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$begingroup$

In the paper written by Man Duen Choi "Completely Positive Linear Maps on Complex Matrices", there was a criterion mentioned/theorem. For reference, I have written it below.

Let $ϕ:M_n→M_m$. Then ϕ is extreme in CP[$M_n,M_m$], iff $ϕ$ has an expression $ϕ(x)=∑_iV_i^*xV_i$ for all $x∈M_n$ and $V_i^*V_j$i,j is a linearly independent set.

I have two questions:

how do you show that $V_i^*V_j$i,j is linearly independent?

Does it imply that $V_i$i is also linearly independent?

I tried searching for ideas to prove it, but I wasn't successful. Thank you!

matrices functional-analysis

share|cite|improve this question

edited Mar 24 at 15:56

YuiTo Cheng

2,1863937

asked Mar 21 at 7:06

SantaSanta

12

$endgroup$

add a comment | 

0

0

$begingroup$

In the paper written by Man Duen Choi "Completely Positive Linear Maps on Complex Matrices", there was a criterion mentioned/theorem. For reference, I have written it below.

Let $ϕ:M_n→M_m$. Then ϕ is extreme in CP[$M_n,M_m$], iff $ϕ$ has an expression $ϕ(x)=∑_iV_i^*xV_i$ for all $x∈M_n$ and $V_i^*V_j$i,j is a linearly independent set.

I have two questions:

how do you show that $V_i^*V_j$i,j is linearly independent?

Does it imply that $V_i$i is also linearly independent?

I tried searching for ideas to prove it, but I wasn't successful. Thank you!

matrices functional-analysis

share|cite|improve this question

edited Mar 24 at 15:56

YuiTo Cheng

2,1863937

asked Mar 21 at 7:06

SantaSanta

12

$endgroup$

add a comment | 

$begingroup$

In the paper written by Man Duen Choi "Completely Positive Linear Maps on Complex Matrices", there was a criterion mentioned/theorem. For reference, I have written it below.

Let $ϕ:M_n→M_m$. Then ϕ is extreme in CP[$M_n,M_m$], iff $ϕ$ has an expression $ϕ(x)=∑_iV_i^*xV_i$ for all $x∈M_n$ and $V_i^*V_j$i,j is a linearly independent set.

I have two questions:

how do you show that $V_i^*V_j$i,j is linearly independent?

Does it imply that $V_i$i is also linearly independent?

I tried searching for ideas to prove it, but I wasn't successful. Thank you!

matrices functional-analysis

share|cite|improve this question

edited Mar 24 at 15:56

YuiTo Cheng

2,1863937

asked Mar 21 at 7:06

SantaSanta

12

$endgroup$

share|cite|improve this question

edited Mar 24 at 15:56

YuiTo Cheng

2,1863937

asked Mar 21 at 7:06

SantaSanta

12

share|cite|improve this question

edited Mar 24 at 15:56

YuiTo Cheng

2,1863937

asked Mar 21 at 7:06

SantaSanta

12

edited Mar 24 at 15:56

YuiTo Cheng

2,1863937

edited Mar 24 at 15:56

YuiTo Cheng

2,1863937

asked Mar 21 at 7:06

SantaSanta

12

asked Mar 21 at 7:06

SantaSanta

12