Mixing
decomposition of general unitary
1
$begingroup$
Does anybody know how to prove the following:
Given a unitary matrix $U$, prove that:
$$U = V sigma_x V^dagger W sigma_x W^dagger$$
For the proper $V$ and $W$.
Any help is greatly appreciated.
matrices matrix-decomposition
edited Jan 24 at 22:27
Robert Lewis
48.1k23167
$endgroup$
add a comment |
1
$begingroup$
Does anybody know how to prove the following:
Given a unitary matrix $U$, prove that:
$$U = V sigma_x V^dagger W sigma_x W^dagger$$
For the proper $V$ and $W$.
Any help is greatly appreciated.
matrices matrix-decomposition
edited Jan 24 at 22:27
Robert Lewis
48.1k23167
$endgroup$
$begingroup$
What is $sigma_x$?
$endgroup$
– Robert Lewis
Jan 24 at 22:28
1
$begingroup$
@RobertLewis Presumably they're referring to the Pauli matrix $$ sigma_x = pmatrix{0&1\1&0} $$
$endgroup$
– Omnomnomnom
Jan 24 at 22:33
1
$begingroup$
If I have interpreted your question correctly, I think this is possible if and only if $U$ is unitary with determinant $1$.
$endgroup$
– Omnomnomnom
Jan 24 at 22:34
$begingroup$
@Omnomnomnom; so either $U$ is of size $2$ or $V, W$ are $n times 2$?
$endgroup$
– Robert Lewis
Jan 24 at 22:43
1
$begingroup$
@RobertLewis I think $U$ is of size $2$
$endgroup$
– Omnomnomnom
Jan 25 at 0:53
add a comment |
1
1
1
$begingroup$
Does anybody know how to prove the following:
Given a unitary matrix $U$, prove that:
$$U = V sigma_x V^dagger W sigma_x W^dagger$$
For the proper $V$ and $W$.
Any help is greatly appreciated.
matrices matrix-decomposition
edited Jan 24 at 22:27
Robert Lewis
48.1k23167
$endgroup$
Does anybody know how to prove the following:
Given a unitary matrix $U$, prove that:
$$U = V sigma_x V^dagger W sigma_x W^dagger$$
For the proper $V$ and $W$.
Any help is greatly appreciated.
matrices matrix-decomposition
matrices matrix-decomposition
edited Jan 24 at 22:27
Robert Lewis
48.1k23167
edited Jan 24 at 22:27
Robert Lewis
48.1k23167
edited Jan 24 at 22:27
Robert Lewis
48.1k23167
edited Jan 24 at 22:27
Robert Lewis
48.1k23167
edited Jan 24 at 22:27
Robert Lewis
48.1k23167
48.1k23167
asked Jan 24 at 22:20
user638016
$begingroup$
What is $sigma_x$?
$endgroup$
– Robert Lewis
Jan 24 at 22:28
1
$begingroup$
@RobertLewis Presumably they're referring to the Pauli matrix $$ sigma_x = pmatrix{0&1\1&0} $$
$endgroup$
– Omnomnomnom
Jan 24 at 22:33
1
$begingroup$
If I have interpreted your question correctly, I think this is possible if and only if $U$ is unitary with determinant $1$.
$endgroup$
– Omnomnomnom
Jan 24 at 22:34
$begingroup$
@Omnomnomnom; so either $U$ is of size $2$ or $V, W$ are $n times 2$?
$endgroup$
– Robert Lewis
Jan 24 at 22:43
1
$begingroup$
@RobertLewis I think $U$ is of size $2$
$endgroup$
– Omnomnomnom
Jan 25 at 0:53
add a comment |
$begingroup$
What is $sigma_x$?
$endgroup$
– Robert Lewis
Jan 24 at 22:28
1
$begingroup$
@RobertLewis Presumably they're referring to the Pauli matrix $$ sigma_x = pmatrix{0&1\1&0} $$
$endgroup$
– Omnomnomnom
Jan 24 at 22:33
1
$begingroup$
If I have interpreted your question correctly, I think this is possible if and only if $U$ is unitary with determinant $1$.
$endgroup$
– Omnomnomnom
Jan 24 at 22:34
$begingroup$
@Omnomnomnom; so either $U$ is of size $2$ or $V, W$ are $n times 2$?
$endgroup$
– Robert Lewis
Jan 24 at 22:43
1
$begingroup$
@RobertLewis I think $U$ is of size $2$
$endgroup$
– Omnomnomnom
Jan 25 at 0:53
$begingroup$
What is $sigma_x$?
$endgroup$
– Robert Lewis
Jan 24 at 22:28
$begingroup$
What is $sigma_x$?
$endgroup$
– Robert Lewis
Jan 24 at 22:28
1
1
$begingroup$
@RobertLewis Presumably they're referring to the Pauli matrix $$ sigma_x = pmatrix{0&1\1&0} $$
$endgroup$
– Omnomnomnom
Jan 24 at 22:33
$begingroup$
@RobertLewis Presumably they're referring to the Pauli matrix $$ sigma_x = pmatrix{0&1\1&0} $$
$endgroup$
– Omnomnomnom
Jan 24 at 22:33
1
1
$begingroup$
If I have interpreted your question correctly, I think this is possible if and only if $U$ is unitary with determinant $1$.
$endgroup$
– Omnomnomnom
Jan 24 at 22:34
$begingroup$
If I have interpreted your question correctly, I think this is possible if and only if $U$ is unitary with determinant $1$.
$endgroup$
– Omnomnomnom
Jan 24 at 22:34
$begingroup$
@Omnomnomnom; so either $U$ is of size $2$ or $V, W$ are $n times 2$?
$endgroup$
– Robert Lewis
Jan 24 at 22:43
$begingroup$
@Omnomnomnom; so either $U$ is of size $2$ or $V, W$ are $n times 2$?
$endgroup$
– Robert Lewis
Jan 24 at 22:43
1
1
$begingroup$
@RobertLewis I think $U$ is of size $2$
$endgroup$
– Omnomnomnom
Jan 25 at 0:53
$begingroup$
@RobertLewis I think $U$ is of size $2$
$endgroup$
– Omnomnomnom
Jan 25 at 0:53
add a comment |
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$begingroup$
What is $sigma_x$?
$endgroup$
– Robert Lewis
Jan 24 at 22:28
1
$begingroup$
@RobertLewis Presumably they're referring to the Pauli matrix $$ sigma_x = pmatrix{0&1\1&0} $$
$endgroup$
– Omnomnomnom
Jan 24 at 22:33
1
$begingroup$
If I have interpreted your question correctly, I think this is possible if and only if $U$ is unitary with determinant $1$.
$endgroup$
– Omnomnomnom
Jan 24 at 22:34
$begingroup$
@Omnomnomnom; so either $U$ is of size $2$ or $V, W$ are $n times 2$?
$endgroup$
– Robert Lewis
Jan 24 at 22:43
1
$begingroup$
@RobertLewis I think $U$ is of size $2$
$endgroup$
– Omnomnomnom
Jan 25 at 0:53