Transcript Griffith University Quantum Theory Seminar

GRIFFITH QUANTUM THEORY SEMINAR
10 NOVEMBER 2003
Entanglement, correlation, and errorcorrection in the ground states of manybody systems
Henry Haselgrove
School of Physical Sciences
University of Queensland
Michael Nielsen - UQ
Tobias Osborne – Bristol
Nick Bonesteel – Florida State
quant-ph/0308083
quant-ph/0303022 – to appear in PRL
When we make basic assumptions about the
interactions in a multi-body quantum system,
what are the implications for the ground state?

Basic assumptions --- simple general assumptions
of physical plausibility, applicable to most
physical systems.
 Nature gets by with just 2-body interactions
 Far-apart

things don’t directly interact
Implications for the ground state --- using the
concepts of Quantum Information Theory.
 Error-correcting
properties
 Entanglement properties
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
Part 1: Two-local interactions

N interacting quantum
systems, each d-level

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?
1
3
2
4
…
N
Two-local interactions
2
1
4
3

Classically:

Quantum-mechanically:
Two-local Hamiltonians

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,

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.

Example
is two-local, but
is not.

Why two-locality restricts ground states: parameter counting
argument
2
O(N )
O(2N) parameters
Necessary condition for |> to be twolocal ground state

We have
and

Take E=0

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
error-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
don’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 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
correlations in the ground state.
Theory usually applies to infinite quantum systems.
 long-range

Non-relativistic Goldstone Theorem.
 Diverging
correlations imply vanishing energy gap.
 Applies to infinite systems, and typically requires
additional symmetry assumptions
Extreme case: maximum entanglement
A

B
C
Assume the ground state has maximum
entanglement between A and C
or
A
B
C

That is, whenever you have couplings of the form
A
B
C
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):
0.1X
X
0.1X
= 0.1 (XX + YY + ZZ)
… has a unique ground state having an
entanglement of formation of 0.96
Can we prove a general trade-off
between ground-state entanglement
and the gap?
1.4000
1.0392
1.0000
0.6485
-1.0000
-1.0000
-1.0392
-1.0485
General result
A

B
C
Have a “target state” |i that we want “close” to
being the ground state |E0i
--- 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.
2.
Assuming two-locality affects the errorcorrecting abilities
Assuming that two parts don’t directly
interact, introduces a correlation-gap
trade-off.