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).

 

[paranormal]

Thank you for your solution attempt.

a) You are correct, the condition for a density matrix is that it is Hermitian (equal to its conjugate transpose), has a trace of 1, and all eigenvalues are non-negative. In this case, the matrix M(a) is already Hermitian and has a trace of 1, so we just need to ensure that all eigenvalues are non-negative. This is true when a = 1/2, as you have correctly determined.

b) The condition for a system to be in a mixed state is that the matrix is not a pure state, meaning it cannot be written as a rank-1 matrix. In this case, M(a) is not a rank-1 matrix, as it has two non-zero entries, so it is indeed in a mixed state.

c) To purify M(a), we need to find a pure state matrix that, when traced over, gives us back M(a). This can be done by taking the square root of M(a) and then normalizing it to have a trace of 1. In this case, the pure state matrix that purifies M(a) is:

P(a) = (sqrt(a) 0; 0 sqrt(1-a))

This matrix is Hermitian and has a trace of 1, making it a pure state matrix. When traced over, it gives us back M(a):

tr(P(a)) = sqrt(a) + sqrt(1-a) = a + (1-a) = 1 = tr(M(a))

Therefore, P(a) is the purified version of M(a).