University of California at San Diego – Department of Physics – Prof. John McGreevy
Quantum Mechanics C (130C) Winter 2014
Assignment 6
Posted February 20, 2014
Due 2pm Thursday, March 6, 2014
Problem Set 6
1. Decoherence by phase damping with non-orthogonal states [from Preskill]
Suppose that a heavy particle A begins its life in outer space in a superposition of two
positions
|ψ0 iA = a|x0 i + b|x1 i.
These positions are not too far apart. The particle interacts with the electromagnetic
field, and in time dt, the whole system evolves according to
p
√
UAE |x0 iA ⊗ |0iE = 1 − p|x0 iA ⊗ |0iE + p|x0 iA ⊗ |γ0 iE
p
√
UAE |x1 iA ⊗ |0iE = 1 − p|x1 iA ⊗ |0iE + p|x1 iA ⊗ |γ1 iE
But because x0 and x1 are close, the (normalized) photon states |γ0 i, |γ1 i have a large
overlap:
hγ0 |γ1 iE = 1 − , with 0 < 1.
(a) Find the Kraus operators describing the time evolution of the reduced density
matrix ρA .
(b) How long does it take the superposition to decohere? More precisely, at what
time t is (ρA )01 (t) = 1e (ρA )01 (t = 0)?
2. Decoherence on the Bloch sphere [from Preskill]
Parametrize the density matrix of a single qubit as
1
~
~ .
ρA =
1 +P ·σ
2
(a) Polarization-damping channel.
Consider the (unitary) evolution of a qbit A coupled to a 4-state environment via
3
X
p
p
σAi ⊗ 1 E |φiA ⊗ |iiE
UAE |φiA ⊗ |0iE = 1 − p|φiA ⊗ |0iE + p/3
i=1
1
Show that this evolution can be accomplished with the Kraus operators
p
p
M0 = 1 − p1, Mi = p/3σ i ,
and show that they obey the completeness relation requred by unitarity of UAE .
Show that the polarization Pi of the qbit evolves according to
4p ~
~
P → 1−
P.
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[Hint: use the identity σi σj σi = 2σj δij − σj .]
Describe this evolution in terms of what happens to the Bloch ball.
What happens if p > 3/4?
(b) Two-Pauli channel.
Consider the (unitary) evolution of a qbit A coupled to a three-state environment
via
2
X
p
p
σAi ⊗ 1 E |φiA ⊗ |iiE
UAE |φiA ⊗ |0iE = 1 − p|φiA ⊗ |0iE + p/2
i=1
Show that this evolution can be accomplished with the Kraus operators
p
p
M0 = 1 − p1, Mi = p/2σ i , i = 1, 2
and show that they obey the completeness relation requred by unitarity of UAE .
Describe this evolution in terms of what happens to the Bloch ball.
(c) Phase-damping channel.
For the evolution of problem 1,
p
√
1 − p|0iA ⊗ |0iE + p|0iA ⊗ |γ0 iE
p
√
UAE |1iA ⊗ |0iE = 1 − p|1iA ⊗ |0iE + p|1iA ⊗ |γ1 iE
UAE |0iA ⊗ |0iE =
now thinking of A as a qbit, describe the evolution of its polarization vector on
the Bloch ball.
3. Purity test. [from Chuang and Nielsen] (This problem is not directly associated with
our current obsessions in 130C, but it is a useful reminder about the notions of pure
and mixed states.)
Show that for any density matrix ρ:
(a) trρ2 ≤ 1
(b) the inequality is saturated only if ρ is a pure state.
[Hint: don’t forget that the trace operation is basis-independent.]
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4. Polarization. [from Boccio]
Recall that in the expression for the general density matrix of a qbit
1
~
ρ=
1 + P~ · σ
2
we called P~ the polarization.
(a) To justify this name, show that
P~ = h~
σ i.
(b) Subject the qbit to a external magnetic field, which couples by
~
~ = − γ σ · B.
H = −~
µ·B
2
(γ is called the gyromagnetic ratio.) Assuming the qbit is isolated, so that its
time evolution is unitary, what is the time evolution of the polarization, ∂t P~ ?
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