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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.