Homework Set 1

Problem 1: Expectation values of spin in pure vs. mixed quantum states

A researcher in spintronics is investigated two devices in order to generate spin-polarized currents. One of those devices has spins comprising the current described by the density matrix:

${\hat {\rho }}_{1}={\frac {|\!\!\uparrow \rangle \langle \uparrow \!\!|+|\!\!\downarrow \rangle \langle \downarrow \!\!|}{2}}$,

while the spins comprising the current in the other device are described by the density matrix

${\hat {\rho }}_{2}=|u\rangle \langle u|$ , where $\ |u\rangle ={\frac {e^{i\alpha }|\!\!\uparrow \rangle +e^{i\beta }|\!\!\downarrow \rangle }{\sqrt {2}}}$.

Here $|\!\!\uparrow \rangle$ and $|\!\!\downarrow \rangle$ are the eigenstates of the Pauli spin matrix ${\hat {\sigma }}_{z}$:

${\hat {\sigma }}_{z}|\!\!\uparrow \rangle =+1|\!\!\uparrow \rangle ,\ {\hat {\sigma }}_{z}|\!\!\downarrow \rangle =-1|\!\!\downarrow \rangle$.

What is the spin-polarization of these two currents? Comment on the physical meaning of the difference between the spin state transported by two currents.

HINT: Compute the x, y, and z components of the spin polarization vector using both of these density matrices following the quantum-mechanical definition of expectation value $P_{x,y,z}=\langle {\hat {\sigma }}_{x,y,z}\rangle =\mathrm {Tr} \,[{\hat {\rho }}{\hat {\sigma }}_{x,y,z}]$. The colloquial "spin-polarization" discussed in spintronics literature is $|\mathbf {P} |$ in this rigorous description.

Problem 2: Dynamics of the Bloch vector

The Hamiltonian of a single spin of an electron in external magnetic field $\mathbf {B}$ is given by (assuming that gyromagnetic ration is unity):

${\hat {H}}=-{\frac {\hbar }{2}}\mathbf {B} \cdot {\hat {\boldsymbol {\sigma }}}$

where ${\boldsymbol {\sigma }}=({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ is the vector of the Pauli matrices. Show that the von Neumann equation of motion

${\frac {\partial {\hat {\rho }}}{\partial t}}=-{\frac {i}{\hbar }}[{\hat {H}},{\hat {\rho }}]$

for the density matrix of spin-1/2

${\hat {\rho }}={\frac {1}{2}}\left({\hat {I}}+\mathbf {P} \cdot {\hat {\boldsymbol {\sigma }}}\right)$

can be recast into the equation of motion for the Bloch (or spin-polarization) vector because ${\hat {\mathbf {\rho } }}$ and $\mathbf {P}$ are in one-to-one correspondence in the case of spin-1/2. Find explicitly the right hand side of such an equation:

${\frac {d\mathbf {P} }{dt}}=?$

HINT: Use the following property of the Pauli matrices:

${\hat {\sigma }}_{\alpha }{\hat {\sigma }}_{\beta }-{\hat {\sigma }}_{\beta }{\hat {\sigma }}_{\alpha }=2i\epsilon _{\alpha \beta \gamma }{\hat {\sigma }}_{\gamma }$.

Problem 3: Does entropy increase in a closed quantum system?

In classical Hamiltonian systems the nonequilibrium entropy

$S=-k_{B}\int \rho \ln \rho$

is constant in time. In this problem we want to demonstrate that in microscopic evolution of an isolated quantum system, the entropy is also time independent, even for general time-dependent density matrix ${\hat {\rho }}$. That is, using the equation of motion:

$i\hbar {\frac {\partial {\hat {\rho }}}{\partial t}}=[{\hat {H}},{\hat {\rho }}]$

prove that von Neumann entropy

$S(t)=-k_{B}\mathrm {Tr} [{\hat {\rho }}(t)\ln {\hat {\rho }}(t)]$

is time independent for arbitrary density matrix ${\hat {\rho }}(t)$.

HINT: Use $\mathrm {Tr} ({\hat {A}}{\hat {B}}{\hat {C}})=\mathrm {Tr} ({\hat {C}}{\hat {A}}{\hat {B}})$ for any operators ${\hat {A}}$, ${\hat {B}}$, ${\hat {C}}$, as well as that an operator ${\hat {M}}$ commutes with any function $f({\hat {M}})$:

$[{\hat {M}},f({\hat {M}})]=0$.

Problem 4: Successive measurements on subsystems of composite bipartite quantum system

Consider a quantum system composed of two spins, labeled as subsystem A and B. The quantum state of the composite system is described by the following density matrix in the Hilbert space ${\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}$:

${\hat {\rho }}={\frac {1}{8}}{\hat {I}}+{\frac {1}{2}}|\Psi \rangle \langle \Psi |$

where ${\hat {I}}$ denotes the $4\times 4$ unit matrix in ${\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}$ and

$|\Psi \rangle ={\frac {1}{\sqrt {2}}}\left(|\!\!\uparrow \rangle \otimes |\!\!\downarrow \rangle -|\!\!\downarrow \rangle \otimes |\!\!\uparrow \rangle \right)$

is entangled state (in the context of spins also called "singlet") of two spins.

Suppose we measure the first spin (subsystem A) along the axis described by the unit vector $\mathbf {n}$, and the second spin (subsystem B) along the axis described by the unit vector $\mathbf {m}$, where $\mathbf {n} \cdot \mathbf {m} =\cos \theta$. What is the probability that both spins are "spin-up" along their respective axes?

HINT: In general, the probability to measure eigenvalue $\lambda$ of a physical quantity in the quantum state described by the density matrix ${\hat {\rho }}$ is given by $\mathrm {prob} =\mathrm {Tr} [{\hat {\rho }}{\hat {P}}_{\lambda }]$. Here ${\hat {P}}_{\lambda }$ is the projection operator on the eigensubspace corresponding to eigenvalue $\lambda$. To find the probability of measurement on the subsystem, one should use the density matrix of that subsystem, obtained by partial trace over the states of the second subsystem. This means that the probability asked in the problem is defined by:

$\mathrm {prob} =\mathrm {Tr} _{B}\left\{{\hat {P}}_{\mathbf {m} }^{B}{\hat {\rho }}^{B}\right\}=\mathrm {Tr} _{B}\left\{{\hat {P}}_{\mathbf {m} }^{B}\mathrm {Tr} _{A}\left[({\hat {P}}_{\mathbf {n} }^{A}\otimes {\hat {I}}^{B}){\hat {\rho }}\right]\right\}$.

The eigenprojector for the "spin-up" (i.e., +1) eigenvalue along the $\mathbf {n}$-axis is simply:

${\hat {P}}_{\mathbf {n} }^{A}=|\!\!\uparrow _{\mathbf {n} }\rangle \langle \uparrow _{\mathbf {n} }\!\!|={\frac {1}{2}}\left({\hat {I}}^{A}+\mathbf {n} \cdot {\hat {\boldsymbol {\sigma }}}^{A}\right)$.

We also use the fact that resulting state of the composite system after the selective measurement on subsystem A is described by the density matrix ${\hat {P}}_{\mathbf {n} }^{A}{\hat {\rho }}{\hat {P}}_{\mathbf {n} }^{A}$, so that subsystem B after the measurement on system A is "collapsed" onto the state described by the density matrix ${\hat {\rho }}^{B}=\mathrm {Tr} _{A}[{\hat {P}}_{\mathbf {n} }^{A}{\hat {\rho }}{\hat {P}}_{\mathbf {n} }^{A}]=\mathrm {Tr} _{A}[{\hat {P}}_{\mathbf {n} }^{A}{\hat {\rho }}]$, where we use cyclic property of the trace and $({\hat {P}}_{\mathbf {n} }^{A})^{2}={\hat {P}}_{\mathbf {n} }^{A}$.

Problem 5: Ambiguity of ensemble decomposition of density operator

(a) Suppose we have a statistical mixture of spin-1/2 particles that consists of the state $|\uparrow _{z}\rangle$ with probability $(1+1/{\sqrt {2}})/2$ and the state $|\downarrow _{z}\rangle$ with probability $(1-1/{\sqrt {2}})/2$. Find the matrix representation of the density operator (i.e., the density matrix) ${\hat {\rho }}$ in the basis of states $\{|\uparrow _{z}\rangle ,|\downarrow _{z}\rangle \}$, as well as in the basis of eigenstates of the Pauli matrix ${\hat {\sigma }}_{x}$ denoted as $\{|\uparrow _{x}\rangle ,|\downarrow _{x}\rangle \}$. Compute purity $\mathrm {Tr} \,{\hat {\rho }}^{2}$ of this density matrix and its Bloch vector, and comment if this quantum state is pure or mixed.

(b) Now suppose that we have a mixed state with $1/2$ probability to have spin pointing along $\mathbf {e} _{n_{1}}=(\mathbf {e} _{z}+\mathbf {e} _{x})/{\sqrt {2}}$ and $1/2$ probability to have spin along $\mathbf {e} _{n_{1}}=(\mathbf {e} _{z}-\mathbf {e} _{x})/{\sqrt {2}}$.

(c) Compare state in (a) to the one in (b), including comparing their Bloch vectors. Are they the same or different?

NOTE: The general theorem on when two different mixtures of pure quantum states represent the same density matrix <math> \hat{\rho} <\math>:

<math> \hat{\rho} = \sum_i p_i |\Phi_i\rangle \langle \Phi_i| = \sum_i q_i |\Psi_i\rangle \langle \Psi_i| <\math>

such as that

was proven by Schrödinger in 1936 (he commented "this theorem was one for which I claim no priority but the permission of deducing it in the following section, for it is certainly not well known"), and then rediscovered by Hughston, Jozsa, and Wootters (HJW) in 1993. In 1998, Mermin simplified a portion of HJW’s proof, commenting that on it as "a pertinent theorem which deserves to be more widely known", while none of them knew about the original Schrödinger work (so, today it is proper to denote it as Schrödinger-HJW theorem).