Entanglement and Area

Imperial College London Cambridge, 17th December 2004 Entanglement and Area Martin Plenio Imperial College London Sponsored by: QUPRODIS Royal Society Senior Research Fellowship

Imperial College London Cambridge, 17th December 2004 Entanglement theory Provide efficient methods to • decide which states are entangled and which are disentangled(Characterize) • decide which LOCC entanglement manipulations are possible and provide the protocols to implement them(Manipulate) • decide how much entanglement is in a state and how efficient entanglement manipulations can be(Quantify) Mathematical characterization of all multi-party states

Imperial College London Cambridge, 17th December 2004 Entanglement theory Multi-particle entanglement is difficult Consider properties of natural states of physical systems

Imperial College London Cambridge, 17th December 2004 M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer, quant-ph/0405142

Imperial College London Cambridge, 17th December 2004

Imperial College London Cambridge, 17th December 2004 How does entanglement scale with size of region?

Imperial College London Cambridge, 17th December 2004 Entanglement properties of the harmonic chain Arrange n harmonic oscillators on a ring and let them interact harmonically by springs. . . . n - 1 n . . . 1 2 . . . } V

Imperial College London Cambridge, 17th December 2004 Basic Techniques • Characteristic function

Imperial College London Cambridge, 17th December 2004 Basic Techniques • Characteristic function • A state is called Gaussian, if and only if its characteristic function is a Gaussian

Imperial College London Cambridge, 17th December 2004 Basic Techniques • Characteristic function • A state is called Gaussian, if and only if its characteristic function is a Gaussian • Ground states of Hamiltonians quadratic in X and P are Gaussian

Imperial College London Cambridge, 17th December 2004 Basic Techniques • Characteristic function • A Gaussian with vanishing firstmoments • Ground states of Hamiltonians quadratic in X and P are Gaussian

Imperial College London Cambridge, 17th December 2004 Entanglement Measures Entropy of Entanglement: Logarithmic Negativity: Distillable Entanglement: Efficiently computable directly from the properties of covariance matrix Tutorial Review: J. Eisert and M.B. Plenio, Int. J. Quant. Inf. 1, 479 (2003)

Imperial College London Cambridge, 17th December 2004 D-dimensional lattices: Entanglement and Area Entanglement per unit length of boundary red square and environment versus length of side of inner square on a 30x30 lattice of oscillators.

Imperial College London Cambridge, 17th December 2004 D-dimensional lattices: Entanglement and Area ViaGLOCC Number of entangled pairs is proportional to volume. G. Giedke, J. Eisert, J.I. Cirac, and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003); A. Botero and B. Reznik, Phys. Rev. A 67, 052311 (2003)

Imperial College London Cambridge, 17th December 2004 Why should the entropy proportional to the area? Can prove this exactly: Intuition from squared interaction.

Imperial College London Cambridge, 17th December 2004 Why should the entropy proportional to the area? Obtain a simple normal form Disentangle oscillators except on surface Disentangle ViaGLOCC Need to bound entanglement of number of entangled pairs linear in surface area. Find area independent bound Entanglement proportional to area

Imperial College London Cambridge, 17th December 2004 Disentangling also works for thermal states! Disentangle ViaGLOCC Now decoupled oscillators are in mixed state, but they are NOT entangled to any other oscillator (only to environment). Then make eigenvalue estimates to find bounds on entanglement.

Imperial College London Cambridge, 17th December 2004 D-dimensional lattices: Entanglement and Area For general interaction: Correlations decrease with distance, contribution bounded Disentangle ViaGLOCC M.B. Plenio, J. Eisert, J. Dreißig and M. Cramer, quant-ph/0405142

Imperial College London Cambridge, 17th December 2004 k = (3,2) s(k,l) =

Imperial College London Cambridge, 17th December 2004 k = (3,2) l = (5,6) s(k,l) = (5-3) + (6-2)

Imperial College London Cambridge, 17th December 2004 The upper bound: Outline Tricks from Matrix Analysis yield: 2cd A 1-2cd

Imperial College London Cambridge, 17th December 2004 The upper bound: Outline Tricks from Matrix Analysis yield: 2 2cd A (2cd) (A+4) + 1-2cd 1-2cd

Imperial College London Cambridge, 17th December 2004 The upper bound: Outline Tricks from Matrix Analysis yield: 2 3 2cd A (2cd) (A+4) (2cd) (A+4) + + 1-2cd 1-2cd 1-2cd This approach is independent of the shape of the region Works for very general V even when correlation length is infinite

Imperial College London Cambridge, 17th December 2004 Where does the entanglement sit?

Imperial College London Cambridge, 17th December 2004 Correlations and Area in Classical Systems g denotes phase space point

Imperial College London Cambridge, 17th December 2004 Correlations and Area in Classical Systems g denotes phase space point Entropy depends on fine-graining in phase space but mutual information is independent of fine-graining Correlations proportional to surface area

Imperial College London Cambridge, 17th December 2004 D-dimensional lattices: Entanglement and Area Can prove: • Upper and lower bound on entropy of entanglement that are proportional to the number of oscillators on the surface • For ground state for general interactions • For thermal states for ‘squared interaction’ • General shape of the regions a a a • Classical harmonic oscillators in thermal state: Classical correlations are bounded from above and below by expressions proportional to number of oscillators on surface. • Proof via quantum systems with ‘squared interactions’ a

Entanglement and Area