thermalization and quantum information fernando g.s.l. brandão university college london new...
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Thermalization and Quantum Information
Fernando G.S.L. BrandãoUniversity College London
New Perspectives on Thermalization, Aspen 2014
partially based on joint work with Aram Harrow and Michal Horodecki
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Plan
1. Equivalence of microcanonical and canonical ensembles
quantum information-theoretic proof
2. Equilibration by random unitary evolutions
relation to unitary designs, quantum circuit complexity
Quantum Information Thermalization
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Microcanonical and Canonical Ensembles
Given a Hamiltonian of n particles :
Microcanonical:
Canonical:
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Microcanonical and Canonical Ensembles
When should we use each?
Micro: System in isolation Macro: System in equilibrium with a heat bath at temperature 1/β
What if we are only interestedin expectation values of local observables?
Is the system an environment for itself?
i.e. For every e, is there a β(e) s.t.
A
X
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Previous Results• Equivalence when A interacts weakly with Ac
• Non-equivalence for critical systems (e.g. 2D Ising model)
• Equivalence for infinite lattices in the “unique phase region” (i.e. only one KMS state). But no bounds on the size of A.
(Goldstein, Lebowitz, Tumulka, Zanghi ’06; Riera, Gogolin, Eisert ‘12)
(Desermo ’04)
(Lima ‘72; Muller, Adlam, Masanes, Wiebe ‘13)
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Equivalence of Ensembles Away from Criticality
thm Let H be a Hamiltonian of n particles on a d-dimensional lattice. Let β be such that ρβ has a correlation length ξ. Then for most regions A of size at most nO(1/d),
Ad = 2 Hij
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Equivalence of Ensembles Away from Criticality
thm Let H be a Hamiltonian of n particles on a d-dimensional lattice. Let β be such that ρβ has a correlation length ξ. Then for most regions A of size at most nO(1/d),
Ad = 2 Hij
Correlation length ξ: For all X, Z
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Equivalence of Ensembles Away from Criticality
thm Let H be a Hamiltonian of n particles on a d-dimensional lattice. Let β be such that ρβ has a correlation length ξ. Then for most regions A of size at most nO(1/d),
Ad = 2 Hij
Obs1: e(β) given by mean energy of ρβ
Obs2: Equivalent to
for all observables X in A
Obs3: For every H, ρβ has finite ξ for β sufficiently small
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Entropy and Relative Entropy
Entropy:
Subaditivity:
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Entropy and Relative Entropy
Entropy:
Subaditivity:
Relative Entropy:
S(ρ || σ) measures the distinguishability of ρ and σ
Positivity:
Data Processing Inequality:
Pinsker Ineq:
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Free Energy and Relative Entropy
Free energy:
Relative Entropy:
Easy-to-derive identity: For every state ρ,
Obs: That ρβ minimizes free energy follows directly from positivity of relative entropy
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Proof by Information-Theoretic Ideas I
A1 B1 A2 B2 …
size(Ai) = O(n1/3), size(Bi) = O(ξn1/3)
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Proof by Information-Theoretic Ideas I
A1 B1 A2 B2 …
size(Ai) = O(n1/3), size(Bi) = O(ξn1/3)
(energy window has spread O(n1/2))
((Simon ’93) proved o(n) bound.
To obtain O(n1/2) consider H without boundary terms between Ai and Bi
and bound error in entropy)
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Proof by Information-Theoretic Ideas I
A1 B1 A2 B2 …
size(Ai) = O(n1/3), size(Bi) = O(ξn1/3)
(energy window has spread O(n1/2))
((Simon ’93) proved o(n) bound.
To obtain O(n1/2) consider H without boundary terms between Ai and Bi
and bound error in entropy)
What does imply?
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Proof by Information-Theoretic Ideas II
A1 B1 A2 B2 …size(Ai) = O(n1/3), size(Bi) = O(ξn1/3)
data processing previous slide
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Proof by Information-Theoretic Ideas II
A1 B1 A2 B2 …size(Ai) = O(n1/3), size(Bi) = O(ξn1/3)
data processing previous slide
Claim 1: Correlation length ξ implies:
Claim 2:
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Proof by Information-Theoretic Ideas III
A1 B1 A2 B2 …size(Ai) = O(n1/3), size(Bi) = O(ξn1/3)
To finish
subadditivity entropy previous slide
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Proof by Information-Theoretic Ideas III
A1 B1 A2 B2 …size(Ai) = O(n1/3), size(Bi) = O(ξn1/3)
To finish
subadditivity entropy previous slide
Since m = O(n2/3):
By Pinsker inequality:
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Further Implications
(Popescu, Short, Winter ’05; Goldstein, Lebowitz, Timulka, Zanghi ‘06, …) Let H be a Hamiltonian and Se the subspace of states with energy (en, en + n1/2). Then for almost every state |ψ> in Se, and region A sufficiently small,
Consequence: If ρβ(e) has a correlation finite correlation length,
Obs: Even stronger statement is conjectured to hold in some cases
Eigenstate Thermalization Hypothesis: For most eigenstates |Ek> in Se
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Plan
1. Equivalence of microcanonical and canonical ensembles
quantum information-theoretic proof
2. Equilibration by random unitary evolutions
relation to unitary designs, quantum circuit complexity
Quantum Information Thermalization
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Dynamical Equilibrationn
State at time t:
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Dynamical Equilibrationn
State at time t:
Will equilibrate?
I.e. for most t ?
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Dynamical Equilibrationn
State at time t:
Will equilibrate?
I.e. for most t ? NO!
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Relative Equilibrationn
How about relative to particular kind of measurements?
• “macroscopic” measurements (e.g. magnetization)
• local measurements (e.g. measurements in the 3 first sites)
Examples
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Equilibration is generic(Linden, Popescu, Short, Winter ’08)Almost* any Hamiltonian H equilibrates:
with
and
S E*Almost: (Ei + Ej – Ek - El) all non-zero and |0n> spread over energy eigenstates
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Time Scale of Equilibration
The previous general approach only gives exponentially long convergence bounds (in of particles )♯
Goal: Show that
Generic local dynamics leads to rapid equilibration
Caveat: time-dependent Hamiltonians…
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Random Quantum CircuitsLocal Random Circuit: in each step an index i in {1, … ,n} is chosen uniformly at random and a two-qubit Haar unitary is applied to qubits i e i+1
Random Walk in U(2n) (Another example: Kac’s random walk)
Introduced in (Hayden and Preskill ’07) as a toy model for the dynamics of a black hole
Example of random circuit (Emerson et al ’03, Oliveira et al ‘07, …)
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Parallel Random Quantum CircuitsParallel Local Random Circuit: in each step n/2 independent Haar two-qubit gates are applied to either ((1, 2), (3, 4), …,(n-1,n)) or ((2, 3), (4, 5), …,(n-2,n-1))
Discrete version of
with random H(t) = H12(t) + H23(t) + … + Hnn-1(t)
(i.e. “brownian quantum circuits” (Lashkari, Stanford, Hastings, Osborne, Hayden ‘12))
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How fast 1D random circuits equilibrate?
(B., Harrow, Horodecki ‘12) Let RCt be the set of all parallel circuits of depth t ≥ 100n. For almost all U in RCt
For every region S, |S| ≤ n/4:
with
As fast as possible:
S
(Brown and Fawzi ‘13, ‘14) Generalization to any dim (t ≥ O(n1/dpolylog(n)))
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Warm-up: Equilibration for Haar Random Unitaries
Let and
We have
But
So for
S Sc
(Page ‘93)
only second moments needed
Haar measure
Is there a similar argument for random circuits?
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Warm-up: Equilibration for Haar Random Unitaries
Let and
We have
But
So for
S Sc
(Page ‘93)
only second moments needed
Haar measure
Is there a similar argument for random circuits?
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Unitary k-designs
Def. An ensemble of unitaries {μ(dU), U} in U(d) is an ε-approximate unitary k-design if for every monomial M = Up1, q1…Upk, qkU*r1, s1…U*rk, sk,
|Eμ(M(U)) – EHaar(M(U))|≤ ε
• First k moments are close to the Haar measure
• Natural quantum generalization of k-wise independent distributions
• Many applications in quantum information
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Random Quantum Circuits as k-designs?
thm (B., Harrow, Horodecki ‘12) Parallel Local Random Circuits of size O(k5n + k4log(1/ε)) are an ε-approximate unitary k-design
• Can replace Page’s calculation with k=2 and ε = 2-O(n) to get
• What does k > 2 gives?
• Previous results (Oliveira et al ‘07, Harrow&Low ‘08, Znidaric ‘08, Brown&Viola ‘09, …)
Random circuits of size O(n(n + log(1/ε))) ε-approximate designs
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Equilibration for low complexity measurements
Given an observable M (POVM element 0 < M < id) in we define its circuit complexity as the minimum number of two-qubit gates needed to measure {M, id-M}.
{0,1}
Let Mk := { M : 0 ≤ M ≤ id, M has gate complexity k }. For almost all U in RCt with t ≥ O(k6) and n ≤ k ≤ 2O(n),
• Shows U|0n> is quantum pseudo random.
• Circuit complexity of U is at least Ω(k).
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Outline Proof Random Circuits are Polynomial Designs
1. Mapping the problem to bounding spectral gap of a Local Hamiltonian
2. Technique for bounding spectral gap (Nachtergaele ’94) + representation theory (reduces the problem to obtaining an exponentially small lower bound on the spectral gap)
3. Path Coupling applied to the unitary group (to prove convergence of the random walk in exponential time)
4. Use detectability Lemma (Arad et al ‘10) to go from local random circuits to parallel local random circuits
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Conclusions• Quantum Information theory provides new tools for studying
thermalization/equilibration and poses new questions about them
Two examples:
• Info-theoretical proof of equivalence of ensembles for non-critical systems. What are the conditions for critical systems (diverging correlation length)?
• Equilibration of random quantum circuits. Can we prove equilibration for random time-independent Hamiltonians?
Thanks!