Entanglement, ground states, energy gap

1
Entanglement, correlations, and error-correction
in the ground states of many-body systems
Henry L. Haselgrove1,2, Michael A. Nielsen1, and
Tobias J. Osborne3 1. School of Physical
Sciences, University of Queensland, Australia 2.
Information Sciences Laboratory, DSTO,
Australia 3. School of Mathematics, University of
Bristol, United Kingdom
Summary Are there any generic properties which
are shared by the ground states of all
physically-realistic many-body quantum systems?
We consider two very simple notions of
physically realistic. First is that
interactions between bodies can be expressed as a
sum of two-body interactions (this is true, for
example, for particles interacting via the
electromagnetic force). We have proved 1 that
ground states of all such systems are provably
far away from an important class of states known
as nondegenerate quantum error-correcting
codes. The second notion is that far-apart
objects dont directly interact, rather they
indirectly interact via other objects. We have
show that this imposes strict conditions on the
type of correlations and entanglement that can
appear in the ground state, as a function of the
spectral energy gap 2. Thus, we are connecting
simple physical notions of locality with abstract
concepts such as entanglement and fault-tolerance.

• Importance of ground states
• Physically, ground states are interesting because
• T0 is only thermal state that can be a pure
  quantum state (vs. mixed state)
• Pure states are the most quantum.
• Important examples of ground states
  superconductivity, superfluidity, quantum hall
  effect,
• Ground states are important in Quantum
  Information Processing. For instance
• Naturally fault-tolerant systems
• Adiabatic quantum computing

Part 2 Indirect interactions We assume that two
quantum objects A and C interact only via
another object (or objects) B. No other
assumptions are made about the interactions.
Result The ground state of all such
systems can only have large amounts of
correlation (and thus entanglement) between A and
C if there is a small energy gap (between ground
and first-excited states). Specifically, we
express the ground state, without loss of
generality, as where jji and are jki are a
basis for the systems A and C. Then a measure of
correlation between A and C is Then the energy
gap and correlations are then related as
follows
General system
A
B
C
C
A
Example, A C could be two sites on a lattice
B

• Part 1 Two-local interactions
• In nature, many-body interactions tend to be
  two-local, that is they are the sum of two-body
  interactions. How are quantum ground states
  affected by this fact alone?
• Consider
• N interacting quantum systems, each d-level.
• Interactions may only be one- and two-body.
• So, the Hamiltonian is a sum of operators that
  each act on at most two bodies
• Consider the whole state space. Which of these
  states could be the ground state of some
  (nontrivial) two-local Hamiltonian?
• Result
• In state space, centred around every
  nondegenerate quantum error-correcting code
  state, is a region of states than cannot be the
  ground state of any physically-plausible
  Hamiltonian.

(C1 means maximum correlations)
(an operator basis for two-body interactions)
(energy eigenvalues)
(total energy scale)
The energy gap of a system is an important
quantity because it determines how sensitively a
system responds to perturbations. In a pure
quantum state (such as a unique ground state),
there is entanglement present whenever there are
correlated measurement results between different
bodies. Entanglement is known to be important for
fast quantum computation, and in the formation of
superconductivity and other quantum effects.
(Stylised depiction of regions in the state space
that contain physically-impossible ground states)
Radius of the holes is
Frustration and entanglement Dawson and Nielsen
3 have found that when two objects interact
directly, the ground state entanglement is
bounded by a measure of the extent to which the
interaction frustrates the single-body terms in
the Hamiltonian. Thus, nature uses quantum
entanglement to find compromise between competing
interactions in a system.
Nondegenerate QECCs
Glossary A quantum error-correcting code (QECC)
is a vector space of states on several bodies
such that, for any state in that space, the state
can be recovered perfectly if an error occurs
on one of the bodies. The code is nondegenerate
if the type of error can be identified during
correction.
1 H. L. Haselgrove, M. A. Nielsen, and T. J.
Osborne, Phys. Rev. Lett. 91, 210401 (2003) 2
H. L. Haselgrove, M. A. Nielsen, and T. J.
Osborne, Phys. Rev. A 69 (3), 032303 (2004) 3
C. M. Dawson and M. A. Nielsen, Phys. Rev. A 69
(5), 052316 (2004)