Fernando G.S.L. Brandão
University College London -> Workshop -> Microsoft Research
Washington DC, February 2015

Quantum Information Theory
Goal : Lay down the theory for future quantum-based technology
(quantum computers, quantum cryptography, …)
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.

Quantum Information Theory
Goal : Lay down the theory for future quantum-based technology
(quantum computers, quantum cryptography, …)
Ultimate limits to information transmission
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.

Quantum Information Theory
Goal : Lay down the theory for future quantum-based technology
(quantum computers, quantum cryptography, …)
Ultimate limits to information transmission
Entanglement as a resource
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.

Quantum Information Theory
Goal : Lay down the theory for future quantum-based technology
(quantum computers, quantum cryptography, …)
Ultimate limits to information transmission
Entanglement as a resource
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.
Quantum computers are digital

Quantum Information Theory
Goal : Lay down the theory for future quantum-based technology
(quantum computers, quantum cryptography, …)
Ultimate limits to information transmission
Entanglement as a resource
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.
Quantum algorithms with exponential speed-up
Quantum computers are digital

Quantum Information Theory
Goal : Lay down the theory for future quantum-based technology
(quantum computers, quantum cryptography, …)
Ultimate limits to information transmission
Entanglement as a resource
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.
Quantum algorithms with exponential speed-up
Quantum computers are digital
Ultimate limits for efficient computation

QIT Connections
QIT
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.

Condensed Matter
QIT Connections
Strongly corr. systems
Topological order
Spin glasses
QIT
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.

Condensed Matter
QIT Connections
Strongly corr. systems
Topological order
Spin glasses
StatMech
QIT
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.
Thermalization
Thermo@nano scale
Quantum-to-Classical
Transition

Condensed Matter
QIT Connections
HEP/GR
Strongly corr. systems
Topological order
Spin glasses
StatMech
QIT
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.
Thermalization
Thermo@nano scale
Quantum-to-Classical
Transition
Topolog. q. field theo.
Black hole physics
Holography

Condensed Matter
QIT Connections
HEP/GR
Strongly corr. systems
Topological order
Spin glasses
StatMech
QIT
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.
Thermalization
Thermo@nano scale
Quantum-to-Classical
Transition
Topolog. q. field theo.
Black hole physics
Holography
Exper. Phys.
Ion traps, linear optics, optical lattices, cQED, superconduc. devices, many more

QIT Connections
StatMech
Thermalization
Thermo@nano scale
Quantum-to-Classical
Transition
QIT
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.

QIT Connections
StatMech
Thermalization
Thermo@nano scale
Quantum-to-Classical
Transition
QIT
Q. Commun.
Entanglement theo.
Q. error correc. + FT
Q. computation
Q. complexity theo.
This Talk

This Talk
Goal: give examples of these connections in statistical mechanics

Entropy
Q. Statistical Mechanics
Q. Information
(Lieb, Ruskai ‘73) conditional mutual information state redistribution, …
data processing, converses, …
(Uhlmann ‘)
classical capacity q. channel,
(Hiai, Petz ‘91) entanglement theory, …

Entropy
Q. Statistical Mechanics
Q. Information
(Lieb, Ruskai ‘73) conditional mutual information state redistribution, …
data processing, converses, …
(Uhlmann ‘)
classical capacity q. channel,
(Hiai, Petz ‘91) entanglement theory, …

Equilibrium Statistical Mechanics
Given Hamiltonian of n particles :
Microcanonical:
Canonical:

Microcanonical Ensemble and
Concentration of Measure
(Popescu, Short, Winter ’05; Goldstein, Lebowitz, Timulka, Zanghi ‘06, …)
Let H be a Hamiltonian and S e
(en-δn 1/2 , en+δn 1/2 the subspace of states with energy
). Then for almost every state |ψ> in S e
, and region A sufficiently small,
Interpretation I: Microcanonical ensemble represents our subjective description of the system given knowledge only of energy.

Microcanonical Ensemble and
Concentration of Measure
(Popescu, Short, Winter ’05; Goldstein, Lebowitz, Timulka, Zanghi ‘06, …)
Let H be a Hamiltonian and S e
(en-δn 1/2 , en+δn 1/2 the subspace of states with energy
). Then for almost every state |ψ> in S e
, and region A of size ≤ n/2
Interpretation I: Microcanonical ensemble represents our subjective description of the system given knowledge only of energy.
Interpretation II: Microcanonical ensemble represents the objective state of the system at a given energy. Randomness comes from entanglement of A and A c .

Microcanonical Ensemble and
Concentration of Measure
(Popescu, Short, Winter ’05; Goldstein, Lebowitz, Timulka, Zanghi ‘06, …)
Let H be a Hamiltonian and S e
(en-δn 1/2 , en+δn 1/2 the subspace of states with energy
). Then for almost every state |ψ> in S e
, and region A of size ≤ n/2
Almost every state: W.h.p. over the Haar measure in S e
Easy consequence of:
Levy’s Lemma
For a Lipchitz function f : S n -> R, f(x) ≈ <f> for almost every x in S n
(see Hayden’ and Szarek’s talks for more)

Microcanonical Ensemble and
Concentration of Measure
(Popescu, Short, Winter ’05; Goldstein, Lebowitz, Timulka, Zanghi ‘06, …)
Let H be a Hamiltonian and S e
(en-δn 1/2 , en+δn 1/2 the subspace of states with energy
). Then for almost every state |ψ> in S e
, and region A of size ≤ n/2
Haar measure requires a lot of randomness
(exp(O(n)) random bits)
Quantum pseudo-randomness : Can replace Haar measure by a state
2-design in S e
(Dankert et al ‘06)
Open Question:
When can we sample efficiently from a 2-design in S e
?

Equilibrium Statistical Mechanics
Given Hamiltonian of n particles :
Microcanonical:
Canonical:

Canonical State
When should we use each ensemble?
Micro: System in isolation
Macro: System in equilibrium with a heat bath at temperature 1/β
Macro follows from micro by considering the system and bath in the microcanonical ensemble and looking at the reduced state of the system
E
S
Justified whenever the interactions of system and bath are weak.

Canonical State
When should we use each ensemble?
Micro: System in isolation
Macro: System in equilibrium with a heat bath at temperature 1/β
What if we are only interested in expectation values of local observables?
A
X
Is the system an environment for itself? i.e. For every β, is for e(β) = tr(H ρ
β
) ?
Ex. Experiments with cold atoms in optical lattices

Equivalence of Ensembles?
Gibbs 1902 : “For the average square of the anomalies of the energy, we find an expression which vanishes in comparison to the square of the average energy, when the number of degrees of freedom is indefinitely increased. An ensemble (…), if distributed canonically , would therefore appear to human observation as an ensemble of systems in which all have the same energy .”
Energy Variance: var(H) = <H 2 > c,β
Canonical ensemble:
- <H> c,β
2
Microcanonical ensemble:

Equivalence of Ensembles?
Gibbs 1902 : “For the average square of the anomalies of the energy, we find an expression which vanishes in comparison to the square of the average energy, when the number of degrees of freedom is indefinitely increased. An ensemble (…), if distributed canonically , would therefore appear to human observation as an ensemble of systems in which all have the same energy .”
Energy Variance: var(H) = <H 2 > c,β
Canonical ensemble:
- <H> c,β
2
Microcanonical ensemble:
Heat Capacity: C(β)
Var(H) = C(β)/β 2 = O(system size) (when correlation length finite)
<H> c,β
= O(system size) >> Var(H) 1/2 = O(system size) 1/2
To simplistic, microcanonical and canonical states are nearly orthogonal for large systems!

Some Rigorous Results
• Non-equivalence for critical systems
(or symmetric-broken systems )
E.g. 2D Ising model with diverging correlation length
(Desermo ’04)
• Equivalence in the thermodynamical limit in “unique phase region”
(i.e. one KMS state)
(Lebowitz, Lieb ’69, Lima ’72, …)
- equivalence reduced states
Footprint: requires thermo limit, translation invariance and gives no bounds on the size of A

Some Rigorous Results
• Non-equivalence for critical systems
(or symmetric-broken systems )
E.g. 2D Ising model with diverging correlation length
(Desermo ’04)
• Equivalence in the thermodynamical limit in “unique phase region”
(i.e. one KMS state)
- equivalence thermodynamical potentials
- equivalence reduced states
(Lebowitz, Lieb ’69, Lima ’72, …)
(Muller, Adlam, Masanes, Wiebe ‘13)
Footprint: requires thermo limit, translation invariance and gives no bounds on the size of A

Equivalence of Ensembles
(B., Cramer ‘15)
Let H be a Hamiltonian of n particles on a d -dimensional lattice.
Let β be such that ρ
β has a correlation length ξ .
Then for most regions A of size ≈ d = 2
H ij
A on the arXiv today
Correlation length ξ: For all X, Z

Equivalence of Ensembles
(B., Cramer ‘15)
Let H be a Hamiltonian of n particles on a d -dimensional lattice.
Let β be such that ρ
β has a correlation length ξ .
Then for most regions A of size ≈
Works for:
Open Question : How small can δ be?
Sometime δ=0, ie. eigenstate thermalization
But not always: many-body localization
(Srednicki ’94)

Equivalence of Ensembles
(B., Cramer ‘15)
Let H be a Hamiltonian of n particles on a d -dimensional lattice.
Let β be such that ρ
β has a correlation length ξ .
Then for most regions A of size ≈
Proof based on quantum information theory : smoothed (max) relative entropy, substate thm, Pinsker’s inequality
+ new quantum lattice version of Berry-Esseen Theorem
Open questions :
How big can the region be? Can prove it for all regions?
What happens in the symmetry-broken phase?

Quantum Equilibration
State at time t:

Quantum Equilibration
State at time t:
Will equilibrate?
I.e. for most t ?

Quantum Equilibration
State at time t:
Will equilibrate?
I.e. for most t ? NO!

Relative Equilibration
How about relative to particular kind of measurements?
• “macroscopic” measurements (von Neumann ‘29)
• local measurements (Linden, Popescu, Short, Winter)
S S c
• local measurements relative to an external observer
(ie. apply U to part of a state correlated with another)
(Hayden, Preskill ‘07, del Rio, Renner, Wehner ‘13)
S’
S S c
• Low-complexity measurements
(i.e. measurements that require time much less than t)
(B. Harrow, Horodecki ’12)
Tightly connected to decoupling : central primitive in q. comm. theory

Equilibration of subsystems is generic
(Linden, Popescu, Short, Winter ’08)
Any Hamiltonian H (with non-degenerate energy gaps) equilibrates: with and
S S c

Time Scale of Equilibration
The previous approach only gives bounds exponentially long in the number of particles
Time-independent Hamiltonians:
No : many-body localization
(zero Lieb-Robinson velocity, clustering for all eigenstates, area law for all eigenstates, ….)
Time-dependent Hamiltonians:
Yes (in a sense)

Time of Equilibration for Random
Circuits
Random Circuits : Model of quantum evolution in which every time step is given by an application of a local unitary (chosen at random)
Discrete version of
Ex.
with random H(t) = H
12
(t) + H
23
(t) + … + H nn-1
(t )
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))
Similar definitions in higher dimensions, other sets of gates, etc
How fast random circuits equilibrate?

Equilibration in Random Circuits
(Oliveira et al ‘06, Harrow, Low ‘07, …) Bounds on local equilibration and connections to unitary designs and mixing times Markov chains
(B, Harrow, Horodecki ‘12) 1D random circuits locally equilibrate as fast as possible (saturate the speed-of sound bound). Open for >1D
(Hayden, Preskill ‘07, Sekino, Susskind ‘08) Fast-Scrambling Conjecture
No dynamics equilibrate (relative to an external observer) in less than
O(log(n)) time. Random circuits saturate the bound. Open, but:
(Lashkari et al ‘11) Evidence for fast scrambling conjecture
(Brown-Fawzi ‘13) Random circuits equilibrate in O(log^3(n)) time
Anything happens after equilibration?
(Sussking ‘14) Yes, preparation complexity of the state keeps increasing up to times exp(O(n)). Open for time-independent models, but
(B, Harrow, Horodecki ‘12) Can prove it for random circuits

Equilibration in Random Circuits
(Oliveira et al ‘06, Harrow, Low ‘07, …) Bounds on local equilibration and connections to unitary designs and mixing times Markov chains
(B, Harrow, Horodecki ‘12) 1D random circuits locally equilibrate as fast as possible (saturate the speed-of sound bound). Open for >1D
(Hayden, Preskill ‘07, Sekino, Susskind ‘08) Fast-Scrambling Conjecture
No dynamics equilibrate (relative to an external observer) in less than
O(log(n)) time. Random circuits saturate the bound. Open, but:
(Lashkari et al ‘11) Evidence for fast scrambling conjecture
(Brown-Fawzi ‘13) Random circuits equilibrate in O(log^3(n)) time
Anything happens after equilibration?
(Sussking ‘14) Yes, preparation complexity of the state keeps increasing up to times exp(O(n)). Open for time-independent models, but
(B, Harrow, Horodecki ‘12) Can prove it for random circuits

Equilibration in Random Circuits
(Oliveira et al ‘06, Harrow, Low ‘07, …) Bounds on local equilibration and connections to unitary designs and mixing times Markov chains
(B, Harrow, Horodecki ‘12) 1D random circuits locally equilibrate as fast as possible (saturate the speed-of sound bound). Open for >1D
(Hayden, Preskill ‘07, Sekino, Susskind ‘08) Fast-Scrambling Conjecture
No dynamics equilibrate (relative to an external observer) in less than
O(log(n)) time. Random circuits saturate the bound. Open, but:
(Lashkari et al ‘11) Evidence for fast scrambling conjecture
(Brown-Fawzi ‘13) Random circuits equilibrate in O(log^3(n)) time
Anything happens after equilibration?
(Sussking ‘14) Yes, preparation complexity of the state keeps increasing up to times exp(O(n)). Open for time-independent models, but
(B, Harrow, Horodecki ‘12) Can prove it for random circuits

Open Quantum Systems
Sometimes we can model the interaction of a quantum system with an environment by an action on the system only
Quantum Master Equations (aka Liovillians, Linbladians, …)
Canonical example: cavity QED
Lindblad Equation:
(most general Markovian and time homogeneous q. master equation)
Generates completely positive trace-preserving map:
How long does it take to reach equilibrium? (ie. The fixed point of e L )

Open Quantum Systems
Sometimes we can model the interaction of a quantum system with an environment by an action on the system only:
Quantum Master Equations (aka Liovillians, Linbladians, …)
Canonical example: cavity QED
Lindblad Equation:
(most general Markovian and time homogeneous q. master equation)
Generates completely positive trace-preserving map:
How long does it take to reach equilibrium? (ie. The fixed point of e L )

Classical Glauber Dynamics
A stochastic map M = e G is a Glauber dynamics for a (classical)
Hamiltonian if it’s generator G is local and the unique fixed point of M is e -βH /Z(β) (+ detailed balance)
Ex: Metropolis, Heat-bath generator, ….
When is Glauber dynamics effective for sampling from Gibbs state?
(Markov Chain Monte-Carlo, many applications)
(Stroock, Zergalinski ’92;
Martinelli, Olivieri ’94, …)
Gibbs state with finite correlation length
(Sly, … ‘)
Rapidly mixing
Glauber dynamics
Proved only for … model
Gibbs Sampling in P
(vs -hard)

Classical Glauber Dynamics
A stochastic map M = e G is a Glauber dynamics for a (classical)
Hamiltonian if it’s generator G is local and the unique fixed point of M is e -βH /Z(β) (+ detailed balance)
Ex: Metropolis, Heat-bath generator, ….
When is Glauber dynamics effective for sampling from Gibbs state?
(Markov Chain Monte-Carlo, many applications)
(Stroock, Zergalinski ’92;
Martinelli, Olivieri ’94, …)
Gibbs state with finite correlation length
(Sly ‘10)
Rapidly mixing
Glauber dynamics
(Proved only for hard core mode and 2-spin antiferromagetic model)
Gibbs Sampling in P
(vs NP -hard)

Quantum Glauber Dynamics
Do we have quantum analogues of Glauber dynamics? Open
(Davies ‘79)
Davies map : non-local generator
(can we at least implement locally on q. computer?)
(Majewski, Zergalisnki ‘95)
Heat-Bath maps (aka Petz’s transpose channel): non-local generator
(can we at least implement locally on q. computer?)
(Temme et al ‘11)
Quantum Metropolis : non-local generators
Can be implemented efficiently on q. computer
For commuting Hamiltonians (e.g Toric code, Levin-Wen model, …) :
Davies maps and Heat-bath maps are local.
Can we understand the efficiency of q. Glauber dynamics in this case?

Quantum Glauber Dynamics
Do we have quantum analogues of Glauber dynamics? Open
(Davies ‘79)
Davies map : non-local generator
(can we at least implement locally on q. computer?)
(Majewski, Zergalisnki ‘95)
Heat-Bath maps (aka Petz’s transpose channel): non-local generator
(can we at least implement locally on q. computer?)
(Temme et al ‘11)
Quantum Metropolis : non-local generators
Can be implemented efficiently on q. computer
For commuting Hamiltonians (e.g Toric code, Levin-Wen model, …) :
Davies maps and Heat-bath maps are local.
Can we understand the efficiency of q. Glauber dynamics in this case?

Clustering -> Rapid Mixing
(B., Kastoryano ‘15)
If a 2D commuting Hamiltonian satisfies local indistinguishability at temperature 1/β, then the quantum heat bath Gibbs sampler converges to ρ
β in poly(n, 2 O(β) , ξ) time
Local indistinguishability:
X c Z
Equivalent to a form of clustering:
Reduces to finite correlation length in classical models
Proof by using theory of weighted non-commutative L p spaces and characterization of states with zero conditional mutual information
(Hayden, Jozsa, Petz, Winter ‘04)
X

Clustering -> Rapid Mixing
(B., Kastoryano ‘15)
If a 2D commuting Hamiltonian satisfies local indistinguishability at temperature 1/β, then the quantum heat bath Gibbs sampler converges to ρ
β in poly(n, 2 O(β) , ξ) time
Local indistinguishability:
Z X c
Open questions:
3D? Log-sSobolev constant?
Non-commuting Hamiltonians?
Classical Algorithms for Q. Gibbs Sampling?
BQP-complete NP-hard phase-transition?
Applications Q. Gibbs Sampling to machine learning?
X

Can “quantum effects” persist at finite temperature?
q1: Quantum PCP Conjecture (see review: Aharonov, Arad, Vidick ‘13)
Is estimating the mean free energy of quantum models QMA-hard?
(known to be NP-hard by PCP thm) q2 : NTLS Conjecture (Freedman, Hastings ‘13)
Are there local Hamiltonians for which all eigenstates up to energy
εn have topological order?
(i.e. cannot be created by constant-depth circuit? How about poly cirucits?) q3 : Self-Correcting Quantum Memories (Dennis, Kitaev, Landahl, Preskill ‘01)
Are there Hamiltonians robust under static pertubations that take a long time to thermalize?
≥4D: YES ≤2D: NO (probably) 3D: ?????
(Dennis et al ’01, …) (Bravyi, Terhal ’08, …,Temme ‘14) (Haah ‘11, …)

Conclusion
• Quantum Information Theory benefited, and still benefits, greatly from quantum statistical mechanics
• QIT is starting to give back to Q. Statistical Mechanics
• A lot of scope for further and deeper connections
(see Jaksic’s talk)

Resource Theories in Quantum
Information
From J. Oppenheim