How do I test if a matrix is unitary?
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Bryan Acer
on 9 May 2016
Answered: Pawel Tokarczuk
on 9 May 2016
My current test for a unitary matrix is shown in the code, I'd also like to know if U = e^(iH) [i is the complex number] is coded correctly. Thanks!
U = exp(i*H)
Uinverse = inv(U)
UConjTran = U'
if UConjTran == Uinverse
disp('U is unitary')
else
disp('U is NOT unitary')
end
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Accepted Answer
Roger Stafford
on 9 May 2016
Again, e^(i*H) is not the same as exp(i*H). Check matlab's "mpower" operator.
2 Comments
Roger Stafford
on 9 May 2016
To get an accurate value you should have used exp(1) instead of 2.718.
As I mentioned, you should read up on matlab's 'mpower' operator. If you have A^B where either A or B is a matrix, it has a different meaning than A.^B. For example, if
A = [1 2;3 4];
then
A^2 = A*A = [7 10;15 22]
whereas
A.^2 = [1 4;9 16];
In the case of your e^(i*H), the eigenvalues of H must be real, since H is hermitian, so the matrix e^(i*H) will have eigenvalues that all lie in the unit circle of the complex plane and that will yield a unitary matrix.
More Answers (3)
Azzi Abdelmalek
on 9 May 2016
Your code is correct, but when the inverse is calculated, there maybe some errors (even if it's equal to 10^(-6)) caused by the precision allowed by any computer (software and hardware). It's recommended to test using a tolerance.
U = exp(i*H);
Uinverse = inv(U);
UConjTran = U';
tol=10^(-6);
er=abs(UConjTran-Uinverse)
if sum(er(:))<tol
disp('U is unitary')
else
disp('U is NOT unitary')
end
Pawel Tokarczuk
on 9 May 2016
Your notation suggests that what you need is the matrix exponential:
U = expm(i*H);
This is not the same thing (in general) as the element-wise exponential:
V = exp(i*H);
Try it and be convinced. Anyway, the test for a unitary matrix is:
U*U' = U'*U = I, to some floating-point tolerance, where I is the unit matrix.
Finally, bear in mind that the evolution operator U takes on a more complicated (time-ordered) form when Hamiltonians H evaluated at different times do not commute.
See: Merzbacher ("Quantum Mechanics"), Constantinescu and Magyari ("Problems in Quantum Mechanics"), etc.
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Kevin
on 9 May 2016
Maybe you have already thought about this but decided not to use this method.
Another way to test if a matrix is unitary is to check if (U * U') == square identity matrix (with some threshold). This way, you can avoid matrix inversion.
Another way that I can think of is to use SVD (Matlab function svd).
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