Entanglement, correlation, and errorcorrection in the ground states of manybody systems

1
Entanglement, correlation, and error-correction
in the ground states of many-body systems
GRIFFITH QUANTUM THEORY SEMINAR
10 NOVEMBER 2003

• 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
2
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 dont directly interact

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

• Error-correcting properties

• Entanglement properties

3
Why ground states are really cool

• Physically, ground states are interesting
• T0 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

4
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?

5
Two-local interactions
2
1
4
3

• Classically

• Quantum-mechanically

6
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.

7

• Example

is two-local, but
is not.

• Why two-locality restricts ground states
  parameter counting argument

O(N2)
O(2N) parameters
8
Necessary condition for ?gt to be two-local
ground state

• We have and

• Take E0

• Not interested in trivial case where all cn0

So the set must
be linearly dependent for ?i to be a two-local
ground state
9
Nondegenerate quantum error-correcting codes

• A state ?gt 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 ?gt
• 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 cn0
• ) trivial Hamiltonian

10

• 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

11

• 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

12
Nondegenerate QECCs
Radius of the holes is
13
Part 2 When far-apart objects dont 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

14

• 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

15
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

16
Extreme case maximum entanglement
A
B
C

• Assume the ground state has maximum entanglement
  between A and C

17

• 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

18
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
0.1X
0.1X
1.4000 1.0392 1.0000 0.6485 -1.0000 -1.0000 -1
.0392 -1.0485
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?
19
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
20
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?

21
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

• Assuming two-locality affects the
  error-correcting abilities
• Assuming that two parts dont directly interact,
  introduces a correlation-gap trade-off.