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Suppose I have got a unitary matrix and I want to introduce random noise to simulate data corruption. How to introduce the noise in a proper way such that the corrupted matrix is also unitary?

Rodrigo de Azevedo
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Марина Лисниченко
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  • Hi, Marina, what have you tried on it ? Can you further precise the range of your study especially the size of your matrix and the context in general ? – EDX May 12 '23 at 10:38
  • @EDX, hi! Actually I try do find information and did not try anything. Except naively add noise matrix to the unitary matrix. But then corrupteed matrix is not unitry anymore. The matrix can be of size 2x2, 4x4, 8x8, 16x16... So, any power of 2. – Марина Лисниченко May 12 '23 at 10:50
  • You may now have what is essentially a coupled system, comprising your original system embedded in a noise bath. It may be that you need to extend your unitary to this whole system so that you have a unitary there but not necessarily on the original system? My thoughts are only vague on this though! – Paul May 12 '23 at 11:06
  • @Paul I'm afraid I did not uunderstand you correctly, but originally I have a unitary matrix. But it is ideal matrix. I need to corrupt it such that the "unitariness" preserves, but values are shifted. – Марина Лисниченко May 12 '23 at 11:33
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    Never mind, I did say vague! The answer below probably covers what you want. – Paul May 12 '23 at 13:20
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    another approach would be to use any standard corruption scheme (such as additive noise) then to project the resulting matrix onto the set of unitary ones. – Nathan Wycoff May 12 '23 at 18:50
  • @JohnMadden How would one perform this projection? Could you point me to a reference? – NNN May 13 '23 at 02:37
  • @JohnMadden Lagrange multipliers? – NNN May 13 '23 at 03:05
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    Sorry for too many comments. Found something relevant here: https://math.stackexchange.com/questions/2215359/showing-that-matrix-q-uvt-is-the-nearest-orthogonal-matrix-to-a – NNN May 13 '23 at 03:20

1 Answers1

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If I may sum up your problem :

You have an unitary matrix $U$ enventually complex so $U^*U=UU^*=I$

So if we introduce a multiplicative noise $N$ such that the resulting signal is $U'=NU$.

Then if we required $U'$ to be unitary :

$$U^*N^*NU=NUU^*N^*=I $$

Second term implies $NN^*=I$. It is sufficient so to take $N$ such as $N^*N=I$

Then it is sufficient to take a multiplicative noise which is unitary.

Then using this paper https://case.edu/artsci/math/esmeckes/Meckes_SAMSI_Lecture2.pdf can give you idea on how to practically generate it.

EDX
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