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Entanglement, correlation, and error- correction in the ground states of many- body systems Henry Haselgrove School of Physical Sciences University of Queensland GRIFFITH QUANTUM THEORY SEMINAR Michael Nielsen - UQ Tobias Osborne – Bristol Nick Bonesteel – Florida State 10 NOVEMBER 2003 quant-ph/ quant-ph/ – to appear in PRL

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n Basic assumptions n Basic assumptions --- simple general assumptions of physical plausibility, applicable to most physical systems. When we make basic assumptions about the interactions in a multi-body quantum system, what are the implications for the ground state? n Implications for the ground state n Implications for the ground state --- using the concepts of Quantum Information Theory. u Far-apart things dont directly interact u Error-correcting properties u Entanglement properties u Nature gets by with just 2-body interactions

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Why ground states are really cool n Physically, ground states are interesting: u T=0 is only thermal state that can be a pure state (vs. mixed state) u Pure states are the most quantum. u Physically: superconductivity, superfluidity, quantum hall effect, … n Ground states in Quantum Information Processing: u Naturally fault-tolerant systems u Adiabatic quantum computing

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n N interacting quantum systems, each d-level Part 1: Two-local interactions … N n Interactions may only be one- and two-body n Consider the whole state space. Which of these states are the ground state of some (nontrivial) two-local Hamiltonian?

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Two-local interactions n Quantum-mechanically: n Classically:

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Two-local Hamiltonians n Any two-local Hamiltonian is written as where the B n are N-fold tensor products of Pauli matrices with no more than two non-identity terms. n N quantum bits, for clarity n Any imaginable Hamiltonian is a real linear combination of basis matrices A n, n {A n } = All N-fold tensor products of Pauli matrices,

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

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Necessary condition for | > to be two- local ground state n Take E=0 n We have and n Not interested in trivial case where all c n =0 So the set must be linearly dependent for | i to be a two-local ground state

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Nondegenerate quantum error-correcting codes n 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 | > n The {B n } form a basis for errors on up to 2 qubits A QECC that corrects two errors is nondegenerate if each {B n } takes | i to a mutually orthogonal state n Only way you can have is if all c n =0 ) trivial Hamiltonian

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n A nondegenerate QECC can not be the eigenstate of any nontrivial two-local Hamiltonian n In fact, it can not be even near an eigenstate of any nontrivial two-local Hamiltonian

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n H = completely arbitrary nontrivial 2-local Hamiltonian n = nondegenerate QECC correcting 2 errors n E = any eigenstate of H (assume it has zero eigenvalue) n Want to show that these assumptions alone imply that || - E || can never get small

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Nondegenerate QECCs Radius of the holes is

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Part 2: When far-apart objects dont interact n In the ground state, how much entanglement is there between the s? energy gap n We find that the entanglement is bounded by a function of the energy gap between ground and first exited states

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n Energy gap E 1 -E 0 : u Physical quantity: how much energy is needed to excite to higher eigenstate u Needs to be nonzero in order for zero-temperature state to be pure u Adiabatic QC: you must slow down the computation when the energy gap becomes small n Entanglement: u Uniquely quantum property u A resource in several Quantum Information Processing tasks u Is required at intermediate steps of a quantum computation, in order for the computation to be powerful

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Some related results n Theory of quantum phase transitions. At a QPT, one sees both u a vanishing energy gap, and u long-range correlations in the ground state. Theory usually applies to infinite quantum systems. n Non-relativistic Goldstone Theorem. u Diverging correlations imply vanishing energy gap. u Applies to infinite systems, and typically requires additional symmetry assumptions

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Extreme case: maximum entanglement n Assume the ground state has maximum entanglement between A and C A C B A C B or

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n That is, whenever you have couplings of the form A C B it is impossible to have a unique ground state that maximally entangles A and C. n So, a maximally entangled ground state implies a zero energy gap n Same argument extends to any maximally correlated ground state

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Can we get any entanglement between A and C in a unique ground state? n Yes. For example (A, B, C are spin-1/2): X0.1X = 0.1 (X X + Y Y + Z Z) … has a unique ground state having an entanglement of formation of Can we prove a general trade-off between ground-state entanglement and the gap?

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General result Have a target state | i that we want close to being the ground state |E 0 i A C B --- measure of closeness of target to ground --- measure of correlation between A and C

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The future… n At the moment, our bound on the energy gap becomes very weak when you make the system very large. Can we improve this? n 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?

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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. Assuming two-locality affects the error- correcting abilities 2. Assuming that two parts dont directly interact, introduces a correlation-gap trade-off.

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