Understanding the density matrix for pure states vs. mixed states

So, there's a question in my textbook that explains the following scenario:

A physicist runs two experiments A and B to prepare quantum systems in a variety of initial states. In experiment A he uses a probabilistic machine that can prepare a single quantum system in one of $$n$$ possible pure states $$\{\lvert \psi_1 \rangle, \lvert \psi_2 \rangle, ..., \lvert \psi_n \rangle \}$$ with corresponding probabilities $$\{\lvert p_1 \rangle, \lvert p_2 \rangle, ..., \lvert p_n \rangle \}$$. In experiment $$B$$, instead, he generates m non-interacting quantum systems, each of them prepared in its corresponding lower energy state $$\{\lvert \phi_1 \rangle, \lvert \phi_2 \rangle, ..., \lvert \phi_m \rangle \}$$. Let $$\rho_A$$ and $$\rho_B$$ be the density matrix operators for the quantum states prepared in experiments $$A$$ and $$B$$, respectively.

And asks the following:

1. Write down an expression for both $$\rho_A$$ and $$\rho_B$$ and derive an expression for the expected value of $$\hat{O_A}$$ and $$\hat{O_B}$$ (the Hermitian operators describing an observable of the system A and kth system of system B, respectively).

2. Let $$P_j$$ and $$P_k$$ be the projectors associated to the states $$\lvert \psi_j \rangle$$ and $$\lvert \psi_k \rangle$$ produced by $$A$$. Discuss whether the product $$P_jP_k$$ vanishes.

For question 1, I wrote the following for system $$A$$:

$$\rho_A = \lvert \psi_j\rangle\langle \psi_j\rvert \tag{1}$$

$$\hat{O_A} = Tr(\hat{O_A} \rho_A) = Tr(\hat{O_A}\lvert\psi_j\rangle\langle \psi_j\rvert) \tag{2}$$

My justification for equation 1 is that system A produces pure states only and that only "a single" state is generated, so therefore the density matrix only contains one term (here j represents the jth state generated). However, the question mentions their probabilities which I haven't included in my answer because only one state is generated. Should $$p_j$$ be the coefficient in equation (1) as opposed to just the number 1?

For system $$B$$, I have the following:

$$\rho_B = \frac{1}{m}\sum_i\lvert \phi_i\rangle\langle \phi_i\rvert \tag{3}$$

My justification for this is that experiment B describes a mixed state, which produces a total of m states and therefore each state has a probability of $$\frac{1}{m}$$. I am unsure, however, about how to write the expected value for experiment $$B$$.

For the second question, I can express the product as

$$P_jP_k = \lvert \psi_j\rangle\langle \psi_j\rvert \psi_k\rangle\langle \psi_k\rvert \tag{4}$$

But as I'm given no information regarding the orthogonality of the states for experiment $$A$$, how am I supposed to deduce whether this equation vanishes or not?

If anyone could provide any hints or insight into my answers, it would be greatly appreciated. I am not fully confident that I correctly understand the differences between pure and mixed states.

Each pure State $$|\psi_n\rangle$$ of the quantum system is a multi-particle state describing all particles of the system. So your pure states are all possible states of the system. Since it is not prepared in a state $$|\psi\rangle$$ there are plenty of possibilites of states in which your systems could be.

Let us say you prepared your system in $$|\psi\rangle$$, then it is in a pure state. What physicists call a pure state is essentially the density matrix / statistical operator:

$$\hat{\rho} = |\psi\rangle \langle\psi|$$

Since you are given the probabilities of the state, which are just real numbers, you can assume your system to be in a mixed state. Also, probabilites are usually not denoted as kets. For a mixed state you write:

$$\hat{\rho} = \sum_i p_i|\psi_i\rangle \langle \psi_i|$$

Also for the probabilites must hold:

$$1 = \sum_i p_i$$

For your second question, the states are not required to be orthogonal nor form a complete orthonormal system, but they are required to be normalized.

I hope that helps you.

Edit: For question 1 you should probably also consider writing out the trace using a complete orthonormal system of states e.g. the eigenstates of the density matrix.

• Thank you for your answer. I understand the different expressions for a pure state and mixed state but regarding my answers, did I write the correct matrix density for each system? My confusion arises in trying to understand if $\rho_A$ is pure and $\rho_B$ is mixed – s.twenty Nov 10 '19 at 23:07
• I am not 100% sure with this, but I would say that the density matrix in Experiment A is mixed, since it is probabilistic and that in experiment B, since you have m non-interacting quantum states, you have m pure density matricies. – Tera Nov 11 '19 at 4:49