Entanglement, correlation, and error- correction in the ground states of many- body systems Henry...
-
Upload
hayden-scott -
Category
Documents
-
view
215 -
download
1
Transcript of 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.