Download Stability of local quantum dissipative systems

Stability of local quantum dissipative systems
Commun.
Math.
Phys.
337 (2015), 1275-1315
David Pérez-Garcı́a
Universidad Complutense de Madrid
25 September 2015
joint work with
Toby Cubitt, Angelo Lucia, Spyridon Michalakis
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
1 / 18
Main message 1
Happy birthday Manolo.
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
2 / 18
Main message 2
Modeling noise in quantum systems is possible.
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
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Main message 3
Dissipative quantum state engineering is possible.
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
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Dissipative quantum systems
Let H be a finite-dimensional Hilbert space.
A dissipative quantum system is given by a 1-parameter continuous
semigroup (Tt )t>0 of completely positive, trace preserving (CPTP) maps
(also called quantum channels):
Tt : B(H) → B(H)
system
environment
Physically, this models a system weakly coupled with an environment (e.g.
thermal or depolarizing noise on a quantum memory).
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
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Liouvillian: generator of dissipative evolution
Those semigroups have always an infinitesimal generator L which is called
Liouvillian.
d Tt t=0
dt
The properties of Tt force L to have a very particular structure, called the
Lindblad-Kossakowski form:
X
1
L(ρ) = i[H, ρ] +
Ki ρKi† − {Ki Ki† , ρ}
2
Tt = e tL ←→ L =
i
In dissipative dynamics, the Liouvillian plays the analogous role to the
Hamiltonian in unitary dynamics (it encodes all the physical properties of
the system).
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
6 / 18
Liouvillians on many-body quantum systems
Bu (r )
L=
XX
Lu (r );
supp Lu (r ) = Bu (r )
u
u∈Λ r >0
On many-body quantum systems on a lattice Λ, it is natural to assume
locality of the Liouvillian:
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
7 / 18
Liouvillians on many-body quantum systems
Bu (r )
L=
XX
Lu (r );
u
supp Lu (r ) = Bu (r )
u∈Λ r >0
We usually assume either:
Finite range: Lu (r ) = 0 for r > r ∗
Exponential decay: kLu (r )k1→1 6 e −αr
Power law decay: kLu (r )k1→1 6 (1 + r )−α
For the rest of the talk, just consider exponential decay,
but results can be generalised to polynomial decay.
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
7 / 18
Motivation
Why are they interesting?
I
Modelling of noise
I
Dissipative quantum computation
Dissipative state engineering
I
• Theoretical work: [Kraus et al, 2008] [Verstraete, Wolf, Cirac, 2008]
• Experimental implementations: [Barreiro et al, 2010] [Krauteret al, 2011]
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
8 / 18
Stability is crucial for applicability
P
We consider a perturbed evolution given by Le = u,r Leu (r ) such that
e
L
(r
)
−
L
(r
)
u
u
1→1
6 ε kLu (r )k1→1
Let ρ0 be the initial state of the system and ρt (resp. ρet ) the state after
e
time t with respect to the evolution given by L (resp. L).
Let A ⊂ Λ be any subregion of the lattice. Denote by ρA
eA
t ,ρ
t the
corresponding reduced states.
The problem
Under which conditions can we conclude
A
∀t > 0, ρA
−
ρ
e
t
t 6 kA ε ?
1
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
9 / 18
It is not just standard perturbation theory
The problem
e
Lu (r ) − Lu (r )
1→1
kLu (r )k1→1
6ε
?
=⇒
A
ρt − ρeA
t 6 kA ε,
1
∀t
Remark
ε is the microscopic strength of the perturbation, not its global norm:
e
Lu (r ) − Lu (r )
1→1
e
6 ε but L − L
→∞
kLu (r )k1→1
1→1
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
10 / 18
Conditions for stability
Conditions for stability:
I
unique fixed point (not necessary of full rank) and no periodic points
I
rapid mixing
I
bulk interactions are defined independently of the system size
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
11 / 18
Rapid mixing
Let Tt = e tL . We define the contraction of Tt by the number
η(Tt ) =
1
sup kTt (ρ) − T∞ (ρ)k1 .
2 ρ
We say that L satisfies rapid mixing if
η(Tt ) 6 poly(|Λ|)e −γt .
Equivalently:
tmix (ε) 6 O(log N/ε).
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
12 / 18
Rapid mixing
We say that L satisfies rapid mixing if
η(Tt ) 6 poly(|Λ|)e −γt .
Recent work has generalized Logarithmic Sobolev inequalities to the
quantum setting [Kastoryano, Temme, 2012].
A size-independent log-Sobolev constant implies exactly the type of
convergence required by rapid mixing (but it is not needed, i.e. rapid mixing
is well defined if the fixed point is pure).
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
12 / 18
Stability theorem
Theorem
Let L be a local Liouvillian with a unique fixed point, that satisfies rapid
mixing.
P P
Let E = u r Eu (r ) be a local perturbation: kEu (r )k1→1 6 εe(r ), and
Leu (r ) = Lu (r ) + Eu (r )
Then
∀t > 0,
D. Pérez-Garcı́a (UCM)
A
ρt − ρeA
t 6 poly(|A|)ε
1
Stability of dissipative systems
25 September 2015
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Sketch of the proof
Proof
Fix an observable OA supported on A and call OA (t) its evolution by the
dual semigroup Tt∗ . In this dual picture, we are done if we get
e A (t)
OA (t) − O
6 poly(|A|)kOA k ε .
Decompose using the integral representation:
t
e A (t) =
OA (t) − O
XXZ
u
r
e ∗ E ∗ (r )OA (s) ds
T
t−s u
0
Take norms
X X Zt
e
kEu∗ (r )OA (s)k ds
OA (t) − OA (t) 6
u
D. Pérez-Garcı́a (UCM)
r
0
Stability of dissipative systems
25 September 2015
14 / 18
Lieb-Robinson Bounds
In many-body systems (Hamiltonian, dissipative) there is a finite speed of
propagation of information. This is given by the Lieb-Robinson bound.
The support of a local observable spreads linearly in time, up to an
exponentially-small error.
A(t)
A
[Nachtergaele, Vershynina, Zagrebnov, 2011]
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
15 / 18
Lieb-Robinson Bounds
In many-body systems (Hamiltonian, dissipative) there is a finite speed of
propagation of information. This is given by the Lieb-Robinson bound.
The support of a local observable spreads linearly in time, up to an
exponentially-small error.
A
ε
e
A(t)
[Nachtergaele, Vershynina, Zagrebnov, 2011]
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
15 / 18
Short times
A
d
Bu (r )
Proof
For “short” times t 6 t0 we apply Lieb-Robinoson bounds
Zt0
kEu∗ (r )OA (s)k ds 6 εe(r ) |A| e vt0 e −µd
0
where d = dist(A, Bu (r )).
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
16 / 18
Long times
Proof
Long times is the most technical part. We have to use extensively the rapid
mixing hypothesis:
1 To obtain that the evolution is quasi-frustration free.
2 To show that its fixed point has Local Topological Quantum Order.
3 Finally, together with 1 and 2, to get the desired control of the long
times.
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
17 / 18
The end!
Thank you for your attention
D. Pérez-Garcı́a (UCM)
Stability of dissipative systems
25 September 2015
18 / 18