Entanglement, correlation, and error-correction in the ground states of many-body systems

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

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 don’t 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: • T=0 is only thermal state that can be a purestate (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. 1 3 2 … N 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 • Quantum-mechanically: 1 4 3 • Classically:

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. is two-local, but • Example is not. • Why two-locality restricts ground states: parameter counting argument O(N2) O(2N)parameters

8. Necessary condition for |> to be two-local 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

9. 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 Lof 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

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 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

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. or A B C 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): 0.1X X 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 don’t directly interact, introduces a correlation-gap trade-off.