Measure on Lie Algebra “induced” by Haar measure on U(n)
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On the unitary group $U(n)$, the Haar measure provides a suitable notion of uniformity for many applications. The Lie Group $U(n)$ is generated by the Hermitian matrices, i.e., I can write any $A in U(n)$ as $$A = exp(sum_{j,k=1}^n i a_{jk}Omega_{jk})$$
where $a_{jk}$ are real coefficients dependent on A and $Omega_{jk}$ are the standard basis elements of the Lie Algebra $mathfrak{u}(n) = {X in mathbb{C}^{n times n } | X^dagger = X}$.
My question is this: What is the measure "induced" on the Lie Algebra by the Haar measure on $U(n)$? Put another way: What measure on $mathbb{R}^{n times n}$ do I have to use to sample the $a_{jk}$ if I want the corresponding unitary matrices to be distributed according to the Haar measure?
measure-theory lie-groups lie-algebras haar-measure
asked Nov 24 at 10:15
tomet
1495
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up vote
1
down vote
favorite
On the unitary group $U(n)$, the Haar measure provides a suitable notion of uniformity for many applications. The Lie Group $U(n)$ is generated by the Hermitian matrices, i.e., I can write any $A in U(n)$ as $$A = exp(sum_{j,k=1}^n i a_{jk}Omega_{jk})$$
where $a_{jk}$ are real coefficients dependent on A and $Omega_{jk}$ are the standard basis elements of the Lie Algebra $mathfrak{u}(n) = {X in mathbb{C}^{n times n } | X^dagger = X}$.
My question is this: What is the measure "induced" on the Lie Algebra by the Haar measure on $U(n)$? Put another way: What measure on $mathbb{R}^{n times n}$ do I have to use to sample the $a_{jk}$ if I want the corresponding unitary matrices to be distributed according to the Haar measure?
measure-theory lie-groups lie-algebras haar-measure
asked Nov 24 at 10:15
tomet
1495
I do not quite understand what you want, more precisely what the expressions "sample the $a_{jk}$" and "distributed according to the Haar measure" mean exactly. Anyway, it seems that the discussion in the comments to this question is related: math.stackexchange.com/q/1383157/96384
– Torsten Schoeneberg
Nov 25 at 6:57
1
Thanks for your comments. I know how to sample random unitaries that are uniformly distributed w.r.t. the Haar measure (arxiv.org/abs/math-ph/0609050, and implemented in docs.scipy.org/doc/scipy/reference/generated/…). What I would like to do is sample $a_{jk}$ instead of the unitary itself (it's for a numerical application). However, I need to sample $a_{jk}$ in such a way that when I calculate the corresponding unitaries, they are distributed in the same way as if I had sampled them directly as described above. Hope this makes sense...
– tomet
Nov 25 at 7:16
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
On the unitary group $U(n)$, the Haar measure provides a suitable notion of uniformity for many applications. The Lie Group $U(n)$ is generated by the Hermitian matrices, i.e., I can write any $A in U(n)$ as $$A = exp(sum_{j,k=1}^n i a_{jk}Omega_{jk})$$
where $a_{jk}$ are real coefficients dependent on A and $Omega_{jk}$ are the standard basis elements of the Lie Algebra $mathfrak{u}(n) = {X in mathbb{C}^{n times n } | X^dagger = X}$.
My question is this: What is the measure "induced" on the Lie Algebra by the Haar measure on $U(n)$? Put another way: What measure on $mathbb{R}^{n times n}$ do I have to use to sample the $a_{jk}$ if I want the corresponding unitary matrices to be distributed according to the Haar measure?
measure-theory lie-groups lie-algebras haar-measure
asked Nov 24 at 10:15
tomet
1495
On the unitary group $U(n)$, the Haar measure provides a suitable notion of uniformity for many applications. The Lie Group $U(n)$ is generated by the Hermitian matrices, i.e., I can write any $A in U(n)$ as $$A = exp(sum_{j,k=1}^n i a_{jk}Omega_{jk})$$
where $a_{jk}$ are real coefficients dependent on A and $Omega_{jk}$ are the standard basis elements of the Lie Algebra $mathfrak{u}(n) = {X in mathbb{C}^{n times n } | X^dagger = X}$.
My question is this: What is the measure "induced" on the Lie Algebra by the Haar measure on $U(n)$? Put another way: What measure on $mathbb{R}^{n times n}$ do I have to use to sample the $a_{jk}$ if I want the corresponding unitary matrices to be distributed according to the Haar measure?
measure-theory lie-groups lie-algebras haar-measure
measure-theory lie-groups lie-algebras haar-measure
asked Nov 24 at 10:15
tomet
1495
asked Nov 24 at 10:15
tomet
1495
asked Nov 24 at 10:15
tomet
1495
asked Nov 24 at 10:15
tomet
1495
asked Nov 24 at 10:15
tomet
1495
1495
I do not quite understand what you want, more precisely what the expressions "sample the $a_{jk}$" and "distributed according to the Haar measure" mean exactly. Anyway, it seems that the discussion in the comments to this question is related: math.stackexchange.com/q/1383157/96384
– Torsten Schoeneberg
Nov 25 at 6:57
1
Thanks for your comments. I know how to sample random unitaries that are uniformly distributed w.r.t. the Haar measure (arxiv.org/abs/math-ph/0609050, and implemented in docs.scipy.org/doc/scipy/reference/generated/…). What I would like to do is sample $a_{jk}$ instead of the unitary itself (it's for a numerical application). However, I need to sample $a_{jk}$ in such a way that when I calculate the corresponding unitaries, they are distributed in the same way as if I had sampled them directly as described above. Hope this makes sense...
– tomet
Nov 25 at 7:16
add a comment |
I do not quite understand what you want, more precisely what the expressions "sample the $a_{jk}$" and "distributed according to the Haar measure" mean exactly. Anyway, it seems that the discussion in the comments to this question is related: math.stackexchange.com/q/1383157/96384
– Torsten Schoeneberg
Nov 25 at 6:57
1
Thanks for your comments. I know how to sample random unitaries that are uniformly distributed w.r.t. the Haar measure (arxiv.org/abs/math-ph/0609050, and implemented in docs.scipy.org/doc/scipy/reference/generated/…). What I would like to do is sample $a_{jk}$ instead of the unitary itself (it's for a numerical application). However, I need to sample $a_{jk}$ in such a way that when I calculate the corresponding unitaries, they are distributed in the same way as if I had sampled them directly as described above. Hope this makes sense...
– tomet
Nov 25 at 7:16
I do not quite understand what you want, more precisely what the expressions "sample the $a_{jk}$" and "distributed according to the Haar measure" mean exactly. Anyway, it seems that the discussion in the comments to this question is related: math.stackexchange.com/q/1383157/96384
– Torsten Schoeneberg
Nov 25 at 6:57
I do not quite understand what you want, more precisely what the expressions "sample the $a_{jk}$" and "distributed according to the Haar measure" mean exactly. Anyway, it seems that the discussion in the comments to this question is related: math.stackexchange.com/q/1383157/96384
– Torsten Schoeneberg
Nov 25 at 6:57
1
1
Thanks for your comments. I know how to sample random unitaries that are uniformly distributed w.r.t. the Haar measure (arxiv.org/abs/math-ph/0609050, and implemented in docs.scipy.org/doc/scipy/reference/generated/…). What I would like to do is sample $a_{jk}$ instead of the unitary itself (it's for a numerical application). However, I need to sample $a_{jk}$ in such a way that when I calculate the corresponding unitaries, they are distributed in the same way as if I had sampled them directly as described above. Hope this makes sense...
– tomet
Nov 25 at 7:16
Thanks for your comments. I know how to sample random unitaries that are uniformly distributed w.r.t. the Haar measure (arxiv.org/abs/math-ph/0609050, and implemented in docs.scipy.org/doc/scipy/reference/generated/…). What I would like to do is sample $a_{jk}$ instead of the unitary itself (it's for a numerical application). However, I need to sample $a_{jk}$ in such a way that when I calculate the corresponding unitaries, they are distributed in the same way as if I had sampled them directly as described above. Hope this makes sense...
– tomet
Nov 25 at 7:16
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I do not quite understand what you want, more precisely what the expressions "sample the $a_{jk}$" and "distributed according to the Haar measure" mean exactly. Anyway, it seems that the discussion in the comments to this question is related: math.stackexchange.com/q/1383157/96384
– Torsten Schoeneberg
Nov 25 at 6:57
1
Thanks for your comments. I know how to sample random unitaries that are uniformly distributed w.r.t. the Haar measure (arxiv.org/abs/math-ph/0609050, and implemented in docs.scipy.org/doc/scipy/reference/generated/…). What I would like to do is sample $a_{jk}$ instead of the unitary itself (it's for a numerical application). However, I need to sample $a_{jk}$ in such a way that when I calculate the corresponding unitaries, they are distributed in the same way as if I had sampled them directly as described above. Hope this makes sense...
– tomet
Nov 25 at 7:16