University of California at San Diego – Department of Physics – TA: Shauna Kravec
Quantum Mechanics C (Physics 130C) Winter 2014
Worksheet 9 –
Solutions
Please read and work on the following problems in groups of 3 to 4.
Solutions will be posted on the course webpage.
Announcements
• The 130C web site is:
http://physics.ucsd.edu/∼mcgreevy/w14/ .
Please check it regularly! It contains relevant course information!
• This week let’s think about some formal properties of density matrices!
Problems
1. With All Your ρ’s Combined
Quantum states are said to form a convex set with extremal points corresponding
to pure states.
What does that mean?
(a) Prove that if ρ1 and ρ2 are density matrices then ρ = p ∗ ρ1 + (1 − p)ρ2 is as well
where p is a probability.
This implies that for every pair of points in the set the straight line that connects
them is also contained in the object.
For example a disk is a convex set but an annulus is not.
We require ρ = ρ† , which is true by inspection, and both Tr [ρ] = 1 andTr [ρ2 ] ≤ 1
Tr [ρ] = pTr [ρ1 ] + (1 − p)Tr [ρ2 ] = p + (1 − p) = 1
It follows from the metric on m × n matrices and the Cauchy-Schwartz inequality
that if A and B are positive semi-definite matrices (such as density matrices) that:
0 ≤ Tr [AB]2 ≤ Tr [A2 ]Tr [B 2 ] ≤ Tr [A]2 Tr [B]2 which for A = ρ and B = |ψihψ|
implies: 0 ≤ Tr [ρ2 ] ∗ 1 ≤ Tr [ρ]2 ∗ 1 = 1 which proves the result.
(b) A point is said to be extremal if it can’t be written as a (non-trivial) linear
combination of other states.
1
Prove the claim that pure states are extremely in this set.
P
P
Consider a state ρ and its spectral decomposition ρ = i pi |ψi ihψi | with i pi = 1
If this decomposition, itself a linear combination, is to be trivial then it only
includes a single term: ρ = |ψihψ| which is pure
2. Scrambled ρ’s
Suppose you have an operation which takes ρ → p(U1 ρU1† ) + (1 − p)(U2 ρU2† )
For some arbitrary unitary operators U1 , U2 where p again is a probability.
(a) Show that there exists at least one pure state mapped to another pure state by
the operation above. (Hint: Consider |ψi an eigenvector of U2† U1 )
Consider |ψi an eigenvector of U2† U1 (exists as this operator is unitary) with
eigenvalue eiθ by unitarity. Consider the above operation acting on ρ = |ψihψ|
We can relate: U2 |ψihψ|U2† = U2 (e−iθ U2† U1 |ψi)(hψ|U1† U2 eiθ )U2† = U1 ρU1†
Using the above result we substitute:
p(U1 ρU1† ) + (1 − p)(U2 ρU2† ) = p(U1 ρU1† ) + (1 − p)(U1 ρU1† ) = U1 ρU1† = |ψ 0 ihψ 0 |
where |ψ 0 i = U1 |ψi proving the result
P
P
†
(b) Suppose we turn up the scrambling to ρ → m
k=1 pk Uk ρUk where
k pk = 1
Where this operation takes all density matrices to the maximally mixed state 21 1
Show that there must be at least 4 unitaries in the set {Uk } exactly if pk = 14
(Hint: Rule out the case of 3 unitary operators by contradiction with the previous
result. Why can rule out the possibilities of 1 and 2 operators immediately?)
We can rule out just one unitary trivially and two unitaries by the result above
(we explicitly showed this operator takes a pure state to a pure state) so we should
rule out the case of three unitary operators with the above result.
Suppose we could construct this operation with three unitaries. This implies:
P
†
†
1
=
1
k Uk ρUk and note that by U3 1U3 = 1 we can write WLOG
2
= p1 U3† U1 ρU1† U3 + p2 U3† U2 ρU2† U3 + p3 ρ
Using the above result we can choose a pure input |ψihψ| such that the terms
involving U1 and U2 map to a different pure state |ψ 0 ihψ 0 | letting us rewrite:
1
1 = (1 − p3 )|ψ 0 ihψ 0 | + p3 |ψihψ| which is solved by p3 = 12
2
But note there was nothing special about our choice of U3 to conjugate by.
We could just as easily repeat this procedure for U1 and U2 to conclude that
p1 = 12 = p2 as well which contradicts the need they sum to 1
Following this method for 4 unitaries leaves you with an extra term which allows
one to explicitly construct the operator needed.
1
1
2
2