# Quantum state

1. Feb 21, 2013

hi
what is a pure quantum state and a mixed quantum state? I looked up the internet but I did not quite understand. For pure quantum state I have read that they can't be written down as a mixture of other states??? are entangled states pure states?

2. Feb 21, 2013

### G01

The distinction between 'pure' and 'mixed' applies to ensembles of quantum systems, not the possible states of individual quantum systems.

Consider the following examples. You have N spin 1/2 systems:

1. All of your spin systems are identically prepared in a S_z= +1/2 eigenstate. This is a pure ensemble.

2. All of your spins are identically prepared in the same linear combination of + and - Sz eigenstates. This is also a pure ensemble.

3. 25% percent of your spin systems are prepared like in 1 above. 75% are prepared in the state described in 2. This is a mixed ensemble.

Mathematically the difference is easily seen in terms of the density matrix $\hat{\rho}$ of the ensemble if you are familiar with that idea. For a pure ensemble, the trace of the square of the density operator is a maximum at one: $Tr(\hat{\rho}^2)=1$. For any mixed ensemble the trace of $\hat{\rho}^2$ is less than 1. Also, for a pure state, the density operator is indempotent, i.e. $\hat{\rho}^2=\hat{\rho}$ So, if you are able to write down the density matrix of the system, you have an easy way of determining whether a state is pure or mixed.

Last edited: Feb 21, 2013
3. Feb 22, 2013

### Fredrik

Staff Emeritus
The "ensemble" can also consist of the systems that participate in the experiment when you run it multiple times. If you use a preparation procedure that puts the spin of a silver atom in the pure state $|+\rangle$ 1/3 of the time and in the pure state $|-\rangle$ 2/3 of the time, the spin of a single silver atom that's been subjected to that procedure is in a mixed state.

4. Feb 22, 2013

### bhobba

A state in general is defined as a positive operator of unit trace.

By definition a pure state is defined as an operator of the form |u><u|. It can be shown that any state can be written as a convex sum of pure states - but not uniquely. Non pure states are called mixed.

Check out;
http://en.wikipedia.org/wiki/Quantum_state#Mixed_states

Thanks
Bill

5. Feb 22, 2013