the eigenvectors of H and exp(-iH) should have inner products of either 0 or 1. However, the code below yields a different result

Full code:

https://pastebin.com/S50Pw17z

Relevant excerpt:

...

jx = qutip.jmat(N,'x')
H0 = jx * jx
H0_expi = (-(1j) * (H0)).expm()
_, estates_H0 = H0.eigenstates()
_, estates_expi = H0_expi.eigenstates()
identity = [[np.abs(state.overlap(U0state)) for state in estates_H0] for U0state in estates_expi]

...

This code, executed in jupyter, yields some values between 0 and 1.

Is there something wrong with matrix exponentiation, or is my reasoning wrong?

Thanks and Regards,

AR

[Simon Cross]

Hi Analabha,

The eigenvalues of jx * jx are degenerate, so the eigenvectors are not unique.

If there are no repeated eigenvalues, then your expectation is correct, although it is true for any diagonalizable matrix without repeated eigenvalues, not only Hermitian ones.

Regards,

Simon

[Analabha Roy]

Hi,

Thanks. That was it.  So if I break the symmetry that causes the degeneracy, for instance, do sx^2 - \gamma sy^2 for small \gamma, the degeneracy should break...

AR

 

Regards,

Simon

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Analabha Roy

Assistant Professor

Department of Physics

The University of Burdwan

Golapbag Campus, Barddhaman 713104

West Bengal, India

Emails: dan...@utexas.edu, ar...@phys.buruniv.ac.in, harise...@gmail.com

Webpage: http://www.ph.utexas.edu/~daneel/