# Reduced density matrix

During a course on quantum mechanics we've been talking about density matrices. Now I came across the following exercise.

Consider a two spin $\frac{1}{2}$ systems, labeled 1 and 2.
Calculate:
i) The full density matrix $\hat{\rho}$
ii) The reduced density matrix $\hat{\rho}_\textrm{red} = \textrm{Tr}_2[\hat{\rho}]$, where $\textrm{Tr}_2[\ldots]$ denotes taking the trace over the Hilbert space that describes the spin 2.
iii) The entanglement entropy: $$S_E = -\textrm{Tr}_1[\hat{\rho}_\textrm{red}\ \ln( \hat{\rho}_\textrm{red})]$$

For the cases that the system is in the state: $$\newcommand{\k}{|#1 \rangle}$$ a) $\k \psi = \k \uparrow _1\k \uparrow _2$
b) $\k \psi = \left(\dfrac{\k\uparrow _1 + \k \downarrow _1 }{\sqrt{2}}\right)\left(\dfrac{\k \uparrow _2 + \k \downarrow _2 }{\sqrt{2}}\right)$
c) $\k \psi = \dfrac{\k \uparrow _1 \k \downarrow _2 + \k \downarrow _1\k \uparrow _2 }{\sqrt{2}}$
d) $\k \psi = \k \uparrow_1 \dfrac{\k \uparrow _2 + \k \downarrow _2 }{\sqrt{2}}$

For question i), for every case I believe the answer is rather simple. Basically you use the definition of the density matrix. So every answer should be:

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

Now for question ii), I don't understand how I can use the trace of a spin 2 Hilbert space. This would be a 5 dimensional Hilbert space I believe, which doesn't work with the 4 dimensional Hilbert space that we are in. How can I calculate the reduced density matrix for the first example? And what is the physical interpretation of the (reduced) density matrix? A density matrix for me is a different kind of way to write down the state. IS this a correct way to see the density matrix?

• $\mathrm{tr}_2$ denote the trace over the Hilbert space of the spin denoted $2$, which is a spin-$1/2$. – Norbert Schuch Jan 23 '16 at 14:38
• Do you know the definition of partial trace? ii) means that you should do a partial trace over the second spin (not spin=2). Do you know what pure and mixed states are? The interpretation of the reduced density matrix is that it describes the state of the subsystem when you are ignorant to other degrees of freedom – OON Jan 23 '16 at 14:38
• The reduced density matrix of a system just tells you what information is relevant to expectation values of measurements on that system alone. And yes, the density matrix is just a way to write down the state of a system. – alanf Jan 23 '16 at 14:42
• Ok, that is more logical. But I don't know the definition of the partial trace and what I've found online doesn't really help me. About the pure/mixed states, for me mixed states are more or less classical states. So pure state is a quantum mechanical state where it can be in a superposition in a quantum way (thus at the same time) while mixed states is a classical super position of states which you can see by the non-complex number in front of the states – DeanTheMachine Jan 23 '16 at 14:46