Reduced Density Operator and Entanglement

[Sharkey4123]

I'm having a little bit of trouble getting my head around the idea of the reduced density operator being used to tell us about the entanglement of a state.

I understand that if you take the reduced density operator of any of the Bell states, you get a reduced density operator proportional to the identity, and this is defined as being a maximally entangled state.

But why is this a proof that the state is entangled?

I have written problem that asks to solve the reduced density operator for a particular state. I solve for the reduced density operator and I get a 2x2 matrix with elements in each slot. It satisfies the conditions to be a density operator, but what does it tell me about the entanglement of the state?

Any clarification would be much appreciated!

 

[kith]

I have written problem that asks to solve the reduced density operator for a particular state. I solve for the reduced density operator and I get a 2x2 matrix with elements in each slot. It satisfies the conditions to be a density operator, but what does it tell me about the entanglement of the state?

You could calculate the purity. If the reduced state is less pure than the full state, you have entanglement.

 

[stevendaryl]

I'm having a little bit of trouble getting my head around the idea of the reduced density operator being used to tell us about the entanglement of a state.

I understand that if you take the reduced density operator of any of the Bell states, you get a reduced density operator proportional to the identity, and this is defined as being a maximally entangled state.

But why is this a proof that the state is entangled?

I have written problem that asks to solve the reduced density operator for a particular state. I solve for the reduced density operator and I get a 2x2 matrix with elements in each slot. It satisfies the conditions to be a density operator, but what does it tell me about the entanglement of the state?

Any clarification would be much appreciated!

Let's stick to the very simplest type of wave function, which is a 2-component spinor. If we have two spin-1/2 particles, then the most general composite state is:

[itex]|\Psi\rangle = C_{uu} |u\rangle |u\rangle + C_{ud} |u\rangle |d\rangle + C_{du} |d\rangle |u\rangle + C_{dd} |d\rangle |d\rangle[/itex]

where [itex]|u\rangle |u\rangle[/itex] means both particles are spin-up in the z-direction, [itex]|u\rangle |d\rangle[/itex] means the first is spin-up, the second is spin-down, etc.

The composite state is "entangled" if it cannot be written as a product [itex]|\psi\rangle |\phi\rangle[/itex]. If I did the calculation correctly, the state above is "entangled" unless [itex]C_{uu} C_{dd} = C_{ud} C_{du}[/itex]. For example, the following is a maximally entangled state:
[itex]\frac{1}{\sqrt{2}} (|u\rangle |d\rangle - |d\rangle |u\rangle)[/itex]

A density matrix [itex]\rho[/itex] is "pure" if it can be written in the form [itex]|\psi\rangle \langle \psi\rangle[/itex] for some state [itex]|\psi\rangle[/itex]. Otherwise, it is mixed.

So the connection between mixed states and entanglement is this:

• Start with an entangled pure state [itex]|\Psi\rangle[/itex]
  
• Form the corresponding density matrix.
• "Trace out" the degrees of freedom of one of the particles.
• Then what is left will be a mixed reduced density matrix.

So tracing out turns a pure state into a mixed state, if the original pure state is entangled.

 

[Sharkey4123]

Ok, great! Thanks for the help kith and stevendaryl. I've created a couple of simple states for examples and played around with them and I think I'm getting to grips with it.

I understand then that the trace of the square of the density operator tells us about how "pure" the state is - ie if it's 1 then it's a pure state, otherwise it's mixed.
So returning to the Bell states again, I can see that if I calculate the the reduced density operator I get one half of the identity. If I further calculate the purity, I get a value of one half. Since the Bell states are "maximally entangled" does this mean the smallest possible value I can get for the purity (of any density matrix) is one half?

Thanks

 

[kith]

So returning to the Bell states again, I can see that if I calculate the the reduced density operator I get one half of the identity. If I further calculate the purity, I get a value of one half. Since the Bell states are "maximally entangled" does this mean the smallest possible value I can get for the purity (of any density matrix) is one half?

For a two-level system, yes. In general, the lowest purity is [itex]1/d[/itex] where [itex]d[/itex] is the number of degrees of freedom of the system.

The purity isn't necessarily associated with subsystems. You can also look at a single system. Let's say we have a system with two possible energies. If you have the system in a well-prepared state (one of the energy eigenstates or a superposition of them), the purity is zero. If you have a thermal state at a given temperature, the density matrix is given by the Boltzmann distribution and the purity depends on the temperature. In the high-temeperature limit, the density matrix approaches one half times the identity and the purity is minimal.