Stability of local observables in closed and open quantum systems Angelo Lucia anlucia@ucm.es Universidad Complutense de Madrid NSF/CBMS Conference Quantum Spin Systems UAB, June 20, 2014 A. Lucia (UCM) Stability of local observables Quantum Spin Systems 1 / 18
Many-body evolutions Fix a finite metric graph Λ sufficiently regular (e.g., Λ = Z D N, and N is called the system size). Algebra of observables: A Λ = x Λ B(H x), where H x C d. Local observables: A X A Λ, for X Λ independent of N. Definition (Many-body evolution) A continuous (semi)group of completely positive, unital maps γ t : A Λ A Λ Closed systems: γ t is a unitary evolution and d dt γ t(a) = i[h, γ t (A)] Open systems: γ t 1 and d dt γ t(a) = L(γ t (A)) In order for γ t to be c.p. and unital, L has to satisfy the Lindblad condition. A. Lucia (UCM) Stability of local observables Quantum Spin Systems 2 / 18
Lindblad form Definition (Lindblad form) L(A) = i[h, A] + j K j AK j 1 2 {K j K j, A} where H is Hermitian. Remark γ t will not be an automorphism in general. A. Lucia (UCM) Stability of local observables Quantum Spin Systems 3 / 18
Local evolutions Definition (Local evolution) γ t is said to be local if L can be written as a sum of local terms L = L x x Λ where the support of L x is finite: L x = L x 1, and L x : A Bx (r) A Bx (r) and is Lindlad. r is called the range of the interaction. Definition (Quasi-local evolution) γ t is said to be quasi-local if L can be written as a sum of local terms L = L x (r) x Λ r 0 where the support of L x (r) is B x (r) and L x (r) f (r). f (r) is the decay rate and is assumed to be faster than any polynomial. A. Lucia (UCM) Stability of local observables Quantum Spin Systems 4 / 18
Expectation values of local observables Expectation values In closed systems, we are usually interested in the ground states of H At times, we are also interested in other eigenstates of H They are all steady states of γ t : ω γ t = ω In open systems, we are usually interested in the case where γ t has a unique steady state ρ and no periodic points: ρ γ t = ρ and moreover lim t γ t(a) = ρ (A)1 A. Lucia (UCM) Stability of local observables Quantum Spin Systems 5 / 18
General problem Given one such interesting state ρ, how sensitive is the expectation value of local observables ρ(a) with respect to changes in L? Can it tolerate local perturbations, i.e. replacing L by L = L + E, where E = x Λ E x where E x is small (but E is not)? ρ(a) ρ (A) k Y sup E x, x A A Y where k Y is a constant independent of the system size. A. Lucia (UCM) Stability of local observables Quantum Spin Systems 6 / 18
Motivation 1 Quantum phase classification The physicists definition of quantum phase is given in terms of smoothness of expectation values of observables Local observables are what is reasonable to expect to be actually measurable in an experiment The mathematicians definition requires the states to be groundstates of the extremes of a smooth path of gapped local Hamiltonians 2 Noise modelling 3 Using engineered dissipation to prepare interesting quantum states as fixed points of open system dynamics (entangled states, magic states, quantum codes, etc.) A. Lucia (UCM) Stability of local observables Quantum Spin Systems 7 / 18
Closed systems Local observables are stable if H(s) = H + se is uniformly gapped: inf gap H(s) γ > 0 s [0,1] We reduce the problem to stability of the spectral gap (Frustration-free, pbc: [Michalakis, Pytel, 2011]) Open systems with unique fixed point Local observables are stable if γ t is fast mixing, i.e. if { } t ε = inf t > 0 sup γ t (A) ρ (A)1 ε A A scales sub-linearly with system size [Cubitt, L., Michalakis, Perez-Garcia, 2013] A. Lucia (UCM) Stability of local observables Quantum Spin Systems 8 / 18
Quasi-adiabatic evolution [Hastings 2004] Let H(s) be a family of uniformly gapped local Hamiltonians. Let P(s) the projector on the groundstate space of H(s). Quasi-adiabatic evolution There exists a unitary U(s) such that P(s) = U(s)P(0)U(s) and U(s) has a quasi-local structure given by d ds U s = id(s)u(s), U(0) = 1 with D(s) being quasi-local Hermitian operator (i.e. a time-dependent Hamiltonian) Also called spectral flow, since it is not specific of the groundstate. A. Lucia (UCM) Stability of local observables Quantum Spin Systems 9 / 18
L.R. bounds for quasi-adiabatic evolution Let α s (A) = U(s) AU(s) the dual of the spectral flow. Lemma (Bachmann, Michalakis, Nachtergaele, Sims 2011) For A A X, B A Y vs µ dist(x,y ) [α s (A), B] 2 A B e By denoting 1/ε, where ε = sup x E x, we have H(s) = H + s E, s [0, ε] ε in such a way that v, µ are independent of ε. A. Lucia (UCM) Stability of local observables Quantum Spin Systems 10 / 18
Local perturbations pertub locally Remember that α s was generated by D(s) = x D x(s) For Y Λ, let αs Y be the evolution generated by D Y (s) = x Y D x(s) Lemma Let A A X. Let X r = {x dist(x, X ) r} Then α s (A) α Xr (A) A e vs g(r) with g(r) a fast decaying function. Remark α Xr s (A) is supported on X r. s A. Lucia (UCM) Stability of local observables Quantum Spin Systems 11 / 18
Stability Theorem For all A A X it holds that A P(0) A P(s) δ(s) kx A where δ(s) grows polynomially in s and A P(s) = tr AP(s) Proof. By using the spectral flow we obtain A P(s) = tr AP(s) = tr AU(s)P(0)U(s) = tr U(s) AU(s)P(0) = tr α s (A)P(0) = α s (A) P(0) and therefore A P(0) A P(s) αs (A) A α s (A) α Xr (A) + s αs Xr (A) A A. Lucia (UCM) Stability of local observables Quantum Spin Systems 12 / 18
Proof. αs A P(0) A P(s) (A) α Xr (A) + s αs Xr (A) A We can make the first term arbitrarily small by Lieb-Robison bounds, choosing r O(s) Then αs Xr (A) A s 0 x X r [D Xr (τ), A] dτ s 0 [D x (τ), A] dτ δ(s) k X A A. Lucia (UCM) Stability of local observables Quantum Spin Systems 13 / 18
Open system case In the closed system case, we used two important facts related to the spectral flow: 1 α s connects P(0) to P(s) in finite time 2 α s can be localized by L.R. : finite times finite volumes For open systems, spectral flow is not available. (Unless somebody has an idea to prove me wrong...) But if we have a unique fixed point, then η t (A) = γ t(a) ρ (A)1 A 0 A. Lucia (UCM) Stability of local observables Quantum Spin Systems 14 / 18
Let γ t and γ t be the original and the perturbed evolutions, respectively. We can apply Duhamel formula: γ t (A) γ t(a) t γ t s(a)eγ s (A) ds x 0 t 0 E x γ s (A) ds where we have assumed γ t 1. Let us focus on one fixed x Λ. A. Lucia (UCM) Stability of local observables Quantum Spin Systems 15 / 18
We can split the integral at a time t 0 = t 0 (x) to be determined later: 1 short times: dissipative version of Lieb-Robinson bounds t 0 with d = dist(x, supp A) 0 E x γ s (A) ds E x A 1 µv eµ(d vt 0) 2 long times: We use the fact E x (1) = L x (1) L x(1) = 0 t t 0 E x γ s (A) ds = t t 0 E x E x [γ s (A) ρ (A)1] ds t γ s (A) ρ (A)1 ds E x η s (A) ds t 0 t 0 A. Lucia (UCM) Stability of local observables Quantum Spin Systems 16 / 18
By taking t 0 (x) = 2d(x)/v where d(x) = dist(x, supp A) we have that γ t (A) γ t(a) c A (sup E x ) e µd(x) + x x Λ 2d(x)/v η s (A) ds Theorem Let A A Y. If x η s (A) ds is summable over Λ and is independent of system size, then γt (A) γ t(a) ky A sup E x x A. Lucia (UCM) Stability of local observables Quantum Spin Systems 17 / 18
Rapid mixing Remember that t ε = inf{t > 0 sup η t (A) ε} A Theorem If t ε O(log m Λ + log 1 ε ) for some m > 0, then for each A A Y it holds that η t (A) c A e γt A. Lucia (UCM) Stability of local observables Quantum Spin Systems 18 / 18