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Part 2: Indirect interactions

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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 don’t 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
- T=0 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.

(C=1 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)