University of California at San Diego – Department of Physics – TA: Shauna Kravec
Quantum Mechanics C (Physics 130C) Winter 2014
Worksheet 8
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!
Problems
1. Decoherence-Free Subspaces
Consider a two-qubit system living in Hq = H1 ⊗ H2 and let me define the following
operators: X2 ≡ X ⊗ 1 + 1 ⊗ X, Y2 ≡ Y ⊗ 1 + 1 ⊗ Y , and Z2 ≡ Z ⊗ 1 + 1 ⊗ Z
(a) Construct a state |ψi ∈ Hq that satisfies: X2 |ψi = Y2 |ψi = Z2 |ψi = 0
(b) Consider a Hamiltonian H = sx X2 + sy Y2 + sz Z2 . Prove that |ψi is stationary
under time evolution by this operator. That is, show e−iHt |ψi = |ψi
(c) Now we couple the system to a bath, enlarging the Hilbert space to H = Hq ⊗Hbath
Suppose the systems interacts by a Hamiltonian: HC = X2 ⊗Bx +Y2 ⊗By +Z2 ⊗Bz .
Show by a similar argument that |ψi⊗|φi is stationary under this evolution where
|φi ∈ Hbath
We’ve found a one-dimensional subspace protected from decoherence!
In principle one can redo the above for a 4-qubit system and find a two-dimensional
subspace for the appropriately defined coupling.
Such a subspace exists whenever there’s a symmetry in how the environment
couples to the system.
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2. Weak Decoherence
Recall that phase damping of a qubit occurs when a photon scatters off it and is
knocked into one of a two orthogonal states correlated with the qubit. Let’s consider
a model where the photon states aren’t mutually orthogonal {|γi, |ηi}. That is:
|ψiqubit ⊗ |uniphoton →
p
√
1 − p|ψi|uni + p(a|0i|γi + b|1i|ηi)
Where hη|γi = 1 − for some real and |ψiqubit = a|0i + b|1i
(a) Consider the case of = 1. What does it correspond to? What about = 0?
(b) Construct a basis for the space spanned by {|γi, |ηi}. Recall one may do this
by writing a vector, |δi which is normalized, orthogonal to |γi or |ηi, and is a
superposition of the two vectors.
(c) Write the evolved state in terms of this new basis. What’s an expression for the
density matrix ρqubit before this scattering?
2
|a| λb∗ a
0
Show ρqubit =
for λ a function of p and λa∗ b |b|2
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