Quantum mechanics: density matrix purification

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To determine if M(a) is a density matrix, a must be set to 1/2, satisfying the conditions of being Hermitian, having a trace of 1, and being positive semi-definite. The system is confirmed to be in a mixed state since M(a)^2 does not equal M(a). For purification, the discussion highlights confusion regarding the decomposition of the matrix and the conditions for positive semi-definiteness. Clarification is needed on how to ensure that the expression x†M(a)x is non-negative for all vectors x. The responses confirm the correctness of the mixed state determination and seek additional guidance on the purification process.
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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 ?
 
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M(a)≥0 means xM(a)x≥0 for all x.
 
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)
 
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).
 
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