# What is a pure quantum state and a mixed quantum state?

## Main Question or Discussion Point

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?

## Answers and Replies

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G01
Homework Helper
Gold Member
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.

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Fredrik
Staff Emeritus
Science Advisor
Gold Member
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.

bhobba
Mentor
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

Thank you all! I understand it now.