Entanglement, correlation, and error- correction in the ground states of many- body systems Henry...

Entanglement, correlation, and error-correction in the ground states of many-

body systems

Henry HaselgroveHenry HaselgroveSchool of Physical SciencesSchool of Physical SciencesUniversity of QueenslandUniversity of Queensland

GRIFFITH QUANTUM THEORY SEMINARGRIFFITH QUANTUM THEORY SEMINAR

Michael Nielsen - UQMichael Nielsen - UQTobias Osborne – BristolTobias Osborne – BristolNick Bonesteel – Florida StateNick Bonesteel – Florida State

10 NOVEMBER 2003

quant-ph/0308083quant-ph/0303022 – to appear in PRL

Basic assumptionsBasic assumptions --- simple general assumptions of physical plausibility, applicable to most physical systems.

When we make basic assumptions about the When we make basic assumptions about the interactions in a multi-body quantum system, interactions in a multi-body quantum system, what are the implications for the ground state?what are the implications for the ground state?

Implications for the ground stateImplications for the ground state --- using the concepts of Quantum Information Theory.

Far-apart things don’t directly interact

Error-correcting properties Entanglement properties

Nature gets by with just 2-body interactions

Why ground states are really cool Physically, ground states are interesting:

T=0 is only thermal state that can be a pure state (vs. mixed state)

Pure states are the “most quantum”. Physically: superconductivity, superfluidity,

quantum hall effect, …

Ground states in Quantum Information Processing: Naturally fault-tolerant systems Adiabatic quantum computing

N interacting quantum systems, each d-level

Part 1: Two-local interactionsPart 1: Two-local interactions

…

1

2

3

4N

Interactions may only be one- and two-body

Consider the whole state space. Which of these states are the ground state of some (nontrivial) two-local Hamiltonian?

Two-local interactions

Quantum-mechanically:

12

34

Classically:

Two-local Hamiltonians

Any two-local Hamiltonian is written as

where the Bn are N-fold tensor products of Pauli

matrices with no more than two non-identity terms.

N quantum bits, for clarity Any imaginable Hamiltonian is a real linear

combination of basis matrices An,

{An} = All N-fold tensor products of Pauli matrices,

Example

is two-local, but

is not.

O(2N) parameters

O(N2)

Why two-locality restricts ground states: parameter counting argument

Necessary condition for |Necessary condition for |> to be two-> to be two-local ground statelocal ground state

Take E=0

We have and

Not interested in trivial case where all cn=0

So the set must be linearly dependent for |i to be a two-local ground state

Nondegenerate quantum Nondegenerate quantum error-correcting codeserror-correcting codes

A state |> is in a QECC that corrects L errors if in principle the original state can be recovered after any unknown operation on L of the qubits acts on |>

The {Bn} form a basis for errors on up to 2 qubits

A QECC that corrects two errors is nondegenerate if each {Bn} takes |i to a mutually orthogonal state

Only way you can have

is if all cn=0

) trivial Hamiltonian

A nondegenerate QECC can not be the eigenstate of any nontrivial two-local Hamiltonian

In fact, it can not be even near an eigenstate of any nontrivial two-local Hamiltonian

H = completely arbitrary nontrivial 2-local Hamiltonian = nondegenerate QECC correcting 2 errors E = any eigenstate of H (assume it has zero eigenvalue) Want to show that these assumptions alone imply that

|| - E || can never get small

Nondegenerate QECCs

Radius of the holes is

Part 2: When far-apart objects Part 2: When far-apart objects don’t interactdon’t interact

In the ground state, how much entanglement is there between the ●’s?

We find that the entanglement is bounded by a function of the energy gapenergy gap between ground and first exited states

Energy gap E1-E0: Physical quantity: how much energy is needed to excite to

higher eigenstate Needs to be nonzero in order for zero-temperature state to

be pure Adiabatic QC: you must slow down the computation

when the energy gap becomes small

Entanglement: Uniquely quantum property A resource in several Quantum Information Processing

tasks Is required at intermediate steps of a quantum

computation, in order for the computation to be powerful

Some related results

Theory of quantum phase transitions. At a QPT, one sees both a vanishing energy gap, and long-range correlations in the ground state.

Theory usually applies to infinite quantum systems.

Non-relativistic Goldstone Theorem. Diverging correlations imply vanishing energy gap. Applies to infinite systems, and typically requires

additional symmetry assumptions

Extreme case: maximum entanglementExtreme case: maximum entanglement

Assume the ground state has maximum entanglement between A and C

A CB

A CBor

That is, whenever you have couplings of the form

A CB

it is impossible to have a unique ground state that maximally entangles A and C.

So, a maximally entangled ground state implies a zero energy gap

Same argument extends to any maximally correlated ground state

Can we get any entanglement between A and C in a unique ground state?

Yes. For example (A, B, C are spin-1/2):

X 0.1X0.1X

= 0.1 (X X + Y Y + Z Z)

… has a unique ground state having an entanglement of formation of 0.96

1.4000 1.0392 1.0000 0.6485-1.0000-1.0000-1.0392-1.0485Can we prove a general trade-off

between ground-state entanglement and the gap?

General resultGeneral result

Have a “target state” |i that we want “close” to being the ground state |E0i

A CB

--- measure of closeness of target to ground

--- measure of correlation between A and C

The future… At the moment, our bound on the energy gap

becomes very weak when you make the system very large. Can we improve this?

The question of whether a state can be a unique ground state is closely related to the question of when a state is uniquely determined by its reduced density matrices. Explore this question further: what are the conditions for this “unique extended state”?

Conclusions

Simple yet widely-applicable assumptions on the interactions in a many-body quantum system, lead to interesting and powerful results regarding the ground states of those systems

1. Assuming two-locality affects the error-correcting abilities

2. Assuming that two parts don’t directly interact, introduces a correlation-gap trade-off.