Pure mixed entangled confusion

[kop442000]

I have read quite a bit on here and elsewhere, but am still coming to terms with pure, mixed, entangled and product states.

I know a state is either mixed or pure, but then is engtaglement a sub-class of being mixed? Or a sub-class of being pure? Or is entangled another name for pure? Or something different altogether?

Also is a product state another way to say mixed? Is it the opposite to entanged?

As you can see, I am very confused!

Thanks in advance of replies posted.
Kop442000.

 

[Fredrik]

Pure state = a projection operator for a 1-dimensional subspace, i.e. an operator of the form $|\psi\rangle\langle\psi|$
Mixed state = a density operator (a linear operator satisfying 2.6-2.8) that isn't a pure state

One way to see if a density operator $\rho$ is pure or not is to compute $\operatorname{Tr}\rho^2$. The result is =1 if and only if the state is pure. (It's <1 for mixed states).

Entangled state = a state of a composite system that can't be expressed as a tensor product $|\psi_1\rangle\otimes\cdots\otimes|\psi_n\rangle$ (It would therefore have to be expressed as a linear combination of such tensor products).

 

[Entropia]

Dear Kop442000,

I understand your confusion and would be happy to provide some clarification on these concepts. Let's start with the difference between pure and mixed states. A pure state is a state in which the system is in a definite quantum state, meaning that its properties can be precisely determined. This is similar to a classical state, where the system is in a well-defined state that can be described by classical physics. On the other hand, a mixed state is a state in which the system is in a combination of different quantum states, and its properties cannot be precisely determined. This is analogous to a statistical mixture in classical physics, where the system is in a combination of different classical states and its properties are described by probabilities.

Now, let's move on to entangled states. Entangled states are a special type of pure state in which two or more particles are inextricably linked, meaning that the properties of one particle are dependent on the properties of the other particle(s). This is a unique feature of quantum mechanics and cannot be explained by classical physics. So, entanglement is not a sub-class of being mixed or pure, but rather a property that can be present in both mixed and pure states.

As for product states, they are the opposite of entangled states. In a product state, the properties of each particle are independent of the other particles, meaning that there is no entanglement present. This does not necessarily mean that a product state is equivalent to a mixed state. A product state can be either pure or mixed, depending on the properties of each individual particle.

I hope this helps to clear up some of your confusion. If you have any further questions, please don't hesitate to ask. Science can be complex, but it's always worth the effort to understand it better.

Best regards,