How Is Bipartite State Purity Affected by Parameter Changes?

[ma18]

Homework Statement

Consider the bipartite state:

|q> = a/sqrt (2) (|1_A 1_B> +|0_A 0_B>)+sqrt((1-a^2)/2) (|0_A 1_B>+|1_A 0_B>)

where a is between or equal to 0 and 1

a) compute the state of the subsystem p_b

b) compute the purity of p_b as a function of a

c) for what values of a is the purity of p_b at a minimum/maximum

d) Compute the entanglement entropy of the bipartite state, for what value of a is it at a min/max

The Attempt at a Solution

I have done a) and found p_b to be :

p_b = |q_b><q_b|

where |q_b> = (a+sqrt(1-a^2))/sqrt(2) (|0_B>+|1_B>)

the computation for this is long and I don't want to replicate it here...

b)

I simply applied:

p (0_B,0_B) = <0_B|p_b|0_b> for the 4 combinations of |0_b> and |1_b> and got the matrix, then taking the trace of this for the diagonal terms I get 1/2 and 1/2 so the purity was equal to 1 and thus not dependent on a

c) It is unrelated

d) I know that the equation is

S (p_AB) = S (Tr_B p_AB)

but I don't know how to proceed from here

Any help such as how to proceed or checking my previous steps would be greatly appreciated!

 

[ma18]

Could anybody help with this please

 

[DrClaude]

b)

I simply applied:

p (0_B,0_B) = <0_B|p_b|0_b> for the 4 combinations of |0_b> and |1_b> and got the matrix, then taking the trace of this for the diagonal terms I get 1/2 and 1/2 so the purity was equal to 1 and thus not dependent on a

How is purity defined?

 

[ma18]

How is purity defined?

In my notes purity is defined as the square of the trace of a state's density operator, it is between 1/d and 1 where d is the dimension of the Hilbert space. P = 1 for pure states. The states with maximal classical uncertainty (maximally mixed states) have the minimum possible purity P = 1/d, we say that the state is 'maximally mixed'.

 

[DrClaude]

In my notes purity is defined as the square of the trace of a state's density operator,

You have a small mistake there. $\mathrm{Tr}(\rho)$ gives you the normalization of $\rho$, so you will always get $\mathrm{Tr}(\rho)^2 = 1$. To calculate the purity, you need to take the trace of the square of the density operator, $\mathrm{Tr}(\rho^2)$.

 

[ma18]

You have a small mistake there. $\mathrm{Tr}(\rho)$ gives you the normalization of $\rho$, so you will always get $\mathrm{Tr}(\rho)^2 = 1$. To calculate the purity, you need to take the trace of the square of the density operator, $\mathrm{Tr}(\rho^2)$.

Ah, that makes it work!

Alright so the only part I am confused about then is part d) and how to compute the entanglement entropy mathamatically

 

[DrClaude]

Your missing an equation for calculating the entropy of a density matrix.

 

[ma18]

Your missing an equation for calculating the entropy of a density matrix.

Yes I know. That is what I am searching for, I do not have it in my notes and there is no textbook :(

Do you know this equation?

 

[DrClaude]

$$
S(\rho) = -\mathrm{Tr} ( \rho \log_2 \rho)
$$
Have also a look at http://www.av8n.com/physics/thermo/rho.html

 

[ma18]

Thank you for the equation and the link!

Hmm I am getting a negative entropy for some values of a

How come you posted it as log 2 but it simply says log, presumably log 10 on the site?

 

[DrClaude]

Thank you for the equation and the link!

Hmm I am getting a negative entropy for some values of a

How come you posted it as log 2 but it simply says log, presumably log 10 on the site?

I put the 2 there to be explicit, as I guessed this would cause confusion. In this field, log always mean the logarithm in base 2.