# Quantum mechanics: density matrix purification

• vj3336
The purification of a density matrix is a vector in a larger space, i.e., a pure state, of which the given density matrix is the projection.In summary, the conversation discusses determining a value for a matrix to be a density matrix, showing that the system is in a mixed state, and purifying the matrix. The solution involves finding a real value for the matrix, determining that the system is in a mixed state, and using vector decomposition to find a pure state.
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 ?

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.

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

Last edited:

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).

## 1. What is quantum mechanics?

Quantum mechanics is a branch of physics that studies the behavior of matter and energy at the subatomic level. It explains how particles and energy interact with each other and how they behave in different situations.

## 2. What is a density matrix purification?

Density matrix purification is a technique used in quantum mechanics to manipulate the density matrix of a quantum system in order to remove any unwanted or decohered states. This allows for the extraction of useful information from the quantum system.

## 3. How does density matrix purification work?

Density matrix purification involves applying unitary operations on the density matrix of a quantum system in order to remove any unwanted or decohered states. These operations can be designed to target specific states or to remove all off-diagonal elements in the density matrix.

## 4. What are the applications of density matrix purification?

Density matrix purification has various applications in quantum computing, quantum communication, and quantum information processing. It can be used to improve the accuracy and efficiency of quantum algorithms, to reduce errors in quantum communication, and to enhance the security of quantum cryptography protocols.

## 5. Are there any limitations to density matrix purification?

While density matrix purification is a powerful technique, it does have some limitations. It can only be applied to quantum systems that have a well-defined density matrix, and it may not be able to completely remove all decoherence or unwanted states. Additionally, the success of the purification process depends on the accuracy of the initial state preparation and the strength of the unitary operations applied.

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