Quantum mechanics: density matrix purification

[vj3336]

Homework Statement

Given a matrix

M(a) = (a -(1/4)i ; (1/4)i a)

(semicolon separates rows)

a) Determine a so that M(a) is a density matrix.
b) Show that the system is in a mixed state.
c) Purify M(a)

The Attempt at a Solution

a) from conditions for a density matrices

1) M(a)=M(a)*
2) tr(M(a))=1
3) M(a)>=0

form 1) a must be real
from 2) a=1/2
I'm not sure what 3) means, but if it means trace and determinant
must be non-negative, than this is also fulfilled if a=1/2.

b) I think that condition for system to be in mixed state is:
M(a)^2 /= M(a)
since this is true for M(a), system is in mixed state.

c) Don't know how to solve this part, Maybe it has to do something with decomposing a matrix ?

 

[vela]

M(a)≥0 means x^†M(a)x≥0 for all x.

 

[vj3336]

thanks,
if I take x to be a vector with components (x y) then,
x†M(a)x = a(xx† + yy†) + (1/4)i(xy† + x†y)

this then means a(xx† + yy†)≥(1/4)i(xy† + x†y)
but term on the left hand side is real and term on the right hand side may be complex,
so how can I conclude when is x†M(a)x≥0 ?

also is my answer to b) correct ?
and if anyone would give me a little help on c)

 

[vela]

First, you made a sign error. Second, note that x^*y and xy^* are conjugates.

Your answer to (b) is correct.

I'm not sure what they're asking for in (c).