Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
−π
−π/2
1.0
0.5
α(k)
Figure 4.3:
Plot of α(k) = arccos cosφ +f cos(k)sin φ according to Eq.(4.19), for f = 0.4
and φ = π
4 .
states, we concentrate on the nondegenerate case of small couplings |f| < tan(φ/2)
or |f| < cot(φ/2) . The above relations between f and the eigenvalues are illustrated
in Fig. 4.2; for a plot of the resulting α(k) see Fig. 4.3.
We will consider below the time evolution of the initial state, show that it converges
and characterize the possible limit states. The following arguments make use of the
projectors onto the eigenspaces of ˆ Γ, which are provided by
Lemma 4.6:
If ˆ Γ(k) has nondegenerate, complex eigenvalues e ±iα(k) with α(k) ∈ (0, π), the
(nonorthogonal) projectors Pk and Pk onto its eigenspaces in a decomposition
are given by
Pk = 1
2
and Pk as the complex conjugate of Pk.
π/2
ˆΓ(k) = e iα(k) Pk + e −iα(k) Pk
k
π
i + 2 cosα(k) − Γ(k) ˆ −1
sin α(k)
(4.21)
(4.22)
Proof: The operators Pk and Pk = − Pk are projectors onto the disjoint eigenspaces
of ˆ Γ(k). 8 Since Pk + Pk = , the real and imaginary parts of both projectors
are connected via Re Pk = − Re Pk and ImPk = − ImPk. Writing the above
decomposition (4.21) in terms of Re Pk and ImPk yields
ˆΓ(k) = cosα(k) − 2 sin α(k) Im Pk + i sinα(k)(2 RePk − ). (4.23)
By (4.18), ˆ Γ(k) has to be real-valued. Hence the last term of (4.23) has to vanish
and we immediately obtain RePk = /2. Note that we excluded the degenerate
8 Proof of this statement: If ψ− is the eigenvector of ˆ Γ(k) to eigenvalue e−iα(k) , then e−iα(k) ψ− =
ˆΓ(k) · ψ− = ` eiα(k) Pk + e−iα(k) ( − Pk) ´ · ψ−, which implies Pk · ψ− = 0 and Pk · ψ− = ψ−.
Similarly, Pk · ψ+ = ψ+ and Pk · ψ+ = 0 for the eigenvector ψ+ to eigenvalue eiα(k) . Since ˆ Γ(k)
has determinant 1 and thus full rank, the eigenvectors are linearly independent and span the
whole spaceÊ2 . Hence 0 = Pk ·Pk = Pk ·( −Pk) = Pk −P 2 k or P2 k = Pk , i.e. Pk is a projector.
The same holds for Pk.
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